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+{"page_0": [["block_0", [{"image_0": "0_0.png", "coords": [0, 0, 472, 728], "fig_type": 0}]], ["block_1", [{"image_1": "0_1.png", "coords": [0, 4, 106, 274], "fig_type": "figure"}]], ["block_2", [{"image_2": "0_2.png", "coords": [1, 521, 100, 727], "fig_type": "figure"}]], ["block_3", ["E\n.\nPolymer\nC Chemistry\n"]], ["block_4", [{"image_3": "0_3.png", "coords": [3, 632, 94, 719], "fig_type": "figure"}]], ["block_5", [{"image_4": "0_4.png", "coords": [5, 16, 95, 103], "fig_type": "figure"}]], ["block_6", [{"image_5": "0_5.png", "coords": [7, 436, 90, 517], "fig_type": "figure"}]], ["block_7", [{"image_6": "0_6.png", "coords": [7, 235, 89, 317], "fig_type": "figure"}]], ["block_8", [{"image_7": "0_7.png", "coords": [113, 26, 396, 206], "fig_type": "figure"}]], ["block_9", [{"image_8": "0_8.png", "coords": [187, 564, 447, 648], "fig_type": "figure"}]], ["block_10", [". Poul C. Hiemenz\nTimothy P. Lodge\n"]], ["block_11", ["\u2019\n\ufb01\u2018\nQ\nC?\n"]], ["block_12", ["LB\n\u20180\nt\no-\n"]], ["block_13", ["an\na\u00bb\n"]], ["block_14", ["SecondEdition\n"]], ["block_15", ["_\n"]], ["block_16", ["E\n(,9...\n"]], ["block_17", ["E\n"]], ["block_18", ["E\n"]], ["block_19", ["E\n"]], ["block_20", ["E\n"]], ["block_21", ["E\n"]]], "page_1": [["block_0", [{"image_0": "1_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "1_1.png", "coords": [83, 56, 384, 202], "fig_type": "figure"}]], ["block_2", [{"image_2": "1_2.png", "coords": [88, 89, 365, 155], "fig_type": "figure"}]], ["block_3", [{"image_3": "1_3.png", "coords": [93, 17, 350, 160], "fig_type": "figure"}]], ["block_4", [{"image_4": "1_4.png", "coords": [100, 21, 356, 93], "fig_type": "figure"}]], ["block_5", ["Polymer\nChemistry\n"]], ["block_6", ["Second Edition\n"]]], "page_2": [["block_0", [{"image_0": "2_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "2_1.png", "coords": [83, 91, 373, 156], "fig_type": "figure"}]], ["block_2", [{"image_2": "2_2.png", "coords": [84, 58, 384, 204], "fig_type": "figure"}]], ["block_3", [{"image_3": "2_3.png", "coords": [93, 17, 351, 161], "fig_type": "figure"}]], ["block_4", ["Polymer\nChemistry\n"]], ["block_5", [{"image_4": "2_4.png", "coords": [104, 21, 356, 90], "fig_type": "figure"}]], ["block_6", [{"image_5": "2_5.png", "coords": [107, 362, 344, 447], "fig_type": "figure"}]], ["block_7", ["Paul C. Hiemenz\nTimothy R Lodge\n"]], ["block_8", ["Second Edition\n"]], ["block_9", ["Taylor & Francis Group, an informa business\n"]], ["block_10", ["CRC Press is an imprint of the\n"]], ["block_11", ["CRC Press\nTaylor & Francis Group\n"]], ["block_12", ["Boca Raton\nLondon\nNew York\n"]]], "page_3": [["block_0", [{"image_0": "3_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["CRC Press\nTaylor & Francis Group\n6000 Broken Sound Parkway NW, Suite 300\nBoca Raton, FL 33487-2742\n"]], ["block_2", ["This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted\nwith permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to\npublish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of\nall materials or for the consequences of their use.\n"]], ["block_3", ["For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:/l\nwww.copyright.coml) or contact the C0pyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923,\n978-750-8400. CCC is a not\u2014for\u2014profit organization that provides licenses and registration for a variety of users. For orga-\nnizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.\n"]], ["block_4", ["No claim to original U.S. Government works\nPrinted in the United States of America on acid-free paper\n10 9 8 7\n"]], ["block_5", ["No part ofthis book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or\nother means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any informa\u2014\ntion storage or retrieval system, without written permission from the publishers.\n"]], ["block_6", ["\u00a9 2007 by Taylor 8: Francis Group, LLC\nCRC Press is an imprint of Taylor 8: Francis Group, an Informa business\n"]], ["block_7", ["Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for\nidentification and explanation without intent to infringe.\n"]], ["block_8", ["and the CRC Press Web site at\nhttp://www.crcpress.com\n"]], ["block_9", ["Visit the Taylor 8; Francis Web site at\nhttp://www.taylorandfrancis.com\n"]], ["block_10", ["International Standard Book Number-10: 1\u201457444\u2014779\u20143 (Hardcover)\nInternational Standard Book Number\u201413: 978\u20141\u201457444-779\u20148 (Hardcover)\n"]], ["block_11", ["QD381.H52 2007\n547.7--dc22\n2006103309\n"]], ["block_12", ["Hiemenz, Paul C., 1936\u2014\nPolymer chemistry / Paul C. Hiemenz and Tim Lodge. 2nd ed.\np. cm.\nIncludes bibliographical references and index.\nISBN-13: 978-1-57444\u2014779v8 (alk. paper)\nISBN-10: 1\u201457444\u2014779\u20143 (alk. paper)\n1. Polymers. 2. Polymerization. I. Lodge, Tim. 11. Title.\n"]], ["block_13", ["Library of Congress Cataloging-in-Publication Data\n"]]], "page_4": [["block_0", [{"image_0": "4_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["essential, and are introduced as needed.\nThe philosophy underlying the approach in this book is the same as that in the first edition, as\nlaid out in the previous preface. Namely, we endeavor to develop the fundamental principles,\nrather than an encyclopedic knowledge of particular polymers and their applications; we seek to\nbuild a molecular understanding of polymer synthesis, characterization, and properties; we\nemphasize those phenomena (from the vast array of possibilities) that we judge to be the most\ninteresting. The text has been extensively reorganized and expanded, largely to re\ufb02ect the\nsubstantial advances that have occurred over the intervening years. For example, there is now\nan entire chapter (Chapter 4) dedicated to the topic of controlled polymerization, an area that has\nrecently undergone a revolution. Another chapter (Chapter 11) delves into the viscoelastic\nproperties of polymers, a topic where theoretical advances have brought deeper understanding.\nThe book also serves as a bridge into the research literature. After working through the appropriate\nchapters, the student should be able to make sense of a large fraction of the articles published today\nin polymer science journals.\nThere is more than enough material in this book for a full-year graduate level course, but as with\nthe first edition, the level is (almost) always accessible to senior level undergraduates. After an\nintroductory chapter of broad scope, the bulk of the text may be grouped into three blocks of four\nchapters each. Chapter 2 through Chapter 5 describe the many ways in which polymers can be\nsynthesized and how the synthetic route in\ufb02uences the resulting molecular structure. This material\ncould serve as the basis for a single quarter or semester chemistry course that focuses on polymer\nsynthesis. Chapter 6 through Chapter 9 emphasize the solution properties of polymers, including\ntheir conformations, thermodynamics, hydrodynamics, and light scattering properties. Much of this\nmaterial is often found in a quarter or semester course introducing the physical chemistry of\npolymers. Chapter 10 through Chapter 13 address the solid state and bulk properties of polymers:\nrubber elasticity, viscoelasticity, the glass transition, and crystallization. These topics, while\npresented here from a physical chemical point of view, could equally well serve as the cornerstone\nof an introductory course in materials science or chemical engineering.\nThe style of the presentation, as with the previous edition, is chosen with the student in mind. To\nthis end, we may point out the following features:\n"]], ["block_2", ["\u00b0\nThere are over 60 worked example problems sprinkled throughout the book.\n\u00b0\nThere are 15 or more problems at the end of every chapter, to reinforce and develop further\nunderstanding; many of these are based on data from the literature.\n"]], ["block_3", ["Polymer science is today a vibrant field. Its technological relevance is vast, yet fundamental\nscientific questions also abound. Polymeric materials exhibit a wealth of fascinating properties,\nmany of which are observable just by manipulating a piece in your hands. Yet, these phenomena\nare all directly traceable to molecular behavior, and especially to the long chain nature of polymer\nmolecules. The central goal of this book is to develop a molecular level understanding of the\nproperties of polymers, beginning with the underlying chemical structures, and assuming no prior\nknowledge beyond undergraduate organic and physical chemistry. Although such an understand-\ning should be firmly based in chemistry, polymer science is a highly interdisciplinary endeavor;\nconcepts from physics, biology, materials science, chemical engineering, and statistics are all\n"]], ["block_4", ["Preface to the Second Edition\n"]], ["block_5", ["V\n"]]], "page_5": [["block_0", [{"image_0": "5_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Jingshan Dong, Will Edmonds, Sandra Fritz, Carolyn Gamble, Piotr Grzywacz, Jeong-Myeong Ha,\nBenjamin Hamilton, Amanda Haws, Nazish Hoda, Hao Hou, Deanna Huehn, Shengxiang Ji, Karan\nJindal, Young Kang, Aaron Khieu, Byeong-Su Kim, BongSoo Kim, Hyunwoo Kim, Jin-Hong\nKim, Seung Ha Kim, Chunze Lai, Castro Laicer, Qiang Lan, Sangwoo Lee, Zhibo Li, Elizabeth\nLugert, Nate Lynd, Sudeep Maheshwari, Huiming Mao, Adam Meuler, Yoichiro Mori, Randy\nMrozek, Siddharthya Mujumdar, Jaewook Nam, Dan O\u2019Neal, Sahban Ozair, Matt Panzer, Alhad\nPhatak, William Philip, Jian Qin, Benjamin Richter, Scott Roberts, Josh Scheffel, Jessica Schom-\nmer, Kathleen Schreck, Peter Simone, Zach Thompson, Kristianto Tjiptowidjojo, Mehul Vora,\nJaye Warner, Tomy Widya, Maybelle Wu, Jianyan Xu, Dan Yu, Ilan Zeroni, Jianbin Zhang, Ling\nZhang, Yu Zhang, Ning Zhou, Zhengxi Zhu, and John Zupancich. Last but not least the\nlove, support, and tolerance of my family, Susanna, Hannah, and Sam, has been a constant source\nof strength.\n"]], ["block_2", ["An undertaking such as writing a textbook can never be completed without important contributions\nfrom many individuals. Large sections of manuscript were carefully typed by Becky Matsch and\nLynne Johnsrud; Lynne also helped greatly with issues of copyright permissions and figure prepar-\nation. My colleagues past and present in the Polymer Group at Minnesota have been consistently\nencouraging and have provided both useful feedback and insightful examples: Frank Bates, Shura\nGrosberg, Marc Hillmyer, Chris Macosko, Wilmer Miller, David Morse, Steve Prager, and Matt\nTirrell. In large measure the style adopted in this second edition has been inspired by the example set\nby my graduate instructors and mentors at the University of Wisconsin: R. Byron Bird, John Perry,\nArthur Lodge, John Schrag, and Hyuk Yu. In particular, it was in his graduate course Chemistry 664\nthat Hyuk Yu so ably demonstrated that no important equation need come out of thin air.\nI would like extend a special thank you to all of the students enrolled in Chemistry/Chemical\nEngineering/Materials Science 8211 over the period 2002\u20142005, who worked through various\ndrafts of Chapter 6 through Chapter 13, and provided many helpful suggestions: Sayeed Abbas,\nDavid Ackerman, Sachin Agarwal, Saurabh Agarwal, Julie Alkatout, Pedro Arrechea, Carlos\nLopez-Barron, Soumendra Basu, Jeff Becker, Joel Bell, A.S. Bhalla, Michael Bluemle, Paul\nBoswell, Bryan Boudouris, Adam Buckalew, Xiuyu Cai, Neha Chandra, Joon Chatterjee, Liang\nChen, Ying Chen, Juhee Cho, Seongho Choi, Jin-Hwa Chung, Kevin Davis, Michail Dolgovskij,\n"]], ["block_3", ["\u00b0\nThere are almost 200 figures, to illustrate concepts or to present experimental results from\nthe literature.\n0\nStudies chosen for the examples, problems, and figures range in vintage from very recent to\nover 50 years old; this feature serves to give the reader some sense of the historical progression\nof the field.\n\u00b0\nConcise reviews of many topics (such as thermodynamics, kinetics, probability, and various\nexperimental techniques) are given when the subject is first raised.\n\u00b0\nA conscious effort has been made to cross-reference extensively between chapters and sections\nwithin chapters, in order to help tie the various tOpics together.\n\u00b0\nImportant equations and mathematical relations are almost always developed step by step. We\nhave avoided, wherever possible, the temptation to pull equations out of a hat. Occasionally\nthis leads to rather long stretches of algebra, which the reader is welcome to skip. However, at\nsome point the curious student will want to know where the result comes from, and then this\nbook should be a particularly valuable resource. Surprisingly, perhaps, the level of mathemat\u2014\nical sophistication is only about the same as needed in undergraduate chemical thermodynam-\nics. As a further help in this regard, an Appendix reviews many of the important mathematical\ntools and tricks.\n"]], ["block_4", ["vi\nPreface to the 2nd Edition\n"]], ["block_5", ["Tim Lodge\n"]]], "page_6": [["block_0", [{"image_0": "6_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Physical chemistry has been defined as that branch of science that is fundamental, molecular,\nand interesting. I have tried to write a polymer textbook that could be described this way also. To\nthe extent that one subscribes to the former definition and that I have succeeded in the latter\nobjective, then the approach of this book is physical chemical. As a textbook, it is intended for\nstudents who have completed courses in physical and organic chemistry. These are the prerequi\u2014\nsites that de\ufb01ne the level of the book; no special background in physics or mathematics beyond\nwhat is required for physical chemistry is assumed. Since chemistry majors generally study\nphysical chemistry in the third year of the undergraduate curriculum, this book can serve as the\ntext for a senior\u2014level undergraduate or a beginning graduate\u2014level course. Although I use\nchemistry courses and chemistry curricula to describe the level of this book, students majoring\nin engineering, materials science, physics, and various specialties in the biological sciences will\nalso find numerous topics of interest contained herein.\nTerms like \u201cfundamental,\u201d \u201cmolecular,\u201d and \u201cinteresting\u201d have different meanings for\ndifferent people. Let me explain how they apply to the presentation of polymer chemistry in this\ntext.\nThe words \u201cbasic concepts\u201d in the title define what I mean by \u201cfundamental.\u201d This is the\nprimary emphasis in this presentation. Practical applications of polymers are cited frequently\u2014\nafter all, it is these applications that make polymers such an important class of chemicals\u2014but in\noverall content, the stress is on fundamental principles. \u201cFoundational\u201d might be another way to\ndescribe this. I have not attempted to cover all aspects of polymer science, but the topics that have\nbeen discussed lay the foundation\u2014built on the bedrock of organic and physical chemistry\u2014from\nwhich virtually all aspects of the subject are developed. There is an enormous literature in polymer\nscience; this book is intended to bridge the gap between the typical undergraduate background in\npolymers\u2014which frequently amounts to little more than occasional \u201crelevant\u201d examples in other\ncourses\u2014and the professional literature on the subject. Accordingly, the book assumes essentially no\nprior knowledge of polymers, and extends far enough to provide a usable level of understanding.\n\u201cMolecular\u201d describes the perspective of the chemist, and it is this aspect of polymeric\nmaterials that I try to keep in view throughout the book. An engineering text might emphasize\nprocessing behavior; a physics text, continuum mechanics; a biochemistry text, physiological\nfunction. All of these are perfectly valid points of view, but they are not the approach of this\nbook. It is polymer molecules\u2014their structure, energetics, dynamics, and reactions\u2014that are the\nprimary emphasis throughout most of the book. Statistics is the type of mathematics that is natural\nto a discussion of molecules. Students are familiar with the statistical nature of, say, the kinetic\nmolecular theory of gases. Similar methods are applied to other assemblies of molecules, or in the\ncase of polymers, to the assembly of repeat units that comprise a single polymer molecule.\nAlthough we frequently use statistical arguments, these are developed quite thoroughly and do\nnot assume any more background in this subject than is ordinarily found among students in a\nphysical chemistry course.\n"]], ["block_2", ["Preface to the First Edition\n"]], ["block_3", ["vii\n"]]], "page_7": [["block_0", [{"image_0": "7_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["The book is divided into three parts of three chapters each, after an introductory chapter which\ncontains information that is used throughout the book.\nIn principle, the three parts can be taken up in any order without too much interruption in\ncontinuity. Within each of the parts there is more carryover from chapter to chapter, so rearranging\nthe sequence of topics within a given part is less convenient. The book contains more material than\ncan be covered in an ordinary course. Chapter 1 plus two of the three parts contain about the right\namount of material for one term. In classroom testing the material, I allowed the class to decide\u2014\nwhile we worked on Chapter l\u2014which two of the other parts they wished to cover; this worked\nvery well.\nMaterial from Chapter 1 is cited throughout the book, particularly the discussion of statistics. In\nthis connection, it might be noted that statistical arguments are developed in less detail further\nalong in the book as written. This is one of the drawbacks of rearranging the order in which the\ntopics are covered. Chapters 2 through 4 are concerned with the mechanical properties of bulk\npolymers, properties which are primarily responsible for the great practical importance of poly-\nmers. Engineering students are likely to have both a larger interest and a greater familiarity with\nthese topics. Chapers 5 through 7 are concerned with the preparation and properties of several\nbroad classes of polymers. These topics are closer to the interests of chemistry majors. Chapters 8\nthrough 10 deal with the solution properties of polymers. Since many of the techniques described\nhave been applied to biopolymers, these chapters will have more appeal to students of biochem\u2014\nistry and molecular biology.\nLet me conclude by acknowledging the contributions of those who helped me with the\npreparation of this book. I wish to thank Marilyn Steinle for expertly typing the manuscript. My\nappreciation also goes to Carol Truett who skillfully transformed my (very) rough sketches into\neffective illustrations. Lastly, my thanks to Ron Manwill for preparing the index and helping me\nwith the proofreading. Finally, let me acknowledge that some errors and/or obscurities will surely\nelude my efforts to eliminate them. I would appreciate reports about these from readers so that\nthese mistakes can eventually be eliminated.\n"]], ["block_2", ["1.\nOver 50 solved example problems are sprinkled throughout the book.\n2.\nExercises are included at the end of each chapter which are based on data from the original\nliterature.\n3.\nConcise reviews of pertinent aspects of thermodynamics, kinetics, spectrophotometry, etc. are\npresented prior to developing applications of these topics to polymers.\n4.\nTheoretical models and mathematical derivations are developed in enough detail to be\ncomprehensible to the student reader. Only rarely do I \u201cpull results out of a hat,\u201d and I\nscrupulously avoid saying \u201cit is obvious that .. .\u201d\n5.\nGenerous cross\u2014referencing and a judicious amount of repetition have been included to help\nunify a book which spans quite a wide range of topics.\n"]], ["block_3", ["The most subjective of the words which (I hope) describe this book is \n\u2018interesting.\u201d The\nfascinating behavior of polymers themselves, the clever experiments of laboratory researchers, and\nthe elegant work of the theoreticians add up to an interesting total. I have tried to tell about these\ntopics with clarity and enthusiasm, and in such a way as to make them intelligible to students. I can\nonly hope that the reader agrees with my assessment of what is interesting.\nThis book was written with the student in mind. Even though \u201cstudent\u201d encompasses persons\nwith a wide range of backgrounds, interests, and objectives; these are different than the corre\u2014\nsponding experiences and needs of researchers. The following features have been to assist\nthe student:\n"]], ["block_4", ["viii\nPreface to the lst Edition\n"]], ["block_5", ["6.\nSI units have been used fairly consistently throughout, and attention is paid to the matter of\nunits whenever these become more than routine in complexity.\n"]], ["block_6", ["Paul C. Hiemenz\n"]]], "page_8": [["block_0", [{"image_0": "8_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["1\n"]], ["block_2", ["1.7.1\nNumber-, Weight\u2014, and z\u2014Average Molecular Weights .................................. 25\n1.7.2\nPolydispersity Index and Standard Deviation .................................................26\n1.7.3\nExamples of Distributions ...............................................................................28\n1.8\nMeasurement of Molecular Weight............................................................................. 31\n1.8.1\nGeneral Considerations ....................................................................................31\n1.8.2\nEnd Group Analysis .........................................................................................32\n1.8.3\nMALDI Mass Spectrometry ............................................................................ 35\n1.9\nPreview of Things to Come......................................................................................... 37\n1.10\nChapter Summary ........................................................................................................ 38\nProblems .................................................................................................................................. 38\nReferences ................................................................................................................................ 41\nFurther Readings ...................................................................................................................... 41\n"]], ["block_3", ["Step-Growth Polymerization ...................................................................... 43\n"]], ["block_4", ["2.1\nIntroduction .................................................................................................................... 43\n2.2\nCondensation Polymers: One Step at a Time ............................................................... 43\n2.2.1\nClasses of Step\u2014Growth Polymers ..................................................................... 43\n2.2.2\nFirst Look at the Distribution of Products ........................................................44\n2.2.3\nA First Look at Reactivity and Reaction Rates ................................................46\n"]], ["block_5", ["Introduction to Chain Molecules ................................................................. 1\n"]], ["block_6", ["1.4\nAddition, Condensation, and Natural Polymers .......................................................... 11\n1.4.1\nAddition and Condensation Polymers ............................................................. 11\n1.4.2\nNatural Polymers ............................................................................................. 13\n1.5\nPolymer Nomenclature ................................................................................................ 18\n1.6\nStructural Isomerism .................................................................................................... 20\n1.6.1\nPositional Isomerism........................................................................................20\n1.6.2\nStereo Isomerism ............................................................................................. 21\n1.6.3\nGeometrical Isomerism .................................................................................... 22\n1.7\nMolecular Weights and Molecular Weight Averages ................................................. 24\n"]], ["block_7", ["1. 1\nIntroduction .................................................................................................................... 1\n1.2\nHow Big Is Big? ............................................................................................................ 3\n1.2.1\nMolecular Weight .............................................................................................. 3\n1.2.2\nSpatial Extent ..................................................................................................... 5\n1.3\nLinear and Branched Polymers, Homopolymers, and Copolymers ............................. 7\n1.3.1\nBranched Structures ...........................................................................................7\n"]], ["block_8", ["1 .3.2\nCopolymers ........................................................................................................9\n"]], ["block_9", ["Contents\n"]]], "page_9": [["block_0", [{"image_0": "9_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["3.1\nIntroduction .................................................................................................................... 77\n3.2\nChain-Growth and Step-Growth Polymerizations: Some Comparisons ....................... 77\n3.3\nInitiation ......................................................................................................................... 79\n3.3.1\nInitiation Reactions ............................................................................................ 80\n3.3.2\nFate of Free Radicals ......................................................................................... 81\n3.3.3\nKinetics of Initiation .......................................................................................... 82\n3.3.4\nPhotochemical Initiation....................... ............................................................. 84\n3.3.5\nTemperature Dependence of Initiation Rates.................................................... 85\n3.4\nTermination .................................................................................................................... 86\n3.4.1\nCombination and Disproportionation ................................................................ 86\n3.4.2\nEffect of Termination on Conversion to Polymer ............................................ 88\n3.4.3\nStationary-State Radical Concentration ............................................................ 89\n3.5\nPropagation .................................................................................................................... 90\n3.5.1\nRate Laws for Propagation ................................................................................91\n3.5.2\nTemperature Dependence of Propagation Rates ...............................................92\n3.5.3\nKinetic Chain Length.........................................................................................94\n3.6\nRadical Lifetime ............................................................................................................ 96\n3.7\nDistribution of Molecular Weights................................................................................ 99\n3.7.1\nDistribution of i\u2014mers: Termination by Disproportionation .............................99\n3.7.2\nDistribution of i-mers: Termination by Combination ..................................... 102\n3.8\nChain Transfer ............................................................................................................. 104\n3.8.1\nChain Transfer Reactions ................................................................................ 105\n3.8.2\nEvaluation of Chain Transfer Constants ......................................................... 106\n3.8.3\nChain Transfer to Polymer .............................................................................. 108\n3.8.4\nSuppressing Polymerization ............................................................................ 109\n3.9\nChapter Summary ........................................................................................................ 110\nProblems ................................................................................................................................ 110\nReferences.............................................................................................................................. 1 14\n.\nFurther Readings .................................................................................................................... I 15\n"]], ["block_2", ["2.3\nKinetics of Step\u2014Growth Polymerization ...................................................................... 49\n2.3.1\nCatalyzed Step\u2014Growth Reactions ..................................................................... 50\n2.3.2\nHow Should Experimental Data Be Compared with\nTheoretical Rate Laws? ..................................................................................... 52\n2.3.3\nUncatalyzed Step-Growth Reactions .................................................................53\n2.4\nDistribution of Molecular Sizes .................................................................................... 55\n2.4.1\nMole Fractions of Species ................................................................................. 56\n2.4.2\nWeight Fractions of Species .............................................................................. 58\n2.5\nPolyesters ....................................................................................................................... 60\n2.6\nPolyamides ..................................................................................................................... 64\n2.7\nStoichiometric Imbalance .............................................................................................. 67\n2.8\nChapter Summary .......................................................................................................... 71\nProblems .................................................................................................................................. 71\nReferences................................................................................................................................ 76\nFurther Readings ............................ 76\n"]], ["block_3", ["Chain-Growth Polymerization....................................................................77\n"]], ["block_4", ["Contents\n"]]], "page_10": [["block_0", [{"image_0": "10_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["4\nControlled Polymerization ....................................................................... 117\n"]], ["block_2", ["Contents\nxi\n"]], ["block_3", ["5\nCopolymers, Microstructure, and Stereoregularity................................ 165\n"]], ["block_4", ["4.1\nIntroduction ................................................................................................................ 1 17\n4.2\nPoisson Distribution for an Ideal Living Polymerization ......................................... 118\n4.2.1\nKinetic Scheme .............................................................................................. 119\n4.2.2\nBreadth of the Poisson Distribution .............................................................. 122\n4.3\nAnionic Polymerization ............................................................................................. 126\n4.4\nBlock Copolymers, End\u2014Functional Polymers, and Branched\nPolymers by Anionic Polymerization........................................................................ 129\n4.4.1\nBlock Copolymers ......................................................................................... 129\n4.4.2\nEnd-Functional Polymers .............................................................................. 133\n4.4.3\nRegular Branched Architectures .................................................................... 135\n4.5\nCationic Polymerization ............................................................................................ 137\n4.5.1\nAspects of Cationic Polymerization .............................................................. 138\n4.5.2\nLiving Cationic Polymerization .................................................................... 140\n4.6\nControlled Radical Polymerization ........................................................................... 142\n4.6.1\nGeneral Principles of Controlled Radical Polymerization............................ 142\n4.6.2\nParticular Realizations of Controlled Radical Polymerization ..................... 144\n4.6.2.1\nAtom Transfer Radical Polymerization (ATRP)........................... 144\n4.6.2.2\nStable Free-Radical Polymerization (SFRP) ................................. 145\n4.6.2.3\nReversible Addition-Fragmentation Transfer (RAFT)\nPolymerization ............................................................................... 146\n4.7\nPolymerization Equilibrium....................................................................................... 147\n4.8\nRing-Opening Polymerization (ROP) ....................................................................... 150\n4.8.1\nGeneral Aspects ............................................................................................. 150\n4.8.2\nSpecific Examples of Living Ring\u2014Opening Polymerizations ...................... 152\n4.8.2.1\nPoly(ethy1ene oxide) ...................................................................... 152\n4.8.2.2\nPolylactide...................................................................................... 153\n4.8.2.3\nPoly(dimethylsiloxane) .................................................................. 154\n4.8.2.4\nRing-Opening Metathesis Polymerization (ROMP) ..................... 155\n4.9\nDendrimers ................................................................................................................. 156\n4.10\nChapter Summary ...................................................................................................... 160\nProblems ................................................................................................................................ 161\nReferences .............................................................................................................................. 163\nFurther Readings.................................................................................................................... 163\n"]], ["block_5", ["5.1\nIntroduction ................................................................................................................ 165\n5.2\nCopolymer Composition ............................................................................................ 166\n5.2.1\nRate Laws ...................................................................................................... 166\n5.2.2\nComposition versus Feedstock ...................................................................... 168\n5.3\nReactivity Ratios ........................................................................................................ 170\n5.3.1\nEffects of r Values ......................................................................................... 171\n5.3.2\nRelation of Reactivity Ratios to Chemical Structure.................................... 173\n5.4\nResonance and Reactivity.......................................................................................... 175\n5.5\nA Closer Look at Microstructure .............................................................................. 179\n5.5.1\nSequence Distributions .................................................................................. 180\n5.5.2\nTerminal and Penultimate Models ................................................................ 183\n"]]], "page_11": [["block_0", [{"image_0": "11_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["xii\nContents\n"]], ["block_2", ["5.6\nCOpolymer Composition and Microstructure: Experimental Aspects ...................... 185\n5.6.1\nEvaluating Reactivity Ratios from Composition Data ................................. 185\n5.6.2\nSpectroscopic Techniques.............................................................................. 188\n5.6.3\nSequence Distribution: Experimental Determination ................................... 190\n5.7\nCharacterizing Stereoregularity ................................................................................. 193\n5.8\nA Statistical Description of Stereoregularity ............................................................ 196\n5.9\nAssessing Stereoregularity by Nuclear Magnetic Resonance ................................... 200\n5.10\nZiegler\u2014Natta Catalysts ............................................................................................. 205\n5.11\nSingle-Site Catalysts .................................................................................................. 208\n5.12\nChapter Summary ...................................................................................................... 211\nProblems ................................................................................................................................ 212\nReferences .............................................................................................................................. 216\nFurther Readings .................................................................................................................... 216\n"]], ["block_3", ["6.1\nConformations, Bond Rotation, and Polymer Size ..................................................... 217\n6.2\nAverage End\u2014to-End Distance for Model Chains ....................................................... 219\nCase 6.2.1\nThe Freely Jointed Chain ....................................................................... 220\nCase 6.2.2\nThe Freely Rotating Chain ..................................................................... 221\nCase 6.2.3\nHindered Rotation Chain ........................................................................ 222\n6.3\nCharacteristic Ratio and Statistical Segment Length.................................................. 223\n6.4\nSemi\ufb02exible Chains and the Persistence Length ........................................................ 225\n6.4.1\nPersistence Length of Flexible Chains ............................................................ 227\n6.4.2\nWorm-Like Chains ..........................................................................................228\n6.5\nRadius of Gyration....................................................................................................... 230\n6.6\nSpheres, Rods, and Coils ............................................................................................. 234\n6.7\nDistributions for End\u2014to-End Distance and Segment Density .................................... 235\n6.7.1\nDistribution of the End-to\u2014End Vector ............................................................ 236\n6.7.2\nDistribution of the End-to\u2014End Distance.........................................................239\n6.7.3\nDistribution about the Center of Mass ............................................................ 240\n6.8\nSelf-Avoiding Chains: A First Look ........................................................................... 241\n6.9\nChapter Summary ........................................................................................................ 242\nProblems ................................................................................................................................ 242\nReferences .............................................................................................................................. 244\nFurther Readings .................................................................................................................... 245\n"]], ["block_4", ["7.1\nReview of Thermodynamic and Statistical Thermodynamic Concepts ..................... 247\n7.2\nRegular Solution Theory ............................................................................................. 249\n7.2.1\nRegular Solution Theory: Entropy of Mixing .................................................249\n7.2.2\nRegular Solution Theory: Enthalpy of Mixing ............................................... 251\n7.3\nFlory\u2014Huggins Theory ................................................................................................. 254\n7.3.1\nFlory\u2014Huggins Theory: Entropy of Mixing by a Quick Route ......................255\n7.3.2\nFlory\u2014Huggins Theory: Entropy of Mixing by a Longer Route .................... 255\n7.3.3\nFlory\u2014Huggins Theory: Enthalpy of Mixing .................................................. 257\n7.3.4\nFlory\u2014Huggins Theory: Summary of Assumptions ........................................ 258\n"]], ["block_5", ["Thermodynamics of Polymer Solutions ................................................. 247\n"]], ["block_6", ["Polymer Conformations ........................................................................... 217\n"]]], "page_12": [["block_0", [{"image_0": "12_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["8\nLight Scattering by Polymer Solutions ................................................... 289\n"]], ["block_2", ["9\nDynamics of Dilute Polymer Solutions ................................................... 327\n"]], ["block_3", ["Contents\nxiii\n"]], ["block_4", ["8.1\nIntroduction: Light Waves ........................................................................................... 289\n8.2\nBasic Concepts of Scattering ....................................................................................... 291\n8.2.1\nScattering from Randomly Placed Objects .....................................................292\n8.2.2\nScattering from a Perfect Crystal ....................................................................292\n8.2.3\nOrigins of Incoherent and Coherent Scattering ..............................................293\n8.2.4\nBragg\u2019s Law and the Scattering Vector ..........................................................294\n8.3\nScattering by an Isolated Small Molecule .................................................................. 296\n8.4\nScattering from a Dilute Polymer Solution................................................................. 298\n8.5\nThe Form Factor and the Zimm Equation .................................................................. 304\n8.5.1\nMathematical Expression for the Form Factor ...............................................305\n8.5.2\nForm Factor for Isotropic Solutions ................................................................ 306\n8.5.3\nForm Factor as n\u2014>0 .................................................................................... 307\n8.5.4\nZimm Equation ................................................................................................ 307\n8.5.5\nZimm Plot ........................................................................................................ 308\n8.6\nScattering Regimes and Particular Form Factors........................................................ 312\n8.7\nExperimental Aspects of Light Scattering .................................................................. 314\n8.7. 1\nInstrumentation ................................................................................................ 3 16\n8.7.2\nCalibration ........................................................................................................317\n8.7.3\nSamples and Solutions ..................................................................................... 319\n8.7.4\nRefractive Index Increment ............................................................................. 319\n8.8\nChapter Summary ........................................................................................................ 320\nProblems ................................................................................................................................ 321\nReferences .............................................................................................................................. 325\nFurther Readings .................................................................................................................... 325\n"]], ["block_5", ["9.1\nIntroduction: Friction and Viscosity............................................................................ 327\n9.2\nStokes\u2019 Law and Einstein\u2019s Law ................................................................................. 330\n"]], ["block_6", ["7.4\nOsmotic Pressure ......................................................................................................... 258\n7.4.1\nOsmotic Pressure: General Case .....................................................................259\n7.4.1.1\nNumber\u2014Average Molecular Weight ............................................... 261\n7.4.2\nOsmotic Pressure: Flory\u2014Huggins Theory ......................................................263\n7.5\nPhase Behavior of Polymer Solutions ......................................................................... 264\n7.5.1\nOverview of the Phase Diagram...................................................................... 265\n7.5.2\nFinding the Binodal .........................................................................................268\n7.5.3\nFinding the Spinodal ........................................................................................269\n7.5.4\nFinding the Critical Point ................................................................................270\n7.5.5\nPhase Diagram from Flory\u2014Huggins Theory ..................................................271\n7.6\nWhat\u2019s in x? ................................................................................................................. 275\n7.6.1\nX from Regular Solution Theory .....................................................................275\n7.6.2\nX from Experiment...........................................................................................276\n7.6.3\nFurther Approaches to X ..................................................................................278\n7.7\nExcluded Volume and Chains in a Good Solvent ...................................................... 280\n7.8\nChapter Summary ........................................................................................................ 283\nProblems ................................................................................................................................ 284\nReferences .............................................................................................................................. 287\nFurther Readings.................................................................................................................... 288\n"]]], "page_13": [["block_0", [{"image_0": "13_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["xiv\n"]], ["block_2", ["10\n"]], ["block_3", ["9.2.1\nViscous Forces on Rigid Spheres .............................................................331\n9.2.2\nSuspension of Spheres............................................................................... 332\n9.3\nIntrinsic Viscosity ................................................................................................... 334\n9.3.1\nGeneral Considerations ............................................................................. 334\n9.3.2\nMark\u2014Houwink Equation .......................................................................... 336\n9.4\nMeasurement of Viscosity ...................................................................................... 341\n9.4.1\nPoiseuille Equation and Capillary Viscometers .......................................341\n9.4.2\nConcentric Cylinder Viscometers ............................................................. 345\n9.5\nDiffusion Coefficient and Friction Factor .............................................................. 346\n9.5.1\nTracer Diffusion and Hydrodynamic Radius............................................ 347\n9.5.2\nMutual Diffusion and Fick\u2019s Laws ........................................................... 348\n9.6\nDynamic Light Scattering ....................................................................................... 354\n9.7\nHydrodynamic Interactions and Draining .............................................................. 357\n9.8\nSize Exclusion Chromatography (SEC) ................................................................. 360\n9.8.1\nBasic Separation Process .......................................................................... 361\n9.8.2\nSeparation Mechanism .............................................................................. 365\n9.8.3\nTwo Calibration Strategies ....................................................................... 367\n9.8.3.1\nLimitations of Calibration by Standards................................... 367\n9.8.3.2\nUniversal Calibration ................................................................ 368\n9.8.4\nSize Exclusion Chromatography Detectors .............................................. 369\n9.8.4.1\nRI Detector ................................................................................ 369\n9.8.4.2\nUV\u2014Vis Detector ....................................................................... 370\n9.8.4.3\nLight Scattering Detector.......................................................... 371\n9.8.4.4\nViscometer................................................................................. 372\n9.9\nChapter Summary ...................................................................................................... 372\nProblems .............................................................................................................................. 373\nReferences ........................................................................................................................... 378\nFurther Readings ................................................................................................................. 379\n"]], ["block_4", ["Networks, Gels, and Rubber Elasticity\n3\n"]], ["block_5", ["10.6\n"]], ["block_6", ["10.1\n"]], ["block_7", ["10.2\n"]], ["block_8", ["10.3\n10.4\n"]], ["block_9", ["10.5\n"]], ["block_10", ["Formation of Networks by Random Cross-Linking ............................................... 381\n10.1.1\nDefinitions ................................................................................................. 381\n10.1.2\nGel Point.................................................................................................... 383\nPolymerization with Multifunctional Monomers ................................................... 386\n10.2.1\nCalculation of the Branching Coefficient.................................................387\n10.2.2\nGel Point....................................................................................................388\n10.2.3\nMolecular-Weight Averages ..................................................................... 389\nElastic Deformation ................................................................................................ 392\nThermodynamics of Elasticity ................................................................................ 394\n10.4.1\nEquation of State .......................................................................................394\n10.4.2\nIdeal Elastomers ........................................................................................396\n10.4.3\nSome Experiments on Real Rubbers ........................................................397\nStatistical Mechanical Theory of Rubber Elasticity: Ideal Case ........................... 398\n10.5.1\nForce to Extend a Gaussian Chain ...........................................................400\n10.5.2\nNetwork of Gaussian Strands....................................................................402\n10.5.3\nModulus of the Gaussian Network ...........................................................403\nFurther Developments in Rubber Elasticity ........................................................... 406\n10.6.1\nNon-Gaussian Force Law..........................................................................406\n10.6.2\nFront Factor ...............................................................................................407\n"]], ["block_11", ["Contents\n"]]], "page_14": [["block_0", [{"image_0": "14_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Contents\nxv\n"]], ["block_2", ["11\n"]], ["block_3", ["12\n"]], ["block_4", ["1 1.7.1\nReptation Model: Longest Relaxation Time and Diffusivity...................451\n11.7.2\nReptation Model: Viscoelastic Properties ................................................453\n11.7.3\nReptation Model: Additional Relaxation Processes ................................. 456\n11.8\nAspects of Experimental Rheometry ...................................................................... 458\n11.8.1\nShear Sandwich and Cone and Plate Rheometers ....................................45 8\n11.8.2\nFurther Comments about Rheometry ........................................................459\n11.9\nChapter Summary .................................................................................................... 460\nProblems .............................................................................................................................. 461\nReferences ........................................................................................................................... 464\nFurther Readings ................................................................................................................. 464\n"]], ["block_5", ["Glass Transition ...................................................................................... 465\n"]], ["block_6", ["10.6.3\nNetwork Defects ........................................................................................408\n10.6.4\nMooney\u2014Rivlin Equation ..........................................................................409\n10.7\nSwelling of Gels ...................................................................................................... 410\n10.7.1\nModulus of a Swollen Rubber ..................................................................411\n10.7.2\nSwelling Equilibrium ................................................................................412\n10.8\nChapter Summary .................................................................................................... 414\nProblems .............................................................................................................................. 416\nReferences ........................................................................................................................... 4 18\nFurther Readings ................................................................................................................. 418\n"]], ["block_7", ["Linear Viscoelasticity ............................................................................. 419\n"]], ["block_8", ["11.1\nBasic Concepts ........................................................................................................ 419\n11.1.1\nStress and Strain ........................................................................................421\n11.1.2\nViscosity, Modulus, and Compliance .......................................................421\n11.1.3\nViscous and Elastic Responses .................................................................422\n11.2\nResponse of the Maxwell and Voigt Elements ...................................................... 423\n11.2.1\nTransient Response: Stress Relaxation .....................................................423\n11.2.2\nTransient Response: Creep........................................................................425\n11.2.3\nDynamic Response: Loss and Storage Moduli.........................................426\n11.2.4\nDynamic Response: Complex Modulus and Complex Viscosity ............429\n11.3\nBoltzmann Superposition Principle ........................................................................ 430\n11.4\nBead\u2014Spring Model................................................................................................. 432\n11.4.1\nIngredients of the Bead\u2014Spring Model .....................................................432\n11.4.2\nPredictions of the Bead\u2014Spring Model .....................................................434\n11.5\nZimm Model for Dilute Solutions, Rouse Model\nfor Unentangled Melts ............................................................................................ 439\n11.6\nPhenomenology of Entanglement ........................................................................... 444\n11.6.1\nRubbery Plateau ........................................................................................444\n11.6.2\nDependence of Me on Molecular Structure ..............................................447\n11.7\nReptation Model ...................................................................................................... 450\n"]], ["block_9", ["12.1\nIntroduction ............................................................................................................. 465\n12.1.1\nDefinition of a Glass ................................................................................. 465\n12.1.2\nGlass and Melting Transitions ..................................................................466\n12.2\nThermodynamic Aspects of the Glass Transition .................................................. 468\n"]], ["block_10", ["12.2.1\nFirst-Order and Second\u2014Order Phase Transitions .....................................469\n"]]], "page_15": [["block_0", [{"image_0": "15_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["xvi\n"]], ["block_2", ["13\n"]], ["block_3", ["Crystalline Polymers............................................................................... 511\n"]], ["block_4", ["12.2.2\nKauzmann Temperature ............................................................................471\n12.2.3\nTheory of Gibbs and DiMarzio ................................................................472\n12.3\nLocating the Glass Transition Temperature ........................................................... 474\n12.3. 1\nDilatometry ................................................................................................474\n12.3.2\nCalorimetry ................................................................................................ 476\n12.3.3\nDynamic Mechanical Analysis .................................................................478\n12.4\nFree Volume Description of the Glass Transition.................................................. 479\n12.4.1\nTemperature Dependence of the Free Volume ........................................480\n12.4.2\nFree Volume Changes Inferred from the Viscosity .................................481\n12.4.3\nWilliams\u2014Landel\u2014Ferry Equation .............................................................483\n12.5\nTime\u2014Temperature Superposition........................................................................... 486\n12.6\nFactors that Affect the Glass Transition Temperature ........................................... 491\n12.6.1\nDependence on Chemical Structure..........................................................491\n12.6.2\nDependence on Molecular Weight ...........................................................492\n12.6.3\nDependence on Composition ....................................................................492\n12.7\nMechanical Properties of Glassy Polymers ............................................................ 496\n12.7.1\nBasic Concepts ..........................................................................................496\n12.7.2\nCrazing, Yielding, and the Brittle-to-Ductile Transition ........................ .498\n12.7.3\nRole of Chain Stiffness and Entanglements .............................................501\n12.8\nChapter Summary.................................................................................................... 504\nProblems .............................................................................................................................. 505\nReferences ........................................................................................................................... 508\nFurther Readings ................................................................................................................. 508\n"]], ["block_5", ["13.1\nIntroduction and Overview ..................................................................................... 511\n13.2\nStructure and Characterization of Unit Cells ......................................................... 513\n13.2.1\nClasses of Crystals ....................................................................................513\n13.2.2\nX-Ray Diffraction .....................................................................................515\n13.2.3\nExamples of Unit Cells .............................................................................518\n13.3\nThermodynamics of Crystallization: Relation of Melting Temperature\nto Molecular Structure ............................................................................................ 521\n13.4\nStructure and Melting of Lamellae ......................................................................... 526\n13.4.1\nSurface Contributions to Phase Transitions..............................................526\n13.4.2\nDependence of Tm on Lamellar Thickness ...............................................527\n13.4.3\nDependence of Tm on Molecular Weight .................................................530\n13.4.4\nExperimental Characterization of Lamellar Structure ............................. 532\n13.5\nKinetics of Nucleation and Growth ........................................................................ 536\n13.5.1\nPrimary Nucleation ...................................................................................537\n13.5.2\nCrystal Growth ..........................................................................................539\n13.6\nMorphology of Semicrystalline Polymers .............................................................. 545\n13.6.1\nSpherulites .................................................................................................545\n13.6.2\nNonspherulitic Morphologies ....................................................................548\n13.7\nKinetics of Bulk Crystallization ............................................................................. 551\n13.7.1\nAvrami Equation .......................................................................................552\n"]], ["block_6", ["Contents\n"]]], "page_16": [["block_0", [{"image_0": "16_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Appendix ......................................................................................................... 567\n"]], ["block_2", ["A.l\nSeries Expansions ...............................................................................................................567\nA2\nSummation Formulae ..........................................................................................................568\nA3\nTransformation to Spherical Coordinates ...........................................................................569\nA.4\nSome Integrals of Gaussian Functions ...............................................................................570\nA5\nComplex Numbers ..............................................................................................................572\n"]], ["block_3", ["Contents\nxvii\n"]], ["block_4", ["Index ................................................................................................................ 575\n"]], ["block_5", ["13.7.2\nKinetics of Crystallization: EXperimental Aspects ..................................556\n13.8\nChapter Summary .................................................................................................... 561\nProblems .............................................................................................................................. 562\nReferences ........................................................................................................................... 565\nFurther Readings ................................................................................................................. 565\n"]]], "page_17": [["block_0", [{"image_0": "17_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["\u201cI am inclined to think that the development of polymerization is perhaps the biggest thing\nchemistry has done, where it has had the biggest impact on everyday life\u201d [1]. This assessment\nof the significance of polymer chemistry to modern society was offered 25 years ago by Lord Todd\n(President of the Royal Society and 1957 Nobel Laureate in Chemistry), and subsequent deve10p-\nments have only reinforced this sentiment. There is hardly an area of modern life in which polymer\nmaterials do not play an important role. Applications span the range from the mundane\n(e.g., packaging, toys, fabrics, diapers, nonstick cookware, pressure-sensitive adhesives, etc.) to\ndemanding specialty uses (e.g., bulletproof vests, stealth aircraft, artificial hip joints, resorbable\nsutures, etc.). In many instances polymers are the main ingredients, and the ingredients whose\ncharacteristic properties are essential to the success of a particular technology: rubber tires, foam\ncushions and insulation, high\u2014performance athletic shoes, clothing, and equipment are good\nexamples. In other cases, polymers are used as additives at the level of a few percent by volume,\nbut which nevertheless play a crucial role in the properties of the \ufb01nal material; illustrations of this\ncan be found in asphalt (to suppress brittle fracture at low temperature and flow at high tempera-\nture), shampoo and other cosmetics (to impart \u201cbody\u201d), automobile Windshields (to prevent\nshattering), and motor oil (to reduce the dependence of viscosity on temperature, and to suppress\ncrystallization).\nFor those polymer scientists \u201cof a certain age,\u201d the 1967 movie \u201cThe Graduate\u201d [2] provided\nan indelible moment that still resonates today. At his college graduation party, the hero Benjamin\nBraddock (played by Dustin Hoffman) is offered the following advice by Mr. McGuire (played by\nWalter Brooke):\n"]], ["block_2", ["In that period, the term \u201cplastic\u201d was often accompanied by negative connotations, including\n\u201cartificial,\u201d as opposed to \u201cnatural,\u201d and \u201ccheap,\u201d as opposed to \u201cvaluable.\u201d Today, in what we\nmight call the \u201cpost-graduate era,\u201d the situation has changed. To the extent that the advice offered\nto Benjamin was pointing him to a career in a particular segment of the chemical industry, it was\nprobably very sound advice. The volume of polymer materials produced annually has grown\nrapidly over the intervening years, to the point where today several hundred pounds of polymer\nmaterials are produced each year for each person in the United States. More interesting than sheer\nvolume, however, is the breadth of applications for polymers. Not only do they continue to\nencroach into the domains of \u201cclassical\u201d materials such as metal, wood, and glass (note the\ninexorable transformation of polymers from minor to major components in automobiles), but they\nalso play a central role in many emerging technologies. Examples include \u201cplastic electronics,\u201d\n"]], ["block_3", ["1.1\nIntroduction\n"]], ["block_4", ["BENJAMIN. Yes I am.\nMR. MCGUIRE. Plastics.\n"]], ["block_5", ["MR. MCGUIRE. I want to say one word to you. Just one word.\n"]], ["block_6", ["BENJAMIN. Yes, sir.\nMR. MCGUIRE. Are you listening?\n"]], ["block_7", ["Introduction to Chain Molecules\n"]], ["block_8", ["1\n"]]], "page_18": [["block_0", [{"image_0": "18_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["a decade of controversy before this \u201cmacromolecular hypothesis\u201d began to experience widespread\nacceptance. Staudinger was awarded the Nobel Prize in 1953 for his work with polymers. By the\n1930s, Carothers\nbegan synthesizing polymers using\nwell-established reactions of organic\nchemistry such as esterification and amidation. His products were not limited to single ester or\namide linkages, however, but contained many such groups: they were polyesters and polyamides.\nPhysical chemists also got in on the act. Kuhn, Guth, Mark, and others were soon applying\nstatistics and crystallography to describe the multitude of forms a long-chain molecule could\nassume [3].\nOur purpose in this introduction is not to trace the history of polymer chemistry beyond the\nsketchy version above; interesting and extensive treatments are available [4,5]. Rather, the primary\nobjective is to introduce the concept of chain molecules, which stands as the cornerstone of all\npolymer chemistry. In the next few sections we shall explore some of the categories of polymers,\nsome of the reactions that produce them, and some aspects of isomerism which multiply the\nstructural possibilities. A common feature of all synthetic polymerization reactions is the statistical\nnature of the individual polymerization steps. This leads inevitably to a distribution of molecular\nweights, which we would like to describe. As a consequence of these considerations, another\nimportant part of this chapter is an introduction to some of the statistical concepts that also play a\ncentral role in polymer chemistry.\n"]], ["block_2", ["can cure such and such a dreaded disease, but they cannot do anything about the common cold\u201d or\n\u201cwe know more about the surface of the moon than the bottom of the sea.\u201d If such comparisons\nwere popular in the 19203, the saying might have been, \u201cwe know more about the structure of the\natom than about those messy, sticky substances called polymers.\u201d Indeed, Millikan\u2019s determin-\nation of the charge of an electron, Rutherford\u2019s idea of the nuclear atom, and Bohr\u2019s model of the\nhydrogen atom were all well-known concepts before the notion of truly covalent macromolecules\nwas accepted. This was the case in spite of the great importance of polymers to human life and\nactivities. Our bodies, like all forms of life, depend on polymer molecules: carbohydrates, proteins,\nnucleic acids, and so on. From the earliest times, polymeric materials have been employed to\nsatisfy human needs: wood and paper; hides; natural resins and gums; fibers such as cotton, wool,\nand silk.\nAttempts to characterize polymeric substances had been made, of course, and high molecular\nweights were indicated, even if they were not too accurate. Early workers tended to be more\nsuspicious of the interpretation of the colligative properties of polymeric solutions than to accept\nthe possibility of high molecular weight compounds. Faraday had already arrived at C5H3 as the\nempirical formula of \u201crubber\u201d in 1826, and isoprene was identi\ufb01ed as the product resulting from\nthe destructive distillation of rubber in 1860. The idea that a natural polymer such as rubber\nsomehow \u201ccontained\u201d isoprene emerged, but the nature of its involvement was more elusive.\nDuring the early years of the 20th century, organic chemists were enjoying success in deter-\nmining the structures of ordinary-sized organic molecules, and this probably contributed to their\nreluctance to look beyond structures of convenient size. Physical chemists were interested\nin intermolecular forces during this period, and the idea that polymers were the result of some\nsort of association between low molecular weight constituent molecules prevailed for a long while.\nStaudinger is generally credited as being the father of modern polymer chemistry, although a\nforeshadowing of his ideas can be traced through older literature. In 1920, Staudinger proposed\nthe chain formulas we accept today, maintaining that structures are held together by covalent\nbonds, which are equivalent in every way to those in low molecular weight compounds. There was\n"]], ["block_3", ["gene therapy, artificial prostheses, optical data storage, electric cars, and fuel cells. In short, a\nreasonable appreciation of the properties of chain molecules, and how these result in the many\ndesirable attributes of polymer-containing materials, is a necessity for a well-trained chemist,\nmaterials scientist, or chemical engineer today.\nScience tends to be plagued by cliches, which make invidious comparison of its efforts; \u201cthey\n"]], ["block_4", ["2\nIntroduction to Chain Molecules\n"]]], "page_19": [["block_0", [{"image_0": "19_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["The term polymer is derived from the Greek words poly and meros, meaning many parts. We noted\nin the Section 1.1 that the existence of these parts was acknowledged before the nature of the\ninteraction which held them together was known. Today we realize that ordinary covalent bonds\n"]], ["block_2", ["One of the first things we must consider is what we mean when we talk about the \u201csize\u201d of a\npolymer molecule. There are two possibilities: one has to do with the number of repeat units and\nthe other to the spatial extent. In the former case, the standard term is molecular weight (although\nagain the reader must be aware that molar mass is often preferred). A closely related concept, the\ndegree of polymerization is also commonly used in this context. A variety of experimental\ntechniques are available for determining the molecular weight of a polymer. We shall discuss a\nfew such methods in Section 1.8 and postpone others until the appropriate chapters. The expression\nmolecular weight and molar mass should always be modified by the word average. This too is\nsomething we shall take up presently. For now, we assume that a polymer molecule has a\nmolecular weight M, which can be anywhere in the range 103\u2014107 or more. (We shall omit units\nwhen we write molecular weights in this book, but the student is advised to attach the units g/mol\nto these quantities when they appear in problem calculations.)\nSince polymer molecules are made up of chains of repeat units, after the chain itself comes the\nrepeat unit as a structural element of importance. Many polymer molecules are produced by\ncovalently bonding together only one or two types of repeat units. These units are the parts from\nwhich chains are generated; as a class of compounds they are called monomers. Throughout this\nbook, we shall designate the molecular weight of a repeat unit as M0.\nThe degree of polymerization of a polymer is simply the number of repeat units in a molecule.\nThe degree of polymerization N is given by the ratio of the molecular weight of the polymer to the\nmolecular weight of the repeat unit:\n"]], ["block_3", ["are the intramolecular forces that keep the polymer molecule intact. In addition, the usual types of\nintermolecular forces\u2014hydrogen bonds, dipole\u2014dipole interactions, London forces, etc.\u2014hold\nassemblies of these molecules together in the bulk state. The only thing that is remarkable about\nthese molecules is their size, but that feature is remarkable indeed. Another useful term is\nmacromolecule, which of course simply means \u201clarge (or long) molecule.\u201d Some practitioners\ndraw a distinction between the two: all polymers are macromolecules, but not all macromolecules\nare polymers. For example, a protein is not made by repeating one or two chemical units many\ntimes, but involves a precise selection from among 20 different amino acids; thus it is a macro\u2014\nmolecule, but not a polymer. In this text we will not be sticklers for formality, and will use the\nterms rather interchangeably, but the reader should be aware of the distinction.\n"]], ["block_4", ["One type of polymerization reaction is the addition reaction in which successive repeat units add\non to the chain. No other product molecules are formed, so the molecular weight of the monomer\nand that of the repeat unit are identical in this case. A second category of polymerization reaction is\nthe condensation reaction, in which one or two small molecules such as water or HCl are\neliminated for each chain linkage formed. In this case the molecular weight of the monomer and\nthe repeat unit are somewhat different. For example, suppose an acid (subscript A) reacts with an\nalcohol (subscript B) to produce an ester linkage and a water molecule. The molecular weight of\nthe ester\u2014the repeat unit if an entire chain is built up this way\u2014differs from the combined weight\nof the reactants by twice the molecular weight of the water; therefore,\n"]], ["block_5", ["How Big Is Big?\n3\n"]], ["block_6", ["1.2.1\nMolecular Weight\n"]], ["block_7", ["1.2\nHow Big ls Big?\n"]], ["block_8", ["N \nM0\n(1.2.1)\n"]], ["block_9", ["M\nM\nN:\u2014:\nM0\nMA +MB ZMHZO\n(1.2.2)\n"]]], "page_20": [["block_0", [{"image_0": "20_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "20_1.png", "coords": [27, 298, 218, 399], "fig_type": "figure"}]], ["block_2", ["as the number of repeat units in a structure increases. By the time polymeric molecular\nsizes are reached, the error associated with failure to distinguish between segments at the\nend and those within the chain is generally less than experimental error. In Section 1.8.2 we\nshall consider a method for polymer molecular weight determination based on chemical\nanalysis for the end groups in a polymer. A corollary of the present discussion is that the\nmethod of end group analysis is applicable only in the case of relatively low molecular weight\npolymers.\nAs suggested above, not all polymers are constructed by bonding together a single kind of\nrepeat unit. For example, although protein molecules are polyamides in which N amino acid repeat\nunits are bonded together, the degree of polymerization is a less useful concept, since an amino\nacid unit might be any one of the 20\u2014odd molecules that are found in proteins. in this case the\nmolecular weight itself, rather than the degree of polymerization, is generally used to describe the\nmolecule. When the actual content of individual amino acids is known, it is their sequence that is\nof special interest to biochemists and molecular biologists.\n"]], ["block_3", ["Although the difference is almost 5% for propane, it is closer to 0.1% for the case of N m 100,\nwhich is about the threshold for polymers. The precise values of these numbers will be\ndifferent, depending on the specific repeat units and end groups present. For example, if\nM0: 100 and Mend=80, the difference would be 0.39% in a calculation such as that above\nfor N g 100.\n"]], ["block_4", ["As a polymer prototype consider an n-alkane molecule consisting of N\u20142 methylenes and 2 methyl\ngroups. How serious an error is made in M for different Ns if the difference in molecular weight\nbetween methyl and methylene groups is ignored?\n"]], ["block_5", ["The effect of different end groups on M can be seen by comparing the true molecular weight with\nan approximate molecular weight, calculated on the basis of a formula (CH2)N. These Ms and the\npercentage difference between them are listed here for several values of N\n"]], ["block_6", ["a methyl group and the middle parts are methylene groups. Of course, the terminal group does\nnot have to be a hydrogen as in alkanes; indeed, it is often something else. Our interest in end\ngroups is concerned with the question of what effect they introduce into the evaluation of N\nthrough Equation\n(1.2.2).\nThe following\nexample examines\nthis\nthrough\nsome numerical\ncalculations.\n"]], ["block_7", ["4\nIntroduction to Chain Molecules\n"]], ["block_8", ["The end units in a polymer chain are inevitably different from the units that are attached on\nboth sides to other repeat units. We see this situation in the n-alkanes: each end of the chain is\n"]], ["block_9", ["Solution\n"]], ["block_10", ["Example 1.1\n"]], ["block_11", ["12\n170\n168\n1.2\n52\n730\n728\n0.3\n102\n1,430\n1,428\n0.14\n502\n7,030\n7,028\n0.028\n1002\n14,030\n14,028\n0.014\n"]], ["block_12", ["N\nM\nMapprox\n% Difference\n"]], ["block_13", ["The example shows that the contribution of the ends becomes progressively less important\n"]], ["block_14", ["7\n100\n98\n2.0\n"]], ["block_15", ["3\n44\n42\n4.5\n"]]], "page_21": [["block_0", [{"image_0": "21_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["3.\nBecause of the possibility of rotation around carbon\u2014carbon bonds, a molecule possessing\nmany such bonds will undergo many twists and turns along the chain.\n4.\nFully extended molecular length is not representative of the spatial extension that a molecule\nactually displays. The latter is sensitive to environmental factors, however, so the extended\nlength is convenient for our present purposes to provide an idea of the spatial size of polymer\nmolecules.\n"]], ["block_2", ["We began this section with an inquiry into how to de\ufb01ne the size of a polymer molecule. In\naddition to the molecular weight or the degree of polymerization, some linear dimension that\ncharacterizes the molecule could also be used for this purpose. As an example, consider a\nhydrocarbon molecule stretched out to its full length but without any bond distortion. There are\nseveral features to note about this situation:\n"]], ["block_3", ["these triangles, one of these atoms is common to two triangles;\ntherefore the number of\ntriangles is the same as the number of pairs of carbon atoms, except where this breaks down\nat the ends of the molecule. If the chain is sufficiently long, this end effect is inconsequential.\nThe law of cosines can be used to calculate the length of the base of each of these triangles:\n[2(0.154)2(1 cos 109.5\u00b0)] 0.252 nm. If the repeat unit of the molecule contributes two carbon\natoms to the backbone of the polymer\u2014as is the case for vinyl polymers\u2014the fully extended\nchain length is given by N(0.252) nm. For a polymer with N 104, this corresponds to 2.52 pm.\nObjects which actually display linear dimensions of this magnitude can be seen in an ordinary\nmicroscope, provided that they have suitable optical properties to contrast with their surroundings;\nan example will be given in Figure 1.1a. Note that the distance between every other carbon atom\nwe have used here is also the distance between the substituents on these carbons for the fully\nextended chains.\nWe shall see in Chapter 6 that, because of all the twists and turns a molecule undergoes,\nthe actual average end-to\u2014end distance of the jumbled molecules increases as N\u201d2. With the\n"]], ["block_4", ["same repeat distance calculated above, but the square root dependence on N, the actual end-to-\nend distance of the coiled chain with N 104 is closer to (104)\u201d2 X 0.252 nm as 25 nm. If we\npicture one end of this jumbled chain at the origin of a coordinate system,.the other end might be\nanywhere on the surface of a sphere whose radius is given by this end-to-end distance. This\nspherical geometry comes about because the random bends occurring along the chain length can\ntake the end of the chain anywhere in a spherical domain whose radius depends on N2.\nThe above discussion points out the difficulty associated with using the linear dimensions of a\nmolecule as a measure of its size: it is not the molecule alone that determines its dimensions, but\nalso the shape or conformation in which it exists. Fully extended, linear arrangements of the sort\ndescribed above exist in polymer crystals, at least for some distance, although usually not over the\nfull length of the chain. We shall take up the structure of polymer crystals in Chapter 13. In the\nsolution and bulk states, many polymers exist in the coiled form we have also described. Still other\nstructures are important, notably the rod or semi\ufb02exible chain, which we shall discuss in Chapter 6.\nThe overall shape assumed by a polymer molecule can be greatly affected by the environment. The\nshape of a molecule in solution plays a key role in determining many properties of polymer\nsolutions. From a study of these solutions, some conclusions can be drawn regarding the shape of\n"]], ["block_5", ["A fully extended hydrocarbon molecule will have the familiar all\u2014trans zigzag profile in\nwhich the hydrogens extend in front of and in back of the plane containing the carbons, with an\nangle of 109.5\u00b0 between successive carbon\u2014carbon bonds. The chain may be pictured as a row\nof triangles resting comer to comer. The length of the row equals the product of the number\nof triangles and the length of the base of each. Although it takes three carbons to define one of\n"]], ["block_6", ["1.2.2\nSpatial Extent\n"]], ["block_7", ["How Big ls Big?\n5\n"]], ["block_8", ["1.\nThe tetrahedral geometry at the carbon atoms gives bond angles of 109.5\u00b0.\nThe equilibrium bond length of a carbon\u2014carbon single bond is 0.154 nm or 1.54 A.\n"]]], "page_22": [["block_0", [{"image_0": "22_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "22_1.png", "coords": [12, 45, 311, 292], "fig_type": "figure"}]], ["block_2", ["6\nIntroduction to Chain Molecules\n"]], ["block_3", ["Figure 1.1\n(3) Individual molecules of DNA of various sizes, spread on a \ufb02uid positively charged surface,\nimaged by \ufb02uorescence. The scale bar is 10 um. (Reproduced from Maier, B. and Radler, J.O. Macromolecules\n33, 7185, 2000. With permission.) (b) Atomic force microscopy images of three-arm star polymers, where each\narm is a heavily branched Comb. The circles indicate linear molecules. (Reproduced from Matyjaszewski, K.,\nQin, S., Boyce, J.R., Shirvanyants, D., and Sheiko, S.S. Macromolecules 36, 1843, 2003. With permission.)\n"]], ["block_4", ["the molecule in the environment. Relevant aspects of polymer solutions are taken up in Chapter 6\nthrough Chapter 9.\nFigure 1.1a and Figure 1.1b are rather striking images of individual polymer molecules. Figure\n"]], ["block_5", ["1.1a shows single molecules of DNA that have been heavily labeled with \ufb02uorescent dyes; the dyes\n"]]], "page_23": [["block_0", [{"image_0": "23_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["as these.\nWe conclude this section by questioning whether there is a minimum molecular weight or linear\ndimension that must be met for a molecule to qualify as a polymer. Although a dimer is a molecule\nfor which N =2, no one would consider it a polymer. The term oligomer has been coined to\ndesignate molecules for which N < 10. If they require a special name, apparently the latter are not\nfull-\ufb02edged polymers either. At least as a first approximation, we shall take the attitude that there\nis ordinarily no discontinuity in behavior with respect to observed properties as we progress\nthrough a homologous series of compounds with different N values. At one end of the series, we\nmay be dealing with a simple low molecular weight compound, and at the other end with a material\nthat is unquestionably polymeric. The molecular weight and chain length increase monotonically\nthrough this series, and a variety of other properties vary smoothly also. This point of view\nemphasizes continuity with familiar facts concerning the properties of low molecular weight\ncompounds. There are some properties, on the other hand, which follow so closely from the\nchain structure of polymers that the property is simply not observed until a certain critical\nmolecular size has been reached. This critical size is often designated by a threshold molecular\nweight. The elastic behavior of rubber and several other mechanical properties fall into this latter\ncategory. In theoretical developments, large values of N are often assumed to justify neglecting end\neffects, using certain statistical methods and other mathematical approximations. With these ideas\nin mind, M 1000 seems to be a convenient round number for designating a compound to be a\npolymer, although it should be clear that this cutoff is arbitrary (and on the low side).\n"]], ["block_2", ["intercalate between the base pairs along the chain, without seriously altering the conformation of\nthe molecule. Under illumination the resulting \ufb02uorescence provides a good representation of the\nmolecules themselves. In this particular image, the DNA molecules are spread out in two\ndimensions, on a cationically charged imitation lipid membrane. DNA, as it turns out, is an\nexcellent example of a semi\ufb02exible chain, which can actually be inferred from these images; the\nmolecules are not straight rods, but neither are they heavily coiled around themselves. The\nscale bar corresponds to 10 um, indicating that these molecules are of very high molecular\nweight indeed. In Figure 1.1b, the image is of a star-shaped polymer, but one in which each arm\nof the star is a heavily branched comb or \u201cbottlebrush.\u201d The molecule is thus akin to a kind of\nstarfish, with very hairy arms. This picture was obtained by atomic force microscopy (AFM),\none of a series of surface-sensitive analysis techniques with exquisite Spatial resolution. The\nmolecules themselves were deposited from a Langmuir\u2014Blodgett trough onto a mica substrate.\nBoth situations depicted in Figure 1.1a and Figure 1.1b raise the question of the relationship\nbetween the conformation observed on the surface and that at equilibrium in solution. In\nChapter 6 through Chapter 9 we will encounter several ways in which the solution conformation\ncan be determined reliably, which can serve to confirm the impression derived from figures such\n"]], ["block_3", ["Most of the preceding section was based on the implicit assumption that polymer chains are\nlinear (with the striking exception of Figure 1.1b). In evaluating both the degree of polymeriza-\ntion and the extended chain length, we assumed that the chain had only two ends. While linear\npolymers are important, they are not the only type of molecules possible: branched and cross-\nlinked molecules are also common. When we speak of a branched polymer, we refer to the\npresence of additional polymeric chains issuing from the backbone of a linear molecule. (Small\nsubstituent groups such as methyl or phenyl groups on the repeat units are generally not\nconsidered branches, or, if they are, they should be specified as \u201cshort\u2014chain branches\u201d)\nBranching can arise through several routes. One is to introduce into the polymerization reaction\nsome monomer with the capability of serving as a branch. Consider the formation of a polyester.\n"]], ["block_4", ["Linear and Branched Polymers, Homopolymers, and Copolymers\n7\n"]], ["block_5", ["1.3.1\nBranched Structures\n"]], ["block_6", ["1.3\nLinear and Branched Polymers, Homopolymers, and Copolymers\n"]]], "page_24": [["block_0", [{"image_0": "24_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "24_1.png", "coords": [27, 455, 212, 670], "fig_type": "figure"}]], ["block_2", ["The presence of difunctional acids and difunctional alcohols allows the polymer chain to\ngrow. These difunctional molecules are incorporated into the chain with ester linkages at both\nends of each. Trifunctional acids or alcohols, on the other hand, produce a linear molecule by\nthe reaction of two of their functional groups. If the third reacts and the resulting chain continues\nto grow, a branch has been introduced into the original chain. A second route is through\nadventitious branching, for example, as a result of an atom being abstracted from the original\nlinear molecule, with chain growth occurring from the resulting active site. This is quite a\ncommon occurrence in the free-radical polymerization of ethylene, for example. A third route is\ngrafting, whereby pre-formed but still reactive polymer chains can be added to sites along an\nexisting backbone (so-called \u201cgrafting to\u201d), or where multiple initiation sites along a chain\ncan be exposed to monomer (so-called \u201cgrafting from\u201d).\nThe amount of branching introduced into a polymer is an additional variable that must be\nspecified for the molecule to be fully characterized. When only a slight degree of branching is\npresent, the concentration of junction points is sufficiently low that these may be simply related to\nthe number of chain ends. For example, two separate linear molecules have a total of four ends. If\nthe end of one of these linear molecules attaches itself to the middle of the other to form a T, the\nresulting molecule has three ends. It is easy to generalize this result. If a molecule has 12 branches, it\nhas 12 + 2 chain ends if the branching is relatively low. Two limiting cases to consider, illustrated in\nFigure 1.2, are combs and stars. In the former, a series of relatively uniform branches emanate\nfrom along the length of a common backbone; in the latter, all branches radiate from a central\njunction. Figure 1.1b gave an example of both of these features.\nIf the concentration of junction points is high enough, even branches will contain branches.\nEventually a point can be reached at which the amount of branching is so extensive that the\npolymer molecule becomes a giant three-dimensional network. When this condition is achieved,\nthe molecule is said to be cross-linked. In this case, an entire macroscopic object may be\nconsidered to consist of essentially one molecule. The forces that give cohesiveness to such a\nbody are covalent bonds, not intermolecular forces. Accordingly, the mechanical behavior of\ncross-linked bodies is much different from those without cross-linking. This will be discussed at\nlength in Chapter 10. However, it is also possible to suppress cross-linking such that the highly\nbranChed molecules remain as discrete entities, known as hyperbranched polymers (see Figure 1.2).\nAnother important class of highly branched polymers illustrated in Figure 1.2 are dendrimers, or\ntreelike molecules. These are completely regular structures, with well-defined molecular weights,\nthat are made by the successive condensation of branched monomers. For example, begin with a\n"]], ["block_3", [{"image_2": "24_2.png", "coords": [34, 606, 95, 645], "fig_type": "molecule"}]], ["block_4", ["Four-arm star\n"]], ["block_5", ["Figure 1.2\nIllustration of various polymer architectures.\n"]], ["block_6", ["8\nIntroduction to Chain Molecules\n"]], ["block_7", [{"image_3": "24_3.png", "coords": [37, 473, 173, 588], "fig_type": "figure"}]], ["block_8", ["Linear\n"]], ["block_9", ["Comb\nDendrimer\n"]], ["block_10", ["Cycle\n"]], ["block_11", [{"image_4": "24_4.png", "coords": [49, 454, 176, 567], "fig_type": "molecule"}]], ["block_12", [{"image_5": "24_5.png", "coords": [61, 477, 178, 531], "fig_type": "molecule"}]], ["block_13", [{"image_6": "24_6.png", "coords": [87, 537, 185, 648], "fig_type": "molecule"}]], ["block_14", ["Hyperbranched\n"]]], "page_25": [["block_0", [{"image_0": "25_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["The empirical formula gives the relative amounts of A, B, and C in the terpolymer. The total\nmolecular weight of this empirical formula unit is given by adding the molecular weight contri-\nbutions of A, B, and C: 3.44(104) + 2.20(184)+ 1.00(128) =902 g/mol per empirical formula unit.\n"]], ["block_2", ["trifunctional monomer \u201cB3,\u201d or \u201cgeneration 0.\u201d This is reacted with an excess of AB2 monomers,\nleading to a generation\n1 dendrimer with 6 B groups. A second reaction with AB2 leads to\ngeneration 2 with 12 pendant B groups. Eventually, perhaps at generation 6 or 7, the surface of\nthe molecule becomes so congested that addition of further complete generations is impossible.\nNote that the \u201cB\u201d part of the ABg monomer needs to be protected in some way so that only one\ngeneration can be added at one time.\nA final class of nonlinear polymers to consider are cycles or rings, whereby the two ends of the\nmolecule react to close the loop. Such polymers are currently more of academic interest than\ncommercial importance, as they are tricky to prepare, but they can shed light on various aspects of\npolymer behavior. Interestingly, nature makes use of this architecture; the DNA of the Lambda\nbacteriophage reversibly cyclizes and uncyclizes during gene expression.\n"]], ["block_3", ["Just as it is not necessary for polymer chains to be linear, it is also not necessary for all repeat units\nto be the same. We have already mentioned proteins, where a wide variety of different repeat\nunits are present. Among synthetic polymers, those with a single kind of repeat unit are called\nhomopolymers, and those containing more than one kind of repeat unit are capolymers. Note that\nthese definitions are based on the repeat unit, not the monomer. An ordinary polyester is not really\na copolymer, even though two different monomers, acids and alcohols, are its monomers.\nBy contrast, copolymers result when different monomers bond together in the same way to produce\na chain in which each kind of monomer retains its respective substituents in the polymer molecule.\nThe unmodified term copolymer is generally used to designate the case where two different repeat\nunits are involved. Where three kinds of repeat units are present, the system is called a terpolymer;\nwhere there are more than three, the system is called a multicomponent copolymer. The copoly-\nmers we discuss in this book will be primarily two\u2014component molecules. We shall explore aspects\nof the synthesis and characterization of copolymers in both Chapter 4 and Chapter 5.\nThe moment we admit the possibility of having more than one kind of repeat unit, we require\nadditional variables to describe the polymer. First, we must know how many kinds of repeat units\nare present and what they are. To describe the copolymer quantitatively, the relative amounts of the\ndifferent kinds of repeat units must be specified. Thus the empirical formula of a copolymer may\nbe written A,C By, where A and B signify the individual repeat units and x and y indicate the relative\nnumber of each. From a knowledge of the molecular weight of the polymer, the molecular weights\nof A and B, and the values of x and y, it is possible to calculate the number of each kind of\nmonomer unit in the copolymer. The sum of these values gives the degree of polymerization of the\ncopolymer. The following example illustrates some of the ways of describing a copolymer.\n"]], ["block_4", ["A terpolymer is prepared from vinyl monomers A, B, and C; the molecular weights of the repeat\nunits are 104, 184, and 128, respectively. A particular polymerization procedure yields a product\nwith the empirical formula A355 32.20C1.00- The authors of this research state that the terpolymer\nhas \u201can average unit weight of 134\u201d and \u201cthe average molecular weight per angstrom of 53.5.\u201d\nVerify these values.T\n"]], ["block_5", ["TA. Ravve and IT Khamis, Addition and Condensation Polymerization Processes, Advances in Chemistry Series, Vol. 91,\nAmerican Chemical Society Publications, Washington, DC, 1969.\n"]], ["block_6", ["Linear and Branched Polymers, Homopolymers, and Copolymers\n9\n"]], ["block_7", ["Solution\n"]], ["block_8", ["1.3.2\nCopolymers\n"]], ["block_9", ["Example 1.2\n"]]], "page_26": [["block_0", [{"image_0": "26_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["The total amount of chain repeat units possessing this total weight is 3.55 + 2.20 + 1.00 6.75\nrepeat units per empirical formula unit. The ratio of the total molecular weight to the total number\nof repeat units gives the average molecular weight per repeat unit:\n"]], ["block_2", ["With copolymers, it is far from sufficient merely to describe the empirical formula to charac-\nterize the molecule. Another question that must be asked concerns the location of the different\nkinds of repeat units within the molecule. Starting from monomers A and B, the following\ndistribution patterns can be obtained in linear polymers:\n"]], ["block_3", ["Since the monomers are specified to be vinyl monomers, each contributes two carbon atoms to the\npolymer backbone, with the associated extended length of 0.252 mm per repeat unit. Therefore, the\ntotal extended length of the empirical formula unit is\n"]], ["block_4", ["The ratio of the total weight to the total extended length of the empirical formula unit gives the\naverage molecular weight per length of chain:\n"]], ["block_5", ["4.\nGraft. This segregation is often accomplished by first homopolymerizing the backbone. This is\ndissolved in the second monomer, with sites along the original chain becoming the origin of\nthe comonomer side-chain growth:\n"]], ["block_6", ["Note that the average weight per repeat unit could be used to evaluate the overall degree of\npolymerization of this terpolymer. For example, if the molecular weight were 43,000, the corre-\nsponding degree of polymerization would be\n"]], ["block_7", ["The above structure has three blocks, and is called poly(A-block\u2014B\u2014block-A), or an ABA\ntriblock copolymer. If a copolymer is branched with different repeat units occurring in the\n"]], ["block_8", ["branches and the backbone, we can have the following:\n"]], ["block_9", ["Such a polymer could be called poly(A-stat-B) or poly(A-ran\u2014B).\n2.\nAlternating. A regular pattern of alternating repeat units in poly(A\u2014alt\u2014B):\n"]], ["block_10", ["3.\nBlock. Long, uninterrupted sequence of each monomer is the pattern:\n"]], ["block_11", ["10\nIntroduction to Chain Molecules\n"]], ["block_12", ["1.\nRandom (or statistical). The A\u2014B sequence is governed strictly by chance, subject only to the\nrelative abundances of repeat units. For equal proportions of A and B, we might have\nstructures like\n"]], ["block_13", ["5\u201475\n= 134 g/mol per repeat unit\n"]], ["block_14", ["902\n.\n\u2014 = 53 g/mol per A\n17\n"]], ["block_15", ["43 ,000\n"]], ["block_16", ["6.75(0.252 nm) 1.79 nm 17.0 A\n"]], ["block_17", ["902\n"]], ["block_18", ["l\nwAAAAAAAAAAAAAAAAA\u2014\n"]], ["block_19", ["l\nl\n\u2014BBBBBBBBBB\nBBBBBBB\u2014\n"]], ["block_20", ["134\n= 321 repeat units per molecule\n"]], ["block_21", ["\u2014\u2014ABABABABABAB\u2014\n"]], ["block_22", ["\u2014AAAAAAAAAAAAAABBBBBBBBBBBBBBBAAAAAAAAA\u2014\n"]], ["block_23", ["\u2014AAABABAABBABBB\u2014\n"]], ["block_24", ["BBBBBBBBBB\u2014\n"]]], "page_27": [["block_0", [{"image_0": "27_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["In a cross\u2014linked polymer, the junction units are different kinds of monomers than the chain\nrepeat units, so these molecules might be considered to be still another comonomer. While the chemical\nreactions that yield such substances are technically copolymerizations, the productsare\ndescribed as cross-linked rather copolymers. In this instance, the behavior due \ntakes precedence over the presence of an additional type of monomer in the structure.\nIt is apparent from items\n1\u2014-3 above that linear copolymers\u2014even those with the same\nproportions of different kinds of repeat units\u2014can be very different in structure and properties.\nIn classifying a copolymer as random, alternating, or block, it should be realized that we are\ndescribing the average character of the molecule; accidental variations from the basic patterns may\nbe present. Furthermore, in some circumstances, nominally \u201crandom\u201d copolymers have\nsubstantial sequences of one or the other. In Chapter 5, we shall see how an experimental\ninvestigation of the sequence of repeat units in a copolymer is a valuable tool for understanding\ncopolymerization reactions.\n"]], ["block_2", ["In the last section, we examined some of the categories into which polymers can be classified.\nVarious aspects of molecular structure were used as the basis for classification in that section. Next\nwe shall consider the chemical reactions that produce the molecules as a basis for classification.\nThe objective of this discussion is simply to provide some orientation and to introduce some\ntypical polymers. For this purpose, many polymers may be classified as being either addition or\ncondensation polymers; both of these classes are discussed in detail in Chapter 2 and Chapter 3,\nrespectively. Even though these categories are based on the reactions which produce the polymers,\nit should not be inferred that only two types of polymerization reactions exist. We have to start\nsomewhere, and these two important categories are the usual places to begin.\n"]], ["block_3", ["Addition, Condensation, and Natural Polymers\n11\n"]], ["block_4", ["4.\nThe polymer repeat unit arises from reacting together two different functional groups, which\nusually originate on different monomers. In this case, the repeat unit is different from either of\n"]], ["block_5", ["These two categories of polymers can be developed along several lines. For example, in addition-\ntype polymers the following statements apply:\n"]], ["block_6", ["1.\nThe repeat unit in the polymer and the monomer has the same composition, although, of\ncourse, the bonding is different in each.\n2.\nThe mechanism of these reactions places addition polymerizations in the kinetic category of chain\nreactions, with either free radicals or ionic groups responsible for propagating the chain reaction.\n3.\nThe product molecules often have an all-carbon chain backbone, with pendant substituent groups.\n"]], ["block_7", ["the monomers. In addition, small molecules are often eliminated during the condensation\nreaction. Note the words usual and often in the previous statements; exceptions to both\nstatements are easily found.\n5.\nThe mechanistic aspect of these reactions can be summarized by saying that the reactions\noccur in steps. Thus, the formation of an ester linkage between two small molecules is not\nessentially different from that between a polyester and a monomer.\n6.\nThe product molecules have the functional groups formed by the condensation reactions\ninterspersed regularly along the backbone of the polymer molecule:\n"]], ["block_8", ["1.4\nAddition, Condensation, and Natural Polymers\n"]], ["block_9", ["1.4.1\nAddition and Condensation Polymers\n"]], ["block_10", ["In contrast, for condensation polymers:\n"]], ["block_11", ["\u2014C\u2014C\u2014Y\u2014C\u2014C\u2014Y\u2014\n"]]], "page_28": [["block_0", [{"image_0": "28_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["First, the reactive center must be initiated by\na suitable reaction to produce a free radical,\nanionic, or cationic reaction site. Next, this reactive entity adds consecutive monomer to\npropagate the polymer chain. Finally, the active site is capped off, terminating the polymer\nformation. If one assumes that the polymer produced is truly a high molecular weight sub-\nstance, the lack of uniformity at the two ends of the chain\u2014arising in one case from the\ninitiation and in the other from the termination\u2014can be neglected. Accordingly, the overall\nreaction can be written\n"]], ["block_2", ["Next let us consider a few Speci\ufb01c examples of these classes of polymers. The addition\npolymerization of a vinyl monomer CsCHX involves\nthree distinctly different\nsteps.\n"]], ["block_3", ["as the conditions of the reaction and other pertinent information will be discussed when these\nreactions are considered in Chapter 3 and Chapter 4. Table 1.1 lists several important addition\npolymers, showing each monomer and polymer structure in the manner of Reaction (1.A). Also\nincluded in Table 1.1 are the molecular weights of the repeat units and the common names of\nthe polymers. The former will prove helpful in many of the problems in this book; the latter\nwill be discussed in the next section. Poly(ethy1ene oxide) and poly(a-caprolactam) have been\nincluded in this list as examples of the hazards associated with classi\ufb01cation schemes. They\nresemble addition polymers because the molecular weight of the repeat unit and that of the\nmonomer are the same; they resemble condensation polymers because of the heteroatom chain\nbackbone. The reaction mechanism, which might serve as arbiter in this case, can be either of the\nchain or the step type, depending on the reaction conditions. These last reactions are examples of\nring-opening polymerizations, yet another possible category of classification.\nThe requirements for formation of condensation polymers are twofold: the monomers must\npossess functional groups capable of reacting to form the linkage, and they ordinarily require\nmore than one reactive group to generate a chain structure. The functional groups can be\ndistributed such that two difunctional monomers with different functional groups react or a single\nmonomer reacts, which is difunctional with one group of each kind. In the latter case especially,\nbut also with condensation polymerization in general, the tendency to form cyclic products from\nintramolecular reactions may compete with the formation of polymers. Condensation polymeriza-\ntions are especially sensitive to impurities. The presence of monofunctional reagents introduces\nthe possibility of a reaction product forming which would not be capable of further growth. If the\nfunctionality is greater than 2, on the other hand, branching becomes possible. Both of these\nmodifications dramatically alter the product compared to a high molecular weight linear product.\nWhen reagents of functionality less than or greater than 2 are added in carefully measured and\ncontrolled amounts, the size and geometry of product molecules can be manipulated. When such\nreactants enter as impurities, the undesired results can be disastrous. Marvel has remarked that\nmore money has been wasted in polymer research by the use of impure monomers than in any other\nmanner [6].\nTable 1.2 lists several examples of condensation reactions and products. Since the reacting\nmonomers can contain different numbers of carbon atoms between functional groups, there are quite\n"]], ["block_4", ["Again, we emphasize that end effects are ignored in writing Reaction (LA). These effects as well\n"]], ["block_5", ["a lot of variations possible among these basic reaction types. The inclusion of poly(dimethylsiloxane)\nin Table 1.2 serves as a reminder that polymers need not be organic compounds. The physical\nproperties of inorganic polymers follow from the chain structure of these molecules, and the concepts\ndeveloped in this volume apply to them and to organic polymers equally well. We shall not examine\nexplicitly the classes and preparations of the various types of inorganic polymers in this text.\n"]], ["block_6", ["12\nIntroduction to Chain Molecules\n"]], ["block_7", [{"image_1": "28_1.png", "coords": [36, 171, 169, 214], "fig_type": "molecule"}]], ["block_8", ["H\n\u2014 \n#4\"\nLA\nn\nH>\u2014exiglas\u00ae, Lucite\u00ae\ne\ne\nH\no\no\no\u201d\n"]], ["block_9", [{"image_3": "29_3.png", "coords": [35, 62, 315, 528], "fig_type": "figure"}]], ["block_10", [{"image_4": "29_4.png", "coords": [41, 389, 92, 442], "fig_type": "molecule"}]], ["block_11", ["H \\\nH\nn\n104\nPolystyrene\n"]], ["block_12", ["H\nMe\n>=<\n56.0\nn\nPolyisobutylene\nH\nMe\nMe Me\n"]], ["block_13", ["H)\n(H\nm\nPolyacrylonitrile\n"]], ["block_14", ["H\n"]], ["block_15", ["H\nH\n.\nPoly(vmyl\nH>'_ Ag)\u201d \nchloride), \u201cvinyl\"\n"]], ["block_16", ["H\nCl\n.\n.\n.\n>_<\n97.0\nW\u201d\nPoly(vrnylidene chloride)\n"]], ["block_17", ["H\nCI\nCl\nCl\n"]], ["block_18", ["H\nH\n>=<\n28.0\nN\u201d\nPolyethylene\nH\nH\n"]], ["block_19", ["E\nC\n100\nPoly(tetra\ufb02uoroethylene),\n\u2014\nn\nTe\ufb02on\u00ae\nF\nF\nF\nF\n"]], ["block_20", ["F\nF\nF\nF\n"]], ["block_21", ["Me\nMe\nPoly(methyl methacrylate),\n"]], ["block_22", ["H\n"]], ["block_23", ["\u2014\n"]], ["block_24", ["0\nPoly(ethy1ene oxide),\n\\g\n44'0\nW\n9\u2018\npoly(ethylene glycol)\n"]], ["block_25", ["ll\n0\nW\n4\nPoly(e\u2014caprolactam),\nN\n1 13\n50\n\"\nNylon-6\n\\H\n"]], ["block_26", ["CN\n"]], ["block_27", ["53.0\n"]], ["block_28", [{"image_5": "29_5.png", "coords": [209, 88, 401, 272], "fig_type": "figure"}]], ["block_29", [{"image_6": "29_6.png", "coords": [235, 524, 289, 575], "fig_type": "molecule"}]], ["block_30", [{"image_7": "29_7.png", "coords": [237, 339, 295, 392], "fig_type": "molecule"}]], ["block_31", [{"image_8": "29_8.png", "coords": [239, 494, 288, 523], "fig_type": "molecule"}]], ["block_32", ["ON\u201d\n\u2018Cacrylic\u201d\n"]]], "page_30": [["block_0", [{"image_0": "30_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "30_1.png", "coords": [33, 386, 298, 424], "fig_type": "molecule"}]], ["block_2", ["Table 1.2\nReactions by Which Several Important Condensation Polymers Can Be Produced\n"]], ["block_3", ["their molecules. For example, the denaturation of a protein involves an overall conformation\nchange from a \u201cnative\u201d state, often a compact globule, to a random coil. As another example,\nthe elasticity and integrity of a cell membrane is often the result of an underlying network of\nfibrillar proteins, with the origin of the elasticity residing in the same conformational entropy as in\na rubber band. Consequently, although we will not discuss the synthesis by, and contribution to the\nfunction of, living organisms by such biopolymers, many of the principles we will develop in detail\napply equally well to natural polymers.\nAs examples of natural polymers, we consider polysaccharides, proteins, and nucleic acids.\nAnother important natural polymer, polyisoprene, will be considered in Section 1.6. Polysaccharides\n"]], ["block_4", ["4. Polycarbonate\no\n\u201c.49\nre\n9\nn CIJLCI + n HOQcQ\u2014QH {o\u2014Qc @o\u2014c} \n"]], ["block_5", ["14\nIntroduction to Chain Molecules\n"]], ["block_6", ["2. Polyamide\n"]], ["block_7", ["5. Inorganic\n"]], ["block_8", ["3. Polyurethane\n"]], ["block_9", ["1. Polyester\n"]], ["block_10", [{"image_2": "30_2.png", "coords": [52, 91, 317, 130], "fig_type": "molecule"}]], ["block_11", ["HO\nO\nO\nO\nn HO/\\/OH\n+\nn m\n-\u2014I-\u2014\nof\n+\n2n H20\n0\nOH\nO\nn\nPoly(ethylene terephthalate), Terylene\u00ae,\nDacron\u00ae, Mylar\u00ae\n"]], ["block_12", ["nocN\\(\\A,NCO+nH0\u2019(\u201d\\i0H\u2014*\u2014t\"'\u00e9JLJOLo\u2019bio\n"]], ["block_13", [{"image_3": "30_3.png", "coords": [74, 225, 282, 264], "fig_type": "molecule"}]], ["block_14", ["n\nH2N\nNH2\nn M\n)-\nN\nN\nM;\n+\nCl\n4\nCl\n*4;\n4\n+\n2n HCl\n"]], ["block_15", [{"image_4": "30_4.png", "coords": [99, 458, 361, 519], "fig_type": "molecule"}]], ["block_16", [{"image_5": "30_5.png", "coords": [116, 471, 357, 502], "fig_type": "molecule"}]], ["block_17", ["Me\nI\n|\nn\nMe-SI'I\u2014CI + :1 H20\n\u2014\u20141-\u2014\nSli\u2014O}\n+\n2n HCI\n"]], ["block_18", [{"image_6": "30_6.png", "coords": [130, 91, 423, 142], "fig_type": "molecule"}]], ["block_19", [{"image_7": "30_7.png", "coords": [136, 157, 346, 194], "fig_type": "molecule"}]], ["block_20", [{"image_8": "30_8.png", "coords": [137, 225, 400, 275], "fig_type": "molecule"}]], ["block_21", ["n Ho)\\(\\/)/U\\10\n+\nn\nH20\n"]], ["block_22", ["Me\nMe\nn\nPoly(4,4-isopropylidenediphenylene carbonate)\nbisphenol A polycarbonate, Lexan\u00ae\n"]], ["block_23", ["Cl\nn\nPoly(dimethylsiloxane)\n"]], ["block_24", ["Cts\n"]], ["block_25", [{"image_9": "30_9.png", "coords": [184, 467, 345, 515], "fig_type": "molecule"}]], ["block_26", [{"image_10": "30_10.png", "coords": [194, 375, 426, 432], "fig_type": "molecule"}]], ["block_27", ["0\nCts\n"]], ["block_28", [{"image_11": "30_11.png", "coords": [201, 91, 393, 130], "fig_type": "molecule"}]], ["block_29", [{"image_12": "30_12.png", "coords": [218, 303, 369, 339], "fig_type": "molecule"}]], ["block_30", ["Poly(tetramethylenehexamethylene urethane),\nSpandex\u00ae, Perlon\u00ae\n"]], ["block_31", [{"image_13": "30_13.png", "coords": [223, 158, 331, 192], "fig_type": "molecule"}]], ["block_32", ["Poly(hexamethylene adipamide), Nylon-6,6\n"]], ["block_33", ["Poly(12\u2014hydroxystearic acid)\n"]], ["block_34", [{"image_14": "30_14.png", "coords": [225, 229, 389, 268], "fig_type": "molecule"}]], ["block_35", ["0\nO n\n"]]], "page_31": [["block_0", [{"image_0": "31_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "31_1.png", "coords": [27, 159, 375, 280], "fig_type": "figure"}]], ["block_2", [{"image_2": "31_2.png", "coords": [29, 193, 350, 399], "fig_type": "figure"}]], ["block_3", ["are macromolecules which make up a large part of the bulk of the vegetable kingdom. Cellulose\nand starch are, respectively, the first and second most abundant organic compounds in plants. The\nformer is present in leaves and grasses; the latter in fruits, stems, and roots. Because of their\nabundance in nature and because of contemporary interest in renewable resources, there is a great\ndeal of interest in these compounds. Both cellulose and starch are hydrolyzed by acids to D-glucose,\nthe repeat unit in both polymer chains. The con\ufb01guration of the glucoside linkage is different in the\ntwo, however. Structure (1.1) and Structure (1.11), respectively, illustrate that the linkage is a B-acetal\u2014\nhydrolyzable to an equatorial hydroxide\u2014in cellulose and an a-acetal\u2014hydrolyzable to an axial\nhydroxide\u2014in amylose, a starch:\n"]], ["block_4", ["Amylopectin and glycogen are saccharides similar to amylose, except with branched chains.\nThe cellulose repeat unit contains three hydroxyl groups, which can react and leave the chain\nbackbone intact. These alcohol groups can be esterified with acetic anhydride to form cellulose\nacetate; this polymer is spun into the fiber acetate rayon. Similarly, the alcohol groups in cellulose\nreact with CS2 in the presence of strong base to produce cellulose xanthates. When extruded into\nfibers, this material is called viscose rayon, and when extruded into sheets, cellophane. In both the\nacetate and xanthate formation, some chain degradation also occurs, so the resulting polymer\nchains are shorter than those in the starting cellulose. The hydroxyl groups are also commonly\nmethylated, ethylated, and hydroxypropylated for a variety of aqueous applications, including food\nproducts. A closely related polysaccharide is chitin, the second most abundant polysaccharide in\nnature, which is found for example in the shells of crabs and beetles. Here one of the hydroxyls on\neach repeat unit of cellulose is replaced with an \u2014NHCO\u2014CH3 amide group. This is converted to a\nprimary amine \u2014NH2 in chitosan, a derivative of chitin finding increasing applications in a variety of\nfields.\nAs noted above, proteins are polyamides in which a-amino acids make up the repeat units, as\nshown by Structure (1.III):\n"]], ["block_5", ["These molecules are also called polypeptides, especially when M 3 10,000. The various amino\nacids differ in their R groups. The nature of R, the name, and the abbreviation used to represent\n"]], ["block_6", ["Addition, Condensation, and Natural Polymers\n15\n"]], ["block_7", [{"image_3": "31_3.png", "coords": [43, 594, 108, 643], "fig_type": "molecule"}]], ["block_8", ["H\nO\n{/\u201cH\ufb01\n(1.111)\n"]], ["block_9", ["0\n0\n(1.1)\nHO\nHO\n0\nH\nOH\nH\nOH\n"]], ["block_10", ["R\n11\n"]], ["block_11", ["H\nOH\nH OH\n"]], ["block_12", ["H\nH\nH\nH\n"]], ["block_13", [{"image_4": "31_4.png", "coords": [136, 278, 342, 404], "fig_type": "figure"}]], ["block_14", [{"image_5": "31_5.png", "coords": [206, 159, 345, 278], "fig_type": "figure"}]], ["block_15", ["(1.11)\n"]]], "page_32": [["block_0", [{"image_0": "32_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Table 1.3\nName, Abbreviations, and R Group for Common Amino Acids\n"]], ["block_2", ["Lysine\nLys\nVNHz\n"]], ["block_3", ["HOOCYNHQ\nName\nAbbreviation\nR Group\nR\n"]], ["block_4", ["Me\nAlanine\nAla\nJ\u201d\n"]], ["block_5", ["Aspartic acid\nAsp\nKLOH\n"]], ["block_6", ["Glutamine\nGln\nm\n"]], ["block_7", ["H\nGlycine\nGly\n\u201c1\u201c,\n"]], ["block_8", ["NH\n/ x)\nHistidine\nHis\nN\n"]], ["block_9", ["H\nNH\nArginine\nArg\np Y\nNH2\n"]], ["block_10", ["Asparagine\nAsn\n\ufb01 NH2\n"]], ["block_11", ["Glutamic acid\nGlu\nm\n"]], ["block_12", ["Me\nMe\nIsoleucine\nIle\n"]], ["block_13", ["Leucine\nLeu\n\ufb01me\n"]], ["block_14", ["Cysteine\nCys\nNE\n"]], ["block_15", ["16\nIntroduction to Chain Molecules\n"]], ["block_16", [{"image_1": "32_1.png", "coords": [239, 140, 321, 187], "fig_type": "molecule"}]], ["block_17", [{"image_2": "32_2.png", "coords": [247, 620, 316, 644], "fig_type": "molecule"}]], ["block_18", [{"image_3": "32_3.png", "coords": [253, 512, 310, 560], "fig_type": "molecule"}]], ["block_19", [{"image_4": "32_4.png", "coords": [253, 374, 310, 421], "fig_type": "molecule"}]], ["block_20", [{"image_5": "32_5.png", "coords": [254, 185, 307, 239], "fig_type": "molecule"}]], ["block_21", [{"image_6": "32_6.png", "coords": [261, 457, 301, 502], "fig_type": "molecule"}]], ["block_22", [{"image_7": "32_7.png", "coords": [262, 243, 300, 282], "fig_type": "molecule"}]], ["block_23", [{"image_8": "32_8.png", "coords": [264, 330, 311, 368], "fig_type": "molecule"}]], ["block_24", ["0\n"]], ["block_25", ["Me\n"]], ["block_26", ["0\n"]], ["block_27", ["SH\n"]], ["block_28", ["NH2\n"]], ["block_29", ["OH\n"]]], "page_33": [["block_0", [{"image_0": "33_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "33_1.png", "coords": [9, 70, 314, 404], "fig_type": "figure"}]], ["block_2", ["Tyrosine\nTyr\nY\nAL/O/\n"]], ["block_3", ["/\nMethionine\nMet\nM\nf\nS\n"]], ["block_4", ["O\nM\nThreonine\nThr\nT\nI\n"]], ["block_5", ["Tryptophan\nTrp\nW\nI\nE\n"]], ["block_6", ["Valine\nVal\nV\n"]], ["block_7", ["some of the more common amino acids are listed in Table 1.3. In proline (Pro) the nitrogen and the\no-carbon are part of a five-atom pyrrolidine ring. Since some of the amino acids carry substituent\ncarboxyl or amino groups, protein molecules are charged in aqueous solutions, and hence can\nmigrate in electric \ufb01elds. This is the basis of electrOphoresis as a means of separating and\nidentifying proteins.\nIt is conventional to speak of three levels of structure in protein molecules:\n"]], ["block_8", ["Addition, \n17\n"]], ["block_9", ["Table \n"]], ["block_10", [".._\u2014\u2014\n"]], ["block_11", ["1.\nPrimary structure refers to the sequence of amino acids in the polyamide chain.\n2.\nSecondary structure refers to the regions of the molecule that have particular spatial arrange-\nments. Examples in proteins include the o-helix and the B-sheet.\n"]], ["block_12", ["Name\nAbbreviation\nCode\nR Group\n\\Fll/\n"]], ["block_13", ["Phenylalanine\nPhe\nF\nJ/Q\n"]], ["block_14", ["H\nProline\nPro\nP\n"]], ["block_15", ["Serine\nSer\nS\n"]], ["block_16", ["Letter\nHOOC\nN H2\n"]], ["block_17", [{"image_2": "33_2.png", "coords": [246, 423, 308, 476], "fig_type": "molecule"}]], ["block_18", [{"image_3": "33_3.png", "coords": [248, 105, 311, 136], "fig_type": "molecule"}]], ["block_19", [{"image_4": "33_4.png", "coords": [251, 204, 309, 253], "fig_type": "molecule"}]], ["block_20", [{"image_5": "33_5.png", "coords": [255, 488, 299, 515], "fig_type": "molecule"}]], ["block_21", ["H\n"]], ["block_22", ["Mel\n"]], ["block_23", ["'?'\nH\nO\n"]], ["block_24", ["to\u201d\n"]], ["block_25", ["\\\n"]], ["block_26", ["Me\n"]], ["block_27", ["O\n"]], ["block_28", ["N\n"]], ["block_29", ["e\n"]], ["block_30", ["OH\n"]]], "page_34": [["block_0", [{"image_0": "34_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["3.\nThe globular proteins albumin in eggs andfibrinogen in blood are converted to insoluble forms\nby modification of their higher-order structure. The process is called denaturation and occurs,\nin the systems mentioned, with the cooking of eggs and the clotting of blood.\n4.\nActin is a fascinating protein that exists in two forms: G-actin (globular) and F\u2014actin (fibrillar).\nThe globular form can polymerize (reversibly) into very long filaments under the in\ufb02uence of\nvarious triggers. These filaments play a crucial role in the cytoskeleton, i.e., in allowing cells\nto maintain their shape. In addition, the uniaxial sliding of actin filaments relative to filaments\nof a related protein, myosin, is responsible for the working of muscles.\n"]], ["block_2", ["3.\nTertiary structure refers to the overall shape of the molecule, for example, a globule perhaps\nstabilized by disul\ufb01de bridges formed by the oxidation of cysteine mercapto groups. By extension\nthe full tertiary structure implies knowledge of the relative spatial positions of all the residues.\n"]], ["block_3", ["Ribonucleic acid (RNA) and deoxyribonucleic acid (DNA) are polymers in which the repeat\nunits are substituted esters. The esters are formed between the hydrogens of phosphoric acid and\nthe hydroxyl groups of a sugar, D\u2014ribose in the case of RNA and D\u20142-deoxyribose in the case of\nDNA. The sugar rings in DNA carry four different kinds of substituents: adenine (A) and guanine\n(G), which are purines, and thymine (T) and cytosine (C), which are pyramidines. The familiar\ndouble-helix structure of the DNA molecule is stabilized by hydrogen bonding between pairs of\nsubstituent base groups: G\u2014C and A\u2014T. In RNA, thymine is usually replaced by uracil (U). The\nreplication of these molecules, the template model of their functioning, and their role in protein\nsynthesis and the genetic code make the study of these polymers among the most exciting and\nactively researched areas in science. As with the biological function of proteins, we will not discuss\nthese phenomena in this book. However, as indicated previously, DNA plays a very important role\n"]], ["block_4", ["Hydrogen bonding stabilizes some protein molecules in helical forms, and disulfide cross\u2014links\nstabilize some protein molecules in globular forms. Both secondary and tertiary levels of structure\nare also in\ufb02uenced by the distribution of polar and nonpolar amino acid molecules relative to the\naqueous environment of the protein molecules. In some cases, individual proteins associate in\nparticular aggregates, which are referred to as quaternary structures.\nExamples of the effects and modifications of the higher-order levels of structures in proteins are\nfound in the following systems:\n"]], ["block_5", ["1.\nCollagen is the protein of connective tissues and skin. In living organisms, the molecules are\nwound around one another to form a three\u2014stranded helix stabilized by hydrogen bonding.\nWhen boiled in water, the collagen dissolves and forms gelatin, thereby establishing a new\nhydrogen bond equilibrium with the solvent. This last solution sets up to form the familiar gel\nwhen cooled, a result of shifting the hydrogen bond equilibrium.\n2.\nKeratin is the protein of hair and wool. These proteins are insoluble because of the disulfide\ncross-linking between cysteine units. Permanent waving of hair involves the rupture of these\nbonds, reshaping of the hair \ufb01bers, and the reformation of cross-links, which hold the chains in\nthe new positions relative to each other. We shall see in Chapter 10 how such cross-linked\nnetworks are restored to their original shape when subjected to distorting forces.\n"]], ["block_6", ["Considering that a simple compound like CZHSOH is variously known as ethanol, ethyl alcohol,\ngrain alcohol, or simply alcohol, it is not too surprising that the vastly more complicated polymer\nmolecules are also often known by a variety of different names. The International Union of Pure\nand Applied Chemistry (IUPAC) has recommended a system of nomenclature based on the\n"]], ["block_7", ["18\nIntroduction to Chain Molecules\n"]], ["block_8", ["as a prototypical semi\ufb02exible polymer, as it is now readily obtainable in pure molecular fractions\nof varying lengths, and because it is readily dissolved in aqueous solution. It is also a charged\npolymer, or polyelectrolyte, and thus serves as a model system in this arena as well.\n"]], ["block_9", ["1.5\nPolymer Nomenclature\n"]]], "page_35": [["block_0", [{"image_0": "35_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["structure \nwidespread usage; these latter names seem even more resistant to replacement than is the case with\nlow molecular polymers of commercial importance are often \nknown by trade names that have more to do with marketing considerations than with scientific\ncommunication. Polymers of biological origin are often described in terms of some aspect of their\nfunction, preparation, or characterization.\nIf a polymer is formed from a single monomer, as in addition and ring-opening polymerizations,\nit is named by attaching the to the name of the monomer. In the IUPAC system, the\nmonomer is named according to the IUPAC recommendations for organic chemistry, and the name\nof the monomer is set off from the prefix by enclosing the former in parentheses. Variations of this\nbasic system often substitutefor the IUPAC name in designating the monomer.\nWhether or not parentheses are used in the latter case is in\ufb02uenced by the complexity of the\nmonomer name; they become more important\nas the number of words in the monomer\nname increases. Thus the polymer (CHg\u2014CHCI)n is called poly(l-chloroethylene) according to\nthe IUPAC system; it commonly called poly(vinyl chloride) or polyvinyl chloride.\nAcronyms are not particularly helpful but are an almost irresistible aSpect of polymer terminology,\n"]], ["block_2", ["as evidenced by the initials PVC, are widely used to describe the polymer just named. The\ntrio of names poly(l-hydroxyethylene), poly(vinyl alcohol), and polyvinyl alcohol emphasizes that\nthe polymer need not actually be formed from the reaction of the monomer named; this polymer is\nactually prepared by the hydrolysis of poly(l-acetoxyethylene), otherwise known as poly(vinyl\nacetate). These same alternatives are used in naming polymers formed by ring-opening reactions;\nfor example, poly(6-aminohexanoic acid), poly(6-aminocapr0ic acid), and poly(s-caprolactam) are\nall more or less acceptable names for the same polymer.\nThose polymers which are the condensation products of two different monomers are named by\napplying the preceding rules to the repeat unit. For example, the polyester formed by the condensation\nof ethylene glycol and terephthalic acid is called poly(oxyethylene oxyterphthaloyl) according to the\nIUPAC system, but is more commonly referred to as poly(ethylene terephthalate) or polyethylene\nterephthalate. The polyamides poly(hexamethylene sebacamide) and poly(hexamethylene adipamide)\nare also widely known as nylon-6,10 and nylon-6,6, respectively. The numbers following the word\nnylon indicate the number of carbon atoms in the diamine and dicarboxylic acids, in that order. On the\nbasis of this system, poly(s\u2014caprolactam) is also known as nylon-6.\nMany of the polymers in Table 1.1 and Table 1.2 are listed with more than one name. Also listed\nare some of the registered trade names by which these substances\u2014or materials which are mostly\nof the indicated structure~\u2014are sold commercially. Some commercially important cross-linked\npolymers go virtually without names. These are heavily and randomly cross-linked polymers\nwhich are insoluble and infusible and therefore widely used in the manufacture of such molded\nitems as automobile and household appliance parts. These materials are called resins and, at best,\nare named by specifying the monomers that go into their production. Often even this information is\nsketchy. Examples of this situation are provided by phenol-formaldehyde and urea\u2014formaldehyde\nresins, for which typical structures are given by Structure (1.1V) and Structure (1.V), respectively:\n"]], ["block_3", ["Polymer Nomenclature\n19\n"]], ["block_4", [{"image_1": "35_1.png", "coords": [41, 601, 127, 671], "fig_type": "figure"}]], ["block_5", [{"image_2": "35_2.png", "coords": [41, 536, 114, 609], "fig_type": "molecule"}]], ["block_6", ["0H\n"]], ["block_7", ["|\\\n(1.V)\n"]], ["block_8", ["(LIV)\n"]]], "page_36": [["block_0", [{"image_0": "36_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "36_1.png", "coords": [32, 225, 253, 324], "fig_type": "figure"}]], ["block_2", ["20\nIntroduction to Chain Molecules\n"]], ["block_3", ["In this section, we shall consider three types of isomerism that are encountered in polymers. These\n"]], ["block_4", ["Structure (1.VI) and Structure (1.VII), respectively, are said to arise from head-to-tat'l or head-to-\nhead orientations. In this terminology, the substituted carbon is defined to be the head and the\nmethylene is the tail. Tail-to-tail linking is also possible.\nFor most vinyl polymers, head-to-tail addition is the dominant mode of addition. Variations\nfrom this generalization become more common for polymerizations which are carried out at higher\ntemperatures. Head-to\u2014head addition is also somewhat more abundant in the case of halogenated\nmonomers such as vinyl chloride. The preponderance of head-to\u2014tail additions is understood to\narise from a combination of resonance and steric effects. In many cases, the ionic or free-radical\nreaction center occurs at the substituted carbon due to the possibility of resonance stabilization or\nelectron delocalization through the substituent group. Head-to-tail attachment is also sterically\nfavored, since the substituent groups on successive repeat units are separated by a methylene\ncarbon. At higher polymerization temperatures, larger amounts of available thermal energy make\nthe less-favored states more accessible. In vinyl \ufb02uoride, no resonance stabilization is possible and\nsteric effects are minimal. This monomer adds primarily in the head\u2014to-tail orientation at low\ntemperatures and tends toward a random combination of both at higher temperatures. The styrene\nradical, by contrast, enjoys a large amount of resonance stabilization in the bulky phenyl group and\npolymerizes almost exclusively in the head-to\u2014tail mode. The following example illustrates how\nchemical methods can be used to measure the relative amounts of the two positional isomers in a\npolymer sample.\n"]], ["block_5", ["are positional isomerism, stereo isomerism, and geometrical isomerism. We shall focus attention\non synthetic polymers and shall, for the most part, be concerned with these types of isomerism\noccurring singly, rather than in combinations. Some synthetic and analytical aspects of stereo\nisomerism will be considered in Chapter 5. Our present concern is merely to introduce the\npossibilities of these isomers and some of the associated vocabulary.\n"]], ["block_6", ["Positional isomerism is conveniently illustrated by considering the polymerization of a vinyl\nmonomer. In such a reaction, the adding monomer may become attached to the growing chain\nend (designated by =k) in either of two orientations:\n"]], ["block_7", ["1,2-Glycol bonds are cleaved by reaction with periodate; hence poly(vinyl alcohol) chains are\nbroken at the site of head\u2014to-head links in the polymer. The fraction of head\u2014to-head linkages\nin poly(vinyl alcohol) may be determined by measuring the molecular weight before (subscript b)\nand\nafter\n(subscript\na)\ncleavage\nwith\nperiodate\naccording\nto\nthe\nfollowing\nformula:\nFraction =44(1/Ma\u20141/Mb). Derive this expression and calculate the value for the fraction in the\ncase of Mb: 105 and M3,: 103.\n"]], ["block_8", ["1.6\nStructural Isomerism\n"]], ["block_9", ["1.6.1\nPositional lsomerism\n"]], ["block_10", ["Example 1.3\n"]], ["block_11", ["H\nH\nX\nL<\n+\n>=<\n.._._...\n(1.13)\n"]], ["block_12", [{"image_2": "36_2.png", "coords": [56, 249, 171, 303], "fig_type": "molecule"}]], ["block_13", ["X\nH\nH\nX\n"]], ["block_14", [{"image_3": "36_3.png", "coords": [152, 218, 438, 324], "fig_type": "figure"}]], ["block_15", [{"image_4": "36_4.png", "coords": [167, 273, 239, 326], "fig_type": "molecule"}]], ["block_16", ["\u2014\nmy\nX\nX'\n(1.VI)\n"]], ["block_17", ["_\nx\nH\u201d\n(1.VII)\n"]], ["block_18", [{"image_5": "36_5.png", "coords": [176, 229, 243, 268], "fig_type": "molecule"}]]], "page_37": [["block_0", [{"image_0": "37_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Begin by recognizing containing x of the head-to-head links will be cleaved into\nx + 1 molecules upon reaction. Hence if n is the number of polymer molecules in a sample of mass\nw, the following relations apply before and after cleavage: na=(x+1)nb or w/Ma=(x+ 1)\n(w/Mb). Solving for x and dividing the latter by the total number of linkages in the original\npolymer gives the desired ratio. The total number of links in the original polymer is Mb/Mo.\nTherefore M0(1/Ma\u20141/Mb). For poly(vinyl alcohol) M0 is 44, so the desired\nformula has been obtained. For the specific data given, x/nb =44(10_3\u2014-10_5) 0.044, or about\n4% 0f the additions are in the less favorable orientation. We shall see presently that the molecular\nweight of a polymer is an average, which is different depending on the method used for its\ndetermination. The present example used molecular weights as a means for counting the number\nof molecules present. Hence the sort of average molecular weight used should also be one which is\nbased on counting.\n"]], ["block_2", ["Structure (1.VIII) and Structure (LIX) are not equivalent; they would not superimpose if the\nextended chains were overlaid. The difference has to do with the stereochemical con\ufb01guration at\nthe asymmetric carbon atoms. Note that the asymmetry is more accurately described as pseudoa-\nsymmetry, since two sections of chain are bonded to these centers. Except near chain ends, which\nwe ignore for high polymers, these chains provide local symmetry in the neighborhood of the\ncarbon under consideration. The designations of D and L or R and S are used to distinguish these\nstructures, even though true asymmetry is absent.\nWe use the word con\ufb01guration to describe the way the two isomers produced by Reaction (1.C)\ndiffer. It is only by breaking bonds, moving substituents, and reforming new bonds that the two\nstructures can be interconverted. This state of affairs is most readily seen when the molecules are\ndrawn as fully extended chains in one plane, and then examining the side of the chain on\nwhich substituents lie. The con\ufb01gurations are not altered if rotation is allowed to occur around\nthe various bonds of the backbone to change the shape of the molecule to a jumbled coil. We shall\nuse the term conformation to describe the latter possibilities for different molecular shapes.\nThe configuration is not in\ufb02uenced by conformational changes, but the stability of different\nconformations may be affected by differences in configuration. We shall return to these effects\nin Chapter 6.\nIn the absence of any external in\ufb02uence, such as a catalyst that is biased in favor of one\ncon\ufb01guration over the other, we might expect Structure (1.VIII) and Structure (LIX) to occur at\nrandom with equal probability as if the configuration at each successive addition were determined\nby the toss of a coin. Such indeed is the ordinary case. However, in the early 19503, stereospeci\ufb01c\ncatalysts were discovered; Ziegler and Natta received the Nobel Prize for this discovery in 1963.\nFollowing the advent of these catalysts, polymers with a remarkable degree of stereoregularity\nhave been formed. These have such a striking impact on polymer science that a substantial part of\n"]], ["block_3", [{"image_1": "37_1.png", "coords": [34, 278, 294, 380], "fig_type": "figure"}]], ["block_4", ["The second type of isomerism we discuss in this section is stereo isomerism. Again we consider the\nnumber of ways a singly substituted vinyl monomer can add to a growing polymer chain:\n"]], ["block_5", ["Structural \n21\n"]], ["block_6", ["Solution\n"]], ["block_7", ["1.6.2\nStereo Isomerism\n"]], ["block_8", ["X\na,\nH\nX\nc\nc\n"]], ["block_9", [".j\u201c\n+\n>=<\n\u2014'*\n(l.C)\n"]], ["block_10", ["_\n(l.VIII)\nH\nH\n"]], ["block_11", [{"image_2": "37_2.png", "coords": [153, 275, 345, 382], "fig_type": "figure"}]], ["block_12", ["I\n\"\n"]], ["block_13", ["(LIX)\n"]]], "page_38": [["block_0", [{"image_0": "38_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "38_1.png", "coords": [31, 133, 313, 197], "fig_type": "molecule"}]], ["block_2", ["Figure 1.3 shows sections of polymer chains of these three types; the substituent X equals\nphenyl for polystyrene and methyl for polypropylene. The general term for this stereoregularity\nis tacticity, a term derived from the Greek word meaning \u201cto put in order.\u201d Polymers of\ndifferent tacticity have quite different properties, especially in the solid state. As we will\nsee in Chapter 13, one of the requirements for polymer crystallinity is a high degree of\nmicrostructural regularity to enable the chains to pack in an orderly manner. Thus atactic\npolypropylene is a soft, tacky substance, whereas both isotactic and syndiotactic polypropylene\nare highly crystalline.\n"]], ["block_3", ["The \ufb01nal type of isomerism we take up in this section is nicely illustrated by the various possible\nstructures that result from the polymerization of 1,3-dienes. Three important monomers of this type\nare 1,3-butadiene, 1,3-isoprene, and 1,3-chloroprene, Structure (1.X) through Structure (1.XII),\nrespectively:\n"]], ["block_4", ["22\nIntroduction to Chain Molecules\n"]], ["block_5", ["Chapter 5 is devoted to a discussion of their preparation and characterization. For now, only the\nterminology involved in their description concerns us. Three different situations can be distin-\nguished along a chain containing pseudoasymmetric carbons:\n"]], ["block_6", ["1.\nIsotactic. All substituents lie on the same side of the extended chain. Alternatively, the\nstereoconfiguration at the asymmetric centers is the same, say, \u2014~DDDDDDDDD\u2014\u2014.\n2.\nSyndiotactic. Substituents on the fully extended chain lie on alternating sides of the backbone.\nThis alternation of configuration can be represented as \u2014DLDLDLDLDLDL\u2014~.\n"]], ["block_7", ["3.\nAtactic.\nSubstituents\nare\ndistributed\nat\nrandom\nalong\nthe\nchain,\nfor\nexample,\nDDLDLLLDLDLL\u2014.\n"]], ["block_8", ["Figure 1.3\nSections of \u201cpolyvinyl X\u201d chains of differing tacticity: (a) isotactic, (b) syndiotactic, and\n"]], ["block_9", ["(c) atactic.\n"]], ["block_10", ["1.6.3\nGeometrical lsomerism\n"]], ["block_11", [{"image_2": "38_2.png", "coords": [41, 619, 111, 671], "fig_type": "molecule"}]], ["block_12", ["HNH\n(LX)\n"]], ["block_13", [{"image_3": "38_3.png", "coords": [54, 218, 301, 284], "fig_type": "molecule"}]], ["block_14", ["H\n"]], ["block_15", ["d9\nd'\n:3\n.4\n\u20185\n\u20ac-\n\u20185\n\u20185\n:3\nf.\n15\n:\n'5\n\u20185\n1\n9..\n5\n\u00a2\n0\nc\n1'.\nc\n'o,\n\u20185\nz\n\u2018\n1\nd\n0\n"]], ["block_16", ["\u201c\u20180\n\\tw\u201c\nI:\nI:\n.1-\n:\u2014\n.-\n[51:\n"]], ["block_17", [".\u2014\nI:\nI;\n"]], ["block_18", ["I,\n[f-\n"]], ["block_19", [".\no\n"]]], "page_39": [["block_0", [{"image_0": "39_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["To illustrate the possible modes of polymerization of these compounds, consider the following\nreactions of \n"]], ["block_2", ["Structural \n23\n"]], ["block_3", ["1.\n1,2\u2014 and 3,4-Polymerizations. As far as the polymer chain backbone is concerned, these\ncompounds could just as well be mono\u2014olefins, since the second double bond is relegated to\nthe status of a substituent group. Because of the reactivity of the latter, however, it might\nbecome involved in cross-linking reactions. For isoprene, 1,2- and 3,4-polymerizations yield\ndifferent products:\n"]], ["block_4", [{"image_1": "39_1.png", "coords": [37, 248, 319, 321], "fig_type": "figure"}]], ["block_5", [{"image_2": "39_2.png", "coords": [39, 47, 106, 104], "fig_type": "molecule"}]], ["block_6", [{"image_3": "39_3.png", "coords": [43, 507, 119, 601], "fig_type": "molecule"}]], ["block_7", [{"image_4": "39_4.png", "coords": [43, 95, 110, 142], "fig_type": "molecule"}]], ["block_8", [{"image_5": "39_5.png", "coords": [44, 410, 244, 468], "fig_type": "molecule"}]], ["block_9", [{"image_6": "39_6.png", "coords": [47, 255, 214, 307], "fig_type": "molecule"}]], ["block_10", ["These differences do not arise from 1,2\u2014 or 3,4\u2014polymerization of butadiene. Structure (1.X111)\nand Structure (1.XIV) can each exhibit the three different types of tacticity, so a total of six\nstructures can result from this monomer when only one of the olefin groups is involved in the\nbackbone formation.\n1,4-Polymerization. This mode of polymerization gives a molecule with double bonds along\nthe backbone of the chain. Again using isoprene as the example,\n"]], ["block_11", ["As in all double-bond situations, the adjacent chain sections can be either cis or trans\u2014\nStructure (1.XV) and Structure (1.XV1), respectivelymwith respect to the double bond,\nproducing the following geometrical isomers:\n"]], ["block_12", ["Figure 1.4 shows several repeat units of cis\u2014l,4-polyisoprene and trans\u20141,4\u2014polyisoprene.\nNatural rubber is the cis isomer of 1,4\u2014polyisoprene and gutta\u2014percha is the trans isomer.\nPolymers of Chloroprene (Structure (1.X11)) are called neoprene and copolymers of butadiene and\nstyrene are called SBR, an acronym for styrene\u2014butadiene rubber. Both are used for many of the\nsame applications as natural rubber. Chloroprene displays the same assortment ofpossible isomers\n"]], ["block_13", ["Me\nH\nm\n(1 .XV)\n"]], ["block_14", ["Me\nMe\nH\nn\nn\nn\nHNH\n\u2014+\n\\\nH\nor\nMe \nH\n(LD)\n"]], ["block_15", ["n\nM\n"]], ["block_16", ["H\nH\nH\n"]], ["block_17", ["(LE)\nE3i\n"]], ["block_18", ["H\nn\n"]], ["block_19", ["H\n(1 .X1)\n"]], ["block_20", ["H\n(1 .XII)\n"]], ["block_21", [{"image_7": "39_7.png", "coords": [102, 423, 235, 459], "fig_type": "molecule"}]], ["block_22", [{"image_8": "39_8.png", "coords": [162, 246, 309, 335], "fig_type": "molecule"}]], ["block_23", ["(1 .X111)\n(1.X1V)\n"]], ["block_24", ["(1.XVI)\n"]]], "page_40": [["block_0", [{"image_0": "40_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "40_1.png", "coords": [22, 54, 341, 100], "fig_type": "molecule"}]], ["block_2", [{"image_2": "40_2.png", "coords": [25, 55, 153, 107], "fig_type": "molecule"}]], ["block_3", ["This example, as simplistic as it is, actually underscores two important points. First, polymer\nchemists have to get used to the idea that while all carbon atoms are identical, and all 1,3-butadiene\nmolecules are identical, polybutadiene actually refers to an effectively in\ufb01nite number of distinct\nchemical structures. Second, almost all synthetic polymers are heterogeneous in more than one\nvariable: molecular weight, certainly; isomer and tacticity distribution, probably; composition and\nsequence distribution, for copolymers; and branching structure, when applicable.\n"]], ["block_4", ["Almost every synthetic polymer sample contains molecules of various degrees of polymerization.\nWe describe this state of affairs by saying that the polymer shows polydispersity with respect to\nmolecular weight or degree of polymerization. To see how this comes about, we only need to think\nof the reactions between monomers that lead to the formation of polymers in the first place.\nRandom encounters between reactive species are responsible for chain growth, so statistical\n"]], ["block_5", ["as isoprene; the extra combinations afforded by copolymer composition and structure in SBR\noffset the fact that Structure (1 .XHI) and Structure (LXIV) are identical for butadiene.\n4.\nAlthough the conditions of the polymerization reactions may be chosen to optimize the\nformation of one specific isomer, it is typical in these systems to have at least some\ncontribution of all possible isomers in the polymeric product, except in the case of polymers\nof biological origin, like natural rubber and gutta-percha.\n"]], ["block_6", [". . - x 3 = 3N, where N is the degree of polymerization. (Recall that the combined probability of a\nsequence of events is equal to the products of the individual probabilities.) In this case N 54,000/\n54: 1000, and thus there are about 31000 m 10500 possible structures. Now we need to count\nhow many molecules we have. Assuming for simplicity that the tank car is 3.3 m\nX\n3.3 m x\n10 m 100 m3 108 cm3, and the density of the polymer is 1 g/cm3 (it is actually closer to 0.89\ng/cm3), we have 108 g of polymer. As M 54,000 g/mol, we have about 2000 moles, or 2000 X 6\nx\n1023 a 1027 molecules. Clearly, therefore, there is essentially no chance that any two molecules\nhave the identical structure, even without taking the molecular weight distribution into account.\n"]], ["block_7", ["We will not attempt to provide a precise answer to such an artificial question; what we really want\nto know is whether the probability is high (approximately 1), vanishing (approximately 0), or finite.\nFrom the discussion above, we recognize three geometrical isomers: trans-1,4, cis-1,4, and 1,2.\nWe will ignore the stereochemical possibilities associated with the 1,2 linkages. Assuming all three\nisomers occur with equal probability, the total number of possible structures is 3 X 3 x 3 x\n"]], ["block_8", ["Suppose you have just ordered a tank car of polybutadiene from your friendly rubber company. By\nsome miracle, all the polymers in the sample have M 254,000. The question we would like to\nconsider is this: what are the chances that any two molecules in this sample have exactly the same\nchemical structure?\n"]], ["block_9", ["24\nIntroduction to Chain Molecules\n"]], ["block_10", ["Solution\n"]], ["block_11", ["(a)\n(b)\n"]], ["block_12", ["1.7\nMolecular Weights and Molecular Weight Averages\n"]], ["block_13", ["Example 1.4\n"]], ["block_14", ["Figure 1.4\n1,4-Polyisoprene (a) all-cis isomer (natural rubber) and (b) all\u2014trans isomer (gutta-percha).\n"]], ["block_15", [{"image_3": "40_3.png", "coords": [174, 57, 325, 96], "fig_type": "molecule"}]]], "page_41": [["block_0", [{"image_0": "41_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["The probability x, is the number fraction or mole fraction of i-mer. We can use this quantity to\ndefine a particular average molecular weight, called the number-average molecular weight, Mn.\nWe do this by multiplying the probability of \ufb01nding an i-mer with its associated molecular weight,\nx,M,-, and adding all these up:\n"]], ["block_2", ["descriptions are appropriate for the resulting product. The situation is reminiscent of the distribu-\n"]], ["block_3", ["tion of molecular \nenergy to some reducing the energy of others. Therefore, we talk about the\nmolecular weight of a polymer, we mean some characteristic average molecular weight. It turns out\nthere are several distinct averages that may be defined, and that may be measured experimentally; it\nis therefore spend time on this topic. Furthermore, one might well \nsamples ofin terms of one kind of \nterms of another; this, in turn, can lead to the situation where the two polymers behave identically\nin terms of some important properties, but differently in terms of others.\nIn Chapter 2 through Chapter 4 we shall examine the expected distribution of molecular weights\nfor condensation and addition polymerizations in some detail. For the present, our only concern is\nhow such distribution of molecular weights is described. We will define the most commonly\nencountered averages, and how they relate to the distribution as a whole. We will also relate them\nto the standard parameters used for characterizing a distribution: the mean and standard deviation.\nAlthough these students are familiar with them only results\nprovided by a calculator, and so we will describe them in some detail.\n"]], ["block_4", ["n,- (we could also refer to n,- as the number of moles of i-mer, but again this just involves a factor of\nAvogadro\u2019s number). The first question we ask is this: if we choose a molecule at random from our\nsample, what is the probability of obtaining an i\u2014mer? The answer is straightforward. The total\nnumber of molecules is 2, m, and thus this probability is given by\n"]], ["block_5", ["Molecular Weights Averages\n25\n"]], ["block_6", ["The other expressions on the right-hand side of Equation 1.7.2 are equivalent, and will prove useful\nsubsequently. You should convince yourself that this particular average is the one you are familiar\nwith in everyday life: take the value of the property of interest, M,- in this case, add it up for all the\n(m) objects that possess that value of the property, and divide by the total number of objects.\nSo far, so good. We return for a moment to our hypothetical sample, but instead of choosing\n"]], ["block_7", ["Suppose we have a polymer sample containing many molecules with a variety of degrees of\npolymerization. We will call a molecule with degree of polymerization i an \u201ci-mer\u201d, and the\nassociated molecular weight MiziMo, where M0 is the molecular weight of the repeat unit.\n(Conversion between a discussion couched in terms of i or in terms of M,- is therefore straightfor\u2014\nward, and we will switch back and forth when convenient.) The number of i-mers we will denote as\n"]], ["block_8", ["a molecule at random, we choose a repeat unit or monomer at random, and ask about the molecular\nweight of the molecule to which it belongs. We will get a different answer, as a simple argument\nillustrates. Suppose we had two molecules, one a 10-mer and another a 20\u2014mer. If we choose\nmolecules at random, we would choose each one 50% of the time. However, if we choose mono-\nmers at random, 2/3 of the monomers are in the 20-mer, so we would pick the larger molecule\ntwice as often as the smaller. The total number of monomers in a sample is 2,- in;, and the chance of\n"]], ["block_9", ["1.7.1\nNumber-, Weight-. and z-Average Molecular Weights\n"]], ["block_10", [{"image_1": "41_1.png", "coords": [42, 489, 216, 526], "fig_type": "molecule"}]], ["block_11", [{"image_2": "41_2.png", "coords": [46, 397, 91, 429], "fig_type": "molecule"}]], ["block_12", ["x,- \n(1.7.1)\n\u201d1\nZr \u201d1'\n"]], ["block_13", ["M \u2014:x,-M- =Z\u2014Z':\u2014:I:4:M\nMO%\u2014:\u201c\n(1.7.2)\n"]]], "page_42": [["block_0", [{"image_0": "42_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Although M2 is not directly related to a simple fraction like x, or w,, it does have some experimental\nrelevance. We could continue this process indefinitely, just by incrementing the power of i by one in\nboth numerator and denominator of Equation 1.7.5, but it will turn out that there is no real need to do\nso. However, there is a direct relationship between this process and something well-known in\nstatistical probability, namely the moments of a distribution, as we will see in the next section.\n"]], ["block_2", ["Of course, mass\u2014average would be the preferred descriptor, but it is not in common usage.\nQualitatively, we can say that Mn is the Characteristic average molecular weight of the sample\nwhen the number of molecules is the crucial factor, whereas MW is the characteristic average\nmolecular weight when the size of each molecule is the important feature. Although knowledge of\nMW and Mn is not suf\ufb01cient to provide all the information about a polydisperse system, these two\naverages are by far the most important and most commonly encountered, as we shall see\nthroughout the book.\nComparison of the last expressions in Equation 1.7.2 and Equation 1.7.4 suggests a trend; we\ncan define a new average by multiplying the summation terms in the numerator and denominator\nby t'. The so-called z-average molecular weight, M2, is constructed in just such a way:\n"]], ["block_3", ["The PDI is always greater than 1, unless the sample consists of exactly one value of M, in which case\nthe PDI :1; such a sample is said to be monodisperse. We will see in Chapter 2 and Chapter 3 that\ntypical polymerization schemes are expected to give PDIs near 2, at least in the absence of various\nside reactions; in industrial practice, such side reactions often lead to PDIs as large as 10 or more. In\nChapter 4, in contrast, so-ealled living polymerizations give rise to PDIs of 1.1 or smaller. Thus,\ndistributions for which the PDI < 1.5 are said to be \u201cnarrow,\u201d whereas those for which PDI > 2 are\nsaid to be \u201cbroad,\u201d of course, such designations are highly subjective. As a very simple illustration,\nthe two\u2014molecule example given in the previous section consisting of a lO-mer and a 20\u2014mer has a\nnumber-average degree of polymerization of 15 and a weight average of 16.7; thus its PDI 1.11,\nwhich in polymer terms would be considered \u201cnarrow.\u201d This trivial example actually underscores\nan important point to bear in mind: polymer samples with \u201cnarrow\u201d distributions will still contain\nmolecules that are quite different in size (see Problem 1.8 for another instance).\n"]], ["block_4", ["Although the values MW or MH tell us something useful about a polymer sample, individually they\ndo not provide information about the breadth of the distribution. However, the ratio of the two\nturns out to be extremely useful in this regard, and it is given a special name, the polydispersity\nindex (PDI) or just the polydispersity:\n"]], ["block_5", ["Accordingly, we define the weight\u2014average molecular weight of the sample, MW, by\n"]], ["block_6", ["picking a particular i\u2014mer will be determined by the product 171,-. The resulting ratio is, in fact, the\nweight fraction. or mass fraction of i-mer in the sample, w,\u2014:\n"]], ["block_7", ["26\nIntroduction to Chain Molecules\n"]], ["block_8", ["1.7.2\nPolydispersity Index and Standard Deviation\n"]], ["block_9", [{"image_1": "42_1.png", "coords": [40, 297, 120, 342], "fig_type": "molecule"}]], ["block_10", ["Z; 53\u201d!\u201c\nM2 =M0 2 {212,-\n(1.7.5)\n"]], ["block_11", ["Mw\nMn\nPDI \n(1.7.6)\n"]], ["block_12", ["(1.7.3)\nWf=\n"]], ["block_13", [{"image_2": "42_2.png", "coords": [47, 139, 280, 171], "fig_type": "molecule"}]], ["block_14", ["2\n\u2014Z_WiM 2,a\nZ \u201d\u2018M\u2019\nMow:\nI \"\u2019\n(1.7.4)\n"]], ["block_15", ["23-171;\n"]], ["block_16", ["in,-\n"]], ["block_17", ["ZZZ in,\nZ, ngM-z M02. in,-\n"]]], "page_43": [["block_0", [{"image_0": "43_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "43_1.png", "coords": [26, 199, 259, 252], "fig_type": "molecule"}]], ["block_2", [{"image_2": "43_2.png", "coords": [28, 551, 326, 623], "fig_type": "figure"}]], ["block_3", [{"image_3": "43_3.png", "coords": [32, 110, 202, 147], "fig_type": "molecule"}]], ["block_4", ["deviation. We will now develop relationshipsbetween these quantities and M\u201c, in\n"]], ["block_5", ["so doing justify useful of the breadth \nmean of any distribution (i), is de\ufb01ned as\n"]], ["block_6", ["where both discrete (x,) continuous (P(i)) versions are considered. From this definition, and\nEquation 1.7.2, more than the number-average \nization, thus Mn weight.\nThe standard deviation 0' quantifies the width of the distribution. It is defined as\n"]], ["block_7", ["Note that a2 has the significance of being the mean of the square of the deviations of\nindividual values from the mean. Accordingly, a is sometimes called the root mean square (rms)\ndeviation.\nFrom a computational point of view, the standard deviation may be written in a more conveni-\nent form by carrying out the following operations. First both sides of Equation 1.7.8 are squared,\nand then the difference i\u2014(i) is squared to give\n"]], ["block_8", [{"image_4": "43_4.png", "coords": [35, 454, 134, 476], "fig_type": "molecule"}]], ["block_9", ["Recalling the definition of the mean, we recognize the first term on the right\u2014hand side of Equation\n1.7.9 to be the mean value of i2 and write\n"]], ["block_10", ["Molecular \n27\n"]], ["block_11", ["where in this case the standard deviation will have the units of molecular weight. If we expand\nEquation 1.7.12 we find\n"]], ["block_12", ["It is, of course, important to realize that (1'2) # (1)2. An alternative to Equation 1.7.8 as a definition\nof standard deviation is, therefore,\n"]], ["block_13", ["We can factor out Mn Z,- x,M,- to obtain\n"]], ["block_14", ["Similarly, the standard deviation can be written\n"]], ["block_15", [{"image_5": "43_5.png", "coords": [35, 498, 171, 532], "fig_type": "molecule"}]], ["block_16", [{"image_6": "43_6.png", "coords": [39, 395, 214, 424], "fig_type": "molecule"}]], ["block_17", [{"image_7": "43_7.png", "coords": [39, 328, 191, 362], "fig_type": "molecule"}]], ["block_18", [{"image_8": "43_8.png", "coords": [40, 626, 172, 668], "fig_type": "molecule"}]], ["block_19", ["In most fields of science, distributions are generally characterized by a mean and a standard\n"]], ["block_20", ["1'\u2014\n1\n2 \u201d2\n\u201d2\na E\n(Eingnf\n\u00bb\n)\n:\n(Xxx;\n_\n"]], ["block_21", [".72 (i2) 2(7)2 + (7)2 (72) (7)2\n(1.7.10)\n"]], ["block_22", ["a 2 [Z x,(M,.2 214,114,, + M3,)\n"]], ["block_23", ["a 2 ((1-2) _ (ml/2\n(1.7.11)\n"]], ["block_24", ["0' : [Zia-(M, 114,02] \n(1.7.12)\n"]], ["block_25", ["(i) 2\nix,- 3 J: iP(i)di\n(1.7.7)\n"]], ["block_26", ["2 _ 2,71,?\n. Zinii\n.2\n"]], ["block_27", ["'\n2\n1/2\n111M\n1]\n(1,-1.1...\n(Zl\u2018xiMi)\n"]], ["block_28", ["1/2\n1/2\n:\n[(Zxr\u2018M?)\na\nMg]\n(1.7.13)\n"]], ["block_29", ["(0)2)\n(1.7.8)\n"]]], "page_44": [["block_0", [{"image_0": "44_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "44_1.png", "coords": [11, 539, 453, 673], "fig_type": "figure"}]], ["block_2", [{"image_2": "44_2.png", "coords": [32, 67, 165, 112], "fig_type": "molecule"}]], ["block_3", ["This result shows that the square root of the amount by which the ratio MW/Mn exceeds unity equals\nthe standard deviation of the distribution relative to the number-average molecular weight. Thus if\n"]], ["block_4", ["MW/Mn 1.50, then the standard deviation is 71% of the value of Mn. This shows that reporting the\nmean and standard deviation of a distribution or the values of Mn and MW/Mn gives equivalent\ninformation.\nWe can define the quantities known as moments of a distribution, again either in the discrete or\ncontinuous forms. The k-th moment 11;, is given by\n"]], ["block_5", ["150130) di 1. From Equation 1.7.17 we can also see that the mean is equivalent to the first moment.\nFrom Equation 1.7.4 and Equation 1.7.5 it is apparent that MW and Mz are proportional to the ratio of\nthe second to the \ufb01rst moment and the third to the second, respectively. More generally, moments\n"]], ["block_6", ["The first and second columns of Table 1.4 give the number of moles of polymer in six different\nmolecular weight fractions. Calculate Mw/Mn for this polymer and evaluate 0- using both Equation\n1.7.12 and Equation 1.7.16.\n"]], ["block_7", ["Table 1.4\nSome Molecular Weight Data for a Hypothetical Polymer Used in Example 1.5\n"]], ["block_8", [{"image_3": "44_3.png", "coords": [34, 258, 169, 292], "fig_type": "molecule"}]], ["block_9", ["and, finally, by recognizing from Equation 1.7.4 that\n"]], ["block_10", ["we reach\n"]], ["block_11", ["a distribution is characterized by Mn 10,000 and (r 3000, then MW/Mn 1.09. Alternatively, if\n"]], ["block_12", ["In this definition both x,- and P(i) are normalized distributions, which mean that 2, x,- and\n"]], ["block_13", ["can be referred to a particular value, such as the k\u2014th moment about the mean, 121,:\n"]], ["block_14", ["2:0.049\n2:734\n2: 113\n2:70\n2:28.10\n"]], ["block_15", ["28\nIntroduction to Chain Molecules\n"]], ["block_16", ["First, consider the following numerical example in which we apply some of the equations of this\nsection to hypothetical data.\n"]], ["block_17", ["0.003\n10,000\n30\n3.0\n25\n7.50\n"]], ["block_18", ["0.008\n12,000\n96\n1 1.5\n9\n7.20\n"]], ["block_19", ["0.011\n14,000\n154\n21.6\n1\n1.10\n"]], ["block_20", ["0.009\n18,000\n162\n29.2\n9\n8.10\n"]], ["block_21", ["0.001\n20,000\n20\n4.0\n25\n2.50\n"]], ["block_22", ["From this expression we see that 02 is the second moment about the mean.\n"]], ["block_23", ["0.017\n16,000\n272\n43.5\n1\n1.70\n"]], ["block_24", ["24. (mol)\nM1 (g/mol)\nm.- (g)\n(gz/mol)\n(g\u2018Z/molz)\n(nmol)\n"]], ["block_25", ["Example 1.5\n"]], ["block_26", ["1.7.3\nExamples of Distributions\n"]], ["block_27", [{"image_4": "44_4.png", "coords": [44, 358, 133, 391], "fig_type": "molecule"}]], ["block_28", ["MW\n1/2\n=Mn\n_1\n.7.1\n\u20187\n[Mn\ni\n(1\n6)\n"]], ["block_29", ["00\n,1, 2m\u201c \n(1.7.17)\n"]], ["block_30", ["m. 2w >t\n(1.7.13)\n"]], ["block_31", [".\n.2\n.\n.\n.2\n"]], ["block_32", [{"image_5": "44_5.png", "coords": [129, 573, 445, 658], "fig_type": "figure"}]], ["block_33", ["\u201d1,114,. x 10\u20185\n(Mt\u2014Mn)2 x 10-6\nn,(M,-\u2014Mn)2 x 10\u20144\n"]]], "page_45": [["block_0", [{"image_0": "45_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "45_1.png", "coords": [21, 412, 194, 446], "fig_type": "molecule"}]], ["block_2", ["The proof of Equation 1.7.21 is left to Problem 1.9 at the end of the chapter. Now we can\nuse Equation 1.7.19 and Equation 1.7.20 to generate distributions for specified values of Mn and\nMw/Mn.\nFigure 1.5a shows the mole fraction and weight fraction as a function of M, with the particular\nchoice of Mn: 10,000, M0: 100, and 2: 1. Thus, according to Equation 1.7.21, the PDI=2\nand Mw=20,000. Both MW and MD are indicated by vertical lines on the plot. There are\nseveral remarkable features to point out. First, x, is a continuously decreasing function of\nM (and therefore i). We shall see in Chapter 2 that this is to be expected in step-growth\npolymerizations. It means, for example, that there are more unreacted monomers (i 1) than\nany other particular i-mer. The weight fraction, however, has a distinct but broad maximum.\nNotice also how many different values of M are present in significant amounts. For example, there\nis certainly a significant mass of the sample that is five times smaller than MW, or five times larger\nthan Mn.\n"]], ["block_3", ["Evaluate the product m, for each class; this is required for the calculation of both Mn and\nMW. Values of quantity are listed in the third column of Table 1.4. From 2,111,114,- and 2,11,,\nMn 734/0049 15,000. (The matter of significant figures will not be strictly adhered to in this\nexample. As a general rule, one has to work pretty hard to obtain more than two significant \ufb01gures\nin an experimental determination of M.)\nThe products miM, are mass-weighted contributions and are listed in the fourth column of\nTable 1.4. From 2.1%,. and 2,11,14,14,, 113 x 105/734 15,400.\nThe ratio MW/Mn is found to be 15,400/15,000 1.026 for these data. Using Equation 1.7.16, we\nhave a/a (1.026 1)\u201d2 0.162 or 0': 0.162(15,000) 2430.\nTo evaluate 0' via Equation 1.7.12, differences between M,- and Mn must be considered. The fifth\nand sixth columns in Table 1.4 list (M, Mn)2 and 11,-(M, Mn)2 for each class of data. From 2,11,-\nand 2,n,(M, Mn)2, 02 28.1 x 104/0049 5.73 x 106 and 0' 2390.\nThe discrepancy between the two values of 0' is not meaningful in terms of significant figures;\nthe standard deviation is 2400.\nAs polymers go, this is a very narrow molecular weight distribution.\n"]], ["block_4", ["The utility of this distribution arises in part because of a very simple relationship between the\nparameter 2 and the polydispersity:\n"]], ["block_5", ["When we consider particular polymerization schemes in Chapter 2 through Chapter 4, we\nwill derive explicit expressions for the expected distributions x,- and 142,. For now, however, let us\nconsider a particular mathematical function known as the Schulz\u2014Zimm distribution. It has the virtue\nthat by varying a single parameter, 2, it is possible to obtain reasonable descriptions for typical\nnarrow or moderately broad samples. We will use it to illustrate graphically how the distribution\nmight appear for a given polydispersity. The Schulz\u2014Zimm distribution can be expressed as\n"]], ["block_6", ["where Hz 1) is the so\u2014called gamma function (which is tabulated in many mathematical\nreferences). For integer values ofz, F(z+ 1) 2!, where z! (z factorial) z X (z 1) x (z 2) x x 1.\nFrom Equation 1.7.2 and Equation 1.7.3 we can see that w,- = x,M,/Mn, and thus\n"]], ["block_7", ["Molecular Weights and Molecular Weight Averages\n29\n"]], ["block_8", ["Solution\n"]], ["block_9", [{"image_2": "45_2.png", "coords": [40, 336, 209, 370], "fig_type": "molecule"}]], ["block_10", ["Mw_z+1\n"]], ["block_11", ["P(M\n\u2014\u2014\n22\u201c\nM\u2018H\nZM\u201d\n1719\n"]], ["block_12", ["1 \n--\u2014-\u2014-\u2014---\u2014--\u2014\u2014-\u2014\u2014\u2014\u2014\u2014\nI\n\u2014\nI\n10702\nW\nF(z+ 1) Mg\u201c exP<\nMn)\n(\nO)\n"]], ["block_13", ["Mn\n2\n(1.7.21)\n"]], ["block_14", ["zz+1\nM?\n2M.\n"]]], "page_46": [["block_0", [{"image_0": "46_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "46_1.png", "coords": [23, 118, 294, 394], "fig_type": "figure"}]], ["block_2", [{"image_2": "46_2.png", "coords": [31, 395, 277, 594], "fig_type": "figure"}]], ["block_3", [{"image_3": "46_3.png", "coords": [33, 147, 254, 315], "fig_type": "figure"}]], ["block_4", ["0.002\n"]], ["block_5", ["30\nIntroduction to Chain Molecules\n"]], ["block_6", ["0.004 \n_\nz\u2014 1\nj\n"]], ["block_7", ["Figure 1.5b shows the analogous curves, but now with PDI =4 and 220.333. Although the\nmole fraction looks superficially similar to the previous case, the mass distribution is exceedingly\nbroad, with a long tail on the high M side of Mn. (Note that the coincidence between Mn and the\npeak in w,- is a feature of this distribution, but not a general result in polymers; see Problem 1.10.)\n"]], ["block_8", ["Figure 1.5\nNumber and weight distributions for the Schulz\u2014Zimm distribution with the indicated poly-\ndispersities, and Mn 2 10,000.\n"]], ["block_9", ["0.0083 Mn\nMw\n0\nx,\nJ\nL\u2018\n_\ni\n_\n0\nWI\ni\n0006\nl\u2014\n\u2014\n__\n"]], ["block_10", ["0.008\n"]], ["block_11", ["0\n4x104\n8x104\n0))\nM\n"]], ["block_12", ["0.004 \n"]], ["block_13", ["l\n\u2018\u2014\u2018\n7\nD\n0\n2x104\n4x104\n5x10\u201cl\n8x104\n1x105\n"]], ["block_14", ["\u2014\n1\n"]], ["block_15", ["L\ni\n"]], ["block_16", [".\n4\n"]], ["block_17", [{"image_4": "46_4.png", "coords": [179, 423, 234, 483], "fig_type": "figure"}]], ["block_18", [{"image_5": "46_5.png", "coords": [180, 173, 234, 230], "fig_type": "figure"}]], ["block_19", [{"image_6": "46_6.png", "coords": [195, 525, 244, 563], "fig_type": "molecule"}]], ["block_20", ["I_I\n"]], ["block_21", ["_l_l_l_\n"]]], "page_47": [["block_0", [{"image_0": "47_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "47_1.png", "coords": [16, 45, 324, 330], "fig_type": "figure"}]], ["block_2", [{"image_2": "47_2.png", "coords": [32, 67, 265, 226], "fig_type": "figure"}]], ["block_3", ["Figure 1.5c shows the opposite extreme, or a narrow distribution with z :10 and PDI :1.11. Here\nboth x,- and w,- are much narrower, and have distinct maxima that are also close to Mlrl and MW,\nrespectively. (Note also that the ordinate scale has been truncated.) Even though this is a narrow\ndistribution by polymer standards, there are still significant numbers of molecules that are 50%\nlarger and 50% smaller than the mean.\n"]], ["block_4", ["The measurement of molecular weight is clearly the most important step in characterizing a\npolymer sample, although the previous sections have introduced many other aspects of polymer\nstructure necessary for a full analysis. A rich variety of experimental techniques have been\ndeveloped and employed for this purpose, and the major ones are listed in Table 1.5. As indicated\nin the rightmost column, we will describe end group analysis and matrix-assisted laser desorption/\nionization (MALDI) mass spectrometry in this section, while deferring treatment of size exclu\u2014\nsion chromatography (SEC), osmotic pressure, light scattering, and intrinsic viscometry to\nsubsequent chapters. Other techniques, such as those involving sedimentation, we will omit\nentirely. Before considering any technique in detail, some general comments about the entries in\nTable 1.5 are in order.\nThe first two techniques listed, SEC and MALDI, can provide information on the entire\ndistribution of molecular weights. To the extent that either one can do this reliably, accurately,\nand conveniently, there is a diminished need for any other approach. Over the past 30 years, SEC\nhas unquestionably emerged as the dominant method. Automated analysis of a few milligrams of\nsample dissolved in a good solvent can be achieved in half an hour. It is hard to imagine any\nserious polymer laboratory that does not have SEC capability. Nevertheless, as will be discussed in\n"]], ["block_5", ["0.008 \n00\n.0\n\u2014\n"]], ["block_6", ["J\n0004 \n.0\n0.2),\nPD|=1.11\nd\n"]], ["block_7", ["Measurement of Molecular Weight\n31\n"]], ["block_8", ["(C)\nM\n"]], ["block_9", ["Figure 1.5 (continued)\n"]], ["block_10", ["1.8\nMeasurement of Molecular Weight\n"]], ["block_11", ["1.8.1\nGeneral Considerations\n"]], ["block_12", ["OJ\n0\n1x104\n2x104\n3x104\n4x104\n"]], ["block_13", ["r\u2014\n00\n00\n\ufb02\n\u2014\n"]], ["block_14", ["_\no\noo\n_\n_\n.0\n\u20190\n_\n_\n.0\n0%\nO\nW,\u201c\n_\n.0\n.0\n_\n00\n_\n"]], ["block_15", ["_I\n"]], ["block_16", [":\n3%\n:\n"]], ["block_17", ["\u201c\n08\n0%\no\nX,\n-\n"]], ["block_18", ["_\n.0\n.0\n_\n"]], ["block_19", ["_\n00\n_\n00\noo\n_\no\n.0\n\"\n"]], ["block_20", ["_\n"]], ["block_21", [".0\n.0\n_\n_\no\n_\n"]], ["block_22", ["_\n.0\n:2:\n"]], ["block_23", ["_\n.0\n.00\n_\no\n\u2014\n'o\n'.o\n2:10\n-\n"]], ["block_24", ["_\n.0\ng???\n"]], ["block_25", ["..\n_\n. o\n_\n"]], ["block_26", ["I\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nl\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI-\n"]], ["block_27", [{"image_3": "47_3.png", "coords": [188, 89, 259, 159], "fig_type": "figure"}]]], "page_48": [["block_0", [{"image_0": "48_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "48_1.png", "coords": [17, 64, 431, 253], "fig_type": "figure"}]], ["block_2", ["Chapter 9, SEC has some serious limitations; one of these is a lack of resolution. Resolution in this\ncontext refers to the difference in M values that can be determined; SEC would struggle to help you\ndecide whether your sample was narrowly distributed with Mn 2 50,000, or was bimodal with peak\nmolecular weights at 40,000 and 60,000. In contrast, resolution is the real strength of MALDI, as\nwe will see below. MALDI is a relative newcomer among the techniques listed in Table 1.5, but it\nwill undoubtedly grow in importance as its scope expands.\nThe next group of techniques provides measurements of Mn. They do so by being sensitive to\nthe number of solute molecules in solution, as is inherent in the so-called colligative properties\n(osmotic pressure, freezing point depression, boiling point elevation, etc.). Of these, osmotic\npressure is the most commonly employed. It has the virtue (shared with light scattering) of\nbeing a technique based on equilibrium thermodynamics that can provide an absolute measurement\nwithout resorting to calibration against other polymer samples. End group analysis, to be discussed\nbelow, includes any of a number of analytical tools that can be used to quantify the presence of the\nunique structure of the polymer chain ends. For a linear chain, with two and only two ends,\ncounting the number of ends is equivalent to counting the number of molecules.\nLight scattering is suf\ufb01ciently important that it merits a full chapter; determination of MW is\nonly one facet of the information that can be obtained. Sedimentation experiments will not be\ndiscussed further in this text, although they play a crucial role for biOpolymer analysis in general,\nand proteins in particular. Similarly, gel electrophoresis (not listed in Table 1.5 or discussed further\nhere) is an analytical method of central importance in biological sciences, and especially for the\nseparation and sequencing of DNA in the Human Genome Project.\nThe intrinsic viscosity approach holds a place of particular historical importance; in the\ndays before routine use of SEC (up to about 1970) it was by far the easiest way to obtain\nmolecular weight information. The viscosity average molecular weight defined in Table 1.5 is not\na simple moment of the distribution, but involves the Mark\u2014Houwink exponent a, which needs to be\nknown based on other information. As 0.5 g a S 0.8 for most \ufb02exible polymers in solution, Mn 3 MV\n5 MW. The relation between the viscosity of a dilute polymer solution and the molecular weight of the\npolymer actually rests on some rather subtle hydrodynamics, as we will explore in Chapter 9.\n"]], ["block_3", ["As indicated previously, the end groups of polymers are inherently different in chemical structure\nfrom the repeat units of the chain, and thus provide a possible means of counting the number of\n"]], ["block_4", ["Z. \"5114 ,2\n.\n.\nMW\n'\nLight scattering\n8\na'M\u2018\nSedimentation velocity\n\u2014\n"]], ["block_5", ["2.31504?\n.\n.\n.\n.\n.\nMz\n2\nM2\nSedimentation equilibrium\n~\u2014\n"]], ["block_6", ["\"I\nl+a\n1/0\nMv\n(Li\u20149:7)\nIntrinsic viscosity\n9.3\n"]], ["block_7", ["Table 1.5\nSummary of the Molecular Weight Averages Most Widely Encountered in Polymer Chemistry\n"]], ["block_8", ["Zr :1e\n-\nMn\n081110t pressure\n7.4\n2\u201d\"\nOther colligative properties\n\u2014\n"]], ["block_9", ["1.8.2\nEnd Group Analysis\n"]], ["block_10", ["Information\nDefinition\nMethods\nSections\n"]], ["block_11", ["Full distribution\nx,, w,-\nSize exclusion chromatography\n9.8\nMALDI mass spectrometry\n1.8\n"]], ["block_12", ["32\nIntroduction to Chain Molecules\n"]], ["block_13", [{"image_2": "48_2.png", "coords": [137, 83, 326, 248], "fig_type": "figure"}]], ["block_14", [".n;\nr\n"]], ["block_15", [{"image_3": "48_3.png", "coords": [222, 80, 430, 236], "fig_type": "figure"}]], ["block_16", ["End group analysis\n1.8\n"]]], "page_49": [["block_0", [{"image_0": "49_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "49_1.png", "coords": [28, 617, 232, 678], "fig_type": "figure"}]], ["block_2", ["As an example, consider condensation polymers such as polyesters and polyamides. They are\nespecially well suited to this molecular weight determination, because they tend to have lower\nmolecular weights than addition polymers, and because they naturally have unreacted functional\ngroups at each end. Using polyamides as an example, we can readily account for the following\npossibilities:\n"]], ["block_3", ["molecules in a sample. Any analytical technique that can reliably quantify the concentration of end\ngroups can be used in this manner, and over the years many have been so employed.\nIt should section that will\nmeasure of Mn. The experiment willinvolve preparing\n"]], ["block_4", ["In this case, there is one functional group of each kind per molecule and could be detected for\nexample by titration with a strong base (for \u2014COOH) or strong acid (for \u2014NH2).\n2.\nIf a polyamide is prepared in the presence of a large excess of diamine, the average chain will\nbe capped by an amino group at each end:\n"]], ["block_5", ["In this case, only the amine can be titrated, and two ends are counted per molecule.\n3.\nSimilarly, if a polyamide is prepared in the presence of a large excess of dicarboxylic acid,\nthen the average chain will have a carboxyl group at each end:\n"]], ["block_6", ["a known mass of sample, probably in solution, which given M0 corresponds to a certain number\nof repeat units. The number of end groups is directly proportional to the number of polymers\nand the ratio of the number to the number of polymers is the number-average degree\nof polymerization.\nSeveral general principles apply to end group analysis:\n"]], ["block_7", ["1.\nThe chemical structure of the end group must be sufficiently different from that of the repeat\nunit for the chosen analytical technique to resolve the two clearly.\n2.\nThere must be a well-defined number of end groups per polymer, at least on average. For a linear\npolymer, there will be two and only two end groups per molecule, which may or may not be\ndistinct from each other. For branched polymers, the relation of the number of end groups to the\nnumber of polymers is ambiguous, unless the total number of branching points is also known.\n"]], ["block_8", ["3.\nThe technique is limited to relatively low molecular weights, as the end groups become more\nand more dilute as N increases. This is an obvious corollary of the fact that we can ignore end\ngroups in considering the structure of high molecular weight chains. How low is low in this\ncontext? The answer will depend on the particular system and analytical technique, but as a\nrule of thumb end groups present at the 1% level (corresponding to degrees of polymerization\nof 100 for a single end group, 200 for both end groups) can be reliably determined; those at the\n0.1% level cannot.\n"]], ["block_9", ["Measurement of \n33\n"]], ["block_10", ["I.\nA linear molecule has a carboxyl group at one end and an amino group at the other, such as\npoly(a-caprolactam):\n"]], ["block_11", [{"image_2": "49_2.png", "coords": [37, 615, 187, 676], "fig_type": "molecule"}]], ["block_12", [{"image_3": "49_3.png", "coords": [39, 424, 158, 471], "fig_type": "molecule"}]], ["block_13", [{"image_4": "49_4.png", "coords": [43, 524, 179, 567], "fig_type": "molecule"}]], ["block_14", ["H\nH\nHO\nFl\"\nN,\nI.N\nFl'\nOH\nv {\ufb01r\nR rd v\n0\n0\nOn\n0\n"]], ["block_15", ["3. ii\nRI\n/Rl\n/\nN\nFl\"\nN\nNH\nH2N\n{H\nH\n)1\n2\n"]], ["block_16", ["O\n0\nHOAm\n5 \u201dMn\n"]]], "page_50": [["block_0", [{"image_0": "50_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Peak (c) corresponds to the methylene protons adjacent to the hydroxyl end group; there are\ntwo such protons per polymer. Peak (a) re\ufb02ects the single olefinic proton per 1,4 repeat unit,\nwhereas peak (b) shows the two vinyl protons per 3,4 repeat unit. If we represent the integration of\npeak i as 1,, then the degree of polymerization is pr0portional to (Iat + 113/2) 26.9 + 2.61 29.5.\nThe number of polymers is proportional to 1J2 1. Thus the number-average degree of poly-\nmerization is 29.5, which gives an Mn=29.5\nx\n68 =2,000. Peak ((1) indicates the six methyl\nprotons on the initiator fragment. The peak integration Id 5.95 should be 31., which is within\nexperimental error. For this particular molecular structure, therefore, either end group could be\nused. An additional conclusion is that essentially 100% of the polymers were terminated with a\nhydroxyl group.\n"]], ["block_2", ["In this case, the acid group can be titrated, and again two ends should be counted per molecule.\nThe preceding discussion illustrated how classical acid\u2014base titration could be used for\nmolecular weight determination. In current practice, nuclear magnetic resonance (NNIR) spectroscopy\nis probably the most commonly used analytical method for end group analysis, especially proton\n(1H) NMR. An additional advantage of this approach is the possibility of obtaining further\ninformation about the polymer structure from the same measurement, as illustrated in the following\nexample.\n"]], ["block_3", ["The 1H NMR spectrum in Figure 1.6 corresponds to a sample of polyisoprene containing a sec-\nbutyl initiating group and a hydroxyl terminating end group. The relative peak integrations are (a)\n"]], ["block_4", ["34\nIntroduction to Chain Molecules\n"]], ["block_5", ["26.9, (b) 5.22, (c) 2.00, and (d) 5.95. What is Mn for this polymer? What is the relative percentage\nof 1,4 and 3,4 addition?\n"]], ["block_6", ["Figure 1.6\nlH NMR spectrum of a polyisoprene sample, discussed in Example 1.6. (Data courtesy of\nN. Lynd and MA. Hillmyer.)\n"]], ["block_7", ["Solution\n"]], ["block_8", ["1 HgC\n/\n,OH\n'\n"]], ["block_9", [".4\n.-\nc\nCH3\n"]], ["block_10", ["d\n..\n"]], ["block_11", ["Example 1.6\n"]], ["block_12", ["C\n\u00e9\n_\ni\n.J\na... 1.\nL\nl\nI\nl\nI\nI\n6\n5\n4\n3\n2\n1\n0\n"]], ["block_13", [{"image_1": "50_1.png", "coords": [45, 398, 285, 638], "fig_type": "figure"}]], ["block_14", ["3\n"]], ["block_15", ["I\n|_\n_|_\n_l\nI\n"]], ["block_16", ["E\n"]], ["block_17", ["Ha\nm\n:1\nCH3\n"]], ["block_18", ["ppm\n"]]], "page_51": [["block_0", [{"image_0": "51_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["phase without degradation. However, progress in recent years has been quite rapid, with two\ngeneral approaches being particularly productive. In one, electrospray ionization, a polymer\nsolution is ejected through a small ori\ufb01ce into a vacuum environment; an electrode at the exit\ndeposits a charge onto each drop of solution. The solvent then evaporates, leaving behind a charged\nmacromolecule in the gas phase. In the other, the polymer sample is dispersed in a particular\nmatrix on a solid substrate. An intense laser pulse is absorbed by the matrix, and the resulting\nenergy transfer vaporizes both polymer and matrix. For uncharged synthetic polymers, the\nnecessary charge is usually complexed with the polymer in the gas phase, after a suitable salt\nhas been codissolved in the matrix. This technique, matrix-assisted laser desorption/ionization\nmass spectrometry, or MALDI for short, is already a standard approach in the biopolymer arena,\nand is making substantial inroads for synthetic polymers as well.\nIt is worth recalling the basic ingredients of a mass spectrometric experiment. A sample molecule\nof mass m is first introduced into the gas phase in a high vacuum, and at some point in the process it\nmust acquire a net charge 2. The resulting ion is accelerated along a particular direction by suitably\nplaced electrodes, and ultimately collected and counted by a suitable analyzer. The ion acquires\nkinetic energy in the applied field, which depends on the net charge 2 and the applied voltage. This\nenergy will result in a mass-dependent velocity v, as the kinetic energy v/Z. This allows for the\ndiscrimination of different masses, by a variety of possible schemes. For example, if the ions\nexperience an orthogonal magnetic field B, their trajectories will be curved to different extents, and\nit is possible to tune the magnitude ofB to allow a particular mass to pass through an aperture before\nthe detector. For polymers it turns out to be more effective to use time-of-\ufb02ight (TOF) analysis.\nFor a given applied field and \ufb02ight path to the detector, larger masses will take longer to reach\nthe detector. As long as all the molecules are introduced into the gas phase at the same instant in\ntime, the time of arrival can be converted directly into a value of m/z.\nThe preceding discussion may give the misleading impression that MALDI is rather a straight-\nforward technique. This is not, in fact, the case, especially for synthetic polymers. A great deal\nremains to be learned about both the desorption and ionization processes, and standard practice\nis to follow particular recipes (matrix and salt) that have been found to be successful for a\ngiven polymer. For example, polystyrene samples are most often dispersed in dithranol (1,8,9-\nanthracenetriol) with a silver salt such as silver tri\ufb02uoroacetate. This mixture is co-dissolved in a\nvolatile common good solvent such as tetrahydrofuran, to ensure homogeneity; after depositing\na drop on a sample plate, the solvent is then allowed to evaporate. An intense pulse from a nitrogen\nlaser ()1 = 337 nm) desorbs some portion of the sample, and some fraction of the resulting gas-\nphase polystyrene molecules are complexed with a single silver cation.\nTwo examples of MALDI spectra on narrow distribution of polystyrene samples are shown in\nFigure 1.7a and Figure 1.7b. In the former, the average molecular weight is in the neighborhood\nof 5000, and different i-mers are clearly resolved. Each peak is separated by 104 g/mol, which is\nthe repeat unit molecular weight. The absolute molecular weight of each peak should correspond\n"]], ["block_2", ["This typical prepared by anionic polymerization in a \n"]], ["block_3", ["solvent \n"]], ["block_4", ["Mass spectrometry offers unprecedented resolution in the analysis of gas phase ions, and is a\nworkhorse of chemical analysis. Its application to synthetic polymers has been limited until rather\nrecently, primarily due to the difficulty of transferring high molecular weight species into the gas\n"]], ["block_5", ["Measurement of Molecular Weight\n35\n"]], ["block_6", ["1.8.3\nMALDI Mass Spectrometry\n"]], ["block_7", ["The percent 1,4 addition can be computed as follows:\n"]], ["block_8", ["26.9\n%1,4=\nx100=91.2%.\n[a + (lb/2)\n"]]], "page_52": [["block_0", [{"image_0": "52_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Figure 1.7\nMALDI spectra for anionically prepared polystyrenes with (a) MW 2 4,971, Mn 2 4,620,\nPDI 1.076, and (b) MW 49,306, Mn 48,916, PDI \n1.008. Inset shows that even in (b) there is\nsome resolution of the different i-mers. (Data courtesy of K. Fagerquist, T. Chang, and T.P. Lodge.)\n"]], ["block_2", ["W\n'hMI-um\n"]], ["block_3", ["to the following sum: (1041' + 108 + 1 -|\u2014 57) :1041' + 166, where 108, 1, and 57 are the contribu-\ntions of the silver ion, terminal proton, and sec\u2014butyl initiator fragment, respectively. Using the\nformulas given in Equation 1.7.2 and Equation 1.7.4, Mn and MW can be calculated as 4620 and\n4971, respectively, and the polydispersity is 1.076. This sample, which was prepared by living\nanionic polymerization as described in Chapter 4, is thus quite narrow. Nevertheless, the plot\nshows distinctly how many different i-mers are present, and in what relative proportion. The\nassumption is made that the height of each peak is proportional to its number concentration in\nthe sample, and thus the yards corresponds to an unnormalized form of mole fraction x,. This\nimage, perhaps more than any other, underscores the point we have already made several times:\neven the best of polymer samples is quite heterogeneous. Recalling Example 1.4 and the tank car\nof polybutadiene, it is worth pointing out that each peak in Figure 1.7a corresponds to many\nstructurally different molecules, in terms of the stereochemical sequence along the backbone.\nThe MALDI spectrum in Figure 1.7b corresponds to a sample about 10 times higher in\nmolecular weight. At this point it is not possible to see any structure between different i-mers,\nalthough the expanded version shows that there is still a hint of resolution of distinct molecular\nweights. This serves to point out one limitation with MALDI, namely that its main attribute, high\nresolution, is diminished as M increases. Not apparent from this plot, but even more troublesome,\nis the fact that the absolute amplitude of the signal is greatly reduced compared to Figure 1.7a. It is\n"]], ["block_4", ["I\nI\nI\nI\nl\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\n|\nI\nI\nl\n|\nI\nI\nl\nI\nl\u2014I\nI\nI\nI\nI\n\u2014|\nI\u2018T llllllllllllllll\nI llllllllllll\n40900\n50,000\n60000\n48,000\n49,000\n50,000\n51,000\n"]], ["block_5", ["36\nIntroduction to Chain Molecules\n"]], ["block_6", ["IIIIII\nIIIIIIIIIIIII IIIIIIllillllllllllltlllllll\nII\n(a)\n2,000\n3,000\n4,000\n5,000\n6,000\n7,000\n"]], ["block_7", ["(b)\n"]], ["block_8", [{"image_1": "52_1.png", "coords": [39, 442, 336, 600], "fig_type": "figure"}]], ["block_9", [{"image_2": "52_2.png", "coords": [48, 284, 315, 412], "fig_type": "figure"}]], ["block_10", ["...||111\n,\n,._.11111111111111\u201c..\n"]], ["block_11", ["m/z\n"]], ["block_12", [{"image_3": "52_3.png", "coords": [230, 478, 399, 579], "fig_type": "figure"}]], ["block_13", ["IV;\n"]], ["block_14", ["80\u2018\u201d g no\nv38 :2 2\na? 22\u2018\n"]], ["block_15", ["V\n"]]], "page_53": [["block_0", [{"image_0": "53_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["3.\nMultiply charged ions can present a problem, because one cannot distinguish between a\nmolecule with a charge of l and a molecule of twice the molecular weight but with a charge\nof 2. In fact, if the technique were called \u201cmass-to-charge ratio spectrometry\u201d it would be a\nmouthful, but it would serve as a constant reminder of this important complication.\n4.\nIt is difficult to compare the amplitudes of peaks from one laser pulse to another, and from one\nsample drop to another. This presumably re\ufb02ects the microscopic details of the spot on the\nsample that is actually at the focus of the laser beam. As a consequence any quantitative\ninterpretation should be restricted to a given spectrum.\n"]], ["block_2", ["The contents of this book may be considered to comprise three sections, each containing four\nseparate chapters. The first section, including Chapter 2 through Chapter 5, addresses the synthesis\nof polymers, the various reaction mechanisms and kinetics, the resulting molecular weight\ndistributions, and some aspects of molecular characterization. In particular, Chapter 2 concerns\nstep-growth (condensation) polymerization and Chapter 3 chain\u2014growth (free radical) polymeriza-\ntion. Chapter 4 describes a family of particular polymerization schemes that permit a much higher\ndegree of control over molecular weight, molecular weight distribution, and molecular architecture\nthan those in the preceding two chapters. Chapter 5 addresses some of the factors that control the\nstructural details within polymers, especially copolymers and stereoregular polymers, and aspects\nof their characterization.\nThe second section takes up the behavior of polymers dissolved in solution. The conformations\nof polymers, and especially random coils, are treated in Chapter 6. Solution thermodynamics are\nthe subject of Chapter 7, including the concepts of solvent quality, osmotic pressure, and phase\nbehavior. The technique of light scattering, which provides direct information about molecular\nweight, solvent quality, and chain conformations, is covered in detail in Chapter 8. Chapter 9\nexplores the various hydrodynamic properties of polymers in solution, and especially as they\nimpact viscosity, diffusivity, and SEC.\nThe concluding section addresses the properties of polymers in the bulk, with a particular\nemphasis on the various solid states: rubber, glass, and crystal. Thus Chapter 10 considers polymer\nnetworks and their characteristic and remarkable elasticity. Chapter 11 treats the unusual visco-\nelastic behavior of polymer liquids, in a way that combines central concepts from both Chapter 9\nand Chapter 10. Chapter 12 introduces the phenomenon of the glass transition, which is central to\nall polymer materials yet relatively unimportant in most atomic or small molecule\u2014based materials.\nFinally, the rich crystallization properties of polymers are taken up in Chapter 13. The text\n"]], ["block_3", ["1.\nGenerally, the more polar a polymer, the easier it is to analyze by MALDI. Thus poly(ethylene\noxide) is relatively easy; poly(methyl methacrylate) is easier than polystyrene; polyethylene is\nalmost impossible.\n2.\nAn important, unresolved issue is relating the amplitude of the signal of a particular peak to\nthe relative abundance of that molecule in the sample. For example, are all molecular weights\ndesorbed to the same extent within a given laser pulse (unlikely), and are all molecular\nweights equally likely to be ionized once in the gas phase (no)? Consequently, it can be\ndangerous to extract MW and M,1 as we did for the samples in Figure 1.7a and Figure 1.7b,\nbecause the signals have an unknown sensitivity to molecular weight. (In this instance, this\nproblem is mitigated because the distributions are quite narrow.) In general, lower molecular\nweights have a much higher yield.\n"]], ["block_4", ["data in Figure 1.7b give MW 49,306, and a polydispersity of 1.008. This turns out to\nbe nearly as narrow as the theoretical limit for this class of polymerizations (see Chapter 4), yet it is\nstill obviously quite \nWe conclude this section with some further general observations about MALDI:\n"]], ["block_5", ["simply weight molecules into the gas phase. the\n"]], ["block_6", ["Preview of Things to Come\n37\n"]], ["block_7", ["1.9\nPreview of Things to Come\n"]]], "page_54": [["block_0", [{"image_0": "54_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["To a signi\ufb01cant extent the problems in this book are based on data from the original literature. In\nmany instances the values given have been estimated from graphs, transformed from other\nfunctional representations, or changed in units. Therefore, these quantities do not necessarily\nre\ufb02ect the accuracy of the original work, nor is the given number of significant figures always\njustified. Finally, the data may be usedfor purposes other than were intended in the original study.\n"]], ["block_2", ["lR.E. Cohen and AR. Ramos Macromolecules, 12, 131 (1979).\n"]], ["block_3", ["1.\nRE. Cohen and AR. RamosJr describe phase equilibrium studies of block copolymers of\nbutadiene (B) and isoprene (1). One such polymer is described as having a 2:1 molar ratio of B\nto I with the following microstructure:\nB\u201445% cis-1,4; 45% trans\u20141,4; 10% vinyl.\nI-\u2014-over 92% cis\u20141,4.\nDraw the structure of a portion of this polymer consisting of about 15 repeat units, and having\napproximately the composition of this polymer.\n2.\nHydrogenation of polybutadiene converts both cis and trans isomers to the same linear\nstructure, and vinyl groups to ethyl branches. A polybutadiene sample of molecular weight\n168,000 was found by infrared spectroscopy to contain double bonds consisting of 47.2% cis,\n"]], ["block_4", ["1.\nThe most important feature of a polymer is its degree of polymerization or molecular weight.\nFor example, even though polyethylene has the same chemical formula as the n-alkanes, it has\nremarkably different physical properties from its small molecule analogs.\n2.\nThe statistical nature of polymerization schemes inevitably leads to a distribution of molecular\nweights. These can be characterized via specific averages, such as the number-average and\nweight-average molecular weights, or by the full distribution, which can be determined by\nSEC or MALDI mass spectrometry.\n3.\nPolymers can exhibit many different architectures, such as linear, randomly branched, or\nregularly branched chains, and networks. Homopolymers contain only one type of repeat unit,\nwhereas copolymers contain two or more.\n4.\nThere are many possible variations in local structure along a polymer chain, which we have\nclassified as positional, stereochemical, or geometrical isomers. Given these possibilities, and\nthose identified in the previous point, it is unlikely that any two polymer molecules within a\nparticular sample have exactly the same chemical structure, even without considering differ-\nences in molecular weight.\n5.\nNatural polymers such as polysaccharides, proteins, and nucleic acids share many of the\nattributes of their synthetic analogs, and as such are an important part of the subject of this\nbook. On the other hand, the specific biological functions of these macromolecules, especially\nproteins and nucleic acids, fall outside our scope.\n"]], ["block_5", ["concludes with an Appendix that reviews some of the mathematical manipulations encountered\nthroughout the book.\n"]], ["block_6", ["In this chapter, we have introduced the central concept of chain molecules, and identified various\nways in which polymers may be classified. The importance of molecular weight and its distribution\nwas emphasized, and associated averages defined. Examples were given of the many possible\nstructural variations that commonly occur in synthetic polymers:\n"]], ["block_7", ["Problems\n"]], ["block_8", ["38\nIntroduction to Chain Molecules\n"]], ["block_9", ["1.10\nChapter Summary\n"]]], "page_55": [["block_0", [{"image_0": "55_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["4.\nSome polymers are listed below using either IUPAC (1) names or acceptable trivial (T) names.\nDraw structural formulas for the repeat units in these polymers, and propose an alternative\nname in the system other than the one given:\nPolymethylene (I)\nPolyforrnaldehyde (T)\nPoly(pheny1ene oxide) (T)\nPoly[(2-propy1-1,3-dioxane-4,6-diyl)methylene] (I)\nPoly(1-acetoxyethylene) (I)\nPoly(methy1 acrylate) (T)\n"]], ["block_2", ["5.\nStar polymers are branched molecules with a controlled number of linear arms anchored to\none central molecular unit acting as a branch point. Schaefgen and Flory\u00a7 prepared poly\n(a-caprolactam) four-\nand eight-arm\nstars using cyclohexanone tetrapropionic\nacid\nand\ndicyclohexanone octapropionic acid as branch points. The authors present the following stoi-\nchiometric de\ufb01nitions/relations to relate the molecular weight of the polymer to the concentration\nof unreacted acid groups in the product. Provide the information required for each of the\nfollowing steps:\n(a)\nThe product has the formula R\u2014{\u2014CO[\u2014NH(CH2)5CO\u2014\u2014]y\u2014OH }b. What is the significance\nof R, y, and b?\n(b)\nIf Q is the number of equivalents of multifunctional reactant which react per mole of\nmonomer and L represents the number of equivalents of unreacted (end) groups per mole\nof monomer, then (1\u2014L)/(Q+L). Justify this relationship, assuming all functional\ngroups are equal in reactivity.\n(c)\nIf M0 is the molecular weight of the repeat unit and Mg, is the molecular weight of the\noriginal branch molecule divided by b, then the number-average molecular weight of\nthe star polymer is\n"]], ["block_3", ["3.\nLandel used a commercial material called Vulcollan\n18/40 to study the rubber-to-glass\ntransition of a polyurethane}\u2014 This material is described as being \u201cprepared from a low\nmolecular weight polyester which is extended and cross-linked by reacting it with naphtha-\nlene\u20141,4,-diisocyanate and 1,4-butanediol. The polyester is prepared from adipic acid and a\nmixture of ethylene and propylene glycols.\u201d Draw the structural formula of a portion of the\ncross\u2014linked polymer which includes the various possible linkages that this description\nincludes. Remember that isocyanates react with active hydrogens; use this fact to account\nfor the cross-linking.\n"]], ["block_4", ["Problems\n39\n"]], ["block_5", ["TW.E. Rochefort, (10. Smith, H. Rachapudy, V.R. Raju, and W.W. Graessley, J. Polym. Sci, Polym. Phys, 17, 1197 (1979).\n*R.F. Landel, J. Colloid Scan, 12, 308 (1957).\n\u00a7 J.R. Schaefgen and RI. Flory, J. Am. Chem. Soc., 70, 2709 (1948).\n"]], ["block_6", ["44.9% trans, and 7.9% vinyl.T After hydrogenation, what is the average number of backbone\ncarbon atoms between ethyl side chains?\n"]], ["block_7", ["((1)\nEvaluate MD for the following molecules:\n"]], ["block_8", [{"image_1": "55_1.png", "coords": [58, 479, 182, 520], "fig_type": "molecule"}]], ["block_9", ["l\u2014L\nMn=b M\u2014+M\n{\u00b0Q+L\nb}\n"]], ["block_10", ["Justify this result and evaluate M0 and Mb for the b 4 and b 8 stars.\n"]], ["block_11", ["4\n0.2169\n0.0018\n"]], ["block_12", ["o\nQ\nL\n"]], ["block_13", ["8\n0.134\n0.00093\n"]]], "page_56": [["block_0", [{"image_0": "56_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["40\n"]], ["block_2", ["tH. Batzer, Makromol. Chem, 5, 5 (1950).\nISN. Chinai, J.D. Matlock, A.L. Resnick, and R.J. Samuels, J. Polym. sci, 17, 391 (1955).\n"]], ["block_3", ["12.\n"]], ["block_4", ["10.\n"]], ["block_5", ["11.\n"]], ["block_6", [{"image_1": "56_1.png", "coords": [43, 472, 150, 520], "fig_type": "molecule"}]], ["block_7", [{"image_2": "56_2.png", "coords": [53, 222, 269, 281], "fig_type": "molecule"}]], ["block_8", [{"image_3": "56_3.png", "coords": [54, 486, 221, 613], "fig_type": "figure"}]], ["block_9", ["(a)\nH2CA\\n/OH\nO\n"]], ["block_10", ["(b)\nO\nO\n"]], ["block_11", ["O\n(c)\nHON/LLOH\n"]], ["block_12", ["(d)\nHsN\n"]], ["block_13", ["Consider a set consisting of 4\u20148 family members, friends, neighbors, etc. Try to select a\nvariety of ages, genders, and other attributes. Take the mass of each individual (a rough\nestimate is probably wiser than asking directly) and calculate the number- and weight\u2014\naverage masses for this set. Does the resulting PDI indicate a rather \u201cnarrow\u201d distribution?\nIf you picture this group in your mind, do you imagine them all to be roughly the same size,\nas the PDI probably suggests?\nProve that the polydispersity of the Schulz-Zimm distribution is given by Equation 1.7.21.\nYou may want to look up the general solution for integrals of the type fxae\u2019bxdx.\nIn Figure 1.5a through Figure 1.5c it appears that the maximum in w; corresponds closely to\nM\". Differentiate Equation 1.7.20 with respect to M,- to show why this is the case.\nThe MALDI spectrum in Figure 1.7b resembles a Gaussian or normal distribution. One\nproperty of a Gaussian distribution is that the half\u2014width at half-height of the peak is\napproximately equal to 120\u2018. Use this relation to estimate 0' from the trace, and compare it\nto the value you would get from Equation 1.7.16.\nGive the overall chemical reactions involved in the polymerization of these monomers, the\nresulting repeat unit structure, and an acceptable name for the polymer.\n"]], ["block_14", ["The Mark\u2014Houwink exponent a for poly(methyl methacrylate) at 25\u00b0C has the value 0.69\nin acetone and 0.80 in chloroform. Calculate (retaining more significant figures than\nstrictly warranted) the value of MV that would be obtained for a sample with the following\nmolecular weight distribution if the sample were studied by viscometry in each of these\nsolvents.i Compare the values of MV with MH and MW.\n"]], ["block_15", ["Batzer reported the following data for a fractionated polyester made from sebacic acid and\n1,6-hexanediolff evaluate Mn, MW, and M2.\n"]], ["block_16", ["M,x10\"5(g/mol)\n2.0\n4.0\n6.0\n8.0\n10.0\n"]], ["block_17", ["Mass (g)\n1.15\n0.73\n0.415\n0.35\n0.51\n0.34\n1.78\n0.10\n0.94\nM X 10'4\n1.25\n2.05\n2.40\n3.20\n3.90\n4.50\n6.35\n4.10\n9.40\n"]], ["block_18", ["Fraction\n1\n2\n3\n4\n5\n6\n7\n8\n9\n"]], ["block_19", ["n,x103(mol)\n1.2\n2.7\n4.9\n3.1\n0.9\n"]], ["block_20", [{"image_4": "56_4.png", "coords": [62, 509, 214, 593], "fig_type": "molecule"}]], ["block_21", [{"image_5": "56_5.png", "coords": [71, 515, 211, 560], "fig_type": "molecule"}]], ["block_22", [{"image_6": "56_6.png", "coords": [83, 528, 215, 552], "fig_type": "molecule"}]], ["block_23", [{"image_7": "56_7.png", "coords": [86, 476, 140, 517], "fig_type": "molecule"}]], ["block_24", ["HQN\u2018E/ENH? CIMCI\n"]], ["block_25", [{"image_8": "56_8.png", "coords": [99, 480, 207, 624], "fig_type": "molecule"}]], ["block_26", ["Me\n"]], ["block_27", ["Introduction to Chain Molecules\n"]]], "page_57": [["block_0", [{"image_0": "57_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["H.R. Allcock, F.W. Lampe, and J.E. Mark, Contemporary Polymer Chemistry, 3rd ed., Prentice Hall, Engle\u2014\nwood Cliffs, NJ, 2003.\nR]. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY, 1953.\nA.Y. Grosberg, and AR. Khokhlov, Giant Molecules, Here, There and Everywhere, Academic Press, San\nDiego, CA, 1997.\nAL. Lehninger, Biochemistry, 2nd ed., Worth Publishers, New York, 1975.\nP. Munk and TM. Aminabhavi, Introduction to Macromolecular Science, 2nd ed., Wiley Interscience, New\nYork, 2002.\nG. Odian, Principles of Polymerization, 4th ed., Wiley, New York, 2004.\nRC. Painter and M.M. Coleman, Fundamentals of Polymer Science, Technomic, Lancaster, PA, 1994.\nRB. Seymour and CE. Carraher, Jr., Polymer Chemistry: An Introduction, Marcel Dekker, New York, 1981.\nL.H. Sperling, Introduction to Physical Polymer Science, Wiley Interscience, New York, 1986.\nR]. Young and PA. Lovell, Introduction to Polymers, 2nd ed., Chapman and Hall, London, 1991.\n"]], ["block_2", ["Further Readings\n41\n"]], ["block_3", ["Further Readings\n"]], ["block_4", ["References\n"]], ["block_5", ["l\u2014n 0\nChemical cf: Engineering News, 58, 29 (1980).\nThe Graduate, directed by Mike Nichols, Embassy Pictures Corporation, 1967.\nNo enterprise as rich as polymer science has only one \u201cfather.\u201d Herman Mark is one of those to whom the\ntitle could readily be applied. An interesting interview with Professor Mark appears in the Journal of\nChemical Education, 56, 38 (1979).\nH. Morawetz, Polymers: The Origins and Growth of a Science, Wiley, New York, 1985.\nRB. Seymour, History of Polymer Science and Technology, Marcel Dekker, New York, 1982.\nCS. Marvel, another pioneer in polymer chemistry, reminisced about the early days of polymer chemistry\nin the United States in the Journal of Chemical Education, 58, 535 (1981).\nSee IUPAC Macromolecular Nomenclature Commission, Macromolecules, 6, 149 (1973).\n"]], ["block_6", ["13.\n"]], ["block_7", ["14.\n"]], ["block_8", ["15.\n"]], ["block_9", ["16.\n"]], ["block_10", [{"image_1": "57_1.png", "coords": [44, 38, 232, 130], "fig_type": "figure"}]], ["block_11", [{"image_2": "57_2.png", "coords": [52, 109, 146, 154], "fig_type": "molecule"}]], ["block_12", ["(e)\nQ\n\"'\nHOHOH\n"]], ["block_13", ["(f)\nH2N WOH\n"]], ["block_14", ["A MALDI-TOF analysis of a polystyrene sample exhibited a peak (one of many) at 1206.\nThe sample was prepared in dithranol, with silver nitrate as the salt. Assuming no head-to-\nhead defects, how many distinct chemical structures could this peak represent? Propose\nstructures for the end groups of the polymer as well.\nProton NMR is used to attempt to quantify the molecular weight of a poly(ethylene oxide)\nmolecule with methyoxy end groups at each terminus. If the integration of the methyl protons\nrelative to the methylene protons gave a ratio of 1:20, what can you say about the molecular\nweight?\nWhat would be Mw and Mn for a sample obtained by mixing\n10 g of polystyrene\n"]], ["block_15", ["(MW 100,000, Mn 70,000) with 20 g of another polystyrene (Mw 60,000, Mn 20,000)?\nWhat would MW and Mn be for an equimolar mixture of tetradecane and decane? (Ignore\nisotope effects.)\n"]], ["block_16", [{"image_3": "57_3.png", "coords": [88, 52, 223, 119], "fig_type": "molecule"}]]], "page_58": [["block_0", [{"image_0": "58_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["In Section 1.4, we discussed the classi\ufb01cation of polymers into the categories of addition or\ncondensation. At that time we noted that these classifications could be based on the following:\n"]], ["block_2", ["It is the third of these criteria that offers the most powerful insight into the nature of the\npolymerization process for this important class of materials. We shall sometimes use the terms\nstep-growth and condensation polymers as synonyms, although step-growth polymerization encom\u2014\npasses a wider range of reactions and products than either criteria (1) or (2) above would indicate.\nThe chapter is organized as follows. First, we examine how the degree of polymerization and its\ndistribution vary with the progress of the polymerization reaction, with the latter defined both in\nterms of stoichiometry and time (Section 2.2 through Section 2.4). Initially we consider these\ntopics for simple reaction mixtures, that is, those in which the proportions of reactants agree\nexactly with the stoichiometry of the reactions. After this, we consider two important classes of\ncondensation or step\u2014growth polymers: polyesters and polyamides (Section 2.5 and Section 2.6).\nFinally we consider nonstoichiometric proportions of reactants (Section 2.7). The important case of\nmultifunctional monomers, which can introduce branching and cross-linking into the products, is\ndeferred until Chapter 10.\n"]], ["block_3", ["1.\nStoichiometry of the polymerization reaction (small molecule eliminated?)\n2.\nComposition of the backbone of the polymer (atoms other than carbon present?)\n3.\nMechanism of the polymerization (stepwise or chain reaction?)\n"]], ["block_4", ["As the name implies, step-growth polymers are formed through a series of steps, and high\nmolecular weight materials result from a large number of steps. Although our interest lies in\nhigh molecular weight, long-chain molecules, a crucial premise of this chapter is that these\nmolecules can be effectively discussed in terms of the individual steps that lead to the formation\nof the polymer. Thus, polyesters and polyamides are substances that result from the occurrence of\nmany steps in which ester or amide linkages are formed between the reactants. Central to our\ndiscussion is the idea that these steps may be treated in essentially the same way, whether they\noccur between small molecules or polymeric species. We shall return to a discussion of the\nimplications and justification of this assumption of equal reactivity throughout this chapter.\n"]], ["block_5", ["2.2\nCondensation Polymers: One Step at a Time\n"]], ["block_6", ["2.1\nIntroduction\n"]], ["block_7", ["2.2.1\nClasses of Step-Growth Polymers\n"]], ["block_8", ["Here are examples of important classes of step-growth polymers:\n"]], ["block_9", ["1.\nPolyesters\u2014successive reactions between diols and dicarboxylic acids:\n"]], ["block_10", ["RI\nn Ho/HT'OH n HOJLR'JLOH\n\u2014\"\" {O\u2018H/Om/ \ufb02\n"]], ["block_11", [{"image_1": "58_1.png", "coords": [55, 575, 349, 629], "fig_type": "molecule"}]], ["block_12", ["Step-Growth Polymerization\n"]], ["block_13", ["0\n0\n"]], ["block_14", [{"image_2": "58_2.png", "coords": [146, 584, 337, 621], "fig_type": "molecule"}]], ["block_15", ["2\n"]], ["block_16", ["43\n"]], ["block_17", ["+ 2n H20\n(2A)\n0 n\nO\n"]]], "page_59": [["block_0", [{"image_0": "59_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "59_1.png", "coords": [19, 553, 280, 653], "fig_type": "figure"}]], ["block_2", [{"image_2": "59_2.png", "coords": [29, 297, 172, 331], "fig_type": "molecule"}]], ["block_3", ["aA and B represent two different functional groups and ab is the product\nof their reaction with each other. Consult the text for a discussion of the\nline-by\u2014line development of the reaction.\n"]], ["block_4", ["Of course, in Reaction (2.A) and Reaction (2B) the hydrocarbon sequences R and R\u2019 can be the\nsame or different, contain any number of carbon atoms, be linear or cyclic, and so on. Likewise, the\ngeneral reactions, Reaction (2.C) and Reaction (2E), certainly involve hydrocarbon sequences\nbetween the reactive groups A and B. The notation involved in these latter reactions is particularly\nconvenient, however, and we shall use it extensively in this chapter. It will become clear as we\nproceed that the stoichiometric proportions of reactive groups\u2014A and B in the above notation\u2014\nplay an important role in determining the characteristics of the polymeric product. Accordingly, we\nshall confine our discussion for the present to reactions of the type given by Reaction (2E), since\nequimolar proportions of A and B are assured by the structure of the monomer.\n"]], ["block_5", ["Table 2.1\nHypothetical Step\u2014Growth Polymerization\nof 10 AB Moleculesa\n"]], ["block_6", ["44\nStep-Growth Polymerization\n"]], ["block_7", ["Table 2.] presents a hypothetical picture of how Reaction (2.E) might appear if we examined the\ndistribution of product molecules in detail. Row 1 of Table 2.1 shows the initial pool of monomers,\n"]], ["block_8", ["3.\nAbababaB AbaB AbabaB AB\n4.\nAbababaB AbabababaB AB\n5.\nAbababaB AbababababaB\n6,\nAbababababababababaB\n"]], ["block_9", ["Row\nMolecular species present\n"]], ["block_10", ["1.\nAB AB AB AB AB AB AB AB AB AB\n2.\nAbaB AbaB AbaB AbaB AB AB\n"]], ["block_11", ["2.\nPolyamides\u2014successive reactions between diamines and dicarboxylic acids:\n"]], ["block_12", ["Since the two reacting functional groups can be located in the same reactant molecule, we add\nthe following:\n4.\nPoly(amino acid)\n"]], ["block_13", ["2.2.2\nFirst Look at the Distribution of Products\n"]], ["block_14", ["5.\nGeneral\n"]], ["block_15", ["3.\nGeneral\u2014successive reactions between difunctional monomer A\u2014A and difunctional mono-\n"]], ["block_16", [{"image_3": "59_3.png", "coords": [39, 214, 246, 286], "fig_type": "figure"}]], ["block_17", [{"image_4": "59_4.png", "coords": [48, 223, 245, 281], "fig_type": "molecule"}]], ["block_18", ["mer B\u2014B:\n"]], ["block_19", ["n H,N/'\\c/\u00b0H fN\n+ n H20\n(2D)\n"]], ["block_20", ["n A\u2014B_yl(a\\b/)H+\n(2.13)\n"]], ["block_21", ["0\nO\nH\nH\nR\nN\nHI\nn H2N/\n\\NH2\n+\n\u201dHOJLHIJ%H\n"]], ["block_22", ["n A\u2014A + n B\u2014B\u2014hta\\a/b\\b/)n+\n(2.0\n"]], ["block_23", ["Fl\nF!\n"]], ["block_24", ["H\nH\no\nn\nO\n"]], ["block_25", [{"image_5": "59_5.png", "coords": [145, 79, 336, 117], "fig_type": "molecule"}]], ["block_26", ["_+ H2O\n(2.B)\n00:1\n"]]], "page_60": [["block_0", [{"image_0": "60_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["10 molecules in a possible composition after a certain amount of\nreaction Section that the particular that account\nfor the rows are not highly probable. Our objective here is\nnot to assess the probability of certain reactions, but rather to consider some possibilities.\nStoichiometrically, we can still account for the initial set of 10 A groups and 10 B groups; we\nindicate those that have reacted with each other as ab groups. The same conservation of atom\ngroupings would be obtained if row 2 showed one trimer, two dimers, and three monomers instead\nof the four dimers and two monomers indicated in Table 2.1. Other combinations could also\n"]], ["block_2", ["be assembled. These possibilities indicate one of the questions that we shall answer in this chapter:\nHow do the distribute among the different possible species as the reaction\nproceeds?\nRow 3 of Table 2.1 shows the mixture after two more reaction steps have occurred. Again,\nthe components we have elected to show are an arbitrary possibility. For the monomer system\nwe have chosen, the concentration of A and B groups in the initial monomer sample are equal\nto each other and equal to the concentration of monomer. In this case, an assay of either A groups\nor B groups in the mixture could be used to monitor the progress of the reaction. Choosing the\nnumber of A groups for this purpose, we see that this quantity drops from 10 to 6 to 4, respectively,\nas we proceed through rows 1, 2, and 3 of Table 2.1. What we wish to point out here is the fact that\nthe 10 initial monomers are now present in four molecules, so the number average degree of\npolymerization is only 2.5, even though only 40% of the initial reactive groups remain. Another\nquestion is thus raised: In general, how does the average molecular weight vary with the extent of\nthe reaction?\nThe reaction mixture in the fourth row of Table 2.1 is characterized by a number average degree\nof polymerization Nn 10/3 3.3, with only 30% of the functional groups remaining. This means\nthat 70% of the possible reactions have already occurred, even though we are still dealing with a\nvery low average degree of polymerization. Note that the average degree of polymerization would\n"]], ["block_3", ["Condensation Polymers: One Step at a Time\n45\n"]], ["block_4", ["be the same if the 70% reaction of functional groups led to the mixture AbababababababaB and\ntwo AB\u2019s. This is because the initial 10 monomers are present in three molecules in both instances,\nand we are using number averages to talk about these possibilities. The weight average would be\ndifferent in the two cases. This poses still another question: How does the molecular weight\ndistribution vary with the extent of reaction?\nBy the fifth row, the reaction has reached 80% completion and the number average value of the\ndegree of polymerization Nn is 5. Although we have considered this slowly evolving polymer\nin terms of the extent of reaction, another question starts to be worrisome: How long is this going\nto take?\nThe sixth row represents the end of the reaction as far as linear polymer is concerned. Of the 10\ninitial A groups, 1 is still unreacted, but this situation raises the possibility that the decamer shown\nin row 6\u20140r for that matter, some other i-mer, including monomer\u2014might form a ring or cyclic\ncompound, thereby eliminating functional groups without advancing the polymerization. Through-\nout this chapter we will assume that the extent of ring formation is negligible.\nIt is an easy matter to generalize the procedure we have been following and express the number\naverage degree of polymerization in terms of the extent of reaction, regardless of the initial sample\nsize. We have been dividing the initial number of monomers present by the total number of\nmolecules present after any extent of reaction. Each molecular species\u2014whether monomer or\npolymer of any length\u2014contains just one A group. The total number of monomers is therefore\nequal to the initial (superscript 0) number of A groups, 12%; the total number of molecules at any\nextent of reaction (no superscript) is equal to the number of A groups, VA, present at that point. The\nnumber average degree of polymerization is therefore given by\n"]], ["block_5", ["V0\nNn : A\n(2.2.1)\nVA\n"]]], "page_61": [["block_0", [{"image_0": "61_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Most of the questions raised in the past few paragraphs will be answered during the course of this\nchapter, some for systems considerably more involved than the one considered here. Before\nproceeding further, we should reemphasize one premise that underlies the entire discussion of\nTable 2.1: How do the chemical reactivities of A and B groups depend on the degree of\npolymerization of the reaction mixture? In Table 2.1, successive entries were generated by simply\nlinking together at random those species present in the preceding row. We have thus assumed that,\nas far as reactivity is concerned, an A reacts as an A and a B reacts as a B, regardless of the size of\nthe molecule to which the group is attached. If this assumption of equal reactivity is valid, it results\nin a tremendous simplification; otherwise we shall have to characterize reactivity as a function of\ndegree of polymerization, extent of reaction, and so on.\nOne of the most sensitive tests of the dependence of chemical reactivity on the size of the\nreacting molecules is the comparison of the rates of reaction for compounds that are members of a\nhomologous series with different chain lengths. Studies by Flory and others on the rates of\nesteri\ufb01cation and saponification of esters were the first investigations conducted to clarify the\ndependence of reactivity on molecular size [1]. The rate constants for these reactions are observed\nto converge quite rapidly to a constant value that is independent of molecular size, after an initial\ndependence on molecular size for small molecules. In the esterification of carboxylic acids, for\nexample, the rate constants are different for acetic, propionic, and butyric acids, but constant for\ncarboxylic acids with 4\u201418 carbon atoms. This observation on nonpolymeric compounds has been\n"]], ["block_2", ["This expression is consistent with the analysis of each of the rows in Table 2.1 as presented above\nand provides a general answer to one of the questions posed there. It is often a relatively easy\nmatter to monitor the concentration of functional groups in a reaction mixture; Equation 2.2.4\nrepresents a quantitative summary of an end group method for determining Nn. We reiterate that\nEquation 2.2.4 assumes equal numbers of A and B grOUps, with none of either lost in nonpolymer\nreactions.\nFrom row 6 in Table 2.1, we see that Nn=10 when p=0.9. The fact that this is also the\nmaximum value for N is an artifact of the example. In a larger sample of monomers higher average\ndegrees of polymerization are attainable. Equation 2.2.4 enables us to calculate that Nn becomes\n20, 100, and 200 for extents of reaction 0.950, 0.990, and 0.995, respectively. These results reveal\nwhy condensation polymers are often of relatively modest molecular weight: it may be very\ndifficult to achieve the extents of reaction required for very high molecular weights. As p increases\nthe concentration of H20 (or other small molecule product) will increase, and the law of mass\naction will Oppose further polymerization. Consequently, steps must be taken to remove the small\nmolecule as it is formed, if high molecular weights are desired.\n"]], ["block_3", ["It is convenient to define the fraction of reacted functional groups in a reaction mixture by a\nparameter p, called the extent of reaction. Thus, p is the fraction of A groups that have reacted at\nany stage of the process, and 1\u2014 p is the unreacted fraction:\n"]], ["block_4", ["46\nStep-Growth Polymerization\n"]], ["block_5", ["Comparison of Equation 2.2.1 and Equation 2.2.2 enables us to write very simply:\n"]], ["block_6", ["2.2.3\nA First Look at Reactivity and Reaction Rates\n"]], ["block_7", ["01\u2018\n"]], ["block_8", ["p \n(2.2.3)\n\u201dA\n"]], ["block_9", ["N11 \u2014\u20141\u2014\u2014\n(2.2.4)\n1 p\n"]], ["block_10", ["1 p 3\u20193\n(2.2.2)\nVA\n"]]], "page_62": [["block_0", [{"image_0": "62_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "62_1.png", "coords": [25, 29, 337, 182], "fig_type": "figure"}]], ["block_2", [{"image_2": "62_2.png", "coords": [26, 46, 183, 165], "fig_type": "figure"}]], ["block_3", ["where the parentheses represent the caged pair, as in Figure 2.1, and the ks are the rate constants for\nthe individual steps: k, and k0 for diffusion into and out of the cage, respectively, and k, for the\nreaction itself.\nSince this is the first occasion we have had to examine the rates at which chemical reactions\noccur, a few remarks about mechanistic steps and rate laws seem appropriate. The reader who feels\nthe need for additional information on this topic should consult any introductory physical chem-\nistry text.\nAs a brief review we recall the following:\n"]], ["block_4", ["generalized to polymerization reactions as well. The latter are subject to several complications that\nare not involved in the study of simple model compounds, but when these complications are\nproperly considered, the independence of reactivity on molecular size has been repeatedly veri\ufb01ed.\nThe foregoing conclusion does not mean that a constant rate of reaction persists throughout\nTable 2.1. The rate of reaction depends on the concentrations of reactive groups, as well as on their\nreactivities. Accordingly, rate of the reaction decreases as the extent of reaction progresses.\nWhen the rate law for the reaction is extracted from proper kinetic experiments, speci\ufb01c reactions\nare found to be characterized by \ufb01xed rate constants over a range of M, values.\nAmong the further complications that can interfere with this conclusion is the possibility that\nthe polymer becomes insoluble beyond a critical molecular weight or that the low molecular\nweight by-product molecules accumulate and thereby shift the equilibrium to favor reactants. It is\nalso possible that the transport of reactants will be affected by the increasing viscosity of the\npolymerization medium, which is a very complicated issue.\nFigure 2.] suggests that reactive end-groups may be brought into contact by rotation around\nonly a few bonds, an effect which is therefore independent of chain length. Once in close\nproximity, the A and B groups may be thought of as being in the same \u201ccage\u201d defined by near\nneighbors. It may take some time for the two reactive groups to diffuse together, but it will also\ntake some time for them to diffuse apart; this provides the opportunity to react. The rate at which\nan A and a B group react to form an ab linkage therefore depends on the relative rates of three\nprocesses: the rate to diffuse together, the rate at which they diffuse apart, and the rate at which\n\u201ctrapped\u201d A and B groups react. These considerations can be expressed more quantitatively by\nwriting the process in terms of the following mechanism:\n"]], ["block_5", ["Figure 2.1\nThe reaction of A and B groups at the ends of two different chains. Note that rotations around\nonly a few bonds will bring A and B into the same cage of neighboring groups, indicated by the dashed-line\nenclosure.\n"]], ["block_6", [{"image_3": "62_3.png", "coords": [35, 500, 208, 531], "fig_type": "molecule"}]], ["block_7", ["Condensation \n47\n"]], ["block_8", ["1.\nThe rate of a process is expressed by the derivative of a concentration (square brackets) with\nrespect to time, d[A]/dt. If the concentration of reaction product is used, this quantity is\n"]], ["block_9", ["3 . ~\\\\u201c\\'\\X\\\n.\u2018\n\\\\:-\\\\\"\u2018-_I\n\\\\\\\\\\\\\\\\\\\u00a7\n\\\n'\n\u20183.\n.\n..'*\n\\\\\n\\ '-\n\u2018\n\\\\u2018uVR\nu x \\ ~\n.\u2018\\\\ ..\n\\\n.\n-_u\u2018\n-\nx.\nk\n-\\\nn\n'-\n.o'\n\\\\u2018 \\\\\u2018~. \u2018A\u2018\nQ\n\\ mu o.:.\\\\\\.. o...\u2018x\\\\\nu~_. you u o;\n.\nuu\u2018\\\\\u201c~\\\\\u2018\\\\Q\u2018\\\u00a7\u2018t\\.\\\\2u\\\\u2018uu\\yu\u20183\\\\\u201c\\\n{on ' i}.-.\\<=.u\\\u00a7oo.- . .\\ \u2018i'u {Q\\ \ufffd\\.\\. \\.\\~:~:u\\ \u2018nuu \\\\\\\\u2018 -\\\\u2018 xxx\u00bb.-\n\\A. . 3-.\\\"\\\\\\\n"]], ["block_10", ["Qk\u201cubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ub>\u2018ub>mubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>u2018ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>\n.. ..ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>u2018ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>~ubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>\n:ubub>>ub>\\ubub>>ub>\na\nubub>>ub>-ubub>>ub>\nNo\n\u201c ub>\u2018ub>\nubub>>ub>um:ubub>>ub>\nI\u201c\nubub>>ub>\\ubub>>ub>\nub>>ub>\n\u201dH.\nub>\u2018ub>\nubub>>ub>\\ubub>>ub>ubub>>ub>\\ubub>>ub>\nu\nub>\u2018ub>ubub>>ub>\\ubub>>ub> \u201c\n"]], ["block_11", ["\u2014A + \u2014B\n$11;\n(\u2014A + B\u2014) \u00a35 \n(2F)\n"]], ["block_12", ["0\n"]], ["block_13", ["..\nx\u2018\n\\\\u2018\n\\\n\"._ \\\\_\\\n. \n"]], ["block_14", ["Q.\n"]], ["block_15", [{"image_4": "62_4.png", "coords": [196, 56, 372, 162], "fig_type": "figure"}]]], "page_63": [["block_0", [{"image_0": "63_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["positive; if a reactant is used, it is negative and a minus sign must be included. Also, each\nderivative d[A]/dt should be divided by the coefficient of that component in the chemical\nequation that describes the reaction, so that a single rate is described whichever component in\nthe reaction is used to monitor it.\n2.\nA rate law describes the rate of reaction as the product of a constant k, called the rate constant,\nand various concentrations, each raised to specific powers. The power of an individual\nconcentration term in a rate law is called the order with respect to that component, and the\nsum of the exponents of all concentration terms gives the overall order of the reaction. Thus in\nthe rate law Rate :k[X]1[Y]2, the reaction is first order in X, second order in Y, and third\norder overall.\n3.\nA rate law is determined experimentally and the rate constant evaluated empirically. There\nis no necessary connection between the stoichiometry of a reaction and the form of the\nrate law.\n4.\nA mechanism is a series of simple reaction steps that, when added together, account for the\noverall reaction. The rate law for the individual steps of the mechanism may be written by\ninspection of the mechanistic steps. The coefficients of the reactants in the chemical equation\ndescribing the step become the exponents of these concentrations in the rate law for that step.\n5.\nFrequently it is possible to write more than one mechanism that is compatible with an\nobserved rate law. Thus, the ability to account for an experimental rate law is a necessary\nbut not a sufficient criterion for the correctness of the mechanism.\n"]], ["block_2", ["48\nStep-Growth Polymerization\n"]], ["block_3", ["These ideas are readily applied to the mechanism described by Reaction (2.F). To begin with, the\nrate at which ab links are formed is first order with reSpect to the concentration of entrapped pairs.\nIn this sense, the latter behaves as a reaction intermediate or transition state according to this\nmechanism. Therefore\n"]], ["block_4", ["We shall return to this type of kinetic analysis in Chapter 3 where we discuss chain-growth\npolymerization.\nAccording to the mechanism provided by Reaction (2F) and the analysis given by Equation\n2.2.8, the rate of polymerization is dependent upon the following:\n"]], ["block_5", ["1.\nThe concentrations of both A and B, hence the reaction slows down as the conversion to\npolymer progresses.\n2\nThe three constants associated with the rates of the individual steps in Reaction (2.F).\n"]], ["block_6", ["The concentration of entrapped pairs is assumed to exist at some stationary\u2014state (subscript 5) level\nin which the rates of formation and loss are equal. In this stationary state d[(FA + B\ufb02)]/dt 0 and\nEquation 2.2.6 becomes\n"]], ["block_7", ["where the subscript s reminds us that this is the stationary\u2014state value. Substituting Equation 2.2.7\ninto Equation 2.2.5 gives\n"]], ["block_8", ["These entrapped pairs, in turn, form at a rate given by the rate at which the two groups diffuse\ntogether minus the rate at which they either diffuse apart or are lost by reaction:\n"]], ["block_9", ["k0 + kr\nRate of ab formation :\n[A][B]\n(22.8)\n"]], ["block_10", ["Rate of ab formation :kr[(\u2014A + B\u2014)]\n(2.2.5)\n"]], ["block_11", ["d[(eA + B\u2014)]\ndt\n= kilAllB] a Icon\u2014A + 3\u2014)] \u2014 kin\u2014A + 3\u2014)]\n(2.2.6)\n"]], ["block_12", ["k1\n[(\u2014A + 3\u2014)]5 \nk0 +16: \n(22-7)\n"]], ["block_13", ["kik,\n"]]], "page_64": [["block_0", [{"image_0": "64_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["4.\nThe two constants k0 describe exactly the same kind of diffusional process, and differ\nonly in direction. Hence they should have the same dependence on molecular size, whatever\nthat might be, and that dependence therefore cancels out.\n5.\nThe mechanism in Reaction (2.F) is entirely comparable to the same reaction in low molecular\nweight systems. Such reactions involve considerably larger activation energies than physical\nprocesses like diffusion and, hence, do proceed slowly.\n"]], ["block_2", ["Note that the rate only on and any size dependence for this constant\nwould not cancel out.\nBoth Equation 2.2.9 and Equation 2.2.10 predict rate laws that are first order with respect to the\nconcentration of each of the reactive groups; the proportionality constant has a different signifi-\ncance in the two cases, however. The observed rate laws, which suggest a reactivity that is\nindependent of molecular size and the a priori expectation cited in item 5 above regarding\nthe magnitudes of different kinds of k values, lend credibility to the version presented in\nEquation 2.2.9.\nOur objective in the preceding argument has been to justify the attitude that each ab linkage\nforms according to the same rate law, regardless of the extent of the reaction. While our attention is\nfocused on the rate laws, we might as well consider the question, raised above, about the actual\nrates of these reactions. This is the topic of the next section.\n"]], ["block_3", ["In this section we consider the experimental side of condensation kinetics. The kinds of ab links\nthat have been most extensively studied are ester and amide groups, although numerous additional\nsystems could also be cited. In many of these systems the carbonyl group is present and believed to\nplay an important role in stabilizing the actual chemical transition state involved in the reactions.\nThe situation can be represented by the following schematic reaction:\n"]], ["block_4", ["3.\nIf the rate of chemical reaction is slow compared to the rate of group diffusion (kr << ki, k0),\nthen Equation 2.2.8 reduces to\n"]], ["block_5", ["6.\nIf k, >> ki, k0, then Equation 2.2.8 reduces to\n"]], ["block_6", ["2.3\nKinetics of Step-Growth Polymerization\n"]], ["block_7", ["in which the intermediate is stabilized by coordination with protons, metal ions, or other Lewis\nacids. The importance of this is to emphasize that the kinds of reactions we are considering are\noften conducted in the presence of an acid catalyst, frequently something like a sulfonic acid or a\nmetal oxide. The purpose of a catalyst is to modify the rate of a reaction, so we must be attentive to\nthe situation with respect to catalysts. At present, we assume a constant concentration of catalyst\nand attach a subscript c to the rate constant to remind us of the assumption. Accordingly, we write\n"]], ["block_8", ["Kinetics of Step-Growth Polymerization\n49\n"]], ["block_9", ["which is consistent with both Equation 2.2.9 and Equation 2.2.10. We expect the constant kc to be\ndependent on the concentration of the catalyst in some way which means that Equation 2.3.1 may\n"]], ["block_10", [{"image_1": "64_1.png", "coords": [38, 484, 300, 536], "fig_type": "molecule"}]], ["block_11", ["0\no\n_\n0\nRJLX\n+\nY\u201c\n\u2014\u2014h-\nR\u2014l\u2014Y\n-\u2014-\u2014\u2014-)-\nJL\n+\nx\u2014\n(Z\u2018G)\nx\nFl\nY\n"]], ["block_12", ["\u2014 99\u2014] = kc[A][B]\n(2.3.1)\ndt\n"]], ["block_13", ["Rate of ab formation ki[A][B]\n(2.2.10)\n"]], ["block_14", ["Rate of ab formation \nk\u2014kr[A][B]\n(2.2.9)\n"]], ["block_15", [{"image_2": "64_2.png", "coords": [179, 486, 281, 530], "fig_type": "molecule"}]]], "page_65": [["block_0", [{"image_0": "65_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["that after cancellation and rearrangement gives\n"]], ["block_2", ["be called a pseudo-second\u2014order rate law. We shall presently consider these reactions in the\nabsence of external catalysts. For now it is easier to proceed with the catalyzed case.\n"]], ["block_3", ["now differentiate with respect to t, noting that only x is a function of r:\n"]], ["block_4", ["Neither [A]0 nor [B]0 are functions of 1, although both [A] and [B] are. We write the latter two as\n[A] [A]0 x and [B] [B]0 x. Substitute these results into Equation 2.3.3 and rearrange:\n"]], ["block_5", ["Since d[A]/dt= Fdx/dt by the definition of x, this proves Equation 2.3.3 to be a solution to\nEquation 2.3.1. Equation 2.3.3 is undefined in the event [A]0 [B]0, but in this case the expression\nis anyhow inapplicable. Since A and B react in a 1:1 proportion, their concentrations are identical\nat all stages of reaction if they are equal initially. In this case, Equation 2.3.1 would reduce to a\nsimpler second-order rate law, which integrates to Equation 2.3.2.\n"]], ["block_6", ["We shall now proceed on the assumption that [A]0 and [B]0 are equal. As noted above, having\nboth reactive groups on the same molecule is one way of enforcing this condition. Accordingly, we\n"]], ["block_7", ["Both of these results are readily obtained; we examine the less obvious relationship in Equation\n2.3.3 in the following example. The consequences of different A and B concentrations on the\nmolecular weight of the polymer will be discussed in Section 2.7.\n"]], ["block_8", ["Equation 2.3.1 is the differential form of the rate law that describes the rate at which A groups are\nused up. To test a proposed rate law and evaluate the rate constant it is preferable to work with the\nintegrated form of the rate law. The integration of Equation 2.3.1 yields different results, depend\u2014\ning on whether the concentrations of A and B are the same or different:\n"]], ["block_9", ["2.3.1\nCatalyzed Step-Growth Reactions\n"]], ["block_10", ["the reaction, and [A]0 and [B]0 as the concentrations of these groups at I: 0.\n2.\nIf [A]0 [B]0, the integration of Equation 2.3.1 yields\n"]], ["block_11", ["By differentiation, verify that Equation 2.3.3 is a solution to Equation 2.3.1 for the conditions\ngiven.\n"]], ["block_12", ["3.\nIf [Ale 75 [B]O, \n"]], ["block_13", ["Solution\n"]], ["block_14", ["50\nStep\u2014Growth Polymerization\n"]], ["block_15", ["Example 2.1\n"]], ["block_16", ["1.\nWe define [A] and [B] as the instantaneous concentrations of these groups at any time t during\n"]], ["block_17", ["dx\nd? \n"]], ["block_18", ["[A10 + 1,12% = ([A]0 \u2014 [310) kcr\n1\n_______\n\u201c[Bn\u2014x\n[Ah\n"]], ["block_19", ["([B]0\n"]], ["block_20", ["1\n1\n\u201d\u2014 _ \u2014 : k\u00b0\n2.3.2\n[AllAb\n\u2019\n(\n)\n"]], ["block_21", ["1\n[AllBlo\nl\n= kc\n2.3.3\n[No [Blo n([A]0[B])\nI\n(\n)\n"]], ["block_22", ["'\u2014\nI) (\u201c([Blo\n- x) + ([No \u2014 x)\ndx\n) dz: ([Alo [1310) kc\n[Ab-x\nawe\u2014n?\n"]]], "page_66": [["block_0", [{"image_0": "66_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["a plot for adipic acid reacted with 1,10-decamethylene glycol, catalyzed by p-toluene sulfonic\nacid. The reaction had already been run to consume 82% of the reactive groups before this\nexperiment was conducted. Interpreting the slope of the line in terms of Equation 2.3.6 and in\nthe light of actual initial concentrations gives a value of kc20.097 kg eq\u20141 min\u2018l. Note that\n"]], ["block_2", ["Figure 2.2\nPlot of l/(l\u2014p) versus time for the late stages of esterification of adipic acid with 1,10-\ndecamethylene glycol at 161\u00b0C, catalyzed by p-toluene sulfonic acid. The reaction time t 0 corresponds\nto a previous extent of reaction in which 82% of the COOH groups had been consumed. (Data from Hamann,\nS.D., Solomon, DH, and Swift, J.D., J. Macromol. Sci. Chem, A2, 153, 1968.)\n"]], ["block_3", ["At this point, itto recall the extent of reaction parameter, p, defined by Equation\n2.2.3. If we and Equation 2.3.4, we obtain\n\u201cFm\n(2'35)\n"]], ["block_4", ["01'\n"]], ["block_5", ["where we incorporated Equation 2.2.4 into the present discussion. These last expressions provide\ntwo very useful views of the progress of a condensation polymerization reaction with time.\nEquation 2.3.4 describes how the concentration of A groups asymptotically approaches zero at\nlong times; Equation 2.3.6 describes how the number average degree of polymerization increases\nlinearly with time.\nEquation 2.3.6 predicts a straight line when 1/(1\u2014p) is plotted against 1\u2018. Figure 2.2 shows such\n"]], ["block_6", ["rearrange Equation 2.3.2 to give the instantaneous concentrations of unreacted A groups as a\nfunction \n"]], ["block_7", ["Kinetics \n51\n"]], ["block_8", ["WHO)\n40\n"]], ["block_9", ["\u2014\u20141\u2014 :NIn 1 + kC[A]Ot\n(2.3.6)\n1 p\n"]], ["block_10", ["[A] \n[A10\n\u2014\n\u2014\u2014\u2014\u2014-\u20141\n+ kc[A]0t\n(2.3.4)\n"]], ["block_11", [{"image_1": "66_1.png", "coords": [48, 354, 290, 594], "fig_type": "figure"}]], ["block_12", ["1 00\n"]], ["block_13", ["80\n"]], ["block_14", ["20\n"]], ["block_15", ["60\n"]], ["block_16", ["0\nI\nI\nl\nI\nI\nI_I_\nI\nI\nI\nI_\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\n"]], ["block_17", ["0\n20\n40\n60\n30\n100\n120\n140\n"]], ["block_18", ["1\u2018 (min)\n"]]], "page_67": [["block_0", [{"image_0": "67_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["these units imply group concentrations expressed as equivalents per kilogram. Mass rather than\nvolume units are often used for concentration, as substantial volume changes may occur during\npolymerization.\n"]], ["block_2", ["a wide portion of the concentration change that occurs during a reaction. Each of these criteria\nseeks to maximize the region of fit, but the former emphasizes maximizing the range of while the\nlatter maximizes the range of p. Both standards tolerate deviations from their respective ideals at\nthe beginning or the end of the experiment. Deviations at the beginning of a process are\nrationalized in terms of experimental uncertainties at the point of mixing or modelistic difficulties\non attainment of stationary-state conditions.\nThe existence ofthese two different standards for success would be only of academic interest ifthe\nanalysis we have discussed applied to experimental results over most of the time range and over most\nextents of reactions as well. Unfortunately, this is not the case in all of the systems that have been\ninvestigated. Ref. [2], for example, shows one particular set of data\u2014adipic acid and diethylene\nglycol at 166\u00b0C, similar reactants as the system in Figure 2.2\u2014analyzed according to two different\nrate laws. This system obeys one rate law between p 0.50 and 0.85 that represents 15% of the\nduration of the experiment, and another rate law between p 0.80 and 0.93, which spans 45% of\nthe reaction time. These would be interpreted differently by the two standards above. This sort of\ndilemma is not unique to the present problem, but arises in many situations where one variable\nundergoes a large percentage of its total change while the other variable undergoes only a small\nfraction of its change. In the present context one way out of the dilemma is to take the view that only\nthe latter stages of the reaction are significant, as it is only beyond, say,p 0.80, that it makes sense to\nconsider the process as one of polymerization. Thus, it is only at large extents of reaction that\npolymeric products are formed and, hence, the kinetics of polymerization should be based on a\ndescription of this part of the process. This viewpoint intentionally focuses attention on a relatively\nmodest but definite range of p values. Since the reaction is necessarily slow as the number of\nunreacted functional groups decreases, this position tends to maximize the time over which the\nrate law fits the data. Calculation from the ordinate of Figure 2.2 shows that the data presented there\nrepresent only about the last 20% of the range of p values. The zero of the timescale has thus been\nshifted to pick up the analysis of the reaction at this point.\nWe commented above that the deviations at the beginning or the end of kinetic experiments can\nbe rationalized, although the different schools of thought would disagree as to what constitutes\n\u201cbeginning\u201d and \u201cend.\u201d Now that we have settled upon the polymer range, let us consider\nspecifically why deviations occur from a simple second-order kinetic analysis in the case of\ncatalyzed polymerizations. At the beginning of the experiment, say, up to p m 0.5, the concentra-\ntions of A and B groups change dramatically, even though the number average degree of\npolymerization has only changed from monomer to dimer. By ordinary polymeric standards, we\nare still dealing with a low molecular weight system that might be regarded as the solvent medium\nfor the formation of polymer. During this transformation, however, 50% of the very\u2014polar A groups\nand 50% of the very-polar B groups have been converted to the less\u2014polar ab groups. Thus, a\nsignificant change in the polarity of the polymerization medium occurs during the first half of the\nchange in p, even though an insignificant amount of true polymer has formed. In view of the role of\n"]], ["block_3", ["Although the results presented in Figure 2.2 appear to verify the predictions of Equation 2.3.6, this\nverification is not free from controversy. This controversy arises because various workers in this\nfield employ different criteria in evaluating the success of the relationships we have presented in\nfitting experimental polymerization data. One school of thought maintains that an adequate kinetic\ndescription of a process must apply to the data over a large part of the time of the experiment.\nA second point of view maintains that a rate law correctly describes a process when it applies over\n"]], ["block_4", ["2.3.2\nHow Should Experimental Data Be Compared with Theoretical Rate Laws?\n"]], ["block_5", ["52\nStep-Growth Polymerization\n"]]], "page_68": [["block_0", [{"image_0": "68_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "68_1.png", "coords": [17, 214, 270, 271], "fig_type": "molecule"}]], ["block_2", [{"image_2": "68_2.png", "coords": [34, 466, 155, 518], "fig_type": "figure"}]], ["block_3", ["on the assumption that A represents carboxyl groups. In this case, ku is the rate constant for the\nuncatalyzed reaction. This differential rate law is the equivalent of Equation 2.3.1 for the catalyzed\nreaction. Equation 2.3.7 is readily integrated when [A]0 = [B]0, in which case it becomes\n"]], ["block_4", ["Until now we have been discussing the kinetics of catalyzed reactions. Losses due to volatility and\nside reactions also raise questions as to the validity of assuming a constant concentration of\ncatalyst. Of course, one way of avoiding this issue is to omit an outside catalyst; reactions\ninvolving carboxylic acids can be catalyzed by these compounds themselves. Experiments\nconducted under these conditions are informative in their own right and not merely as a means\nof eliminating errors in the catalyzed case. As noted in connection with the discussion of Reaction\n(2G), the intermediate is stabilized by coordination with a proton from the catalyst. In the case of\nautOprotolysis by the carboxylic intermediate,\n"]], ["block_5", ["as this intermediate involves an additional equivalent of acid functional groups, the rate law for the\ndisappearance of A groups becomes\n"]], ["block_6", ["In order to achieve large p\u2019s and high molecular weights, it is essential that these equilibria be\nshifted to the right by removing the by-product molecule, water in these reactions. This may be\naccomplished by heating, imposing a partial vacuum, or purging with an inert gas, or some\ncombination of the three. These treatments also open up the possibility of reactant loss due to\nvolatility, which may accumulate to a significant source of error for reactions that are carried out to\nlarge values of p.\n"]], ["block_7", ["ionic ofmight \nwell in\ufb02uence ofreaction.\nAt the other end of the reaction, deviations from idealized rate laws are attributed to secondary\nreactions alcohols, amines through decarboxylation, dehydration,\nand deamination, respectively. The step-growth polymers that have been most widely studied are\nsimple condensation suchas polyesters and polyamides. Although we shall take up these\nclasses of polyamides\u2014specifically in Section 2.5 and Section 2.6,\nrespectively, it is appropriate to mention here that these are typically equilibrium reactions.\n"]], ["block_8", ["2.3.3\nUncatalyzed Step-Growth Reactions\n"]], ["block_9", ["and\n"]], ["block_10", ["Kinetics \n53\n"]], ["block_11", [{"image_3": "68_3.png", "coords": [40, 157, 251, 204], "fig_type": "molecule"}]], ["block_12", [{"image_4": "68_4.png", "coords": [43, 227, 256, 256], "fig_type": "molecule"}]], ["block_13", ["amJLOH + HO/Rzm \ufb01 ,. Rio/Hz + H20\n(2H)\n"]], ["block_14", ["d A\n__[_] __k[A]2[3]\n(2.3.7)\ndt\n"]], ["block_15", ["\u2014[__A]3\nd[A]\u2014\n(2.3.8)\n"]], ["block_16", ["4.\nOH\nR1\u2014l\u2014OH\n_ \n"]], ["block_17", ["o\no\n2\n"]], ["block_18", ["0...\nR2\n"]], ["block_19", [{"image_5": "68_5.png", "coords": [101, 475, 145, 513], "fig_type": "molecule"}]]], "page_69": [["block_0", [{"image_0": "69_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "69_1.png", "coords": [22, 27, 300, 295], "fig_type": "figure"}]], ["block_2", ["6i\n- 95.9 I\nO\nO\n0.1\nO\no\nc\n\u2018-\n.9\nX\n_\n*6\n"]], ["block_3", ["This integrates to\n"]], ["block_4", ["which shows that Nn increases more gradually with t than in the catalyzed case, all other things\nbeing equal.\nFigure 2.3 shows data for the uncatalyzed polymerization of adipic acid and 1,10~decamethylene\nglycol at 161\u00b0C plotted according to Equation 2.3.10. The various provisos of the catalyzed\ncase apply here also, so it continues to be appropriate to consider only the final stages of the\nconversion to polymer. From these results, ku is about 4.3 x 10\u20183 kg2 eq\u20182 min\u20181 at 161\u00b0C.\nWe conclude this section with a numerical example that serves to review and compare some of\nthe important relationships we have considered.\n"]], ["block_5", ["Thus for the uncatalyzed reaction, we have the following:\n"]], ["block_6", ["o.\nE\nI\nC\nCD\n3.:\n2\nCD\n0.\n\u2014 92.9\n2 ..\n"]], ["block_7", ["Assuming that 117C210\"1 kg eq\u2018l min\u2014l, ku=10_3 kg2 eq\u2018l min\"1, and [A]0:: 10 eq kg\u2018l,\ncalculate the time required for p to reach values 0.2, 0.4, 0.6, and so on, for both catalyzed and\n"]], ["block_8", ["Figure 2.3\nPlot of 1/(1\u201419)2 (left ordinate) and p (right ordinate) versus time for an uncatalyzed esterifi\u2014\ncation. (Data from Hamann, S.D., Solomon, DH, and Swift, J.D., J. Macromol. Sci. Chem, A2, 153, 1968.)\n"]], ["block_9", ["and this shows that a plot of (1\u2014,1'))_2 increases linearly with t.\n3.\nSince [A]/[A]O UN\u201c, Equation 2.3.10 becomes\n"]], ["block_10", ["Example 2.2\n"]], ["block_11", ["54\nStep-Growth Polymerization\n"]], ["block_12", ["1.\nThe rate law is third order.\nSince [A]/[A]O l\ufb02p, Equation 2.3.9 may be rewritten as\n"]], ["block_13", ["8 i-\n\u2014 96.5\n"]], ["block_14", ["N\ufb01 1 + 2ku[A]021\n(2.3.11)\n"]], ["block_15", ["(1 ~10)2\n"]], ["block_16", ["1\n1\n1\n1\nl\nJ\u2014 _L\n0\n200\n400\n600\n800\n1000\n1200\n1400\nr(min)\n"]], ["block_17", ["1\n1\n"]], ["block_18", ["1\n"]], ["block_19", ["\u2014[\nl\u2014\n\u2014i_\ni\nI\nl\nl\n"]], ["block_20", ["= 1 + 2ku[A]022\u2018\n(2.3.10)\n"]], ["block_21", ["\u2014 82.0\n"]]], "page_70": [["block_0", [{"image_0": "70_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "70_1.png", "coords": [0, 236, 339, 382], "fig_type": "figure"}]], ["block_2", ["p\n[A]/[A]0\nNn\nTime (min) catalyzed\nTime (min) uncatalyzed\n"]], ["block_3", ["1.\nEquation 2.2.4:\nNn 1/(1\u2014p).\n2.\nEquation 2.2.2:\n[A]/[A]0 l\u2014p.\n3.\nEquation 2.3.6:\nr=(1v,,\u20141)/k,[A]0 Nn\u2014l if catalyzed, since 10\u201c1(10) =1.\n4.\nEquation 2.3.11:\n2\u2018: (Nn2\u20141)/2 ku[A]02 (Nn2\u20141)(5) if uncatalyzed, since\n2(10\u20143) (10)2= 0.2.\n"]], ["block_4", ["uncatalyzed 2.3.2 and Equation 2.3.9, respectively, \n"]], ["block_5", ["In this section we turn our attention to two other questions raised in Section 2.2, namely, how do\nthe molecules distribute themselves among the different possible species, and how does this\ndistribution vary with the extent of reaction? Since a range of species is present at each stage of\npolymerization, it is apparent that a statistical answer is required for these questions. This time, our\nanswer begins, \u201cOn the average. . . .\u201d\nWe shall continue basing our discussion on the step-growth polymerization of the hypothetical\nmonomer AB. In Section 2.7, we shall take a second look at this problem for the case of unequal\nconcentrations of A and B groups. For now, however, we assure this equality by considering a\nmonomer that contains one group of each type. In a previous discussion of the polymer formed\nfrom this monomer, we noted that remnants of the original functional groups are still recognizable,\nalthough modified, along the backbone of the polymer chain. This state of affairs is emphasized by\nthe notation Ababa. . . abaB in which the as and b\u2019s of the ab linkages are groups of atoms carried\nover the initial A and B reactive groups. In this type of polymer molecule, then, there are i\u2014l a\u2019s\n"]], ["block_6", ["Using these relationships, the accompanying table is developed.\n"]], ["block_7", ["0.2\n0.8\n1.25\n0.25\n2.8\n0.4\n0.6\n1.67\n0.67\n8.9\n0.6\n0.4\n2.50\n1.5\n26\n0.8\n0.2\n5.00\n4.0\n120\n0.9\n0.1\n10.0\n9.0\n500\n0.95\n0.05\n20.0\n19\n2.0 x 103\n0.99\n0.01\n100\n99\n5.0 x 104\n0.992\n0.008\n120\n119\n7.2 x 104\n0.998\n0.002\n500\n499\n1.3 x 106\n"]], ["block_8", ["Since for specific values of \nsummarize the following relationships:\n"]], ["block_9", ["to the entire ofof\nand the fraction of unreacted A groups as a function of time.\n"]], ["block_10", ["A graphical comparison of the trends appearing here is presented in Figure 2.4. The importance\nof the catalyst is readily apparent in this hypothetical but not atypical system: To reach Nl1l 5\nrequires 4 min in the catalyzed case and 120 min without any catalyst, assuming that the\nappropriate rate law describes the entire reaction in each case.\n"]], ["block_11", ["Distribution of Molecular Sizes\n55\n"]], ["block_12", ["The question posed in Section 2.2\u2014how long will it take to reach a certain extent of reaction or\ndegree of polymerization?\u2014is now answered. As is often the case, the answer begins, \u201cIt all\ndepends. . .\n"]], ["block_13", ["Solution\n"]], ["block_14", ["2.4\nDistribution of Molecular Sizes\n"]]], "page_71": [["block_0", [{"image_0": "71_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "71_1.png", "coords": [22, 527, 180, 561], "fig_type": "molecule"}]], ["block_2", ["We now turn to the question of evaluating the fraction of i\u2014mers in a mixture as a function of\np. The fraction of molecules of a particular type in a population is just another way of describing\nthe probability of such a molecule. Hence our restated objective is to find the probability of an\ni\u2014mer in terms of p; we symbolize this quantity as the mole fraction x,(p). Since the i-mer consists\nof i\u2014l a\u2019s and 1 A, its probability is the same as the probability of finding i\u2014l a\u2019s and 1 A in the\nsame molecule. Recalling from Chapter 1 how such probabilities are compounded, we write\n"]], ["block_3", ["where pa and pA are the probabilities of individual a and A groups, respectively, and pa=p and\npA=1\u2014p. Equation 2.4.1 is known as the most probable distribution, and it arises in several\ncircumstances in polymer science, in particular free radical polymerization (see Chapter 3). The\nprobability of an i-mer can be converted to the number of i-mer molecules in the reaction mixture,\nm, by multiplying by the total number of molecules m in the mixture after the reaction has occurred\nto the extent p:\n"]], ["block_4", ["and 1 A if the degree of polymerization of the polymer is i. The a\u2019s differ from the A\u2019s precisely\nin that the former have undergone reaction whereas the latter have not. At any point during the\npolymerization reaction the fraction of the initial number of A groups that have reacted to become\na\u2019s is given by p, and the fraction that remains as A\u2019s is given by 1\u20141:). In these expressions p is\nthe same extent of reaction defined by Equation 2.2.3.\n"]], ["block_5", ["Figure 2.4\nComparison of catalyzed (solid lines) and uncatalyzed (dashed lines) polymerizations using\nresults calculated in Example 2.2. Here l\u2014p (left ordinate) and Nn (right ordinate) are plotted versus time.\n"]], ["block_6", ["2.4.1\nMole Fractions of Species\n"]], ["block_7", ["56\nStep-Growth Polymerization\n"]], ["block_8", ["2\n<1\n.3\n- 30\ni\nN\"\n"]], ["block_9", [{"image_2": "71_2.png", "coords": [45, 31, 269, 295], "fig_type": "figure"}]], ["block_10", ["x:(p) \ufb01lm pi\"(1 1))\n(2.4.1)\n"]], ["block_11", ["n,- m pi\u2014\u2018(l p)\n(2.4.2)\n"]], ["block_12", ["\\ \\ a\n\u201c20\n\u201ca ____. -...,_\n"]], ["block_13", ["\u2014 \u2014 \u2014 : Uncatalyzed\n\u2014: Catalyzed\n-40\n"]], ["block_14", ["150\n20\n250\n"]], ["block_15", ["_l\nI\n"]], ["block_16", ["- 50\n"]], ["block_17", ["- 10\n"]]], "page_72": [["block_0", [{"image_0": "72_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "72_1.png", "coords": [16, 421, 279, 641], "fig_type": "figure"}]], ["block_2", ["where m0 is present initially; m0: [A]0 \n"]], ["block_3", ["elect itanswer \nposed earlier.\nFigure 2.5 is a plot of the ratio fig/m versus1' for several values of p. Several features are apparent\nfrom Figure of\npresent:\n"]], ["block_4", ["Note that \n"]], ["block_5", ["Incorporating \n"]], ["block_6", ["result \n"]], ["block_7", ["seen before, contains\n"]], ["block_8", ["1.\nOn a number basis, the fraction of molecules always decreases with increasing regardless of\nthe value of p. The distributions in Table 2.1 are unrealistic in this regard.\n2.\nAs p increases, the proportion of molecules with smaller 1' values decreases and the proportion\nwith larger i values increases.\n"]], ["block_9", ["each multiplied by appropriate factor by Equation \n"]], ["block_10", ["Note that the upper limit of the second summation has been shifted from to 00 for mathematical\nreasons, namely that the answer is simple and known (see Appendix). The change is of little\npractical significance, since Equation 2.4.1 drops off for very large values of i. In particular, the\nresult derived in the Appendix is\n"]], ["block_11", ["003\nIlllll\u2014l'lIllllllllllllllllllll\n"]], ["block_12", ["The number average degree of polymerization for these mixtures is easily obtained by recalling\nthe de\ufb01nition of this average from Section 1.7. It is given by the sum of all possible i values, with\n"]], ["block_13", ["3.\nThe combination of effects described in item (2) tends to \ufb02atten the curves as p increases, but\nnot to the extent that the effect of item (1) disappears.\n"]], ["block_14", ["Distribution \n57\n"]], ["block_15", ["032\n"]], ["block_16", ["031\n"]], ["block_17", ["Figure 2.5\nMole fraction of i\u2014mer as a function of i for several values of p.\n"]], ["block_18", [{"image_2": "72_2.png", "coords": [40, 322, 202, 354], "fig_type": "molecule"}]], ["block_19", ["m p\u201c'(1 P)2m0\n(2.4.3)\n"]], ["block_20", ["mo\n00\n.\nNn Z 3x10?) Z arr\u2014la p)\n(2.4.4)\ni=1\ni=l\n"]], ["block_21", ["o\n100\nf\n200\n300\n"]], ["block_22", ["p=095\n"]], ["block_23", ["JllllllIlIIIIIIIIIIIIIIIIIJI\n"]]], "page_73": [["block_0", [{"image_0": "73_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "73_1.png", "coords": [29, 50, 138, 87], "fig_type": "molecule"}]], ["block_2", ["The weight fraction of i-mers is plotted as a function of i in Figure 2.6 for several large values of\np. Inspection of Figure 2.6 and comparison with Figure 2.5 reveals the following:\n"]], ["block_3", ["Next we turn our attention to the distribution of the molecules by weight among the various\nspecies. This will lead directly to the determination of the weight average molecular weight and the\nratio MW/Mn.\nWe begin by recognizing that the weight fraction w; of i\u2014mers in the polymer mixture at any\nvalue of p equals the ratio of the mass of i-mer in the mixture divided by the mass of the total\nmixture. The former is given by the product i rig-M0, where M0 is the molecular weight of the repeat\nunit; the latter is given by mOMO. Therefore we write\n"]], ["block_4", ["0.02\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\ni\n|\nI T\n|\n[TI\nI\u2014l\u2014I\nI\u2014I\u2014I\n'\n1\np 0.95\n"]], ["block_5", ["Of course, this is the same result that was obtained more simply in Equation 2.2.4. The earlier\nresult, however, was based on purely stoichiometric considerations and not on the detailed\ndistribution as is the present result.\n"]], ["block_6", ["i\n0.01 \n"]], ["block_7", ["into which Equation 2.4.3 may be substituted to give\n"]], ["block_8", ["Figure 2.6\nWeight fraction of i-mer as a function of i for several values of p.\n"]], ["block_9", ["Simplification of the summation in Equation 2.4.4 thus yields\n"]], ["block_10", ["2.4.2\nWeight Fractions of Species\n"]], ["block_11", ["58\nStep-Growth Polymerization\n"]], ["block_12", ["-\n1\nT\n\u201cl\nW;\n'\n-\n-\ni\n"]], ["block_13", [{"image_2": "73_2.png", "coords": [42, 431, 241, 640], "fig_type": "figure"}]], ["block_14", ["0\n"]], ["block_15", ["NI] \n(2.4.5)\n"]], ["block_16", ["w.- 53\n(2.4.6)\nm0\n"]], ["block_17", ["w.- ipirla p)2\n(2.4.7)\n"]], ["block_18", ["gm.\u2014 :(1\u2014p)2\n"]], ["block_19", ["0\n100\nI\nJ300\n"]], ["block_20", ["1\ni\np = 0.97\n_\ni\nr\np 0.99\n"]], ["block_21", ["-\n"]], ["block_22", ["I\nI\na\na\nI\nI\nI\nI\n|\ni\n-\n.\nI\n"]], ["block_23", ["p 0.995\n1\n"]]], "page_74": [["block_0", [{"image_0": "74_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "74_1.png", "coords": [25, 184, 221, 259], "fig_type": "figure"}]], ["block_2", [{"image_2": "74_2.png", "coords": [27, 285, 143, 324], "fig_type": "molecule"}]], ["block_3", [{"image_3": "74_3.png", "coords": [34, 496, 288, 681], "fig_type": "figure"}]], ["block_4", ["P\nNn\nNW\nNw/Nn\n"]], ["block_5", ["The weight by averaging the contributions \n"]], ["block_6", ["where the upper limit on i has been extended to infinity as before. The new summation that we\nneed is also evaluated in Appendix:\n"]], ["block_7", ["Table 2.2\nValues of N\u201c, NW, and NW/Nn for Various Large\nValues of p\n"]], ["block_8", ["which is the desired result.\nWe saw in Chapter 1 that the ratio Mw/Mn, or polydispersity index, is widely used in polymer\nchemistry as a measure of the width of a molecular weight distribution. If the effect of chain ends is\ndisregarded, this ratio is the same as the corresponding ratio of i values:\n"]], ["block_9", ["where the ratio of Equation 2.4.9 to Equation 2.2.4 has been used. Table 2.2 lists values of NW, N\u201c,\n"]], ["block_10", ["Using this in Equation 2.4.8 gives\n"]], ["block_11", ["0.90\n10.0\n19.0\n1.90\n0.92\n12.5\n24.0\n1.92\n0.94\n16.7\n32.3\n1.94\n0.96\n25.0\n49.0\n1.96\n0.98\n50.0\n99.0\n1.98\n0.990\n100\n199\n1 .990\n0.992\n125\n249\n1.992\n0.994\n167\n332\n1.994\n0.996\n250\n499\n1.996\n0.998\n500\n999\n1.998\n"]], ["block_12", ["1,\nAt any p, very small and very large values of i contribute a lower weight fraction to the\nmixture than do intermediate values of i. This arises because of the product in,- in Equation\n2.4.6: n; is large for monomers, in which case i is low, and then n,- decreases as i increases.\nAt intermediate values of i, w,- goes through a maximum.\n2,\nAs p increases, the maximum in the curves shifts to larger i values and the tail of the curve\nextends to higher values of i.\n"]], ["block_13", ["Distribution \n59\n"]], ["block_14", ["1' values using in the averaging procedure:\n"]], ["block_15", ["3.\nThe effect in item (2) is not merely a matter of shifting curves toward higher i values as p\nincreases, but re\ufb02ects a distinct broadening of the distribution of i values as p increases.\n"]], ["block_16", ["and NW/Nn for a range of high p values. Note that NW/Nn \u2014> 2 as p \u2014+ 1; this is a characteristic result\n"]], ["block_17", [{"image_4": "74_4.png", "coords": [42, 529, 265, 648], "fig_type": "table"}]], ["block_18", [{"image_5": "74_5.png", "coords": [43, 187, 203, 255], "fig_type": "molecule"}]], ["block_19", ["mo\n00\nZiw;\n2310910 -19)2\n_ \n{:1\nNw\n(2.4.8)\n00\nin\u00bb.\nZipi\u2014lu \u2014p)2\n"]], ["block_20", ["MW\nNW\n\u2014\u2014 = \u2014 = 1 +\n2.4.10\nM,\nNn\n\u20180\n(\n)\n"]], ["block_21", ["_1+p\nNW\u2014 l\u2014p\n(2.4.9)\n"]], ["block_22", ["0\u00b0.\n,-_\n1+1)\n:12? \n(lvp)\ni=1\n"]], ["block_23", ["121\ni=1\n"]]], "page_75": [["block_0", [{"image_0": "75_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["We have not attempted to indicate the conditions of temperature, catalyst, solvent, and so on, for\nthese various reactions. For this type of information, references that deal specifically with synthetic\npolymer chemistry should be consulted. In the next few paragraphs we shall comment on\nthe various routes to polyester formation in the order summarized above and followed in Table\n2.3. The studies summarized in Figure 2.2 and Figure 2.3 are examples of Reaction 2 in Table 2.3.\n"]], ["block_2", ["of the most probable distribution. In light of Equation 1.7.16, the standard deviation of the\nmolecular distribution is equal to Mn for the polymer sample produced by this polymerization.\nIn a manner of speaking, the molecular weight distribution is as wide as the average is high. The\nbroadening of the distribution with increasing p is dramatically shown by comparing the values\nin Table 2.2 with the situation at a low p value, say p205. At p205, Nn=2, Nw=3, and\nNW/Nn 1.5.\nSince Equation 2.3.5 and Equation 2.3.11 give p as a function of time for the catalyzed and\nuncatalyzed polymerizations, respectively, the distributions discussed in the last few paragraphs\ncan also be expressed with time as the independent variable instead of p.\nThe results we have obtained on the basis of the hypothetical monomer AB are also applicable\nto polymerizations between monomers of the AA and BB type, as long as the condition [A] [B]\nis maintained. In Section 2.7, we shall extend the arguments of this section to conditions in which\n[A] 79 [B]. In the meanwhile, we interrupt this line of reasoning by considering a few particular\ncondensation polymers as examples of step-growth systems. The actual systems we discuss will\nserve both to verify and reveal the limitations of the concepts we have been discussing. In addition,\nthey point out some of the topics that still need clarification. We anticipate some of the latter points\nby noting the following:\n"]], ["block_3", ["Polyesters and polyamides are two of the most-studied step-growth polymers, as well as\nsubstances of great commercial importance. We shall consider polyesters in Section 2.5 and\npolyamides in Section 2.6.\n"]], ["block_4", ["The preceding discussions of the kinetics and molecular weight distributions in the step-growth\npolymerizations of AB monomers are exemplified by esterification reactions between such\nmonomers as glycolic acid and w\u2014hydroxydecanoic acid. Therefore one method of polyester\nsynthesis is the following:\n"]], ["block_5", ["Several other chemical reactions are also widely used for the synthesis of these polymers. This\nlist enumerates some of the possibilities, and Table 2.3 illustrates these reactions by schematic\nchemical equations.\n"]], ["block_6", ["Esteri\ufb01cation of a diacid and a diol\nEster interchange with alcohol\nEster interchange with ester\nEsterification of acid chlorides\nLactone polymerization\n9\u201899?!\u201d\n"]], ["block_7", ["2.5\nPolyesters\n"]], ["block_8", ["1.\nWhen [A] 7\u00e9 [B], both ends of the growing chain tend to be terminated by the group that is\npresent in excess. Subsequent reaction of such a molecule involves reaction with the limiting\ngroup. The effect is a decrease in the maximum attainable degree of polymerization.\n2.\nWhen a monofunctional reactant is present\u2014-one containing a single A or B group\u2014the effect\nis also clearly a decrease in the average degree of polymerization. It is precisely because this\ntype of reactant can only react once that it is sometimes introduced into polymer formulations,\nthereby eliminating the possibility of long-term combination of chain ends, and/or restricting\nthe average molecular weight.\n"]], ["block_9", ["60\nStep-Growth Polymerization\n"]], ["block_10", ["1.\nEsterification of a hydroxycarboxylic acid\n"]]], "page_76": [["block_0", [{"image_0": "76_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["While up to now we have emphasized bifunctional reactants, both monofunctional compounds and\nmonomers with functionality greater than 2 are present in some polymerization processes, either\nintentionally or adventitiously. The effect of the monofunctional reactant is clearly to limit chain\ngrowth. As noted above, a functionality greater than 2 results in branching. A type of polyester that\nincludes mono-, di-, and trifunctional monomers is the so-called alkyd resin. A typical example is\nbased on the polymerization of phthalic acid (or anhydride), glycerol, and an unsaturated mono-\ncarboxylic acid. The following suggests the structure of a portion of such a polyester:\n0 \n0 \n\u201d\u20194.\nHo/Y\\0H + H0\n+ H0\nm \u2014..\nO\n2.J\nOH\nI\ng/\\O(\\O \n(\n)\n0\n\ufb02\n"]], ["block_2", ["Table 2.3\nSome Schematic Reactions for the Formation of Polyesters\n"]], ["block_3", ["2. Esterification \n"]], ["block_4", ["6. Lactone polymerization:\n"]], ["block_5", ["4. Ester interchange with ester (\u201ctransesterification\u201d):\n"]], ["block_6", ["3. Ester \n"]], ["block_7", ["5. Esterification of acid chlorides (Schotten-Baumann reaction):\n0 \u00b0\n9 it \nR1\n1\nn HO/\n\\OH\n+\nCIJLRELQ =\n{0/\n\\0\nHrs/\u201c9n +\n2n HCI\n"]], ["block_8", ["Polyesters\n51\n"]], ["block_9", ["1. Esterification \n"]], ["block_10", [{"image_1": "76_1.png", "coords": [40, 588, 255, 643], "fig_type": "molecule"}]], ["block_11", [{"image_2": "76_2.png", "coords": [45, 435, 260, 488], "fig_type": "molecule"}]], ["block_12", [{"image_3": "76_3.png", "coords": [71, 89, 324, 138], "fig_type": "figure"}]], ["block_13", [{"image_4": "76_4.png", "coords": [76, 231, 328, 281], "fig_type": "molecule"}]], ["block_14", [{"image_5": "76_5.png", "coords": [81, 442, 245, 473], "fig_type": "molecule"}]], ["block_15", [{"image_6": "76_6.png", "coords": [87, 93, 321, 142], "fig_type": "molecule"}]], ["block_16", [{"image_7": "76_7.png", "coords": [87, 302, 351, 354], "fig_type": "molecule"}]], ["block_17", [{"image_8": "76_8.png", "coords": [89, 303, 209, 344], "fig_type": "molecule"}]], ["block_18", [{"image_9": "76_9.png", "coords": [95, 372, 274, 412], "fig_type": "molecule"}]], ["block_19", ["n HO/ 1\\OH\n+\nn HOJLHQJLOH\n{0/ \\D/ILHQ/K)\u201d 20 \n"]], ["block_20", ["0\nO\nR1\non2\n+\nFia\u2014OH\nR1/L|\\OR3\n+\nR2_0H\n"]], ["block_21", ["R1/U\\OR2\n+\nR3/U\\0Fi4\n\u2014_'\"\nR1/U\\0Fi4\nRa/LkOFiz\n"]], ["block_22", [{"image_10": "76_10.png", "coords": [96, 167, 326, 204], "fig_type": "molecule"}]], ["block_23", ["O\nn\n"]], ["block_24", ["Ft\n0\n"]], ["block_25", ["O\n0\nO\n"]], ["block_26", ["H\no\n0\nH1\n0\no\n"]], ["block_27", [{"image_11": "76_11.png", "coords": [144, 589, 368, 672], "fig_type": "molecule"}]], ["block_28", ["0\nO\n|\n"]], ["block_29", [{"image_12": "76_12.png", "coords": [144, 434, 235, 483], "fig_type": "molecule"}]], ["block_30", [{"image_13": "76_13.png", "coords": [147, 449, 225, 476], "fig_type": "molecule"}]], ["block_31", [{"image_14": "76_14.png", "coords": [150, 235, 314, 273], "fig_type": "molecule"}]], ["block_32", [{"image_15": "76_15.png", "coords": [194, 304, 339, 348], "fig_type": "molecule"}]], ["block_33", [{"image_16": "76_16.png", "coords": [202, 165, 415, 206], "fig_type": "molecule"}]], ["block_34", [{"image_17": "76_17.png", "coords": [222, 368, 403, 420], "fig_type": "molecule"}]], ["block_35", [{"image_18": "76_18.png", "coords": [235, 378, 400, 405], "fig_type": "molecule"}]]], "page_77": [["block_0", [{"image_0": "77_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["It has been hypothesized that cross\u2014linked polymers would have better mechanical properties if\ninterchain bridges were located at the ends rather than the center of chains. To test this, low\n"]], ["block_2", ["The ethylene glycol liberated by Reaction (2L) is removed by lowering the pressure or purging\nwith an inert gas. Because the ethylene glycol produced by Reaction (2L) is removed, proper\nstoichiometry is assured by proceeding via the intermediate bis(2\u2014hydroxyethyl) terephthalate;\notherwise the excess glycol used initially would have a deleterious effect on the degree of\npolymerization. Poly(ethylene terephthalate) is more familiar by some of its trade names: Mylar\nas a film and Dacron, Kodel, or Terylene as fibers; it is also known by the acronym PET.\nEster interchange reactions like that shown in Reaction 4 in Table 2.3 (transesterification) can\nbe carried out on polyesters themselves to produce a scrambling between the two polymers.\nStudies of this sort between high and low molecular weight prepolymers result in a single polymer\nwith the same molecular weight distribution as would have been obtained from a similarly\nconstituted diol\u2014diacid mixture by direct polymerization. This is true when the time-catalyst\nconditions allow the randomization to reach equilibrium. If the two prepolymers are polyesters\nformed from different monomers, the product of the ester interchange reaction will be a copolymer\nof some sort. If the reaction conditions favor esterification, the two chains will merely link together\nand a block copolymer results. If the conditions favor the ester interchange reaction, then a\nscrambled 00polymer molecule results. These possibilities underscore the idea that the derivations\nof the preceding sections are based on complete equilibrium among all molecular species present\nduring the condensation reaction.\n"]], ["block_3", ["The presence of the unsaturated substituent along this polyester backbone gives this polymer cross-\nlinking possibilities through a secondary reaction of the double bond. These polymers are used in\npaints, varnishes, and lacquers, where the ultimate cross-linked product results from the oxidation of\nthe double bond as the coating cures. A cross-linked polyester could also result from Reaction (2.1)\nwithout the unsaturated carboxylic acid, but the latter would produce a gel in which the entire\nreaction mass solidified, and is therefore not as well suited to coating applications as a polymer that\ncross-links upon \u201cdrying.\u201d\nMany of the reactions listed at the beginning of this section are acid catalyzed, although a number\nof basic catalysts are also employed. Esterifications are equilibrium reactions, and often carried out\nat elevated temperatures for favorable rate and equilibrium constants and to shift the equilibrium in\nfavor of the polymer by volatilization of the by-product molecules. An undesired feature of higher\npolymerization temperatures is the increased possibility of side reactions, such as the dehydration of\nthe diol or the pyrolysis of the ester. Basic catalysts tend to produce fewer undesirable side reactions.\nEster exchange reactions are valuable, since, say, methyl esters of dicarboxylic acids are often\nmore soluble and easier to purify than the diacid itself. The methanol by-product is easily removed\nby evaporation. Poly(ethylene terephthalate) is an example of a polymer prepared by double\napplication of Reaction 4 in Table 2.3. The first stage of the reaction is conducted at temperatures\nbelow 200\u00b0C and involves the interchange of dimethyl terephthalate with ethylene glycol.\n"]], ["block_4", ["The rate of this reaction is increased by using excess ethylene glycol, and removal of the methanol\nis assured by the elevated temperature. Polymer is produced in the second stage after the\ntemperature is raised above the melting point of the polymer, about 260\u00b0C.\n"]], ["block_5", ["Example 2.3\n"]], ["block_6", [{"image_1": "77_1.png", "coords": [35, 278, 239, 317], "fig_type": "molecule"}]], ["block_7", ["62\nStep-Growth Polymerization\n"]], ["block_8", [{"image_2": "77_2.png", "coords": [39, 373, 368, 417], "fig_type": "molecule"}]], ["block_9", ["Me\u2014O\nO\u2014Me\n"]], ["block_10", ["0\no\no\no\nn HO_\\_0>_\u00a9_n\n"]], ["block_11", ["0\nO\nO\nO\nW\n+\n2 \n+ 2 MeOH\n"]], ["block_12", [{"image_3": "77_3.png", "coords": [190, 274, 428, 321], "fig_type": "molecule"}]], ["block_13", [{"image_4": "77_4.png", "coords": [296, 375, 439, 411], "fig_type": "molecule"}]], ["block_14", ["(2K)\n"]]], "page_78": [["block_0", [{"image_0": "78_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "78_1.png", "coords": [15, 98, 411, 264], "fig_type": "figure"}]], ["block_2", ["-\u2014_\u2014\n"]], ["block_3", ["On the basis of these facts, do the following:\n"]], ["block_4", ["2.\n3.\n"]], ["block_5", ["molecular weight polyesters were synthesized from a diol and two different diacids: one saturated\n"]], ["block_6", ["Endene\nCentrene\nStep 1:\n8 h at about 150\u00b0C\u2014200\u00b0C\nMaleic anhydride (mol)\n0\n2.0\nSuccinic anhydride (mol)\n2.0\n0\nDiethylene glycol (mol)\n3.0\n3.0\nStep 2:\nAbout 1/2 h at about 120\u00b0C\u2014130\u00b0C\nMaleic anhydride (mol)\n2.0\n0\nSuccinic anhydride (mol)\n0\n2.0\nCatalyst\n0\n0\nStep 3:\n30% Styrene + catalyst\n16 h at 55\u00b0C -l- 1 h at 110\u00b0C\n"]], ["block_7", ["located \n"]], ["block_8", ["and the other unsaturated. The synthetic procedure was such that the unsaturated acid units were\n"]], ["block_9", ["aspects of the overall experiment are listed below:\n"]], ["block_10", ["Polyesters\n63\n"]], ["block_11", ["Solution\n"]], ["block_12", ["1.\nSince the reaction conditions are mild in step 2 (only 6% as much time allowed as in step 1 at a\nlower temperature) and no catalyst is present, it is unlikely that any significant amount of ester\nscrambling occurs. Isomerization of maleate to fumarate is also known to be insigni\ufb01cant\nunder these conditions.\nThe idealized structures of these molecules are\n"]], ["block_13", ["1.\n"]], ["block_14", [{"image_2": "78_2.png", "coords": [39, 382, 284, 601], "fig_type": "figure"}]], ["block_15", [{"image_3": "78_3.png", "coords": [48, 504, 285, 554], "fig_type": "molecule"}]], ["block_16", [{"image_4": "78_4.png", "coords": [50, 516, 262, 598], "fig_type": "molecule"}]], ["block_17", [{"image_5": "78_5.png", "coords": [51, 413, 282, 499], "fig_type": "figure"}]], ["block_18", ["Comment on the likelihood that the comonomers are segregated as the names of these\npolymers suggest.\nSketch the structure of the average endene and centrene molecules.\nComment on the results in terms of the initial hypothesis.\n"]], ["block_19", ["Endene\n"]], ["block_20", ["A cross-linked product with unsaturation at the chain ends does, indeed, have a higher modulus.\nThis could be of commercial importance and indicates that industrial products might be formed\nby a nonequilibrium process precisely for this sort of reason. A fuller discussion of the factors\nthat contribute to the modulus will be given in Chapter 10 and Chapter 12.\n"]], ["block_21", ["Centrene\n"]], ["block_22", ["O\nO\nO\nO\nl\\\\u2018\nHOJJWOwOA/ONo/VO\n"]], ["block_23", ["O\nO\nO\nO\n\\\nHowowom/ONONO\n"]], ["block_24", [{"image_6": "78_6.png", "coords": [62, 412, 279, 495], "fig_type": "molecule"}]], ["block_25", [{"image_7": "78_7.png", "coords": [65, 418, 271, 458], "fig_type": "molecule"}]], ["block_26", ["HO\n"]], ["block_27", ["Elastic modulus (Pa)\n21,550\n16,500\n"]], ["block_28", [{"image_8": "78_8.png", "coords": [89, 107, 307, 233], "fig_type": "figure"}]], ["block_29", ["O\nO\n0\nO\n"]], ["block_30", ["0\nO\n"]], ["block_31", ["O\nO\n"]], ["block_32", ["O\nO\n"]]], "page_79": [["block_0", [{"image_0": "79_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Acid chlorides are generally more reactive than the parent acids, so polyester fonnation via\nReaction 5 in Table 2.3 can be carried out in solution and at lower temperatures, in contrast with\nthe bulk reactions of the melt as described above. Again, the by-product molecules must be\neliminated either by distillation or precipitation. The method of interfacial condensation, described\nin Section 2.6, can be applied to this type of reaction.\nThe formation of polyesters from the polymerization of lactones (Reaction 6 in Table 2.3) is a\nring\u2014opening reaction that may follow either a step-growth or chain mechanism, depending on\nconditions. For now our only concern is to note that the equilibrium representing this reaction in\nTable 2.3 describes polymerization by the forward reaction and ring formation by the back reaction.\nRings clearly compete with polymers for monomer in all polymerizations. Throughout the chapter\nwe have assumed that all competing side reactions, including ring formation, could be neglected.\n"]], ["block_2", ["Table 2.4\nSome Schematic Reactions for the Formation of Polyamides\n"]], ["block_3", ["64\nStep-Growth Polymerization\n"]], ["block_4", ["Additional synthetic routes that closely resemble polyesters are also available. Several more of\nthese are listed below and are illustrated by schematic reactions in Table 2.4:\n"]], ["block_5", ["The discussion of polyamides parallels that of polyesters in many ways. To begin with, polyamides\nmay be fonned from an AB monomer, in this case amino acids:\n"]], ["block_6", ["2.6\nPolyamides\n"]], ["block_7", ["1.\nAmidation of amino acids\n"]], ["block_8", ["4. Amidation of acid chlorides:\n"]], ["block_9", ["3. Interchange reactions:\n"]], ["block_10", ["1. Amidation of amino acids:\nj:\n..\nO\n"]], ["block_11", ["2. Amidation of diamine and diacid:\n"]], ["block_12", ["5. Lactam polymerization:\n"]], ["block_13", [{"image_1": "79_1.png", "coords": [58, 605, 264, 656], "fig_type": "molecule"}]], ["block_14", [{"image_2": "79_2.png", "coords": [80, 335, 312, 397], "fig_type": "figure"}]], ["block_15", [{"image_3": "79_3.png", "coords": [102, 613, 252, 646], "fig_type": "molecule"}]], ["block_16", [{"image_4": "79_4.png", "coords": [102, 338, 313, 393], "fig_type": "molecule"}]], ["block_17", ["n\nH2N._R\nOH\n\u2014'__\u2018_'\u2014\"\"-\n\u2018(N\u2018R/Ugn\n4..\n\u201dH20\n"]], ["block_18", ["n H2N\u2019R1\u201cNH2 \nn HOJ-LRZJkOH = {HERE/Lag\u201c H20\n"]], ["block_19", [".Fi\n.\nn H2N\n1\"NH2\n4'\nn CIJLRELCI\n\u2014--'--...\u2014\u2014\n{NRLH\nR\ufb01n+\n2n HCI\n"]], ["block_20", [{"image_5": "79_5.png", "coords": [113, 543, 312, 582], "fig_type": "molecule"}]], ["block_21", [{"image_6": "79_6.png", "coords": [119, 477, 394, 523], "fig_type": "molecule"}]], ["block_22", ["R1/LSHR2\n+\nPlain/R4\nmin/R4\n+\nF\u2018s/\u201c\\H\u2019Fb\n"]], ["block_23", [{"image_7": "79_7.png", "coords": [137, 346, 303, 379], "fig_type": "molecule"}]], ["block_24", ["0\nO\nO\nO\n"]], ["block_25", ["O\n0\nj:\n0\n"]], ["block_26", ["O\nO\nO\nO\n"]], ["block_27", [{"image_8": "79_8.png", "coords": [217, 474, 378, 514], "fig_type": "molecule"}]], ["block_28", [{"image_9": "79_9.png", "coords": [224, 411, 420, 452], "fig_type": "molecule"}]], ["block_29", [{"image_10": "79_10.png", "coords": [233, 543, 417, 583], "fig_type": "molecule"}]]], "page_80": [["block_0", [{"image_0": "80_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "80_1.png", "coords": [21, 356, 243, 413], "fig_type": "molecule"}]], ["block_2", ["Reaction (2.N) describes the nylon salt $ nylon equilibrium. Reaction (2.0) and Reaction (2.P)\nshow proton transfer with water between carboxyl and amine groups. Since proton transfer\nequilibria are involved, the self\u2014ionization of water, Reaction (2.Q), must also be included.\nEspecially in the presence of acidic catalysts, Reaction (2R) and Reaction (2.8) are the equilibria\nof the acid-catalyzed intermediate described in general in Reaction (2.G). The main point in\nincluding all of these equilibria is to indicate that the precise concentration of A and B groups in\n"]], ["block_3", ["compounds first number after the name gives the number ofatoms in\nthe diamine, of carbons in the diacid.\nThe diacid\u2014diamine amidation described in Reaction 2 in Table 2.4 has been widely studied in\nthe melt, solution, and state. When equal amounts of two functional groups are present,\nboth the rate laws and weight distributions are given by the treatment of the\npreceding sections. The stoichiometric balance between reactive groups is readily obtained\nby precipitating the 1:1 ammonium salt from ethanol:\n"]], ["block_4", ["We only need to recall the trade name of synthetic polyamides, nylon, to recognize the importance\nof these to prepare them. Recall the System naming these\n"]], ["block_5", ["This compound is sometimes called a nylon salt. The salt :3 polymer equilibrium is more\nfavorable to the production of polymer than in the case of polyesters, so this reaction is often\ncarried out in a sealed tube or autoclave at about 200\u00b0C until a fairly high extent of reaction is\nreached; then the temperature is raised and the water driven off to attain high molecular weight\npolymer. Also in contrast to polyesters, Reaction 1 and Reaction 2 in Table 2.4 can be conducted\nrapidly without an acid catalyst.\nThe process represented by Reaction 2 in Table 2.4 actually entails a number of additional\nequilibrium reactions. Some of the equilibria that have been considered include the following:\n"]], ["block_6", ["a diacid-diamine reaction mixture is a complicated function of the moisture content and the pH, as\nwell as the initial amounts of reactants introduced. Because of the high affinity for water of the\n"]], ["block_7", ["Amidation of a diacid and a diamine\nInterchange reactions\nAmidation of acid chlorides\nLactam polymerization\nP\u2018P\u2018E\u2019JP\n"]], ["block_8", ["Polyamides\n65\n"]], ["block_9", ["r\"JQOH\n+ HZNM\n~\u2014.\u2014\u2014\u2014\u2014-=\nALNJ\u2018\"\n+ H30+\n(Tl-S)\nH\n"]], ["block_10", ["H\u2018JLo'\n+\nHal-5M\n\u201cAw\u2014~\u2014\nFJLNF\n+\nH2O\n(2N)\nH\n"]], ["block_11", ["0\n0\n2.0\nH\u2018JLOH\n+\nH20\n._..__.._a...._\nrpi\u2018H\\O-\n+\nH30+\n"]], ["block_12", ["2 H20 \ufb02\nH30+\n+ \u2018OH\n(2Q)\n"]], ["block_13", ["WNH2\n+\nH20 \nnHs , -0H\n(2.P)\n"]], ["block_14", ["0\nO\nO\nO\nR\nHQN/ 1\\NH2\n+ HO/lLRQ/LkOH\n\u2014\u2014-\u2014\u2014r-\nHera/RKIQHC,\n+ -0J\\R{LL0\u2014\n(2M)\n"]], ["block_15", ["o\no\n"]], ["block_16", ["OH\n0\n"]], ["block_17", ["0\nOH\n(2R)\n"]], ["block_18", [{"image_2": "80_2.png", "coords": [54, 407, 227, 442], "fig_type": "molecule"}]], ["block_19", [{"image_3": "80_3.png", "coords": [62, 221, 187, 256], "fig_type": "molecule"}]], ["block_20", [{"image_4": "80_4.png", "coords": [77, 524, 234, 558], "fig_type": "molecule"}]], ["block_21", [{"image_5": "80_5.png", "coords": [78, 365, 234, 402], "fig_type": "molecule"}]], ["block_22", [{"image_6": "80_6.png", "coords": [113, 218, 373, 264], "fig_type": "molecule"}]], ["block_23", [{"image_7": "80_7.png", "coords": [115, 221, 177, 256], "fig_type": "molecule"}]], ["block_24", [{"image_8": "80_8.png", "coords": [237, 220, 356, 258], "fig_type": "molecule"}]], ["block_25", ["(\n)\n"]]], "page_81": [["block_0", [{"image_0": "81_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "81_1.png", "coords": [21, 399, 191, 635], "fig_type": "figure"}]], ["block_2", ["Figure 2.7\nSketch of an interfacial polymerization with the collapsed polymer film being withdrawn from\nthe surface between the immiscible phases. (From Morgan, P.W. and Kwolek, S.L. J. Chem. Educ, 36, 182,\n1959. With permission.)\n"]], ["block_3", ["various functional groups present, the complete removal of water is impossible: the equilibrium\nmoisture content of molten nylon-6,6 at 290\u00b0C under steam at\n1 atm is 0.15%. Likewise, the\nvarious ionic possibilities mean that at both high and low pH values the concentration of unionized\ncarboxyl or amine groups may be considerably different from the total concentration\u2014without\nregard to state of ionization\u2014of these groups. As usual, upsetting the stoichiometric balance of the\nreactive groups lowers the degree of polymerization attainable. The abundance of high-quality\nnylon products is evidence that these complications have been overcome in practice.\nAmide interchange reactions of the type represented by Reaction 3 in Table 2.4 are known to\noccur more slowly than direct amidation; nevertheless, reactions between high and low molecular\nweight polyamides result in a polymer of intermediate molecular weight. The polymer is initially a\nblock copolymer of the two starting materials, but randomization is eventually attained.\nAs with polyesters, the amidation reaction of acid chlorides may be carried out in solution because\nof the enhanced reactivity of acid chlorides compared with carboxylic acids. A technique known as\ninterfacial polymerization has been employed for the formation of polyamides and other step-growth\npolymers, including polyesters, polyurethanes, and polycarbonates. In this method, the polymeriza-\ntion is carried out at the interface between two immiscible solutions, one of which contains one of the\ndissolved reactants, while the second monomer is dissolved in the other. Figure 2.7 shows a\npolyamide film forming at the interface between layers of an aqueous diamine solution and a solution\nof diacid chloride in an organic solvent. In this form, interfacial polymerization is part of the standard\nrepertoire of chemical demonstrations. It is sometimes called the \u201cnylon rope trick\u201d because of the\nfilament of nylon that can be produced by withdrawing the collapsed film.\nThe amidation of the reactive groups in interfacial polymerization is governed by the rates at\nwhich these groups can diffuse to the interface where the growing polymer is deposited. Accord-\ningly, new reactants add to existing chains rather than interacting to form new chains. This is\n"]], ["block_4", ["66\nStep-Growth Polymerization\n"]], ["block_5", ["4-1\u201c- Collapsed\n"]], ["block_6", ["at interface\n"]]], "page_82": [["block_0", [{"image_0": "82_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "82_1.png", "coords": [34, 227, 154, 268], "fig_type": "molecule"}]], ["block_2", ["By definition of the problem, this ratio cannot exceed unity.\n"]], ["block_3", ["We now turn to one of the problems we have sidestepped until now\u2014the polymerization of\nreactants in which a stoichiometric imbalance exists in the numbers of reactive groups A and B.\nIn earlier sections dealing with the quantitative aspects of step-growth polymerization, we focused\nattention on monomers of the AB type to assure equality of reactive groups. The results obtained\nabove also apply to AA and BB polymerizations, provided that the numbers of reactive groups are\nequal. There are obvious practical difficulties associated with the requirement of stoichiometric\nbalance. Rigorous puri\ufb01cation of monomers is difficult and adds to the cost of the final product.\nThe effective loss of functional groups to side reactions imposes restrictions on the range of\nexperimental conditions at best and is unavoidable at worst. These latter considerations apply even\nin the case of the AB monomer. We have already stated that the effect of the imbalance of A and\nB groups is to lower the eventual degree of polymerization of the product. A quantitative\nassessment of this limitation is what we now seek.\nWe define the problem by assuming that the polymerization involves AA and BB monomers\nand that the B groups are present in excess. We define 12A and V3 to be the numbers of A and B\nfunctional groups, respectively. The number of either of these quantities in the initial reaction\nmixture is indicated by a superscript o; the numbers at various stages of reaction have no\nsuperscript. The stoichiometric imbalance is defined by the ratio r, where\n"]], ["block_4", ["The combination of strong intermolecular forces and high chain-stiffness accounts for the high\nmelting points of polyamides (see Chapter 13). The remarks of this section and Section 2.5\nrepresent only a small fraction of what could be said about these important materials. We have\ncommented on aspects of the polymerization processes and of the polymers themselves that have a\ndirect bearing on the concepts discussed throughout this volume. This material provides an\nexcellent example of the symbiosis between theoretical and application-oriented viewpoints.\nEach stimulates and reinforces the other with new challenges, although it must be conceded that\nmany industrial processes reach a fairly high degree of empirical refinement before the conceptual\nbasis is quantitatively deve10ped.\n"]], ["block_5", ["different elsewhere in it\nis evident that polymer should result from this difference. The HCl\nby-product of the amidation reaction is neutralized by also dissolving an inorganic base in the\naqueous layer The choice of the organic solvent plays a role in\ndetermining \nquality for Since this reaction is carried out at low temperatures, the\ncomplications be kept to a minimum. yield\nincreased between the two solutions by stirring.\nLactam polymerization represented by Reaction 5 in Table 2.4 is another example of a ring-\nopening reaction, the reverse of which is a possible competitor with polymer for reactants. The\nvarious mechanical properties of polyamides may be traced in many instances to the possibility of\nintermolecular between the polymer molecules, and to the relatively stiff chains\nthese substances possess. latter, in turn, may be understood by considering still another\nequilibrium, along the chain backbone:\n"]], ["block_6", ["2.7\nStoichiometric Imbalance\n"]], ["block_7", ["Stoichiometric Imbalance\n67\n"]], ["block_8", ["O\nr\n_=_\n3%\n(2.7.1)\n\u201dB\n"]], ["block_9", ["O\n0\u2014\nMIL...\u00bb =2: .3}s\n(2T)\n"]], ["block_10", ["H\nH\n"]]], "page_83": [["block_0", [{"image_0": "83_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "83_1.png", "coords": [26, 424, 198, 465], "fig_type": "molecule"}]], ["block_2", ["4.\nThe total number of repeat units distributed among these chains is the number of monomer\nmolecules present initially:\n"]], ["block_3", ["As a check that we have done this correctly, note that Equation 2.7.7 reduces to the previously\nestablished Equation 2.4.5 when r 1.\nOne distinction that should be pointed out involves the comparison of Equation 2.2.1 and\nEquation 2.7.7. In the former we considered explicitly the AB monomer, whereas the latter is based\non the polymerization of AA and BB monomers. In both instances Nn is obtained by dividing the\ntotal number of monomer molecules initially present by the total number of chains after the\nreaction has occurred to extent p. Following the same procedure for different reaction mixtures\nresults in a different de\ufb01nition of the repeat unit. In the case of the AB monomer, the repeat unit is\nthe ab entity, which differs from AB by the elimination of the by-product molecule. In the case\nof the AA and BB monomers, the repeat unit in the polymer is the aabb unit, which differs from\nAA + BB by two by\u2014product molecules. Equation 2.2.1 counts the number of ab units in the\npolymer directly. Equation 2.7.7 counts the number of aa + bb units. The number of an + bb units\nis twice the number of aabb units. Rather than attempting to formalize this distinction by\nintroducing more complex notation, we simply point out that application of the formulas of this\nchapter to Specific systems must be accompanied by a re\ufb02ection on the precise meaning of the\ncalculated quantity for the system under consideration.\n"]], ["block_4", [{"image_2": "83_2.png", "coords": [34, 240, 187, 274], "fig_type": "molecule"}]], ["block_5", ["As with other problems with stoichiometry, it is the less-abundant reactant that limits the\nproduct. Accordingly, we define the extent of reaction p to be the fraction of A groups that have\nreacted at any point. Since A and B groups react in a 1:1 proportion, the number of B groups that\nhave reacted when the extent of reaction has reached p equals p123, which in turn equals prv\u2018\ufb01. The\nproduct pr gives the fraction of B groups that have reacted at any point. With these definitions in\nmind, the following relationships are readily obtained:\n"]], ["block_6", ["The number average degree of polymerization is given by dividing the number of repeat units by\nthe number of chains, or\n"]], ["block_7", ["3.\nThe total number of chains is half the number of chain ends:\n"]], ["block_8", ["2.\nThe total number of chain ends is given by the sum of Equation 2.7.2 and Equation 2.7.3:\n"]], ["block_9", ["68\nStep-Growth Polymerization\n"]], ["block_10", ["1.\nThe number of unreacted functional groups after the reaction reaches extent p is\n"]], ["block_11", [{"image_3": "83_3.png", "coords": [45, 291, 181, 326], "fig_type": "molecule"}]], ["block_12", ["1+1/r\n_\nl+r\nNu:\n1+1/rh2p \n1+r-\u20142pr\n(2.7.7)\n"]], ["block_13", ["and\n"]], ["block_14", ["1\n1\n0\nVchains \nE\n1 + ;\n"]], ["block_15", ["I\nl\nl\nl\nI\"repeat units \n'2\u2018\nVOA + 5\nV2, \n'2\u2018 (I + ;) V:\n(2.76)\n"]], ["block_16", ["1 _\nVends : (I p +\npr) VOA\n(2.7.4)\nr\n"]], ["block_17", ["VB (1 mpg (1 \n(2.7.3)\n,.\n"]], ["block_18", ["VA (1 \u2014p)v\u2019\u00b0A\n(2.7.2)\n"]], ["block_19", [{"image_4": "83_4.png", "coords": [54, 362, 225, 394], "fig_type": "molecule"}]], ["block_20", ["\u201c\u2014 2p\nVA\n(2H75)\n"]]], "page_84": [["block_0", [{"image_0": "84_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Table 2.5\nSome Values of NH Calculated by Equation 2.7.7\nfor Values of r and p Close to Unity\n"]], ["block_2", ["0.95\n13.5\n13.2\n23.3\n39.0\n0.97\n15.5\n22.3\n39.9\n65.7\n0.99\n13.3\n23.7\n66.3\n199\n1.00\n20.0\n33.3\n100\n00\n"]], ["block_3", ["The parameter r\u2019 continues to measure the ratio of the number of A and B groups; the factor 2\nenters since the monofunctional reagent has the same effect on the degree of polymerization as a\ndifunctional molecule with two B groups, hence, is doubly effective compared to the latter. With\nthis modification taken into account, Equation 2.7.7 enables us to evaluate quantitatively the effect\nof stoichiometric imbalance or monofunctional reagents, whether these are intentionally intro-\nduced to regulate Nn or whether they arise from impurities or side reactions.\nThe parameter r varies between 0 and 1; as such it has the same range as p. Although the\nquantitative effect of r and p on Nn is different, the qualitative effect is similar for each: the closer\neach of these fractions is to unity, higher degrees of polymerization are obtained. Table 2.5 shows\nsome values of NH calculated from Equation 2.7.7 for several combinations of (larger values of)\nr and p. Inspection of Table 2.5 reveals the following:\n"]], ["block_4", ["The distinction pointed out in the last paragraph carries over to the evaluation of MD from N\u201c.\nWe assume that the chain of the polymer is great enough to render unnecessary any\ncorrection case, the molecular ofpolymer\nis obtained the of polymerization by multiplying the latter by the molecular weight of\nthe repeat unit. The following examples illustrate the distinction under consideration:\n"]], ["block_5", ["Equation 2.7.7 also applies to the case when some of the excess B groups present are in the form\nof monofunctional reagents. In this latter situation the definition of r is modified somewhat\n(and labeled with a prime) to allow for the fact that some of the B groups are in BB-type monomers\n(unprimed) and some are in monofunctional (primed) molecules:\n"]], ["block_6", ["r\np 0.95\np 0.97\np 0.99\np 1.00\n"]], ["block_7", ["and M0 58. Neglecting end groups, we have Mn 58 ND with Nn given by Equation 2.2.1.\n2.\nPolymerization of AA and BB monomers is illustrated by butane-1,4-diol and adipic acid. The\naabb repeat unit in the polymer has an M0 value of 200. If Equation 2.2.4 is used to evaluate\nNn, it gives the number of aa + bb units; therefore Mn (200 Nn)/2.\n3.\nAn equivalent way of looking at the conclusion of item (2) is to recall that. Equation 2.7.7 gives\nthe (number average) number of monomers of both kinds in the polymer; we should multiply\nthis quantity by the average molecular weight of the two kinds of units in the structure: (88 +\n112)/2 100.\n"]], ["block_8", ["Stoichiometric \n59\n"]], ["block_9", ["1.\nPolymerization of an AB monomer is illustrated by the polyester formed from glycolic acid.\nThe repeat unit in this polymer has the structure\n"]], ["block_10", ["\u2019 \u2014 \u2014\u2014\u2014V3\n2 7 3\n_vg+2v%,\u2019\n(H)\n"]], ["block_11", ["(05.7\n"]]], "page_85": [["block_0", [{"image_0": "85_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["4.\nThe monofunctional reagent B\u2019 is the acetic acid in this case and the number of monofunc\u2014\ntional carboxyl groups is 2(0.010) 0.020 125. The number of B groups in BB monomers is\n"]], ["block_2", ["The various expressions we have developed in this section relating p to the size of the polymer\nare all based on N\u201c. Accordingly, we note that the average reactant molecule in this mixture\nhas a molecular weight of 100 as calculated above. Therefore the desired polymer has a value of\nNH 50.\n"]], ["block_3", ["1.\nCalculate the value of p at which the reaction should be stopped to obtain this polymer,\nassuming perfect stoichiometric balance and neglecting end group effects on Mn.\n2.\nAssuming that 0.5 mol% of the diol is lost to polymerization by dehydration to ole\ufb01n, what\nwould be the value of Mn if the reaction were carried out to the same extent as in (1)?\nHow could the loss in (2) be offset so that the desired polymer is still obtained?\n4.\nSuppose the total number of carboxyl groups in the original mixture is 2 mol, of which 1.0% is\npresent as acetic acid to render the resulting polymer inert to subsequent esterification. What\nvalue of p would be required to produce the desired polymer in this case, assuming no other\nstoichiometric imbalance?\n"]], ["block_4", ["1.\nWe use Equation 2.4.5 for the case of equal numbers of A and B groups and find that\np 1\u2014 I/Nn 0.980. Even though Equation 2.4.5 was derived for an AB monomer, it applies\nto this case with the \u201caverage monomer\u201d as the repeat unit.\n2.\nComponent AA is the diol in this case and v3, 0.995 mol; therefore r: 0.995/ 1.00 0.995.\nWe use Equation 2.7.7 and solve for Nn with p 0.980 and r= 0.995:\n"]], ["block_5", ["1.\nFor any value of r, Nn is greater for larger values of p; this conclusion is the same whether the\nprOportions of A and B are balanced or not.\n2.\nThe final 0.05 increase in p has a bigger effect on Nn at r values that are closer to unity than for\nless-balanced mixtures.\n3.\nFor any value of p, Nn is greater for larger values of r; stoichiometric imbalance lowers the\naverage chain-length for the preparation.\n4.\nAn 0.05 increase in r produces a much bigger increase in NH at p 1 than in mixtures that have\nreacted to a lesser extent.\n"]], ["block_6", ["The following example illustrates some of the concepts developed in this section.\n"]], ["block_7", ["and therefore Mn 44.5 x 100 4450 g mol\u201c.\n3.\nThe effect of the lost hydroxyl groups can be offset by carrying out the polymerization to a\nhigher extent of reaction. We use Equation 2.7.7 and solve for p with Nn 50 and r = 0.995:\n"]], ["block_8", ["5-\u201d\n"]], ["block_9", ["It is desired to prepare a polyester with MH 5000 by reacting 1 mol of butane\u20141,4\u2014diol with 1 mol\nof adipic acid.\n"]], ["block_10", ["Solution\n"]], ["block_11", ["70\nStep-Growth Polymerization\n"]], ["block_12", ["Example 2.4\n"]], ["block_13", ["1\n1 + r\n1\n1.995\n=\n1 -\u2014 _\n=\n1 \u2014 \u2014 \u2014 =\n.\n2\np\n(\nN)( 2r )\n(\n50) 1.990\n098 5\n"]], ["block_14", ["Nn\n= 44.5\n"]], ["block_15", ["Nn \nl\u2014r\n(2.7.9)\n"]], ["block_16", ["An interesting special case occurs when p =1; Equation 2.7.7 then becomes\n"]], ["block_17", ["_\n1.995\n\u2014\n1.995 2 (0.995) (0.980)\n"]]], "page_86": [["block_0", [{"image_0": "86_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["1.\nIn the simplest case of stoichiometric balance, that is, equal numbers of A and B reactive\ngroups, the number average degree of polymerization N,1 is given by 1/(1\u2014p), where p is the\nextent of reaction, equal to the fraction of A (or B) groups reacted. In general, therefore,\nthe reaction must be driven far toward products (p\u2014il) before appreciable molecular weights\ncan be attained.\n2.\nThe resulting distribution of molecular sizes is called the most probable distribution and the\nassociated polydispersity index approaches 2 as p\u2014il. Two important features of this distri-\nbution are that there are always more i-mers present than (i+l)-mers, for any value of i, but\nthere is an intermediate value of i for which the weight fraction w,- is maximum.\n3.\nIf the reaction is run in the presence of a catalyst (the usual situation), then N,1 should grow\nlinearly in time, whereas for the uncatalyzed case, N,1 will grow with the square root of time.\n4.\nIn reality N will almost always be lower than the theoretical value for a given p, due to a\ncombination of side reactions, including ring formation, contamination by monofunctional\nreagents, and stoichiometric imbalance.\n5.\nThe analysis of these reactions builds on the principle of equal reactivity, the assumption that\nthe reactivity of a given functional group is independent of the molecular weight of the\npolymer to which it is attached. This assumption is quite reliable in most cases of interest.\n"]], ["block_2", ["Remember from Section 2.3 that a progressively longer period of time is required to shift the\nreaction ofand \ntional reactants polymerization times, but are accepted in the\nform of lower \n"]], ["block_3", ["In this chapter we have considered step-growth or condensation polymerization, one of the two\nmain routes to synthetic polymers. Our emphasis has been on the description of the distribution of\npolymer sizes as a function of the extent of reaction and the concentration of reactants, and on the\nassociated kinetics. In addition, we have given an introduction to the two major classes of\ncommercial condensation polymers, polyesters and polyamides, and the different ways they may\nbe produced. The principal results are as follows:\n"]], ["block_4", ["Problems\n"]], ["block_5", ["*GJ. Howard, J. Polym. sci, 37, 310 (1959).\n"]], ["block_6", ["Equation 2.7.7 p usingNn 50 and r\u2019 0.990:\n"]], ["block_7", ["2.8\nChapter Summary\n"]], ["block_8", ["Problems\n71\n"]], ["block_9", ["1.\nHoward describes a model system used to test the molecular weight distribution of a conden-\nsation polymer.)r \u201cThe polymer sample was an acetic acid\u2014stabilized equilibrium nylon-\n6,6. Analysis showed it to have the following end group composition (in equivalents per\n"]], ["block_10", [{"image_1": "86_1.png", "coords": [44, 137, 267, 173], "fig_type": "molecule"}]], ["block_11", ["1\n1 + r\u2019\n1\n1.990\n=\n1\u2014\u2014\n=\n1\u2014\u2014\u2014\u2014 \u2014=0.9849\np\n ( \nNH) ( 2r\u2019 )\n ( \n50) 1.980\n"]], ["block_12", ["1.980: We use Equation 2.7.8 to define r\u2019 for this situation, assuming the number of\nhydroxyl mol:\n"]], ["block_13", ["Ir\n= 0.990\n"]], ["block_14", ["106 g): acetyl=28.9, amine=35.3, and carboxyl=96.5. The number average degree of\n"]], ["block_15", ["_\n2.00\n_\n1.980 + 2 (0.020)\n"]]], "page_87": [["block_0", [{"image_0": "87_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["72\nStep-Growth Polymerization\n"]], ["block_2", ["polymerization is, therefore, 110 and the conversion degree ( extent of reaction) 0.9909.\u201d\nVerify the self-consistency of those numbers.\n2.\nHaward et al. have reported some research in which a copolymer of styrene and hydro-\nxyethylmethacrylate was cross-linked by hexamethylene di-isocyanate.\u2019r Draw the structural\nformula for a portion of this cross-linked polymer and indicate what part of the molecule is\nthe result of a condensation reaction and what part results from addition polymerization.\nThese authors indicate that the cross-linking reaction is carried out in sufficiently dilute\nsolutions of copolymer that the cross\u2014linking is primarily intramolecular rather than intermo-\nlecular. Explain the distinction between these two terms and why concentration affects the\nrelative amounts of each.\n"]], ["block_3", ["4.\nExamination of Figure 2.5 shows that N,/N is greater for i240 at p=0.97 than at either\np=0.95 or p=0.99. This is generally true: various i\u2014mers go through a maximum in\nnumerical abundance as p increases. Show that the extent of reaction at which this maximum\noccurs varies with i as follows: pmax =(i \n1)/z'. For a catalyzed AB reaction, extend this\nexpression to give a function for the time required for an i-mer to reach its maximum\nnumerical abundance. If k0 2.47 X 10\u2018\"4 L mol\u20181 3'1 at 160.5\u00b0C for the polymerization of\n12-hydroxystearic acid,\u00a7 calculate the time at which 15-mers show their maximum abundance\nif the initial concentration of monomer is 3.0 M.\n"]], ["block_4", ["3.\nThe polymerization of B-carboxymethyl caprolactam has been observed to consist of initial\nisomerization via a second-order kinetic process followed by condensation of the isomer to\npolymer:\n"]], ["block_5", ["*RN. Haward, B.M. Parker, and E.F.T. White, Adv. Chem... 91, 49s (1969).\n111x. Reimschuessel, Adv. Chem, 91, 717 (1969).\n9 C.E.H. Bawn and MB. Huglin, Poiymer, 3, 257 (1962).\n"]], ["block_6", [{"image_1": "87_1.png", "coords": [38, 410, 287, 503], "fig_type": "figure"}]], ["block_7", [{"image_2": "87_2.png", "coords": [50, 219, 267, 290], "fig_type": "figure"}]], ["block_8", [{"image_3": "87_3.png", "coords": [51, 224, 118, 292], "fig_type": "molecule"}]], ["block_9", ["The rates of polymerization are thus of \ufb01rst order in VNH2 and in V(c0),o or second order\noverall. Since VNH2= V(C0)20, the rate=kc2, if catalyzed; third order is expected under\nuncatalyzed conditions. The indirect evaluation of c was accomplished by measuring the\namount of monomer reacted, and the average degree of polymerization of the mixture was\ndetermined by viscosity at different times. The following data were obtained at 270\u00b0C; the\nearly part of the experiment gives nonlinear results.1E Graphically test whether these data\nindicate catalyzed or uncatalyzed conditions, and evaluate the rate of constant for polymer-\nization at 270\u00b0C. Propose a name for the polymer.\n"]], ["block_10", ["20\n0.042\n90\n0.015\n30\n0.039\n1 10\n0.013\n40\n0.028\n120\n0.012\n50\n0.024\n150\n0.0096\n60\n0.021\n180\n0.0082\n80\n0.018\n"]], ["block_11", ["NH\nNH2\n0\nO\nO\n.\n.\n.\n.\nIsomenzatlon :\nO\nPolymerizatlonr:\n3\nN\nHO\nO\nO\nO\n"]], ["block_12", ["1' (min)\n6 (Mole fraction)\nt (min)\n6 (Mole fraction)\n"]], ["block_13", [{"image_4": "87_4.png", "coords": [97, 218, 383, 293], "fig_type": "figure"}]], ["block_14", [{"image_5": "87_5.png", "coords": [152, 224, 245, 282], "fig_type": "molecule"}]], ["block_15", [{"image_6": "87_6.png", "coords": [215, 226, 364, 288], "fig_type": "molecule"}]]], "page_88": [["block_0", [{"image_0": "88_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "88_1.png", "coords": [12, 77, 340, 148], "fig_type": "figure"}]], ["block_2", ["TGt). Cooper and A. Katchman, Adv. Chem, 91, 660 (1969).\n1GB. Taylor, J. Am. Chem. 806., 69, 638 (1947).\n"]], ["block_3", [{"image_2": "88_2.png", "coords": [35, 83, 308, 142], "fig_type": "molecule"}]], ["block_4", ["5.\nIn the the polymerization occurs:\n"]], ["block_5", ["Problems\n73\n"]], ["block_6", [{"image_3": "88_3.png", "coords": [36, 278, 338, 392], "fig_type": "figure"}]], ["block_7", [{"image_4": "88_4.png", "coords": [49, 177, 160, 233], "fig_type": "molecule"}]], ["block_8", ["as the x\u2014axis. Evaluate w,- by Equation 2.4.7 for 1'21, 2, 3, and 4 and 0.1 S p S 0.9 in\nincrements of 0.1. Plot these results (w,- on the y-axis) on a separate graph drawn to the same\nscale as the experimental results. Compare your calculated curves with the experimental\ncurves with respect to each of the following points: (1) coordinates used, (2) general shape\nof curves, and (3) labeling of curves.\nThe polymer described in the last problem is commercially called poly(phenylene oxide),\nwhich is not a \u201cproper\u201d name for a molecule with this structure. Propose a more correct name.\nUse the results of the last problem to criticize or defend the following proposition: The\nexperimental data for dimer polymerization can be understood if it is assumed that one\nmolecule of water and one molecule of monomer may split out in the condensation step.\nSteps involving incorporation of the monomer itself (with only water split out) also occur.\nTaylor carefully fractionated a sample of nylon-6,6 and determined the weight fraction of\ndifferent i\u2014mers in the resulting mixtureft The results obtained are given below. Evaluate NW\nfrom these data, then use Equation 2.4.9 to calculate the corresponding value of p. Calculate\nthe theoretical weight fraction of i-mers using this value of p and a suitable array of i values.\nPlot your theoretical curve and the above data points on the same graph. Criticize or defend the\n"]], ["block_9", [",\nWeight percent composition in reaction mixture\nPercent of theoretical\n02 absorbed\nMonomer\nDimer\nTrimer\nTetramer\n"]], ["block_10", ["was used as a starting material. The composition of the mixture was studied as the reaction\nprogressed and the accompanying results were obtained:1\n"]], ["block_11", ["In an investigation to examine the mechanism of this reaction, the dimer (1' 2)\n"]], ["block_12", ["2O\n3\n38 .5\n23\n9\n35\n6\n26\n21\n1 1\n60\nl l\n4\n4\n1\n80\n1\nO\n0\n0\n"]], ["block_13", ["Me\nMe\nHQOQW\n"]], ["block_14", ["n\nOH\n+\u2014g\u2014og\u2014\u2014+H\nOH+nH20\n"]], ["block_15", ["9\n1\n69\n15\n9\n12\n1 .5\n68\n24\n9\n"]], ["block_16", ["Plot a family of curves, each of different i, with composition as the y-axis and 02 absorbed\n"]], ["block_17", ["Me\nMe\n"]], ["block_18", ["Me\nMe\nn\n"]], ["block_19", ["Me\nMe\n"]], ["block_20", [{"image_5": "88_5.png", "coords": [113, 306, 325, 374], "fig_type": "figure"}]]], "page_89": [["block_0", [{"image_0": "89_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["l\u2018C.W. Ayers, J. Appl. Chem, 4, 444 (1954).\ntH.L. Reimschuessel and OJ. Dege, J. Polym. sax. 14-1. 2343 (1971).\n"]], ["block_2", ["74\nStep-Growth Polymerization\n"]], ["block_3", ["10.\nIn the study described in the last problem, caprolactam was polymerized for 24 h at 225\u00b0C in\nsealed tubes containing various amounts of water. Mn and MW were measured for the\n"]], ["block_4", ["On the basis of these observations, criticize or defend the following prOposition: the fact that\nthe separate spots fuse into a single spot of intermediate Rf value proves that block copoly\u2014\nmers form between the two species within the blend upon heating.\n9.\nReimschuessel and Dege polymerized caprolactam in sealed tubes containing about 0.0205\nmol H20 per mole caprolactam.1 In addition, acetic acid (V), sebacic acid (S), hexamethylene\ndiamine (H), and trimesic acid (T) were introduced as additives into separate runs. The\nfollowing table lists (all data per mole caprolactam) the amounts of additive present and the\nanalysis for end groups in various runs. Neglecting end group effects, calculate Mn for each of\nthese polymers from the end group data. Are the trends in molecular weight qualitatively what\nwould be expected in terms of the role of the additive in the reaction mixture? Explain brie\ufb02y.\n"]], ["block_5", ["8.\nPaper chromatograms were developed for 50:50 blends of nylon-6,6 and nylon-6,10 after the\nmixture had been heated to 290\u00b0C for various periods of time.\u2019r The following observations\ndescribe the chromatograms after the indicated times of heating:\n"]], ["block_6", [{"image_1": "89_1.png", "coords": [45, 81, 264, 276], "fig_type": "figure"}]], ["block_7", [{"image_2": "89_2.png", "coords": [53, 510, 323, 595], "fig_type": "figure"}]], ["block_8", ["following proposition: although the fit of the data points is acceptable with this value of p, it\nappears that a slightly smaller value of p would give an even better fit.\n"]], ["block_9", ["12\n6.5\n31 1\n15.2\n35\n19.6\n334\n14.1\n53\n29.4\n357\n13.0\n31\n33.0\n330\n11.5\n104\n35.4\n403\n1 1.0\n127\n36.5\n426\n9.1\n150\n33.0\n449\n7.2\n173\n27.6\n472\n6.5\n196\n25.2\n495\n4.9\n219\n22.9\n513\n4.3\n242\n19.4\n541\n3.9\n265\n13.5\n564\n3.3\n233\n16.3\n"]], ["block_10", ["S\n0.0102\n21.1\n2.3\nH\n0.0102\n1.4\n19.7\nT\n0.0067\n22.0\n2.5\n"]], ["block_11", ["0 h\u2014two spots with Rf values of individual polymers.\n1/4 h\u2014two distinct spots, but closer together than those of 0 h.\n1/2 h\u2014spots are linked together.\n3/4 h\u2014one long, diffuse spot.\n11/2 h\u2014one compact spot, intermediate Rf value.\n"]], ["block_12", ["None\n\u2014\n5.40\n4.99\nV\n0.0205\n19.8\n2.3\n"]], ["block_13", ["Additives\nMoles additive\n\u2014-\u2014COOH (mEq)\n\u2014\u2014NH2 (mEq)\n"]], ["block_14", ["i\nw, x 10\u20144\nr\nw, x 10\u20144\n"]]], "page_90": [["block_0", [{"image_0": "90_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Problems\n75\n"]], ["block_2", ["WKV. Korshak and S.V. Vinogradova, Polyester-s, Pergamon, Oxford, 1965.\n"]], ["block_3", ["11.\nAt 270\u00b0C adipic acid decomposes to the extent of 0.31 mol% after 1.5 h.Jr Suppose an initially\nequimolar mixture of adipic acid and diol achieves a value of p 0.990 after 1.5 h, compare\nthe expected and observed values of ND in this experiment. Criticize or defend the following\nproposition: the difference between the observed and expected values would be even greater\nthan calculated above if, instead of the extent of reaction being measured analytically, the\nvalue of p expected (neglecting decomposition) after 1.5 h was calculated by an appropriate\nkinetic equation.\n12.\nShow the reaction sequence and the structure of the resulting polymer from the polyconden-\nsation of these two monomers; note that the reaction (a) has two distinct steps, and that (b) it\nis base-catalyzed.\n"]], ["block_4", ["14.\nFor the most probable distribution, it is clear that there is always more i-mer present than\n(i+1)-mer, at any 0 < p < 1. However, the absolute amount of an i-mer should go through a\nmaximum with time, as the reaction progresses; there is zero to start, but at late enough stages\ni-mer will have mostly reacted to contribute to all the larger species. Use the chain rule and any\n"]], ["block_5", ["13.\nA polyester is prepared under conditions of stoichiometric balance, but no attempt is made to\nremove water. Eventually, the reaction comes to equilibrium with equilibrium constant K. If\n[COOH]0 is the initial concentration of carboxylic acid groups, show that the equilibrium\nwater concentration is\n"]], ["block_6", [{"image_1": "90_1.png", "coords": [37, 382, 317, 449], "fig_type": "figure"}]], ["block_7", [{"image_2": "90_2.png", "coords": [47, 85, 284, 174], "fig_type": "figure"}]], ["block_8", ["Use the molecular weight ratio to calculate the apparent extent of reaction of the caprolactam\nin these systems. Is the variation in p qualitatively consistent with your expectations of the\neffect of increased water content in the system? Plot p versus moisture content and estimate\nby extrapolation the equilibrium moisture content of nylon-6 at 255\u00b0C. Does the apparent\nequilibrium moisture content of this polymer seem consistent with the value given in Section\n2.6 for nylon-6,6 at 290\u00b0C?\n"]], ["block_9", ["resulting mixture by osmometry (see Chapter 7) and light scattering (see Chapter 8),\nrespectively, and the following results were obtained:\n"]], ["block_10", ["Moles H20 (x 103)/mole\nCaprolactam\nMn x 10'3\nMD x 10\"3\n"]], ["block_11", ["49.3\n13.4\n20.0\n34.0\n16.4\n25.6\n25.6\n17.9\n29.8\n20.5\n19.4\n36.6\n"]], ["block_12", ["o\no\nA\n_\n(a)\no\n|\\%\no\n+\nNH2QO\u2014QNH2\n"]], ["block_13", ["[COOH]0\n[H20] KNnavn \u2014 1)\n"]], ["block_14", [{"image_3": "90_3.png", "coords": [58, 384, 152, 445], "fig_type": "molecule"}]], ["block_15", [{"image_4": "90_4.png", "coords": [68, 455, 278, 491], "fig_type": "molecule"}]], ["block_16", [{"image_5": "90_5.png", "coords": [81, 386, 311, 445], "fig_type": "molecule"}]], ["block_17", ["o\no\n"]], ["block_18", [{"image_6": "90_6.png", "coords": [155, 395, 299, 436], "fig_type": "molecule"}]], ["block_19", ["R\n"]]], "page_91": [["block_0", [{"image_0": "91_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["2.\nSolomon, D.H. (Ed), Step Growth Polymerization, Marcel Dekker, New York, 1972.\n"]], ["block_2", ["Allcock, HR. and Lampe, F.W., Contemporary Polymer Chemistry, 2nd ed., Prentice Hall, Englewood Cliffs,\nNJ, 1990.\nOdian, G. Principles of Polymerization, 4th ed., Wiley, New York, 2004.\nRempp, P. and Merrill, E.W., Polymer Synthesis, 2nd ed., Hiithig & Wepf, Basel, 1991.\n"]], ["block_3", ["References\n"]], ["block_4", ["76\nStep-Growth Polymerization\n"]], ["block_5", ["1.\nFlory, P.J., Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY, 1953.\n"]], ["block_6", ["Further Readings\n"]], ["block_7", ["16.\nHydrolysis of an aromatic polyamide with M,l 24,116 gives 39.31% by weight m-amino-\naniline, 59.81% terephthalic acid, and 0.88% benzoic acid. Draw the repeat unit structure of\nthe polymer. Calculate the degree of polymerization and the extent of reaction. Calculate\nwhat the degree of polymerization would have been if the amount of benzoic acid were\ndoubled.\n17.\nCalculate the feed ratio of adipic acid and hexamethylene diamine necessary to achieve a\nmolecular weight of approximately 10,000 at 99.5% conversion. What would the identity of\nthe end groups be in the resulting polymer?\n"]], ["block_8", ["suitable simplifications (k[A]0t >> 1\u2018?) to find the degrees of conversion at which the mole\nfraction and the absolute concentration of i-mer have their maximum in time. Compare this to the\nnumber average degree of polymerization at the same conversion; does the answer make sense?\n15.\nFor the polymerization of succinic acid and 1,4\u2014butanediol under stoichiometric balance in\nxylene:\n"]], ["block_9", ["(a)\nDraw the chemical structures of the reactants, products, and important intermediates for\nboth the strong acid\u2014catalyzed and self\u2014catalyzed case.\n(b)\nGenerate a quantitative plot of NI, versus time for the self\u2014catalyzed case up to 28,000 s,\ngiven k: 6\nX\n10\u201d3 mol\u20182 L2 s\"2 and 3 mol L\u20181 starting concentration of each\nmonomer. How many hours would it take to make a polymer with NI] 300\u2018?\n(c)\nDo the same for the catalyzed case, with k 6 x 10\u20182 mol\"1 L s\u201c1 and the same starting\nconcentration. How many hours would it take to make a polymer with NI, 300\u2018?\n(d)\nQualitatively explain the origin of the different shapes of the curves in the two plots.\n"]]], "page_92": [["block_0", [{"image_0": "92_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["In Chapter\n1 we indicated that the category of addition polymers is best characterized by the\nmechanism \nto be chain \nmolecules. thing chain in this discussion. \nchain continues to offer the best description of large polymer molecules. A chain reaction, on the other\nhand, describes a whole of successive events triggered by some initial occurrence. We\nsometimes encounter this description of highway accidents in which one traffic mishap on a fogbound\nhighway results in pileup colliding that can extend for miles. In nuclear reactorsa\ncascade of fission reactions occurs, which is initiated by the capture of the first neutron. In both of\n"]], ["block_2", ["Our primary focus in this section is to point out some of the similarities and differences between\nstep-growth and chain-growth polymerizations. In so doing we shall also have the opportunity to\nindicate some of the different types of chain-growth polymerization systems.\nIn Chapter 2 we saw that step-growth polymerizations occur, one step at a time, through a series of\nrelatively simple organic reactions. By treating the reactivity of the functional groups as independent\n"]], ["block_3", ["automobile collisions that result from the \ufb01rst accident, or the number of fission reactions follow\nfrom the first neutron capture. When we think about the number of monomers that react as a result of a\nsingle initiation step, we are led directly to the degree of polymerization of the resulting molecule. In\nthis way the chain mechanism and the properties of the polymer chains are directly related.\nChain reactions do not go on forever. The fog may clear and the improved visibility ends the suc-\ncession of accidents. Neutron-scavenging control rods may be inserted to shut down a nuclear reactor.\nThe chemical reactions that terminate polymer chain growth are also an important part of the poly-\nmerization mechanism. Killing off the reactive intermediate that keeps the chain going is the essence\nof a termination reaction. Some interesting polymers can be formed when this termination process\nis suppressed; these are called living polymers, and will be discussed extensively in Chapter 4.\nThe kind of reaction that produces a \u201cdead\u201d polymer from a growing chain depends on the\nnature of the reactive intermediate. These intermediates may be free radicals, anions, or cations.\nWe shall devote the rest of this chapter to a discussion of the free-radical mechanism, as it readily\nlends itself to a very general treatment. Furthermore, it is by far the most important chain-growth\nmechanism from a commercial point of view; examples include polyethylene (specifically,\nlow-density polyethylene, LDPE), polystyrene, poly(vinyl chloride), and poly(acry1ates) and\npoly(methacrylates). Anionic polymerization plays a central role in Chapter 4, where we discuss\nthe so-called living polymerizations. In this chapter we deal exclusively with homopolymers. The\nimportant case of copolymers formed by chain-growth mechanisms is taken up in Chapter 4 and\nChapter 5; block copolymers in the former, statistical or random copolymers in the latter.\n"]], ["block_4", ["these examples some initiating event is This is also true in chain-growth polymerization.\nIn the above examples the size of the chain can be measured by considering the number of\n"]], ["block_5", ["3.1\nIntroduction\n"]], ["block_6", ["3.2\nChain-Growth and Step-Growth Polymerizations: Some Comparisons\n"]], ["block_7", ["Chain-Growth Polymerization\n"]], ["block_8", ["3\n"]], ["block_9", ["77\n"]]], "page_93": [["block_0", [{"image_0": "93_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Elsewhere in this chapter we shall see that other reactions\u2014notably, chain transfer and chain\ninhibition\u2014also need to be considered to give a more fully developed picture of chain-growth\npolymerization, but we shall omit these for the time being. Most of this chapter examines the\nkinetics of these three mechanistic steps. We shall describe the rates of the three general kinds of\nreactions by the notation Ri, RP, and Rt for initiation, propagation, and termination, respectively.\nIn the last chapter we presented arguments supporting the idea that reactivity is independent of\nmolecular size. Although the chemical reactions are certainly different between this chapter and\nthe last, we shall also adopt this assumption of equal reactivity for addition polymerization. For\nstep-growth polymerization this assumption simplified the discussion tremendously and at the\nsame time needed careful qualification. We recall that the equal reactivity premise is valid only\nafter an initial size dependence for smaller molecules. The same variability applies to the\npropagation step of addition polymerizations for short-chain oligomers, although things soon\nlevel off and the assumption of equal reactivity holds. We are thus able to treat all propagation\nsteps by the single rate constant kp. Since the total polymer may be the product of hundreds or\neven thousands of such steps, no serious error is made in neglecting the variation that occurs in the\nfirst few steps.\nIn Section 2.3 we rationalized that, say, the first 50% of a step-growth reaction might be different\nfrom the second 50% because the reaction causes dramatic changes in the polarity of the reaction\n"]], ["block_2", ["of the size of the molecule carrying the group, the entire course of the polymerization is described by\nthe conversion of these groups to their condensation products. Two consequences of this are that\nboth high yield and high molecular weight require the reaction to approach completion. In contrast,\nchain-growth polymerization occurs by introducing an active growth center into a reservoir of\nmonomer, followed by the addition of many monomers to that center by a chain-type kinetic\nmechanism. The active center is ultimately killed off by a termination step. The (average) degree\nof polymerization that characterizes the system depends on the frequency of addition steps relative to\ntermination steps. Thus high-molecular\u2014weight polymer can be produced almost immediately. The\nonly thing that is accomplished by allowing the reaction to proceed further is an increased yield of\npolymer. The molecular weight of the product is relatively unaffected. (This simple argument tends\nto break down at high extents of conversion. For this reason we shall focus attention in this chapter on\nlow to moderate conversions to polymer, except where noted.)\nStep-growth polymerizations can be schematically represented by one of the individual reaction\nsteps A + B \u2014> ab, with the realization that the species so connected can be any molecules\ncontaining A and B groups. Chain-growth polymerization, by contrast, requires at least three\ndistinctly different kinds of reactions to describe the mechanism. These three types of reactions\nwill be discussed in the following sections in considerable detail; for now our purpose is just to\nintroduce some vocabulary. The principal steps in the chain-growth mechanism are the following:\n"]], ["block_3", ["78\nChain-Growth Polymerization\n"]], ["block_4", ["2.\nPropagation. The initiator fragment reacts with a monomer M to begin the conversion to\npolymer; the center of activity is retained in the adduct. Monomers continue to add in the same\nway until polymers P,- are formed with the degree of polymerization i:\n"]], ["block_5", ["If i is large enough, the initiator fragment\u2014an endgroup\u2014need not be written explicitly.\n3.\nTermination. By some reaction, generally involving two polymers containing active centers,\nthe growth center is deactivated, resulting in dead polymer:\n"]], ["block_6", ["1.\nInitiation. An active species 1* is formed by the decomposition of an initiator molecule I:\n"]], ["block_7", ["1\u2014) 1*\n(3A)\n"]], ["block_8", ["P?\" + Pj\u201c \u2014> PM (dead polymer)\n(3.C)\n"]], ["block_9", ["1* + M \u2014> IM* J\u2014> IMM* \u2014)\u2014\u00bb\u2014> 19:\"\n(3B)\n"]]], "page_94": [["block_0", [{"image_0": "94_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["mixture. We shall that, under certain circumstances, the rate of addition polymerization\naccelerates as the extent of conversion to polymer increases due to a composition-dependent effect\non termination. the assumption of equal reactivity at all extents of\nreaction, because of the simplification it allows. We will then\nseek to explain the deviations from this ideal or to \ufb01nd experimental conditions\u2014low conversions to\npolymer\u2014under which the assumptions apply. This approach is common in chemistry; for example,\nmost discussions ideal gas law and describe real gases as deviating from the\nideal at high and approaching the ideal as pressure approaches zero.\nIn the last chapter we saw that two reactive groups per molecule are the norm for the formation\nof linear step-growth polymers. A pair of monofunctional reactants might undergo essentially the\nsame reaction, but no polymer is produced because no additional functional groups remain to react.\nOn the other hand, if a molecule contains more than two reactive groups, then branched or cross-\nlinked products can result from step-growth polymerization. By comparison, a wide variety\nof unsaturated monomers undergo chain-growth polymerization. A single kind of monomer\nsuffices\u2014more than one yields a copolymer\u2014and more than one double bond per monomer\nmay result in branching or cross-linking. For example, the 1,2-addition reaction of butadiene\nresults in a chain that has a substituent vinyl group capable of branch formation. Divinyl benzene is\nan example of a bifunctional monomer, which is used as a cross-linking agent in chain-growth\npolymerizations. We shall be primarily concerned with various alkenes or olefins as the monomers\nof interest; however, the carbon\u2014oxygen double bond in aldehydes and ketones can also serve as\nthe unsaturation required for addition polymerization. The polymerization of alkenes yields a\ncarbon atom backbone, whereas the carbonyl group introduces carbon and oxygen atoms into the\nbackbone, thereby illustrating the inadequacy of backbone composition as a basis for distinguish-\ning between addition and condensation polymers.\nIt might be noted that most (butnot all) alkenes are polymerizable by the chain mechanism involving\nfree-radical intermediates, whereas the carbonyl group is generally not polymerized by the free-\nradical mechanism. Carbonyl groups and some carbon\u2014carbon double bonds are polymerized by\nionic mechanisms. Monomers display far more speci\ufb01city where the ionic mechanism is involved\nthan with the free-radical mechanism. For example, acrylamide will polymerize through an anionic\nintermediate but not a cationic one, N\u2014vinyl pyrrolidones by cationic but not anionic intermediates, and\nhalogenated ole\ufb01ns by neither ionic species. In all ofthese cases free-radical polymerization is possible.\nThe initiators used in addition polymerizations are sometimes called \u201ccatalysts,\u201d although\nstrictly speaking this is a misnomer. A true catalyst is recoverable at the end of the reaction,\nchemically unchanged. This is not true of the initiator molecules in most addition polymerizations.\nMonomer and polymer are the initial and final states of the polymerization process, and these\ngovern the thermodynamics of the reaction; the nature and concentration of the intermediates in\nthe process, on the other hand, determine the rate. This makes initiator and catalyst synonyms\nfor the same material. The former term stresses the effect of the reagent on the intermediate, and\nthe latter its effect on the rate. The term catalyst is particularly common in the language of ionic\npolymerizations, but this terminology should not obscure the importance of the initiation step in\nthe overall polymerization mechanism.\nIn the next three sections (Section 3.3 through Section 3.5) we consider initiation, termination,\nand propagation steps in the free-radical mechanism for addition polymerization. As noted above\ntwo additional steps, inhibition and chain transfer, are being ignored at this point. We shall take up\nthese latter topics in Section 3.8.\n"]], ["block_2", ["In this section we shall discuss the initiation step of free-radical polymerization. This discussion is\ncentered around initiators and their decomposition behavior. The first requirement for an initiator\nis that it be a source of free radicals. In addition, the radicals must be produced at an acceptable rate\n"]], ["block_3", ["Initiation\n79\n"]], ["block_4", ["3.3\nInitiation\n"]]], "page_95": [["block_0", [{"image_0": "95_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Table 3.1\nExamples of Free\u2014Radical Initiation Reactions\n"]], ["block_2", ["1.\nOrganic peroxides or hydroperoxides\n2.\nA20 compounds\n"]], ["block_3", ["Some of the most widely used initiator systems are listed below, and Table 3.1 illustrates their\nbehavior by typical reactions:\n"]], ["block_4", ["3.3.1\nInitiation Reactions\n"]], ["block_5", ["at convenient temperatures; have the required solubility behavior; transfer their activity to mono-\nmers efficiently; be amenable to analysis, preparation, purification, and so on.\n"]], ["block_6", ["80\nChain-Growth Polymerization\n"]], ["block_7", ["2. A20 compounds\n"]], ["block_8", ["3. Redox systems\n"]], ["block_9", ["4. Electromagnetic radiation\n"]], ["block_10", ["1. Organic peroxides or hydroperoxides\n"]], ["block_11", [{"image_1": "95_1.png", "coords": [42, 74, 393, 212], "fig_type": "figure"}]], ["block_12", [{"image_2": "95_2.png", "coords": [51, 93, 307, 168], "fig_type": "figure"}]], ["block_13", [{"image_3": "95_3.png", "coords": [67, 95, 281, 137], "fig_type": "molecule"}]], ["block_14", ["O/EO\u2019O\n(DE/O\n,\n2 (Zia\nBenzoyl peroxide\none?)\ncri-\n"]], ["block_15", [{"image_4": "95_4.png", "coords": [79, 241, 356, 282], "fig_type": "molecule"}]], ["block_16", [{"image_5": "95_5.png", "coords": [79, 317, 312, 378], "fig_type": "figure"}]], ["block_17", [{"image_6": "95_6.png", "coords": [79, 488, 317, 542], "fig_type": "molecule"}]], ["block_18", [{"image_7": "95_7.png", "coords": [80, 375, 314, 563], "fig_type": "figure"}]], ["block_19", [{"image_8": "95_8.png", "coords": [81, 230, 360, 292], "fig_type": "figure"}]], ["block_20", [".00\n"]], ["block_21", [{"image_9": "95_9.png", "coords": [94, 321, 321, 377], "fig_type": "molecule"}]], ["block_22", ["Me\nMe\nMe\nNC>CN \nCqMe\n"]], ["block_13", ["Me \\\nMe\n"]], ["block_14", [{"image_4": "96_4.png", "coords": [121, 586, 196, 613], "fig_type": "molecule"}]], ["block_15", [{"image_5": "96_5.png", "coords": [123, 573, 201, 625], "fig_type": "molecule"}]], ["block_16", ["Mekc\u201d\nN $111:\n"]], ["block_17", [{"image_6": "96_6.png", "coords": [132, 517, 199, 571], "fig_type": "molecule"}]], ["block_18", ["Me\n"]], ["block_19", ["Me;\nCN\n"]], ["block_20", ["\u2019l'.\n"]], ["block_21", ["CN\n"]], ["block_22", ["Me\n"]]], "page_97": [["block_0", [{"image_0": "97_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "97_1.png", "coords": [27, 520, 300, 589], "fig_type": "figure"}]], ["block_2", ["We recall some of the ideas of kinetics from the summary given in Section 2.2 and recognize that\nthe rates of initiator decomposition can be developed in terms of the reactions listed in Table 3.1.\n"]], ["block_3", ["The diphenylpicrylhydrazyl radical itself is readily followed spectrophotometrically, as it loses an\nintense purple color on reacting. Unfortunately, this reaction is not always quantitative.\n"]], ["block_4", ["The initiator efficiency is not an exclusive property of the initiator, but depends on the conditions of\nthe polymerization experiment, including the solvent. In many experimental situations,f lies in the\nrange of 0.3\u20140.8. The efficiency should be regarded as an empirical parameter whose value is\ndetermined experimentally. Several methods are used for the evaluation of initiator ef\ufb01ciency, the\nbest being the direct analysis for initiator fragments as endgroups compared to the amount of initiator\nconsumed, with proper allowances for stoichiometry. As an endgroup method, this procedure is\ndif\ufb01cult in addition polymers, where molecular weights are higher than in condensation polymers.\nResearch with isotopically labeled initiators is particularly useful in this application. Since the quantity\nis so dependent on the conditions of the experiment, it should be monitored for each system studied.\nScavengers such as diphenylpicrylhydrazyl radicals [II] react with other radicals and thus\nprovide an indirect method for analysis of the number of free radicals in a system:\n"]], ["block_5", ["While this reaction with solvent continues to provide free radicals, these may be less reactive\nspecies than the original initiator fragments. We shall have more to say about the transfer of free\u2014\nradical functionality to solvent in Section 3.8.\nThe significant thing about these, and numerous other side reactions that could be described, is\nthe fact that they lower the ef\ufb01ciency of the initiator in promoting polymerization. To quantify this\nconcept we de\ufb01ne the initiator e\ufb01i\u2018ciencyf to be the following fraction:\n"]], ["block_6", ["In both of these examples, initiator is consumed, but no polymerization is started.\nOnce the radicals diffuse out of the solvent cage, reaction with monomer is the most\nprobable reaction in bulk polymerizations, since monomers are the species most likely to be encoun-\ntered. Reaction with polymer radicals or initiator molecules cannot be ruled out, but these are less\nimportant because of the lower concentration of the latter species. In the presence of solvent, reactions\nbetween the initiator radical and the solvent may effectively compete with polymer initiation. This\ndepends very much on the speci\ufb01c chemicals involved. For example, carbon tetrachloride is quite\nreactive toward radicals because of the resonance stabilization of the solvent radical produced:\n"]], ["block_7", ["82\nChain-Growth Polymerization\n"]], ["block_8", ["3.3.3\nKinetics of Initiation\n"]], ["block_9", [{"image_2": "97_2.png", "coords": [37, 519, 258, 587], "fig_type": "molecule"}]], ["block_10", [{"image_3": "97_3.png", "coords": [41, 226, 318, 267], "fig_type": "molecule"}]], ["block_11", [{"image_4": "97_4.png", "coords": [44, 51, 229, 114], "fig_type": "molecule"}]], ["block_12", ["0\nJL\n,Me\n+\n002\no\n/\nMe\n0\n/IL\n(3B)\n2\nMe\nO \\\nMe\u2014Me +\n2(302\n"]], ["block_13", ["(3|\n(3|\n(3|\n(3|\n(3|\n(3|\n(3|\n(3|\u2018\n"]], ["block_14", ["_ \nRadicals formed by initiator\n(33.1)\n"]], ["block_15", ["(3|\n(3|\n(3|\n(3|\n"]], ["block_16", ["N\u2014N\nN02\n4-\nR\n0\nadduct\n(3 .F)\n"]], ["block_17", [{"image_5": "97_5.png", "coords": [83, 54, 227, 92], "fig_type": "molecule"}]], ["block_18", [{"image_6": "97_6.png", "coords": [103, 62, 221, 83], "fig_type": "molecule"}]], ["block_19", [{"image_7": "97_7.png", "coords": [151, 227, 296, 263], "fig_type": "molecule"}]], ["block_20", ["[H]\n"]], ["block_21", ["[1]\n"]]], "page_98": [["block_0", [{"image_0": "98_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["where [abs is the intensity of the light absorbed and the constant (15\u2019 is called the quantum yield. The\nfactor 2 is again included for reasons of stoichiometry.\nSince (1/2) d[ I\u00b0]/dt\u2014d[I]/dt in the case of the azo initiators, Equation 3.3.2 can also be\nwritten as \u2014d[I]/dt= kd[l] or, by integration, 1n([I]/[I]0) \u2014kdt, where [Hg is the initiator concen\u2014\ntration at t 0. Figure 3.1 shows a test of this relationship for AIBN in xylene at 77\u00b0C. Except for a\nshort induction period, the data points fall on a straight line. The evaluation of kd from these data is\npresented in the following example.\n"]], ["block_2", ["The decomposition of AIBN in xylene at 77\u00b0C was studied by measuring the volume of N2 evolved\n"]], ["block_3", ["as a function of time. The volumes obtained at time t and r: 00 are V, and V00, respectively. Show\nthat the manner of plotting used in Figure 3.1 is consistent with the integrated \ufb01rst\u2014order rate law\nand evaluate kd.\n"]], ["block_4", ["Figure 3.1\nVolume of nitrogen evolved from the decomposition of AIBN at 77\u00b0C plotted according to the\nfirst\u2014order rate law as discussed in Example 3.1. (Reprinted from Amett, L.M., J. Am. Chem. Soc, 74, 2027,\n1952. With permission.)\n"]], ["block_5", ["2.\nFor redox systems\n"]], ["block_6", ["Using monitorthe rates, we write the\nfollowing:\n"]], ["block_7", ["3.\nFor photochemical initiation\n"]], ["block_8", ["Initiation\n33\n"]], ["block_9", ["Example 3.1\n"]], ["block_10", ["1.\nFor peroxides and compounds\n"]], ["block_11", ["A\nS\" >3\nJ-\n-\u20140.4\n\u2014\nv\nD)\n.9\n_08\n_.\n"]], ["block_12", ["9\u2014371q\u2019 abs\n(3.3.4)\n"]], ["block_13", ["d[I-] _\n\u20195\" \n(3.3.2)\n"]], ["block_14", ["d\u2014E? k[0x][Red]\n(3.3.3)\n"]], ["block_15", ["where kd is the rate constant for the homolytic decomposition of the initiator and [I] is the\nconcentration of the initiator. The factor 2 appears because of the stoichiometry in these\nparticular reactions.\n"]], ["block_16", ["where the bracketed terms describe the concentrations of oxidizing and reducing agents and k\nis the rate constant for the particular reactants.\n"]], ["block_17", ["0.0\n\u2014\n"]], ["block_18", [{"image_1": "98_1.png", "coords": [62, 465, 220, 620], "fig_type": "figure"}]], ["block_19", ["J_\nl\n_l\nl\n0\n80\n160\n240\n320\nTime (min)\n"]]], "page_99": [["block_0", [{"image_0": "99_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["where we have combined the factors off and qfr\u2019 into a composite quantum yield qb, since both of\nthe separate factors are measures of efficiency.\nAny one of these expressions gives the rate of initiation R, for the particular catalytic system\nemployed. We shall focus attention on the homolytic decomposition of a single initiator as the\nmode of initiation throughout most of this chapter, since this reaction typifies the most widely used\nfree-radical initiators. Appropriate expressions for initiation that follow Equation 3.3.6 are readily\nderived.\n"]], ["block_2", ["An important application of photochemical initiation is in the determination of the rate constants\nthat appear in the overall analysis of the chain-growth mechanism. Although we outline this\n"]], ["block_3", ["1.\nV0 0 at t: 0, when no decomposition has occurred.\n2.\nV00 at t: 00, when complete decomposition has occurred.\n3.\nV, at time t, when some fraction of initiator has decomposed.\n"]], ["block_4", ["Next we assume that only a fractionf of these initiator fragments actually reacts with monomer\nto transfer the radical functionality to monomer:\n"]], ["block_5", ["As indicated in the last section, we regard the reactivity of the species IP; to be independent of\nthe value of 1'. Accordingly, all subsequent additions to IM3 in Reaction (3.G) are pr0pagation steps\nand Reaction (3.G) represents the initiation of polymerization. Although it is premature at this\npoint, we disregard endgroups and represent the polymeric radicals of whatever size by the symbol\n"]], ["block_6", ["84\nChain-Growth Polymerization\n"]], ["block_7", ["The ratio [I]/[I]0 gives the fraction of initiator remaining at time t. The volume of N2 evolved is:\n"]], ["block_8", ["3.\nBy photochemical initiation\n"]], ["block_9", ["3.3.4\nPhotochemical Initiation\n"]], ["block_10", ["2.\nBy redox systems\n"]], ["block_11", ["The fraction decomposed\nat t is given by (V,- V0)/(VOO\u2014 V0)\nand the fraction remaining\nat t is l\n\u2014 (V,\u2014V0)/(VOO\u2014V0)= (VOO\u2014V,)/(VOO\u2014-VO). Since VO=0, this becomes (VOO\u2014V,)/VOO or\n[I]/[I]O l V,/VOO. Therefore a plot of ln(l \u2014-V,/VOO) versus t is predicted to be linear with slope\n\u2014kd. (If logarithms to base 10 were used, the slope would equal -\u2014kd/2.303.)\nFrom Figure 3.1,\n"]], ["block_12", ["Po. Accordingly, we write the following for the initiation of polymer radicals:\n"]], ["block_13", ["Solution\n"]], ["block_14", ["1.\nBy peroxide and azo compounds\n"]], ["block_15", ["kd = 5.8 x 10\u20143 min\u20141\n"]], ["block_16", ["I. + M L M\n(3.G)\n"]], ["block_17", ["\u2014\u20140.4 (\u2014-0.8) _\n\u2018kd\n81\n=\u2014 _ \u20142.5\n10\u20183\n' \u20181 =\n0136\n160 320\nX\nmm\n2.303\n"]], ["block_18", ["@ 2f\u2018ie\u2019rlabs 2(iif\u2019labs\n(33\u00b0?)\ndt\n"]], ["block_19", ["5:? \ufb02c[0x][Red]\n(3.3.6)\n"]], ["block_20", ["d[P\u00b0]\nT\n= kd\ufb02]\n(3.3.5)\n"]]], "page_100": [["block_0", [{"image_0": "100_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["method in Section develop Equation 3.3.7 somewhat It\nis not feasible \npertinent desires \nanalytical or physical \n"]], ["block_2", ["2.\nAbsorbance A as measured by spectrophotometers is defined as\n"]], ["block_3", ["Initiation\n85\n"]], ["block_4", ["3.\nSince lab, equals the difference 10 It\n"]], ["block_5", ["4.\nSubstituting this result into Equation 3.3.7 gives\n"]], ["block_6", ["Note that although Equation 3.3.5 and Equation 3.3.12 are both first-order rate laws, the physical\nsigni\ufb01cance of the proportionality factors is quite different in the two cases. The rate constants\nshown in Equation 3.3.5 and Equation 3.3.6 show a temperature dependence described by the\nArrhenius equation:\n"]], ["block_7", ["where 13* is the activation energy, which is interpreted as the height of the energy barrier to a\nreaction, and A is the prefactor. Activation energies are evaluated from experiments in which rate\nconstants are measured at different temperatures. Taking logarithms on both sides of Equation\n3.3.13 gives In k In A \u2014E*/RT. Therefore 13* is obtained from the slope of a plot of In k against\n1/T. As usual, T is in kelvin and R and E* are in (the same) energy units.\nSince 8* is positive according to this picture, the form of the Arrhenius equation assures that\nk gets larger as T increases. This means that a larger proportion of molecules have sufficient energy\nto surmount the energy barrier at higher temperatures. This assumes, of course, that thermal\nenergy is the source of 13*, something that is not the case in photoinitiated reactions. The effective\n\ufb01rst-order rate constants k and Io8b\u2014fOI' thermal initiation and photoinitiation, respectively\u2014do\n"]], ["block_8", ["where [c] is the concentration of monomer or initiator for the two reactions shown in Table 3.1.\n"]], ["block_9", ["3.3.5\nTemperature Dependence of Initiation Rates\n"]], ["block_10", ["1.\nIntensity of light transmitted (subscript t) through a sample It depends on the intensity of the\nincident (subscript 0) light 10, the thickness of the sample b, and the concentration [c] of\nthe absorbing species\n"]], ["block_11", ["k Ae_E*/RT\n(3.3. 13)\n"]], ["block_12", ["1, = 106\u2014616\u201c)\n(3.3.8)\n"]], ["block_13", ["where the proportionality constants is called the absorption coefficient (or molar absorptivity\nif [c] is in moles/liter) and is a property of the absorber. The reader may recognize this\nequation as a form of the famous Beer\u2019s law.\n"]], ["block_14", ["10\nA loglo (X)\n(3.3.9)\n"]], ["block_15", ["labs Io(8[clb)\n(3.3.11)\n"]], ["block_16", ["% 2\u00a2106[c]b\n(3.3.12)\n"]], ["block_17", ["labs M1 e\u2018elcl\u201d)\n(3.3.10)\n"]], ["block_18", ["The variation in absorbance with wavelength re\ufb02ects the wavelength dependence of .9.\n"]], ["block_19", ["If the exponent in Equation 3.3.10 is small\u2014which in practice means dilute solutions, since\nmost absorption experiments are done where .9 is large\u2014then the exponential can be expanded\n(see Appendix), e)r g 1 +x + o, with only the leading terms retained to give\n"]]], "page_101": [["block_0", [{"image_0": "101_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "101_1.png", "coords": [30, 583, 315, 628], "fig_type": "molecule"}]], ["block_2", ["This mode of termination produces a negligible effect on the molecular weight of the reacting\nspecies, but it does produce a terminal unsaturation in one of the dead polymer molecules.\nEach polymer molecule contains one initiator fragment when termination occurs by disproportionation.\n"]], ["block_3", ["The formation of initiator radicals is not the only process that determines the concentration of free\nradicals in a polymerization system. Polymer propagation itself does not change the radical\nconcentration; it merely converts one radical to another. Termination steps also occur, however,\nand these remove radicals from the system. We shall discuss combination and disproportionation\nreactions as the two principal modes of termination.\n"]], ["block_4", ["The degrees of polymerization i andj in the two combining radicals can have any values, and the\nmolecular weight of the product molecule will be considerably higher on the average than the\nradicals so terminated. The polymeric product molecule contains two initiator fragments per\nmolecule by this mode of termination. Note also that for a vinyl monomer, such as styrene or\nmethyl methacrylate, the combination reaction produces a single head-to-head linkage, with the\nside groups attached to adjacent backbone carbons instead of every other carbon.\nTermination by disproportionation comes about when an atom, usually hydrogen, is transferred\nfrom one polymer radical to another:\n"]], ["block_5", ["Table 3.2\nRate Constants (at the Indicated Temperature) and Activation Energies for Some Initiator\nDecomposition Reactions\n"]], ["block_6", ["Source: Data from Masson, J.C. in Polymer Handbook, 3rd ed., Brandrup, J. and Immergut, E.H. (Eds), Wiley, New York,\n1989.\n"]], ["block_7", ["Termination by combination results in the simultaneous destruction of two radicals by direct\ncoupling:\n"]], ["block_8", ["not show the same temperature dependence. The former follows the Arrhenius equation, whereas\nthe latter cluster of terms in Equation 3.3.12 is essentially independent of T.\nThe activation energies for the decomposition (subscript (1) reaction of several different\ninitiators in various solvents are shown in Table 3.2. Also listed are values of kd for these systems\nat the temperature shown. The Arrhenius equation can be used in the form In (kd,1/kd,2)= \u2014(E*/R)\n(l/Tl HR) to evaluate kd values for these systems at temperatures different from those given in\nTable 3.2.\n"]], ["block_9", ["3.4.1\nCombination and Disproportionation\n"]], ["block_10", ["2,2\u2019-Azobisisobutyronitrile\nBenzene\n70\n3.17 x 10\u20145\n123.4\nccn\n40\n2.15 x 10-7\n128.4\nToluene\n100\n1.60 x 10-3\n121.3\nt-Buty] peroxide\nBenzene\n100\n8.8 x 10\u20147\n146.9\nBenzoyl peroxide\nBenzene\n70\n1.48 x 10\u20145\n123.8\nCumene\n60\n1.45 x 10'6\n120.5\nt\u2014Butyl hydroperoxide\nBenzene\n169\n2.0 x 10\u20145\n170.7\n"]], ["block_11", ["Initiator\nSolvent\nr (\u201d(3)\nk, (3\")\n53 (kJ moi\u201c)\n"]], ["block_12", ["86\nChain-Growth Polymerization\n"]], ["block_13", ["3.4\nTermination\n"]], ["block_14", [{"image_2": "101_2.png", "coords": [37, 74, 438, 203], "fig_type": "figure"}]], ["block_15", [{"image_3": "101_3.png", "coords": [43, 589, 211, 625], "fig_type": "molecule"}]], ["block_16", ["Pf. + 'Pj --> Pg+j\n(3.H)\n"]], ["block_17", ["X\nX\nX\nX\nPom\n+\nrpm Pi\u20141/\\/\n+\n \\fpj \ufb01 \n(3.1)\nH\nH\n"]], ["block_18", [{"image_4": "101_4.png", "coords": [140, 588, 302, 619], "fig_type": "molecule"}]]], "page_102": [["block_0", [{"image_0": "102_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Since the disproportionation reaction requires bond breaking, which is not required for combin-\nation, Ei'fd is expected to be greater than E\ufb01fc. This causes the exponential to be large at low\ntemperatures, making combination the preferred mode of termination under these circumstances.\nNote that at higher temperatures this bias in favor of one mode of termination over another\ndecreases as the difference in activation energies becomes smaller relative to the thermal energy\nRT. Experimental results on modes of termination show that this qualitative argument must be\napplied cautiously. The actual determination of the partitioning between the two modes of\ntermination is best accomplished by analysis of endgroups, using the difference in endgroup\ndistribution noted above.\nTable 3.3 lists the activation energies for termination (these are overall values, not identified as\nto mode) of several different radicals. The rate constants for termination at 60\u00b0C are also given. We\nshall see in Section 3.6 how these constants are determined.\n"]], ["block_2", ["Combination and disproportionation are competitive processes and do not occur to the same\nextent for all polymers, although in general combination is more prevalent. For poly(methyl\nmethacrylate), both reactions contribute to termination, with disproportionation favored. Both\nrate constants for termination individually follow the Arrhenius equation, so the relative amounts\nof termination by the two modes are given by\n"]], ["block_3", ["where the subscript c specifically indicates termination by combination.\n4.\nFor disproportionation,\n"]], ["block_4", ["Termination\n87\n"]], ["block_5", ["3.\nFor combination,\n"]], ["block_6", ["termination \nfollowing:\n"]], ["block_7", ["where RI and kt are the rate and rate constant for termination (subscript t) and the factor 2\nenters (by convention) because two radicals are lost for each termination step.\n2.\nThe polymer radical concentration in Equation 3.4.1 represents the total concentration of all\nsuch species, regardless of their degree of polymerization; that is,\n"]], ["block_8", ["where the subscript d specifically indicates termination by disproportionation.\n5.\nIn the event that the two modes of termination are not distinguished, Equation 3.4.1 represents\nthe sum of Equation 3.4.3 and Equation 3.4.4, or\n"]], ["block_9", ["1.\nFor general termination,\n"]], ["block_10", ["Termination by combination\n_ _ _ \n\u2014(E\u00a7':c\u2014 Eifd)\n3 4 6\nTermination by disproportionation \nkm\n\u2014\nAt,de*5?,d/RT\n\u2014\nAnd\nCXP(\nRT\n)\n(\n.\n\u00b0 )\n"]], ["block_11", ["Kinetic analysis of the two modes of termination is quite straightforward, since each mode of\n"]], ["block_12", ["d P-\nR. \u2014Q 24,413.12\n(3.4.3)\ndz\n"]], ["block_13", ["d P0\nRt \u2014\u00a5 2kt,d[P~]2\n(3.4.4)\ndz\n"]], ["block_14", ["Rt \u201c(E5 2kt[P']2\n(3.4.1)\ndr\n"]], ["block_15", ["kt kt,c + kt,d\n(3.4.5)\n"]], ["block_16", ["[P] Z [Pr]\n(3.4.2)\n"]], ["block_17", [{"image_1": "102_1.png", "coords": [63, 198, 115, 232], "fig_type": "molecule"}]], ["block_18", ["all i\n"]]], "page_103": [["block_0", [{"image_0": "103_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Table 3.3\nRate Constants at 60\u00b0C and Activation Energies for Some\nTermination Reactions\n"]], ["block_2", ["Source: Data from Korus, R. and O\u2019Driscoll, K.F. in Polymer Handbook, 3rd ed.,\nBrandrup, J. and Irnmergut, E.H. (Eds), Wiley, New York, 1989.\n"]], ["block_3", ["88\nChain-Growth Polymerization\n"]], ["block_4", ["Acrylonitrile\n1.5\n78.2\nMethyl acrylate\n22.2\n0.95\nMethyl methacrylate\n11.9\n2.55\nStyrene\n8.0\n6.0\nVinyl acetate\n21.9\n2.9\n2-Vinyl pyridine\n21.0\n3.3\n"]], ["block_5", ["Monomer\nE; (kJ 1110]\u201c)\nk, x 10\u201c7 (L mor\u2018 s\")\n"]], ["block_6", ["The assumption that k values are constant over the entire duration of the reaction breaks down\nfor termination reactions in bulk polymerizations. Here, as in Section 2.2, we can consider the\ntermination process\u2014\u2014\u2014whether by combination or disproportionation\u2014to depend on the rates at\nwhich polymer molecules can diffuse into (characterized by ki) or out of (characterized by k0) the\nsame solvent cage and the rate at which chemical reaction between them (characterized by k,)\noccurs in that cage. In Chapter 2, we saw that two limiting cases of Equation 2.2.8 could be readily\nidentified:\n"]], ["block_7", ["4.\nThis situation is expected to apply to radical termination, especially by combination, because\nof the high reactivity of the trapped radicals. Only one constant appears that depends on the\ndiffusion of the polymer radicals, so it cannot cancel out and may contribute to a dependence\nof kt on the extent of reaction or the degree of polymerization.\n"]], ["block_8", ["Figure 3.2 shows how the percent conversion of methyl methacrylate to polymer varies with\ntime. These experiments were carried out in benzene at 50\u00b0C. The different curves correspond to\ndifferent concentrations of monomer. Up to about 40% monomer, the conversion varies smoothly\nwith time, gradually slowing down at higher conversions owing to the depletion of monomer. At\nhigh concentrations, however, the polymerization starts to show an acceleration between 20% and\n40% conversion. This behavior, known as the Trommsdorjf e\ufb01\u2018ect [2], is attributed to a decrease in\nthe rate of termination with increasing conversion. This, in turn, is due to the increase in viscosity\nthat has an adverse effect on kt through Equation 3.4.8. Considerations of this sort are important in\nbulk polymerizations where high conversion is the objective, but this complication is something\nwe will avoid. Hence we shall be mainly concerned with solution polymerization and/or low\ndegrees of conversion where Itt may be justifiably treated as a true constant. We shall see in Section\n3.8 that the introduction of solvent is accompanied by some complications of its own, but we shall\nignore this for now.\n"]], ["block_9", ["3.4.2\nEffect of Termination on Conversion to Polymer\n"]], ["block_10", ["2.\nThis situation seems highly probable for step-growth polymerization because of the high\nactivation energy of many condensation reactions. The constants for the diffusion-dependent\nsteps, which might be functions of molecular size or the extent of the reaction, cancel out.\n3.\nRate of reaction > rate of diffusion (Equation 2.2.10):\n"]], ["block_11", ["1.\nRate of diffusion > rate of reaction (Equation 2.2.9):\n"]], ["block_12", [{"image_1": "103_1.png", "coords": [37, 79, 317, 179], "fig_type": "figure"}]], ["block_13", ["kt \u2014kr\n(3.4.7)\n"]], ["block_14", ["lct ki\n(3.4.8)\n"]]], "page_104": [["block_0", [{"image_0": "104_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "104_1.png", "coords": [28, 17, 286, 194], "fig_type": "figure"}]], ["block_2", ["Termination\n89\n"]], ["block_3", ["benzene \npermission.)\n"]], ["block_4", ["Polymer propagation \ntwo opposing processes, willeventually reach point of balance. This\ncondition is called the and characterized by constant total concentration of free\nradicals. Under s) the net rate of initiation must equal the net\nrate of termination. Using Equation 3.3.5 for the rate of initiation (i.e., two radicals per initiator\nmolecule) and Equation 3.4.1 for termination, we write\n"]], ["block_5", ["Calculate [Po]S and the time required for the free-radical concentration to reach 99% of this value\nusing the following as typical values for constants and concentrations: kd =1.0 X 10\"4 s\", kt:\n3 x 107 L mol\u2014l s\u201c,f: 1/2, and [I]O 10\u20183 mol L\u201c. Comment on the assumption [I] [I]0 that\nwas made in deriving this nonstationary-state equation.\n"]], ["block_6", ["3.4.3\nStationary-State Radical Concentration\n"]], ["block_7", ["01'\n"]], ["block_8", ["This important equation shows that the stationary-state free-radical concentration increases with\n[I]\u201d2 and varies directly with kit/2 and inversely with ktm. The concentration of free radicals\ndetermines the rate at which polymer forms and the eventual molecular weight of the polymer,\nsince each radical is a growth site. We shall examine these aspects of Equation 3.4.10 in the next\nsection. We conclude this section with a numerical example illustrating the stationary-state radical\nconcentration for a typical system.\n"]], ["block_9", ["For an initiator concentration that is constant at [I]0, the nonstationary-state radical concentration\nvaries with time according to the following expression:\n"]], ["block_10", ["Figure 3.2\nAcceleration of the polymerization rate for methyl methacrylate at the concentrations shown in\n"]], ["block_11", ["Example 3.2\n"]], ["block_12", [".5\nE\n10%\ne 60\ng\nr\nC\n8 40 \n"]], ["block_13", ["a?\nr\n20 \n"]], ["block_14", ["2mm 2mm:\n(3.4.9)\n"]], ["block_15", ["\ufb02cd \n[13.15: (7;)\n[111/2\n(3.4.10)\n"]], ["block_16", [{"image_2": "104_2.png", "coords": [47, 395, 140, 441], "fig_type": "molecule"}]], ["block_17", ["[12.] = eXP[(4fkdkt[I]0)l/2 t] 1\n[P].\nexp [(4\ufb02cdk,[l]0)'/2 t] + 1\n"]], ["block_18", ["30 \n100%\n80%\n60%\n40%\n"]], ["block_19", ["l\nI\nI\nI\nI\nI\nI\nI\nI\n_L\nI\nI\nI\nI\n_I_\nI\nI\n0\n500\n1000\n1500\nTime (min)\n"]], ["block_20", ["L\n"]]], "page_105": [["block_0", [{"image_0": "105_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["This low concentration is typical of free-radical polymerizations. Next we inquire how long it will\ntake the free-radical concentration to reach 0.99 [P\u00b0]s, or 4.04 x\n10\u20148 mol L\u20181 in this case. Let\na =(4fkdkt[l]0)1/2 and rearrange the expression given to solve for t when [P\u00b0]/[P\u00b0]s 0.99: 0.99\n(ear + 1) e\u2018\u201d\u2014 1, or 1 + 0.99 e\u2018\u201d(1 0.99). Therefore the product at = ln(1.99/0.01) In 199 5.29,\nand a [4(1/2)(1.0 x 10\u20144)(3 x 10500\u2014511\u201d: 2.45 3*. Hence t= 529/245 2.16 s.\nThis short period is also typical of the time required to reach the stationary state. The\nassumption that [I] [He may be assessed by examining the integrated form of Equation 3.3.2\nfor this system and calculating the ratio [I]/[I]0 after 2.16 s:\n"]], ["block_2", ["The propagation of polymer chains is easy to consider under stationary-state conditions. As the\npreceding example illustrates, the stationary state is reached very rapidly, so we lose only a brief\nperiod at the start of the reaction by restricting ourselves to the stationary state. Of course, the\nstationary-state approximation breaks down at the end of the reaction also, when the radical\nconcentration drops toward zero. We shall restrict our attention to relatively low conversion to\npolymer, however, to avoid the complications of the Trommsdorff effect. Therefore deviations\nfrom the stationary state at long times need not concern us.\nIt is worth taking a moment to examine the propagation step more explicitly in terms of the\nreaction mechanism itself. As an example, consider the case of styrene as a representative vinyl\nmonomer. The polystyryl radical is stabilized on the terminal-substituted carbon by resonance\ndelocalization:\nsea~a>s\n"]], ["block_3", ["Use Equation 3.4.10 to evaluate [Po]S for the system under consideration:\n"]], ["block_4", ["Over the time required to reach the stationary state, the initiator concentration is essentially\nunchanged. As a matter of fact, it would take about 100 s for [I] to reach 0.99 [HO and about 8.5\nmin to reach 0.95 me, so the assumption that [I] [Hg is entirely justi\ufb01ed over the short times\ninvolved.\n"]], ["block_5", ["Consequently, the addition of the next monomer is virtually exclusively in a \u201chead-to-tail\u201d arrange-\nment, leading to an all-carbon backbone with substituents (X) on alternating backbone atoms:\n"]], ["block_6", ["90\nChain-Growth Polymerization\n"]], ["block_7", ["3.5\nPropagation\n"]], ["block_8", ["Solution\n"]], ["block_9", [{"image_1": "105_1.png", "coords": [44, 548, 306, 611], "fig_type": "molecule"}]], ["block_10", [{"image_2": "105_2.png", "coords": [45, 92, 360, 138], "fig_type": "molecule"}]], ["block_11", ["ln(\ufb02) \u2014kdt ~(1.0 >< 10\u20144)(2.16)= \u20142.16 X 10*4\n[110\n[I]\n\u2014\u2014 = 0.9998\n[Ila\n"]], ["block_12", ["1/2\n\u20144\n\u20143\nl/2\n[P 1, (@1110)\n\u2014(\n3 X107\n)\n_ (1.67 x 10\n)\n"]], ["block_13", ["= 4.08 x 10\u20148 mol L-1\n"]], ["block_14", [{"image_3": "105_3.png", "coords": [108, 549, 219, 611], "fig_type": "molecule"}]], ["block_15", [{"image_4": "105_4.png", "coords": [179, 552, 291, 603], "fig_type": "molecule"}]]], "page_106": [["block_0", [{"image_0": "106_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["in which the second form reminds us that an experimental study of the rate of polymerization\nyields a single apparent rate constant (subscript app), which the mechanism reveals to be a\ncomposite of three different rate constants. Equation 3.5.3 shows that the rate of polymerization\nis first order in monomer and half order in initiator and depends on the rate constants for each of\nthe three types of steps\u2014initiation, propagation, and termination\u2014\u201cthat make up the chain mech-\nanism. Since the concentrations change with time, it is important to realize that Equation 3.5.3\ngives an instantaneous rate of polymerization at the concentrations considered. The equation can\nbe applied to the initial concentrations of monomer and initiator in a reaction mixture only to\ndescribe the initial rate of polymerization. Unless stated otherwise, we shall assume the initial\nconditions apply when we use this result.\nThe initial rate of polymerization is a measurable quantity. The amount of polymer formed after\nvarious times in the early stages of the reaction can be determined directly by precipitating the\npolymer and weighing. Alternatively, some property such as the volume of the system (or\nthe density, the refractive index, or the viscosity) can be measured. Using an analysis similar to\nthat followed in Example 3.1, we can relate the values of the property measured at r, r: 0 and\nt: 00 to the fraction of monomer converted to polymer. If the rate of polymerization is measured\nunder known and essentially constant concentrations of monomer and initiator, then the cluster\nof constants (fkgkd/ktX/Z can be evaluated from the experiment. As noted earlier, f is best\ninvestigated by endgroup analysis. Even with the factor f excluded, experiments on the rate of\npolymerization still leave us with three unknowns. Two other measurable relationships among\nthese unknowns must be found if the individual constants are to be resolved. In anticipation of this\ndeve10pment, we list values of kp and the corresponding activation energies for several common\nmonomers in Table 3.4.\n"]], ["block_2", ["1.\nThe radical concentration has the stationary\u2014state value given by Equation 3.4.10.\n2.\nkp is a constant independent of the size of the growing chain and the extent of conversion\nto polymer.\n3.\nThe rate at which monomer is consumed is equal to the rate of polymer formation RP:\n"]], ["block_3", ["This should linkage that results from termination by\nrecombination \n"]], ["block_4", ["as the expression converted to polymer. In writing this expression,\nwe assume \n"]], ["block_5", ["Consideration (3.B) leads to\n"]], ["block_6", ["3.5.1\nRate Laws for Propagation\n"]], ["block_7", ["Propagation\n91\nW'\n+ AX\n\u2014\" WYW\n(3.1)\nX\nX\nX\nX\nX\nX\nX\n"]], ["block_8", [{"image_1": "106_1.png", "coords": [38, 331, 232, 367], "fig_type": "molecule"}]], ["block_9", [{"image_2": "106_2.png", "coords": [38, 48, 354, 90], "fig_type": "molecule"}]], ["block_10", [{"image_3": "106_3.png", "coords": [38, 52, 123, 88], "fig_type": "molecule"}]], ["block_11", ["d M\n\ufb01._[_].:kp[M][p.]\n(3.5.1)\ndt\n"]], ["block_12", ["k\u2014t)\n[111/2 kapp[M][I]1/2\n(3-5-3)\nRD kp[M]<\n"]], ["block_13", ["Combining Equation 3.4.10 and Equation 3.5.1 yields\n"]], ["block_14", ["d[M] _ d[polymer] _\ndt\nT\ndr\n_ Rp\n(3.5.2)\n"]], ["block_15", ["d\n1/2\n"]], ["block_16", [{"image_4": "106_4.png", "coords": [190, 51, 334, 86], "fig_type": "molecule"}]]], "page_107": [["block_0", [{"image_0": "107_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Table 3.4\nRate Constants at 60\u00b0C and Activation Energies for Some\nPropagation Reactions\n"]], ["block_2", ["92\nChain-Growth Polymerization\n"]], ["block_3", ["Monomer\nEff (kl moi\u2014l)\nk1, x 10\u20183 (L mol\u2018l s\u2018l)\n"]], ["block_4", ["Source: Data from Korus, R. and O\u2019Driscoll, K.F. in Polymer Handbook, \nBrandrup, J. and Immergut, E.H. (Eds), Wiley, New York, 1989.\n"]], ["block_5", ["Acrylonitrile\n16.2\n1.96\nMethyl acrylate\n29.7\n2.09\nMethyl methacrylate\n26.4\n0.515\nStyrene\n26.0\n0. 165\nVinyl acetate\n18.0\n2.30\n2-Vinyl pyridine\n33.0\n0.186\n"]], ["block_6", ["Figure 3.3 shows some data that constitute a test of Equation 3.5.3. In Figure 3.33, RP and [M] are\nplotted on a log\u2014log scale for a constant level of redox initiator. The slope of this line, which\nindicates the order of the polymerization with respect to monomer, is unity, showing that the\npolymerization of methyl methacrylate is first order in monomer. Figure 3.3b is a similar plot of\nthe initial rate of polymerization\u2014which essentially maintains the monomer at constant concentra\u2014\ntion\u2014versus initiator concentration for two different monomer\u2014initiator combinations. Each of the\nlines has a slope of 1/2, indicating a half-order dependence on [I] as predicted by Equation 3.5.3.\n"]], ["block_7", ["where [M] = [M]0 at t: 0.\n2.\nInstead of using 2\ufb02 \u2014(\nkt\n[HO\nt\n(3.54)\n"]], ["block_19", ["R.\n1/2\n[Pr]. (2\u20141;)\n(3.5.5)\n"]], ["block_20", [{"image_3": "107_3.png", "coords": [64, 399, 147, 449], "fig_type": "molecule"}]]], "page_108": [["block_0", [{"image_0": "108_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["This enables us to identify the apparent activation energy in Equation 3.5.7 with the difference in\nE* values for the various steps:\n"]], ["block_2", ["Equation 3.5.9 allows us to conveniently assess the effect of temperature variation on the rate of\npolymerization. This effect is considered in the following example.\n"]], ["block_3", ["Using typical activation energies from Table 3.2 through Table 3.4, estimate the percent change in\nthe rate of polymerization with a 1\u00b0C change in temperature at 50\u00b0C, for both thermally initiated\nand photoinitiated polymerization.\n"]], ["block_4", ["Figure 3.3\nLog\u2014log plots of RP versus concentration that confirm the kinetic order with reSpect to the\nconstituent varied. (a) Monomer (methyl methacrylate) concentration varied at constant initiator concentra\u2014\ntion. (Data from Sugirnura, T. and Minoura, Y., J. Polym. Sci, A-l, 2735, 1966.) (b) Initiator concentration\nvaried: AIBN in methyl methacrylate (data from Amett, L.M., J. Am. Chem. Soc., 74, 2027, 1952) and\nbenzoyl peroxide in styrene (data from Mayo, F.R., Gregg, R.A., and Matheson, M.S., J. Am. Chem. 806., 73,\n1691, 1951).\n"]], ["block_5", ["The mechanistic analysis of the rate of polymerization and the fact that the separate constants\nindividually follow the Arrhenius equations means that\n"]], ["block_6", ["Propagation\n93\n"]], ["block_7", ["Example 3.3\n"]], ["block_8", [{"image_1": "108_1.png", "coords": [39, 64, 223, 289], "fig_type": "figure"}]], ["block_9", ["T\n__\n03\n\u20180\n\u2018T\n"]], ["block_10", ["1|\u2014\n_\n_l\n\"\u2018\na\nE\ng\ns.\nv\nmn-\nTO.\n10 3\n10-5:\nx\nZ\n:\na:\n\u2018\nStyrene/benzoyl peroxide at 60\u00b0C\n"]], ["block_11", ["E\n10\u20144.\u2014\n,.\nI\n\u20197\"\n'\n"]], ["block_12", ["(a)\n[M] (mol L4)\n(b)\n[He (mol L\"\u2018)\n"]], ["block_13", ["E3;\n5*\nEgg, =E\u00a7+7\u20147\u2018\n(3.5.9)\n"]], ["block_14", [{"image_2": "108_2.png", "coords": [46, 425, 250, 503], "fig_type": "figure"}]], ["block_15", ["1/2\nln km, =ln k, (73,9)\n"]], ["block_16", [{"image_3": "108_3.png", "coords": [54, 426, 243, 498], "fig_type": "molecule"}]], ["block_17", ["100 \n"]], ["block_18", [{"image_4": "108_4.png", "coords": [60, 122, 200, 266], "fig_type": "figure"}]], ["block_19", ["\u20141\nfAd \nEg\u2018+E,\u00a7\u201c/2\u2014E;\"/2\n_ \u201cAP _\n_\nRT\n"]], ["block_20", ["llllll\nI\nIIIIIIII\nI\n10_6\n....|\n........|\n.......|\n........|\n"]], ["block_21", ["1-0\n10\n1041\n10-3\n10-2\n10*1\n"]], ["block_22", [{"image_5": "108_5.png", "coords": [168, 47, 452, 277], "fig_type": "figure"}]], ["block_23", ["10\u20143:\"\"'|\n'\nIIIII\"I\n'\nI\"\"\"l\n\u2018\n\"_'_'_'\"|\n'2'\n"]], ["block_24", [{"image_6": "108_6.png", "coords": [226, 78, 438, 226], "fig_type": "figure"}]], ["block_25", ["Methyl methacrylate/AIBN at 50\u00b0C\n"]], ["block_26", ["(3.5.8)\n"]]], "page_109": [["block_0", [{"image_0": "109_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["under conditions where an initiator yields one radical, where f1, and where the final polymer\ncontains one initiator fragment per molecule. For this set of conditions the ratio gives the number\nof monomers polymerized per chain initiated, which is the average degree of polymerization.\nA more general development of this idea is based on a quantity called the kinetic chain length 17.\nThe kinetic chain length is defined as the ratio of the number of propagation steps to the number of\ninitiation steps, regardless of the mode of termination:\n"]], ["block_2", ["or 1.90% per \u00b0C. Note that the initiator decomposition makes the largest contribution to E*;\ntherefore photoinitiated processes display a considerably lower temperature dependence for the\nrate of polymerization.\n"]], ["block_3", ["3.5.3\nKinetic Chain Length\n"]], ["block_4", ["Suppose we consider the ratio\n"]], ["block_5", ["Finally, we recognize that a 1\u00b0C temperature variation can be approximated as (17' and that\n(dRp/Rp) x 100 gives the approximate percent change in the rate of polymerization. Taking average\nvalues of Efrom the appropriate tables, we obtain E3; 145, E? 16.8, and E3 24.9 k] mol\u2018l.\nFor thermally initiated polymerization\n"]], ["block_6", ["or 10.3% per \u00b0C.\nFor photoinitiation there is no activation energy for the initiator decomposition; hence\n"]], ["block_7", ["Expand d In kapp by means of the Arrhenius equation via Equation 3.5.8:\n"]], ["block_8", ["Write Equation 3.5.3 in the form\n"]], ["block_9", ["Taking the derivative, treating [M] and [I] as constants with respect to T while k is a function of T:\n"]], ["block_10", ["Substitute Equation 3.5.9 for \u00a331,131\n"]], ["block_11", ["94\nChain-Growth Polymerization\n"]], ["block_12", ["Solution\n"]], ["block_13", [{"image_1": "109_1.png", "coords": [42, 631, 109, 678], "fig_type": "molecule"}]], ["block_14", ["Rp/Ri \nd[M]/dt\n\u2014d[I]/dt\n"]], ["block_15", ["d\ufb02 _ (24.9 + 145/2 16.8/2)(103)(1) = 0.103\nRP\n(8.314)(323)2\n"]], ["block_16", ["dRp (24.9 16.8/2)(103)(1)\nRp \n(8.314)(323)2\n= 1.90 x 10-2\n"]], ["block_17", ["dRp\nE?\nE:\n\u2014=d1Aa\n\u2014\u2014d _P =__PP\nRp\n\"\nPP\n(R75\nRTZdT\n"]], ["block_18", ["dRp E; + Iii/2 133/s\nRp \nRT2\n"]], ["block_19", ["dR\n(:11a =R\u2014P= drnk...pp\n"]], ["block_20", ["1\n1n RP = 1n kapp + In [M] + 2 In [I]\n"]], ["block_21", ["= _\n(3.5.10)\n"]], ["block_22", ["P\n"]]], "page_110": [["block_0", [{"image_0": "110_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["As with the rate of polymerization, we see from Equation 3.5.14 that the kinetic chain length\ndepends on the monomer and initiator concentrations and on the constants for the three different\nkinds of kinetic processes that constitute the mechanism. When the initial monomer and initiator\nconcentrations are used, Equation 3.5.14 describes the initial polymer formed. The initial degree of\npolymerization is a measurable quantity, so Equation 3.5.14 provides a second functional rela-\ntionship, distinct from Equation 3.5.3, among experimentally available quantities\u2014ND, [M], [I]\u2014\nand theoretically important parameters\u2014-\u2014kp, kt, and kd. Note that the mode of termination, which\nestablishes the connection between\n17 and Nn, and the value off are both accessible through\nendgroup characterization. Thus we have a second equation with three unknowns; one more\nindependent equation and the evaluation of the individual kinetic constants from experimental\nresults will be feasible.\nThere are several additional points about Equation 3.5.14 that are worthy of comment. First it\nmust be recalled that we have intentionally ignored any kinetic factors other than initiation,\npropagation, and termination. We shall see in Section 3.8 that another process, chain transfer,\nhas significant effects on the molecular weight of a polymer. The result we have obtained,\ntherefore, is properly designated as the kinetic chain length without transfer. A second observation\nis that 1'\u00bb depends not only on the nature and concentration of the monomer, but also on the nature\nand concentration of the initiator. The latter determines the number of different sites competing for\nthe addition of monomer, so it is not surprising that L7 is decreased by increases in either kd or [I].\nFinally, we observe that both kp and kt are properties of a particular monomer. The relative\nmolecular weight that a speci\ufb01c monomer tends toward\u2014all other things being equal\u2014is charac\u2014\nterized by the ratio ftp/kt\u201d2 for a monomer. Using the values in Table 3.3 and Table 3.4, we see that\nkp/kt\u201d2 equals 0.678 for methyl acrylate and 0.0213 for styrene at 60\u00b0C. The kinetic chain length for\npoly(methyl acrylate) is thus expected to be about 32 times greater than for polystyrene if the two\nare prepared with the same initiator (kd) and the same concentrations [M] and [1]. Extension of this\ntype of comparison to the degree of polymerization requires that the two polymers compared show\nthe same proportion of the modes of termination. Thus for vinyl acetate (subscript V) relative to\nacrylonitrile\n(subscript A)\nat 60\u00b0C, with\nthe\nsame\nprovisos\nas\nabove,\n{xv/17A 6 while\nv/NHA = 3 because of the differences in the mode of termination for the two.\n"]], ["block_2", ["where the uses the stationary-state condition R, =Rt. The \ncance of\n"]], ["block_3", ["2.\nFor termination by combination\n"]], ["block_4", ["3.\n17 is an average quantity\u2014indicated by the overbar\u2014since not all kinetic chains are identical\nany more than all molecular chains are.\n"]], ["block_5", ["Propagation\n95\n"]], ["block_6", ["1,\nFor termination by disproportionation\n"]], ["block_7", ["This may be combined with Equation 3.4.10 to give the stationary-state value for 17:\n"]], ["block_8", ["Using Equation 3.5.3 and Equation 3.4.4 for RP and Rt, respectively, we write\n"]], ["block_9", ["where N\" is the number average degree of polymerization.\n"]], ["block_10", ["17 =Nn\n(3.5.11)\n"]], ["block_11", ["Nn\nv 3\n(3.5.12)\n"]], ["block_12", ["k\nM\nk\nM\n17 :\np[\n]\np[\n]\n(3.5.14)\nmums/M\u201d\nZ awash\u201d:\n"]], ["block_13", ["\u2014 = kp[P\u00b0][M] _ kp[M]\n_\n3.5.13\nV\n2k.[Po 12\n2mm\n(\n)\n"]]], "page_111": [["block_0", [{"image_0": "111_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["In the preceding section we observed that both the rate of polymerization and the degree of\npolymerization under stationary-state conditions can be interpreted to yield some cluster of the\nconstants kp, k,, and kd. The situation is summarized diagrammatically in Figure 3.4. The circles at\nthe two bottom corners of the triangle indicate the particular grouping of constants obtainable from\nthe measurement of RF, or N\u201c, as shown. By combining these two sources of data in the manner\nsuggested in the boxes situated along the lines connecting these circles kd can be evaluated, as well\nas the ratio lag/kt. Using this stationary\u2014state data, however, it is not possible to further resolve the\npropagation and termination constants. Another relationship is needed to do this. A quantity called\nthe radical li\ufb01ztime 1\" supplies the additional relationship and enables us to move off the base of\nFigure 3.4.\nTo arrive at an expression for the radical lifetime, we return to Equation 3.5.1, which may be\ninterpreted as follows:\n"]], ["block_2", ["It is interesting to compare the application of this result to thermally initiated and photoinitiated\npolymerizations as we did in Example 3.3. Again using the average values of the constants from\nTable 3.2 through Table 3.4 and taking T= 50\u00b0C, we calculate that 17 decreases by about 6.5% per\n\u00b0C for thermal initiation and increases by about 2% per \u00b0C for photoinitiation. It is clearly the large\nactivation energy for initiator dissociation that makes the difference. This term is omitted in the\ncase of photoinitiation, where the temperature increase produces a bigger effect on propagation\nthan on termination. On the other hand, for thermal initiation an increase in temperature produces a\nlarge increase in the number of growth centers, with the attendant reduction of the average kinetic\nchain length.\nPhotoinitiation is not as important as thermal initiation in the overall picture of free-radical\nchain-growth polymerization. The foregoing discussion reveals, however, that the contrast\nbetween the two modes of initiation does provide insight into, and confirmation of, various aspects\nof addition polymerization. The most important application of photoinitiated polymerization is in\nproviding a third experimental relationship among the kinetic parameters of the chain mechanism.\nWe shall consider this in the next section.\n"]], ["block_3", ["3.6\nRadical Lifetime\n"]], ["block_4", ["The proviso \u201call other things being equal\u201d in discussing the last point clearly applies to\ntemperature as well, since the kinetic constants can be highly sensitive to temperature. To evaluate\nthe effect of temperature variation on the molecular weight of an addition polymer, we follow the\nsame sort of logic as was used in Example 3.3:\n"]], ["block_5", ["2.\nDifferentiate with respect to T, assuming the temperature dependence of the concentrations is\nnegligible compared to that of the rate constants:\n"]], ["block_6", ["96\nChain-Growth Polymerization\n"]], ["block_7", ["3.\nBy the Arrhenius equation (1 ln k = \u2014d(E*/RT) = (E*/ R72) dT; therefore\n"]], ["block_8", ["1.\n\u2014d[M]/dt gives the rate at which monomers enter polymer molecules. This, in turn, is given by\nthe product of number of growth sites, [Po], and the rate at which monomers add to each\n"]], ["block_9", ["1.\nTake logarithms of Equation 3.5.14:\n"]], ["block_10", ["cw __ 13;; Eat/2 Eat/2d],\n"]], ["block_11", ["9; dln k, 1 /2d 1h (ktkd)\n(3.5.16)\n"]], ["block_12", ["1n :7 = 1n kp(ktkd)_1/2 + 1n(%)\n(3.5.15)\n"]], ["block_13", ["I7 \nRT2\n(3.5.17)\n"]]], "page_112": [["block_0", [{"image_0": "112_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Figure 3.4\nSchematic relationship among the various experimental quantities (RP, NH, and 1\") and the rate\nconstants kd, kp, and kt derived therefrom.\n"]], ["block_2", ["Radical Lifetime\n97\n"]], ["block_3", ["Hp\n"]], ["block_4", ["\ufb02) = kd\nN\nkd\n%\n\"\nkp\n\u2014-\u2014>\u2014\nk\n\u2014\n1\n4\u2014 N\n\"\nkt\n(mo/2\n\u201d\n"]], ["block_5", ["The degree of polymerization in Equation 3.6.1 can be replaced with the kinetic chain length,\nand the resulting expression simplified. To proceed, however, we must choose between the\npossibilities described in Equation 3.5.11 and Equation 3.5.12. Assuming termination by\ndisproportionation, we replace N,n by 17, using Equation 3.5.14:\n"]], ["block_6", ["growth site. On the basis of Equation 3.5.1, the rate at which monomers add to a radical is\ngiven by kP[M].\nIf kp[M] gives the number of monomers added per unit time, then 1/kp[M] equals the time\nelapsed per monomer addition.\nIf we multiply the time elapsed per monomer added to a radical by the number of monomers in\nthe average chain, then we obtain the time during which the radical exists. This is the\ndefinition of the radical lifetime. The number of monomers in a polymer chain is, of course,\nthe degree of polymerization. Therefore we write\n"]], ["block_7", ["The radical lifetime is an average quantity, as indicated by the overbar.\n"]], ["block_8", ["kp [M]\n(3.6.1)\n7\u2014:\n"]], ["block_9", ["2(1\u20187ctt\u2019alll)\u201d2 \n_\n2(\ufb02ctkduD1/2\n(3.6.2)\nIr:\n"]], ["block_10", [{"image_1": "112_1.png", "coords": [75, 42, 313, 369], "fig_type": "figure"}]], ["block_11", ["Nn\n"]], ["block_12", ["kpiM]\n1\n_\n1\n"]], ["block_13", [{"image_2": "112_2.png", "coords": [89, 152, 292, 309], "fig_type": "figure"}]], ["block_14", ["+57 M\n1\nes,\n\u201c\\3,\n1rt\n\u00ab3\u2018\n"]], ["block_15", ["[91%\nkt\u2018/&\n"]], ["block_16", ["kn\np\n"]], ["block_17", ["RD Nn kpe/kt\n"]], ["block_18", ["1\u2014\n(ktkdiyz\n"]], ["block_19", ["?\nl\n"]]], "page_113": [["block_0", [{"image_0": "113_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["1\" = 0.5[(1.0)(2.9 x 107)(0.85 x 10\u20185)(10\u20193)]\u20181/2 = 1.01s. This figure contrasts sharply with the\ntime required to obtain high-molecular-weight molecules in step-growth polymerizations.\nSince the radical lifetime provides the final piece of information needed to independently\nevaluate the three primary kinetic constants\u2014remember, we are still neglecting chain transfer\u2014\nthe next order of business is a consideration of the measurement of f. A widely used technique for\nmeasuring radical lifetime is based on photoinitiated polymerization using a light source,\nwhich blinks on and off at regular intervals. In practice, a rotating opaque disk with a wedge\nsliced out of it is interposed between the light and the reaction vessel. Thus the system is in\ndarkness when the solid part of the disk is in the light path and is illuminated when the notch\npasses. With this device, called a rotating sector or chopper, the relative lengths of light and dark\nperiods can be controlled by the area of the notch, and the frequency of the \ufb02ickering by the\nvelocity of rotation of the disk. We will not describe the rotating sector experiments in detail. It is\nsufficient to note that, with this method, the rate of photoinitiated polymerization is studied as a\nfunction of the time of illumination with the rapidly blinking light. The results show the rate of\npolymerization dropping from one plateau value at slow blink rates (\u201clong\u201d bursts of illumination)\nto a lower plateau at fast blink rates (\u201cshort\u201d periods of illumination). A plot of the rate of\npolymerization versus the duration of an illuminated interval resembles an acid\u2014base titration\ncurve with a step between the two plateau regions. Just as the \u201cstep\u201d marks the end point of a\ntitration, the \u201cstep\u201d in rotating sector data identifies the transition between relatively long and\nshort periods of illumination. Here is the payoff: \u201clong\u201d and \u201cshort\u201d times are defined relative to\nthe average radical lifetime. Thus f may be read from the time axis at the midpoint of the transition\nbetween the two plateaus.\nThis qualitative description enables us to see that the radical lifetime described by Equation\n3.6.2 is an experimentally accessible quantity. More precise values of i may be obtained by curve\nfitting since the nonstationary-state kinetics of the transition between plateaus have been analyzed\nin detail. To gain some additional familiarity with the concept of radical lifetime and to see how\nthis quantity can be used to determine the absolute value of a kinetic constant, consider the\nfollowing example:\n"]], ["block_2", ["We can use the constants tabulated elsewhere in the chapter to get an idea of a typical\nradical lifetime. Choosing 10\u201c3 mol L\u20181 AIBN as the initiator (kd=0.85 x 10\u20145 s\u201c at 60\u00b0C)\nand vinyl acetate as the monomer (terminates entirely by disproportionation, k,=2.9 x 107 L\nmol\u2019l\ns\u20181\nat\n60\u00b0C),\nand\ntaking f=1\nfor\nthe\npurpose\nof\ncalculation,\nwe\nfind\n"]], ["block_3", ["The polymerization of ethylene at 130\u00b0C and 1500 atm was studied using different concentrations\nof the initiator, 1-t-butylazo-1-phenoxycyclohexane. The rate of initiation was measured directly\n"]], ["block_4", ["1.\nIn going from the experimental quantities RP, N\u201c, and f to the associated clusters of kinetic\nconstants, it has been assumed that the monomer and initiator concentrations are known and\nessentially constant. In addition, the efficiency factorf has been left out, the assumption being\nthat still another type of experiment has established its value.\n2.\nBy following the lines connecting two sources of circled information, the boxed result in the\nperimeter of the triangle may be established. Thus kl, is evaluated from f and N\u201c.\n3.\nHere kp can be combined with one of the various kP/kt ratios to permit the evaluation of k,.\n"]], ["block_5", ["We shall see presently that the lifetime of a radical can be measured. When such an experiment is\nconducted with a known concentration of initiator, then the cluster of constants (ktkd)_u2 can be\nevaluated. This is indicated at the apex of the triangle in Figure 3.4.\nThere are several things about Figure 3.4 that should be pointed out:\n"]], ["block_6", ["98\nChain-Growth Polymerization\n"]], ["block_7", ["Example 3.4\n"]]], "page_114": [["block_0", [{"image_0": "114_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "114_1.png", "coords": [27, 335, 177, 428], "fig_type": "figure"}]], ["block_2", [{"image_2": "114_2.png", "coords": [32, 83, 194, 170], "fig_type": "figure"}]], ["block_3", ["We begin by writing a kinetic expression for the concentration of radicals of the degree of\npolymerization i, which we designate [Pp]. This rate law will be the sum of three contributions:\n"]], ["block_4", ["Until this point in the chapter we have intentionally avoided making any differentiation among\nradicals on the basis of the degree of polymerization of the radical. Now we seek a description of\nthe molecular weight distribution of addition polymer molecules. Toward this end it becomes\nnecessary to consider radicals with different i values.\n"]], ["block_5", ["Even though the rates of initiation span almost a 10-fold range, the values of kt show a standard\ndeviation of only 4%. which is excellent in view of the inevitable experimental errors. Note that the\nrotating sector method can be used in high-pressure experiments and other unusual situations, a\nhighly desirable characteristic it shares with many optical methods in chemistry.\n"]], ["block_6", ["If the data follow the kinetic scheme presented here, the values of kt calculated for the different\nruns should be constant:\n"]], ["block_7", ["and radical lifetimes using the rotating sector method. The following results were\n.\nobtained.1\u2018\n1\nRun\n1\"(s)\nR,x109(molL\u2014ls\u20141)\n"]], ["block_8", ["3.7.1\nDistribution of i-mers: Termination by Disproportionation\n"]], ["block_9", ["3.7\nDistribution of Molecular Weights\n"]], ["block_10", ["tData from T. Takahashi and P. Ehrlich, Polym. Prepr. Am. Chem. Soc. Polym. Chem. Div., 22, 203 (1981).\n"]], ["block_11", ["Since the rate of initiation is measured, we can substitute R, for the terms (kdm)\n3.6.2 to give\n"]], ["block_12", ["Run\n1ct X 10\u20143 (L mol\u20141 3'1)\n"]], ["block_13", ["13\n3.98\nAverage\n3.89\n"]], ["block_14", ["Demonstrate that the variations in the rate of initiation and\nF? are consistent with free-radical\nkinetics, \n"]], ["block_15", ["Solution\n"]], ["block_16", ["Distribution of Molecular Weights\n99\n"]], ["block_17", ["12\n4.00\n"]], ["block_18", ["12\n0.50\n5.00\n"]], ["block_19", ["13\n0.29\n14.95\n"]], ["block_20", ["5\n3.99\n"]], ["block_21", ["6\n3.64\n"]], ["block_22", ["8\n3.83\n"]], ["block_23", ["5\n0.73\n2.35\n6\n0.93\n1.59\n"]], ["block_24", ["8\n0.32\n12.75\n"]], ["block_25", ["\u2019 \u2014\n1\nor\nk \n1\nT (zany/2\n\u2018 2am,\n"]], ["block_26", ["1\u20192 in Equation\n"]]], "page_115": [["block_0", [{"image_0": "115_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "115_1.png", "coords": [28, 554, 199, 588], "fig_type": "molecule"}]], ["block_2", ["The change in [P3] under stationary\u2014state conditions equals zero for all values of 1'; hence we\n"]], ["block_3", ["The ratio [P1-]/[P-] in Equation 3.7.8 can be eliminated by applying Equation 3.7.1 explicitly to the\n"]], ["block_4", ["Dividing both sides of Equation 3.7.7 by [P-], the total radical concentration, gives the number\n(or mole) fraction of i-rner radicals in the total radical population. This ratio is the same as the\nnumber of i-mers n,- in the sample containing a total of n (no subscript) polymer molecules:\n"]], ["block_5", ["1.\nAn increase that occurs by addition of monomer to the radical P,-_1-\n2.\nA decrease that occurs by addition of a monomer to the radical P).\n3.\nA decrease that occurs by the termination of P,\u00bb with any other radical P-\n"]], ["block_6", ["or\n"]], ["block_7", ["This shows that the number of i-mer radicals relative to the number of the smallest radicals is given\nby multiplying the ratio [P,~]/ [P,-_1\u00b0] by i 2 analogous ratios. Since each of the individual ratios\nis given by 17/(1 + 17), we can rewrite Equation 3.7.4 as\n"]], ["block_8", ["P1 radical:\n"]], ["block_9", ["Since it is more convenient to focus attention on i-mers than (i\u20141)\u2014mers, the corresponding\nexpression for the i\u2014mer is written by analogy:\n"]], ["block_10", ["can write\n"]], ["block_11", ["Dividing the numerator and denominator of Equation 3.7.2 by 2kt [Po] and recalling the de\ufb01nition\nof 17 provided by Equation 3.5.13 enables us to express this result more succinctly as\n"]], ["block_12", ["Next let us consider the following sequence of multiplications:\n"]], ["block_13", ["100\nChain-Growth Polymerization\n"]], ["block_14", ["which can be rearranged to\n"]], ["block_15", ["1.\nWrite Equation 3.7.1 for Pp, remembering in this case that the leading term describes initiation:\n"]], ["block_16", [{"image_2": "115_2.png", "coords": [37, 358, 154, 396], "fig_type": "molecule"}]], ["block_17", [{"image_3": "115_3.png", "coords": [42, 409, 170, 446], "fig_type": "molecule"}]], ["block_18", ["Mn.-_[P.--1_[P1-1\n:7\n\u201c1\nx\u2018 \u2018Z\" [13-] [P-] (1+ :7)\n(3'7\u20188)\n"]], ["block_19", ["d[P.-'l\n \u201821? \n(311)\n"]], ["block_20", ["[Pi\u20141\"]\nh mm + 2kt[P-]\n(3.7.2)\n"]], ["block_21", ["Pi'\nPi\u2014'\nPi\u2014'\nPi\u2014i\u2014\n\u00b0\nPr'\n[ ll\n1][\n2]_\u201d[\n(2)]_[\nl\n(314)\n[Pi\u20141'] [Pi\u20142'] [Pi\u20143\u00b0]\n[Pi\u2014(i\u2014l)'] _[P1']\n"]], ["block_22", ["_\ni\u2014l\n[Pr] [Pi-1 (\ufb01g)\n(3.7.7)\n"]], ["block_23", ["[Pg\u20141'] \n(3.7.6)\n"]], ["block_24", ["[[151]] [15:] (1:12)\n(3.7.5)\n"]], ["block_25", ["[Pr]\nI7\n:\n3.7.3\n[Par]\n1 + 17\n(\n)\n"]], ["block_26", ["[Pi-1 _\nkp[M]\n"]], ["block_27", ["93;? = Ri kp[M][P1-] \u2014 2kt[P1-][P-] : 0\n(3.7.9)\n"]], ["block_28", ["17\n0\u20140\u20141\n"]], ["block_29", ["_\ni\u20142\n"]]], "page_116": [["block_0", [{"image_0": "116_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["This change of notation now expresses Equation 3.7.14 in exactly the same form as its equivalent\nin Section 2.4. In other words, the distribution of chain lengths is the most probable distribution,\njust as was the case for step-growth polymerization! Several similarities and differences should be\nnoted in order to take full advantage of the parallel between this result and the corresponding\nmaterial for condensation polymers in Chapter 2:\n"]], ["block_2", ["This expression gives the number fraction or mole fraction, x,, of i-mers in the polymer and is thus\nequivalent to Equation 2.4.2 for step-growth polymerization.\nThe kinetic chain length 13 may also be viewed as merely a cluster of kinetic constants and\nconcentrations, which was introduced into Equation 3.7.13 to simplify the notation. As an\nalternative, suppose we define for the purposes of this chapter a fraction p such that\n"]], ["block_3", ["4,\nRearrange under stationary\u2014state conditions:\n"]], ["block_4", ["1.\nIn Chapter 2, p was defined as the fraction (or probability) of functional groups that had\nreacted at a certain point in the polymerization. According to the current definition provided\nby Equation 3.7.15, p is the fraction (or probability) of propagation steps among the com-\nbined total of propagation and termination steps. The quantity 1 p is therefore the fraction\n(or probability) of termination steps. An addition polymer with the degree of polymerization\ni has undergone i 1 propagation steps and one termination step. Therefore it makes sense to\ndescribe its probability in the form of Equation 3.7.16.\n2.\nIt is apparent from Equation 3.7.15 that p \u20149\n1 as\n17 \u2014> oo; hence those same conditions\nthat favor the formation of a high-molecular-weight polymer also indicate p values close\nto unlty.\n"]], ["block_5", ["It follows from this definition that 1/(1 + 17) 1 [9, so Equation 3.7.14 can be rewritten as\n"]], ["block_6", ["2,\nRearrange under stationary\u2014state conditions:\n"]], ["block_7", ["3,\nWrite Equation 3.4.9 using the same notation for initiation as in Equation 3.7.9:\n"]], ["block_8", ["5.\nTake the ratio of Equation 3.7.10 to Equation 3.7.12:\n"]], ["block_9", ["Distribution of \n101\n"]], ["block_10", ["\u201di\nr\u2014r\nx,- 2;:\n(1 \u2014p)p\n(3.7.16)\n"]], ["block_11", [":7\n_\nkp[M]\n1+ :7 \nkp1M1+ 2k11P-1\np 2\n(3.7.15)\n"]], ["block_12", [{"image_1": "116_1.png", "coords": [52, 294, 230, 326], "fig_type": "molecule"}]], ["block_13", ["n,-\n1\nr\u00bb\nH\n1\n:7\n\"\n;=\u2014=\n_\n=\u2014\n.z\nx\nn\n1+5<1+v>\na<1+v>\n(3\n14)\n"]], ["block_14", ["Egj=Ri \u20142kt[P-]2:0\n(3.7.11)\n"]], ["block_15", ["P \n2k P-\n1\n\u2014\u2014[\n1 ]\u2014\n\u2018[\n1\n(3.7.13)\n[P-1 \nkp1M1+ \n1 + 17\n"]], ["block_16", ["Combining Equation 3.7.13 with Equation 3.7.8 gives\n"]], ["block_17", ["The total radical concentration under stationary-state conditions can be similarly obtained.\n"]], ["block_18", ["R1\n' =\n3.7.12\n[P 1\nM1,]\n(\n)\n"]], ["block_19", ["R1\n[Pl'] kptM1+ 2k11P-1\n(3.7.10)\n"]]], "page_117": [["block_0", [{"image_0": "117_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "117_1.png", "coords": [26, 464, 178, 506], "fig_type": "molecule"}]], ["block_2", ["3.\nIn Chapter 2 all molecules\u2014whether monomer or i-mers of any i\u2014carry functional groups;\nhence the fraction described by Equation 2.4.] applies to the entire reaction mixture. Equation\n3.7.16, by contrast, applies only to the radical population. Since the radicals eventually end up\nas polymers, the equation also describes the polymer produced. Unreacted monomers are\nspecifically excluded, however.\n4.\nOnly one additional stipulation needs to be made before adapting the results that follow from\nEquation 2.4.1 to addition polymers. The mode of termination must be specified to occur by\ndisproportionation to use the results of Section 2.4 in this chapter, since termination by\ncombination obviously changes the molecular weight distribution. We shall return to the\ncase of termination by combination presently.\n5.\nFor termination by disproportionation (subscript d), we note thatp kp[M]/(kP[M] + 2k d[ P D,\nand therefore by analogy with Equation 2.4.5, Equation 2.4.9, and Equation 2.4.10,\n"]], ["block_3", ["To deal with the case of termination by combination, it is convenient to write some reactions by\nwhich an i-mer might be formed. Table 3.5 lists several specific chemical reactions and the\ncorresponding rate expressions as well as the general form for the combination of an (i j)-mer\nand a j-mer. On the assumption that all km, values are the same, we can write the total rate of\nchange of [Pi]:\n"]], ["block_4", ["The\nfraction\nof\ni-mers\nformed\nby\ncombination\nmay\nbe\nevaluated\nby\ndividing\nd[P,-]/dt by Z, d[P,-]/dt. Assuming that termination occurs exclusively by combination, then\n"]], ["block_5", ["3.7.2\nDistribution of i-mers: Termination by Combination\n"]], ["block_6", ["and the number or mole fraction of i\u2014mers formed by combination (subscript c) is\n"]], ["block_7", ["Equation 3.7.16 can be used to relate [Pi\u20141'] and [Pl-o] to the total radical concentration:\n"]], ["block_8", ["102\nChain-Growth Polymerization\n"]], ["block_9", ["[Pr\u2014f] (1 \u2014p)p\"\u201c\u201d-\u2018[Po]\n(3.7.23)\n"]], ["block_10", ["(3.7.22)\n (iii): \nd[P;]/dt\nzktacz,;:[Pt-j'][Prl\n"]], ["block_11", ["d P,-\ni\u201c\n(El\u2014t1) = kt, Z; [Pr\u20141\"][PJ\u2018]\n(3720)\n[0!\nj:\n"]], ["block_12", ["By virtue of Equation 3.7.15, (Nn)a can also be written as\n1 + 17 \u201c=-\u2019 i! for large 17, which is\nthe result already obtained in Equation 3.5.11. Figure 2.5 and Figure 2.6 also describe the\ndistribution by number and weight of addition polymers, if the provisos enumerated above\nare applied.\n"]], ["block_13", ["1\n(Nn)d\n- to\n(3.7.17)\n"]], ["block_14", ["1\n(Nw)d = _1__'_\"_P\n(3.7.18)\n\u2018P\n"]], ["block_15", ["1'\n"]], ["block_16", ["c\nEdna-yd:\nkart\u00bb?\nl1\n"]], ["block_17", ["Nw\n(\u2014)=1+p\u2014>2\nas\np\u2014>1\n(3.7.19)\nNn\nd\n"]], ["block_18", ["(1 P,-\n[dt] \n(3.7.21)\n"]], ["block_19", [{"image_2": "117_2.png", "coords": [72, 587, 239, 620], "fig_type": "molecule"}]]], "page_118": [["block_0", [{"image_0": "118_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "118_1.png", "coords": [11, 87, 278, 204], "fig_type": "figure"}]], ["block_2", [{"image_2": "118_2.png", "coords": [28, 384, 180, 415], "fig_type": "molecule"}]], ["block_3", ["Table 3.5\nSome Free-Radical Combination Reactions\n"]], ["block_4", ["Which \n"]], ["block_5", ["Therefore\n"]], ["block_6", ["The indexj drops out of the last summation; we compensate for this by multiplying the final result\nby i\u2014l in recognition of the fact that the summation adds up i\u2014l identical terms. Accordingly, the\ndesired result is obtained:\n"]], ["block_7", ["This expression is plotted in Figure 3.5 for several values ofp near unity. Although it shows the\nnumber distribution of polymers terminated by combination, the distribution looks quite different\nfrom Figure 2.5, which describes the number distribution for termination by disproportionation. In\nthe latter x,- decreases monotonically with increasing 1'. With combination, however, the curves go\nthrough a maximum, which re\ufb02ects the fact that the combination of two very small or two very\nlarge radicals is a less probable event than a more random combination.\nExpressions for the various averages are readily derived from Equation 3.7.26 by procedures\nidentical to those used in Section 2.4 (see Problem 6). We only quote the final results for the case\nwhere termination occurs exclusively by combination:\n"]], ["block_8", ["These various expressions differ from their analogs in the case of termination by disproportiona-\ntion by the appearance of occasional 2\u2019s. These terms arise precisely because two chains are\ncombined\nin\nthis\nmode\nof\ntermination.\nAgain\nusing\nEquation\n3.7.15,\nwe\nnote\nthat\n(Nu)c 2(1\n\u2014|\u2014 17) 9\u2014: 217 for large 17, a result that was already given as Equation 3.5.12.\n"]], ["block_9", ["Reaction\nRate law\n"]], ["block_10", [",\n.\no\n\u2014>\n.\nCl Pf\nPM \n1:\"\n%=k.,c[P.-_1.][Pr-i\n"]], ["block_11", ["d P,\nP;_2\u00b0 + Pz\u2018 Pr\n% kt,c[Pi\u20142\u00b0][P2\u00b0]\n"]], ["block_12", ["d P,-\nP,-_3\u00b0 + P3\u00b0 Pr\n% kt,c[Pi\u20143\u00b0][P3\u00b0]\n"]], ["block_13", [":\nd P:\n:\nP54 + P,\u00bb P.-\n% kr,c[P.-_j\u00b0][Py]\n"]], ["block_14", ["and\n"]], ["block_15", [{"image_3": "118_3.png", "coords": [35, 264, 271, 315], "fig_type": "molecule"}]], ["block_16", ["Distribution \n103\n"]], ["block_17", ["2\n(Nw)c 1\u2014H\n(3.7.28)\n\u2014 P\n"]], ["block_18", ["2\n(Nn)c If\n(3.7.27)\n"]], ["block_19", ["[Pj'] (1 p)p0\u2018\u201d[P'1\n(3.7.24)\n"]], ["block_20", ["(3) =x,- (r 1)(1 10)?\u201d\n(3.7.26)\nn\nc\n"]], ["block_21", ["Nw\n_ \n(N\u2014nl\u2014\n2\n(3.7.29)\n"]], ["block_22", ["(m)\nk... 2:1 (1 mp\u2018 1[P\u00b0](1 mp! l[Pu\n"]], ["block_23", ["n c\n\u2014\n1c.,.,[i?\u00ab]2\n"]], ["block_24", ["f\u2014l\n(3.7.25)\n= Z (1 mid\u20142\nj=1\n"]], ["block_25", ["i\u2014l\n-_-_\n._\n"]]], "page_119": [["block_0", [{"image_0": "119_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "119_1.png", "coords": [31, 55, 273, 247], "fig_type": "figure"}]], ["block_2", [{"image_2": "119_2.png", "coords": [32, 357, 185, 387], "fig_type": "molecule"}]], ["block_3", ["Figure 3.5\nMole fraction of i-mers as a function of i for termination by combination, according to Equation\n3.7.26, for various values of p.\n"]], ["block_4", ["One rather different result that arises from the case of termination by combination is seen by\nexamining the limit of Equation 3.7.29 for large values of p:\n"]], ["block_5", ["This contrasts with a limiting ratio of 2 for the case of termination by disproportionation. Since Mn\nand MW can be measured, the difference is potentially a method for determining the mode of\ntermination in a polymer system. In most instances, however, termination occurs by some\nproportion of both modes. Furthermore, other factors in the polymerization such as transfer,\nautoacceleration, etc., will also contribute to the experimental molecular weight distribution, so\nin general it is risky to draw too many conclusions about mechanisms from the measured\ndistributions. Also, we have used p and 17 to describe the distribution of molecular weights, but\nit must be remembered that these quantities are de\ufb01ned in terms of various concentrations and\ntherefore change as the reactions proceed. Accordingly, the results presented here are most simply\napplied at the start of the polymerization reaction when the initial concentrations of monomer and\ninitiator can be used to evaluate p or 17.\n"]], ["block_6", ["The three-step mechanism for free-radical polymerization represented by Reaction (3.A) through\nReaction (3.C) does not tell the whole story. Another type of free\u2014radical reaction, called chain\ntransfer, may also occur. This is unfortunate in the sense that it further complicates the picture\npresented so far. On the other hand, this additional reaction can be turned into an asset in\nactual polymer practice. One consequence of chain transfer reactions is a lowering of the kinetic\nchain length and hence the molecular weight of the polymer, without necessarily affecting the\nrate of polymerization. A certain minimum average molecular weight is often needed to achieve\na desired physical property, but further increases in chain length simply make processing\nmore dif\ufb01cult.\n"]], ["block_7", ["3.8\nChain Transfer\n"]], ["block_8", ["104\nChain-Growth Polymerization\n"]], ["block_9", ["0,004llll\u2018l\u2018l\u2014I\u2014IIIIIIIIIIIIIIIIIIIIII\n"]], ["block_10", ["IIIIIIIIII'IIIIIIIIIIIIIIIIIII\n0.001\n"]], ["block_11", ["_,><\n"]], ["block_12", ["N\u201c.\n2+1\n_)_....\u2014\nNn\n2\n21.5\nas\np\u2014>l\n(3.7.30)\n"]], ["block_13", ["0\n200\n.\n400\n600\nt\n"]], ["block_14", ["p 0.995\n"]]], "page_120": [["block_0", [{"image_0": "120_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["involves Chain transfer \nmolecule in system. reactions specifically describe transfer to initiator, mono-\nmer, solvent, and respectively:\n"]], ["block_2", ["If the rate constant is comparable to kp, the substitution of a polymer radical with a new radical\nhas little or no effect on the rate of polymerization. If kR << kp, the rate of polymerization will be\ndecreased, or even effectively suppressed by chain transfer.\nThe kinetic chain length acquires a slightly different definition in the presence of chain transfer.\nInstead of being simply the ratio Rp/Rt, it is redefined to be the rate of propagation relative to the\nrates of all other steps that compete with propagation; specifically, termination and transfer\n(subscript tr):\n"]], ["block_3", ["4.\nTransfer to polymer, PJ-X:\n"]], ["block_4", ["designated Thus if\n"]], ["block_5", ["The transfer reactions follow second\u2014order kinetics, the general rate law being\n"]], ["block_6", ["3.8.1\nChain Transfer Reactions\n"]], ["block_7", ["Chain transfer arises orsome other atom X is transferred from a molecule in the\nsystem to the growth the original radical but it \n"]], ["block_8", ["a new one: the fragment species from which X was extracted. These latter molecules will be\n"]], ["block_9", ["It is apparent from these reactions how chain transfer lowers the molecular weight of a chain-\ngrowth polymer. The effect of chain transfer on the rate of polymerization depends on the rate at\nwhich the new radicals reinitiate polymerization:\n"]], ["block_10", ["chain Transfer\n105\n"]], ["block_11", ["2.\nTransfer to monomer, MX:\n"]], ["block_12", ["3.\nTransfer to solvent, SX:\n"]], ["block_13", ["where ktr is the rate constant for chain transfer to a specific compound RX. Since chain transfer can\noccur with several different molecules in the reaction mixture, Equation 3.8.1 becomes\n"]], ["block_14", ["5.\nGeneral transfer to RX:\n"]], ["block_15", ["1.\nTransfer to initiator, IX:\n"]], ["block_16", ["Ru kH[P-][RX]\n(3.8.2)\n"]], ["block_17", ["kP\nR- + M 3 RM- \u2014+\u2014>\u2014> RP,--\n(3.P)\n"]], ["block_18", ["RP\n\u2018\nr \n3.8.1\nVt\nRt +RH\n(\n)\n"]], ["block_19", ["a. kplP'HMl/l 2am? + ktr,1[P-][IX] + ku,M[P\u00b0][MX] + ku,s[P'][SX] + km. [Para-X1}\n2\narm\n2d-1 + Z kmalRX]\n"]], ["block_20", ["Pi- + IX \u2014> + I-\n(3.K)\n"]], ["block_21", ["P;- + SX \u2014> PgX \u2014|\u2014 8-\n(3M)\n"]], ["block_22", ["P; + MX \u2014> Pix \u2014|\u2014 M-\n(3L)\n"]], ["block_23", ["P,- + PjX \u2014> PIX + Pj'\n(3N)\n"]], ["block_24", ["Pr + RX Pix + R-\n(3.0)\n"]], ["block_25", ["(3.8.3)\n"]]], "page_121": [["block_0", [{"image_0": "121_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "121_1.png", "coords": [25, 258, 158, 303], "fig_type": "molecule"}]], ["block_2", [{"image_2": "121_2.png", "coords": [32, 82, 168, 123], "fig_type": "molecule"}]], ["block_3", ["Since the first term on the right-hand side is the reciprocal of the kinetic chain length in the absence\nof transfer, this becomes\n"]], ["block_4", ["This notation is simplified still further by defining the ratio of constants\n"]], ["block_5", ["It is apparent from this expression that the larger the sum of chain transfer terms becomes, the\nsmaller will be it\u201d.\nThe magnitude of the individual terms in the summation depends on both the speci\ufb01c chain\ntransfer constants and the concentrations of the reactants under consideration. The former are\ncharacteristics of the system and hence quantities over which we have little control; the latter can\noften be adjusted to study a particular effect. For example, chain transfer constants are generally\nobtained under conditions of low conversion to polymer where the concentration of polymer is low\nenough to ignore the transfer to polymer. We shall return below to the case of high conversions\nwhere this is not true.\n"]], ["block_6", ["If an experimental system is investigated in which only one molecule is significantly involved in\ntransfer, then the chain transfer constant to that material is particularly straightforward to obtain. If\nwe assume that species SX is the only molecule to which transfer occurs, Equation 3.8.7 becomes\n"]], ["block_7", ["This suggests that polymerizations should be conducted at different ratios of [SX]/[M] and the\nresulting molecular weight measured for each. Equation 3.8.8 indicates that a plot of 1 \ufb02y, versus\n[SX]/[M] should be a straight line with s10pe CSX. Figure 3.6 shows this type of plot for the\npolymerization of styrene at 100\u00b0C in the presence of four different solvents. The fact that all show\n"]], ["block_8", ["a common intercept as required by Equation 3.8.8 shows that the rate of initiation is unaffected by\nthe nature of the solvent. The following example examines chain transfer constants evaluated in\nthis situation.\n"]], ["block_9", ["Estimate the chain transfer constants for styrene to isopropylbenzene, ethylbenzene, toluene, and\nbenzene from the data presented in Figure 3.6. Comment on the relative magnitude of these\nconstants in terms of the structure of the solvent molecules.\n"]], ["block_10", ["where the summation is over all pertinent RX species. It is instructive to examine the reciprocal of\nthis quantity:\n"]], ["block_11", ["which is called the chain transfer constant for the monomer in question to molecule RX:\n"]], ["block_12", ["3.8.2\nEvaluation of Chain Transfer Constants\n"]], ["block_13", ["106\nChain-Growth Polymerization\n"]], ["block_14", ["Example 3.5\n"]], ["block_15", [{"image_3": "121_3.png", "coords": [46, 155, 148, 192], "fig_type": "molecule"}]], ["block_16", ["1\n1\n[RX]\n\u2014 = \u2014\nC\n\u2014~\u2014\u2014\n3.8.7\na.\nr\u00bb Z\nRX [M]\n(\n)\nall RX\n"]], ["block_17", ["klt_,R: CRx\n(3.8.6)\nk1:\n"]], ["block_18", ["17\u201c\nf) [M]\n(3.8.8)\n"]], ["block_19", ["1 _1+Zktr,R[RXI\n(3 5)\n17,,\na\nkp[M]\n'\n\u2018\n"]], ["block_20", [".\nk,\nRX\n_i=2kI[P]+Z\n,R[\n]\n(3.8.4)\nVtr\nkl]\nkl]\n"]], ["block_21", ["1\n1\n[SX]\n"]]], "page_122": [["block_0", [{"image_0": "122_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "122_1.png", "coords": [34, 42, 259, 247], "fig_type": "figure"}]], ["block_2", ["In certain commercial processes it is essential to regulate the molecular weight of the polymer either\nfor ease of processing or because low molecular weight products are desirable for particular applica-\ntions such as lubricants or plasticizers. In such cases the solvent or chain transfer agent is chosen and its\nconcentration selected to produce the desired value of \ufb01n. Certain mercaptans have particularly large\nchain transfer constants for many common monomers and are especially useful for molecular weight\nregulation. For example, styrene has a chain transfer constant for n-butyl mercaptan equal to 21 at 60\u00b0C.\nThis is about 107 times larger than the chain transfer constant to benzene at the same temperature.\nChain transfer to initiator or monomer cannot always be ignored. It may be possible, however,\nto evaluate these transfer constants by conducting a similar analysis on polymerizations without\nadded solvent or in the presence of a solvent for which Csx is known to be negligibly small. Fairly\n"]], ["block_3", ["cSX x 104\n2.08\n1.38\n0.55\n0.16\n"]], ["block_4", ["The relative magnitudes of these constants are consistent with the general rule that benzylic\nhydrogens are more readily abstracted than those attached directly to the ring. The reactivity of\nthe benzylic hydrogens themselves follows the order tertiary > secondary > primary, which is a\nwell\u2014established order in organic chemistry. The benzylic radical resulting from hydrogen abstrac-\ntion is resonance stabilized. For toluene, as an example,\n"]], ["block_5", ["SX\nl\"(:31'I'KCsHs)\nC2H5(C6Hs)\nCH3(C6H5)\nH(C6Hs)\n"]], ["block_6", ["The chain transfer constants are given by Equation 3.8.8 the slopes of the lines in Figure 3.6.\nThese are estimated to be as follows (note that X H in this case):\n"]], ["block_7", ["Figure 3.6\nEffect of chain transfer to solvent according to Equation 3.8.8 for polystyrene at 100\u00b0C.\nSolvents used were ethylbenzene isopropylbenzene (O), (A), and benzene (D). (Data from\nGregg, R.A. and Mayo, F.R., Faraday Soc., 2, 328, 1947. permission).\n"]], ["block_8", ["Chain Transfer\n107\n"]], ["block_9", ["Solution\n"]], ["block_10", [{"image_2": "122_2.png", "coords": [37, 42, 148, 235], "fig_type": "figure"}]], ["block_11", ["X105\n"]], ["block_12", ["Nn\n"]], ["block_13", ["H\nH\nH\nH\nH\nH\nH\nH\n"]], ["block_14", ["200\n"]], ["block_15", ["150\n"]], ["block_16", ["100\n"]], ["block_17", ["50\n"]], ["block_18", [{"image_3": "122_3.png", "coords": [98, 481, 296, 527], "fig_type": "molecule"}]], ["block_19", ["[SX]I[M]\n"]]], "page_123": [["block_0", [{"image_0": "123_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["This procedure is used commercially to produce rubber-modi\ufb01ed or high impact polystyrene (HIPS).\nThe polybutadiene begins to segregate from the styrene as it polymerizes (see Chapter 7 to learn\nwhy), but is prevented from undergoing macroscopic phase separation due to the covalent linkages to\npolystyrene chains. Consequently, small (micron\u2014sized) domains of polybutadiene rubber are\ndistributed throughout the glassy polystyrene matrix. These \u201crubber balls\u201d are able to dissipate\nenergy effectively (see Chapter 10 and Chapter 12), and counteract the brittleness of polystyrene.\nA second example of chain transfer to polymer is provided by the case of polyethylene. In this\ncase the polymer product contains mainly ethyl and butyl side chains. At high conversions such\nside chains may occur as often as once every 15 backbone repeat units on the average. These short\nside chains are thought to arise from transfer reactions with methylene hydrogens along the same\npolymer chain. This process is called backbiting and reminds us of the stability of rings of certain\nsizes and the freedom of rotation around unsubstituted bonds:\n"]], ["block_2", ["As noted above, chain transfer to polymer does not interfere with the determination of other transfer\nconstants, since the latter are evaluated at low conversions. In polymer synthesis, however, high\nconversions are desirable and extensive chain transfer can have a dramatic effect on the properties of\nthe product. This comes about since chain transfer to polymer introduces branching into the product:\n"]], ["block_3", ["extensive tables of chain transfer constants have been assembled on the basis of investigations of\nthis sort. For example, the values of CM}; for acrylamide at 60\u00b0C is 6 X 10\u2018s, and that for vinyl\nchloride at 30\u00b0C is 6.3 X 1041. Likewise, for methyl methacrylate at 60\u00b0C, CD; is 0.02 to benzoyl\nperoxide and 1.27 to t-butyl hydroperoxide.\n"]], ["block_4", [{"image_1": "123_1.png", "coords": [34, 179, 254, 212], "fig_type": "molecule"}]], ["block_5", ["A moment\u2019s re\ufb02ection reveals that the effect on 17 of transfer to polymer is different from the\neffects discussed above inasmuch as the overall degree of polymerization is not decreased by such\ntransfers. Investigation of chain transfer to polymer is best handled by examining the extent of\nbranching in the product. We shall not pursue the matter of evaluating the transfer constants, but\nshall consider describing two important specific examples of transfer to polymer.\nRemember from Section 1.3 that graft copolymers have polymeric side chains that differ in the\nnature of the repeat unit from the backbone. These can be prepared by introducing a prepolymer-\nized sample of the backbone polymer into a reactive mixture\u2014i.e., one containing a source of free\nradicals\u2014of the side-chain monomer. As an example, consider introducing 1,4-polybutadiene into\na reactive mixture of styrene:\n"]], ["block_6", ["3.8.3\nChain Transfer to Polymer\n"]], ["block_7", ["108\nChain-Growth Polymerization\n"]], ["block_8", ["\"\"w\nH\n\u00b0\nH\n\u201dI\u201c\nMe\nU\n"]], ["block_9", ["N+P.-\u2014+W+PX\u2014-+\u2014M\u2014F\u2014\u2014+\n"]], ["block_10", [{"image_2": "123_2.png", "coords": [47, 344, 213, 387], "fig_type": "molecule"}]], ["block_11", ["Y\nY\nx\n(3Q)\nYPJ-\n"]], ["block_12", [{"image_3": "123_3.png", "coords": [81, 534, 347, 686], "fig_type": "figure"}]], ["block_13", ["+\n.\n__;...\n+ M __... .S\ufb02\u2019\ufb01ne\u2014\ufb01\n(3.R)\n"]], ["block_14", ["\u2014\u2014\u2014)--\n+\nH20: CH2\u2014\u2014\u2014-I-\n"]], ["block_15", [{"image_4": "123_4.png", "coords": [118, 333, 397, 387], "fig_type": "molecule"}]], ["block_16", ["n\nMe\nO\n"]], ["block_17", [{"image_5": "123_5.png", "coords": [132, 343, 341, 378], "fig_type": "molecule"}]], ["block_18", ["l\nnHQC= CH2\n"]], ["block_19", [{"image_6": "123_6.png", "coords": [210, 182, 370, 211], "fig_type": "molecule"}]], ["block_20", [{"image_7": "123_7.png", "coords": [212, 539, 331, 592], "fig_type": "molecule"}]], ["block_21", ["1\n"]], ["block_22", ["etc.\n"]], ["block_23", [{"image_8": "123_8.png", "coords": [295, 330, 373, 389], "fig_type": "molecule"}]], ["block_24", ["/n\n"]], ["block_25", ["(3.8)\n"]]], "page_124": [["block_0", [{"image_0": "124_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "124_1.png", "coords": [31, 324, 318, 396], "fig_type": "figure"}]], ["block_2", ["However, transfer long-chain branches. The \nknown as low density is formed by mechanism in process \n"]], ["block_3", ["p\u2014benzoquinone is 518, and that for 02 is 1.5 x 104. The Polymer Handbook [1] is an excellent\nsource for these and most other rate constants discussed in this chapter.\n"]], ["block_4", ["at high \ntherefore have \nproperties of\n"]], ["block_5", ["We conclude this an extreme case of chain transfer, a reaction that produces\nradicals suppressed. \naccomplish \nduring inhibitors, depending \nof protection afford. be removed from monomers before use, and failure\n"]], ["block_6", ["and retarders differ in whichthey interfere with polymerization, but not in their\nessential activity. substance blocks polymerization \nuntil it is either removed failure to totally eliminate an inhibitor from puri\ufb01ed\nmonomer is first converted \nform before polymerization A retarder is and merely slows down the\npolymerization process by competing for radicals.\nBenzoquinone [III] is widely used as an inhibitor:\n"]], ["block_7", ["3.8.4\nSuppressing Polymerization\n"]], ["block_8", ["The resulting radical is stabilized by electron delocalization and eventually reacts with either\nanother inhibitor radical by combination (dimerization) or disproportionation or with an inhibitor\nor other radical. Another commonly used inhibitor is 2,6-di-tert\u2014butyl\u20144-methylphenol (butylated\nhydroxytoluene, or BHT):\n"]], ["block_9", ["Chain \n109\n"]], ["block_10", ["to achieve \n"]], ["block_11", ["Inhibitors are characterized by inhibition constants, which are defined as the ratio of the rate\nconstant for transfer to inhibitor to the propagation constants for the monomer, by analogy\nwith Equation 3.8.6 for chain transfer constants. For styrene at 50\u00b0C the inhibition constant of\n"]], ["block_12", ["which is also known as an antioxidant. Such free-radical scavengers often act as antioxidants, in\nthat the first stage of oxidative attack generates a free radical.\nMolecular oxygen contains two unpaired electrons and has the distinction of being capable of\nboth initiating and inhibiting polymerization. Molecular oxygen functions in the latter capacity by\nforming the relatively unreactive peroxy radical:\n(h+M~+M\u2014O\u2014O-\n8U)\n"]], ["block_13", [{"image_2": "124_2.png", "coords": [38, 462, 124, 531], "fig_type": "molecule"}]], ["block_14", ["Me \nOH\nMeMe\nMe\nMe\n"]], ["block_15", [{"image_3": "124_3.png", "coords": [48, 333, 338, 380], "fig_type": "molecule"}]], ["block_16", ["0\nnm\n"]], ["block_17", ["O\n"]], ["block_18", [".\n.\n\u2014.\u2014.\u2014\nO\nO \n\u2014> Inert products\n0-\u201c\n+\n.4o\n"]], ["block_19", ["Me\n"]]], "page_125": [["block_0", [{"image_0": "125_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["1.\nIn comparison with step-growth polymerization, free-radical polymerization can lead to much\nhigher molecular weights and in much shorter times, although the resulting distributions of\nmolecular weight are comparably broad.\n2.\nThere are three essential reaction steps in a chain-growth polymerization: initiation, propagation,\nand termination. A wide variety of free-radical initiators are available; the most common act by\nthermally induced cleavage of a peroxide or azo linkage. Propagation occurs by head-to-tail\naddition of a monomer to a growing polymer radical, and is typically very rapid. Termination\noccurs by reaction between two radicals, either by direct combination or by disproportionation.\n3.\nA fourth class of reactions, termed transfer reactions, is almost always important in practice.\nThe primary effect of transfer of a radical from a growing chain to another molecule is to\nreduce the average degree of polymerization of the resulting polymer chains, but in some cases\nit can also lead to interesting architectural consequences in the \ufb01nal polymer.\n4.\nKinetic analysis of the distribution of chain lengths is made tractable by three key assump-\ntions. The steady-state approximation requires that the net rates of initiation and termination\nbe equal; thus the total concentration of radicals is constant. The same approximation extends\nto the concentration of each radical species individually. The principle of equal reactivity\nasserts that a single rate constant describes each propagation step and each termination step,\nindependent of the degree of polymerization of the radicals involved. Thirdly, transfer\nreactions are assumed to be absent.\n5.\nThe aforementioned assumptions are most successful in describing the early stages of poly-\nmerization, before a host of competing factors become significant, such as depletion of\nreactants, loss of mobility of chain radicals, etc. Under these assumptions explicit expressions\nfor the number and weight distribution of polymer chains can be developed. In the case that\ntermination occurs exclusively by disproportionation, the result is a most probable distribution\nof molecular weights, just as with step-growth polymerization. Termination by recombination,\non the other hand, leads to a somewhat narrower distribution, with MW/Mn m 1.5 rather than 2.\n"]], ["block_2", ["In this chapter we have explored chain growth or addition polymerization, as exemplified by the\nfree-radical mechanism. This particular polymerization route is the most prevalent from a commer-\ncial perspective, and is broadly applicable to a wide range of monomers, especially those containing\ncarbon\u2014carbon double bonds. The main points of the discussion may be summarized as follows:\n"]], ["block_3", ["I\u2018J.C. Bevington, J.I-l. Bradbury, and GM. Burnett, J. Polym. Sal, 12, 469 (1954).\n"]], ["block_4", ["3.9\nChapter Summary\n"]], ["block_5", ["Problems\n"]], ["block_6", ["110\nChain-Growth Polymerization\n"]], ["block_7", ["1.\nThe efficiency of AIBN in initiating polymerization at 60\u00b0C was determinedr by the following\nstrategy. They measured Rp and I7 and calculated R1 RP/17. The constant kd was measured\ndirectly in the system, and from this quantity and the measured ratio RP/ 17 the fractionf could\nbe determined. The following results were obtained for different concentrations of initiator:\n"]], ["block_8", [{"image_1": "125_1.png", "coords": [40, 523, 225, 633], "fig_type": "figure"}]], ["block_9", ["0.0556\n0.377\n0.250\n1.57\n0.250\n1.72\n1.00\n6.77\n1.50\n10.9\n2.50\n17.1\n"]], ["block_10", ["Using kd 0.0388 h\u2018l, evaluate ffl\u2018om these data.\n"]], ["block_11", ["[I] (g L\u201c)\nrep/:7 x 108 (mol L\u201c s\u201c)\n"]]], "page_126": [["block_0", [{"image_0": "126_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["2,\nAIBN was synthesized using 14C\u2014labeled reagents and the tagged compound was used to\ninitiate \nto methyl methacrylate\nand\nstyrene.\nSamples of initiator\nand\npolymers containing initiator fragments were burned to C02. The radioactivity of uniform\n(in sample size and treatment) C02 samples was measured in counts per minute (cpm) by a\nsuitable Geiger counter. A formula for poly(methy1 methacrylate) with its initiator\nfragments is (C5H802),,(C4H6N)m, where n is the degree of polymerization for the polymer\nand m is either or 2, depending on the mode of termination. The specific activity measured\nin the C02, resulting combustion of the polymer relative to that produced by the\ninitiator is\n"]], ["block_2", ["1to. Bevington, H.w. Melville, and RP. Taylor, J. Polym. 5a., 12, 449 (1954).\n"]], ["block_3", ["3.\nIn the research described in Example 3.4, the authors measured the following rates of\npolymerization:\n"]], ["block_4", ["problems\n111\n"]], ["block_5", [{"image_1": "126_1.png", "coords": [40, 466, 215, 569], "fig_type": "figure"}]], ["block_6", [{"image_2": "126_2.png", "coords": [51, 293, 317, 439], "fig_type": "figure"}]], ["block_7", ["Methyl methacrylate\nStyrene\nMn\nCounts per minute\nMn\nCounts per minute\n"]], ["block_8", ["Activity of C in polymer _\n4m\nN \nActivity of C in initiator \n5n + 4m\n\u2014\n5n.\n"]], ["block_9", ["From the ratio of activities and measured values of n, the average number of initiator\nfragments per polymer can be determined.\nCarry out a similar argument for the ratio of activities for polystyrene and evaluate the\naverage number of initiator fragments per molecule for each polymer from the following data.Jr\nFor both sets of data, the radioactivity from the labeled initiator gives 96,500 cpm when\nconverted to C02.\n"]], ["block_10", ["Run number\n1\u2018?p X 104 (H101 1:1 3\u20141)\n"]], ["block_11", ["They also reported a kp value of 1.2 x 104 L mol_l s\ufb02l, but the concentrations of monomer in\neach run were not given. Use these values of RP and kp and the values of 17 and k, given in\nExample 3.4 to evaluate [M] for each run. As a double check, evaluate [M] from these values\nof RI, (and kp) and the values of R, and k, given in the example.\n"]], ["block_12", ["444,000\n20.6\n383,000\n25.5\n312,000\n30.1\n117,000\n86.5\n298,000\n29.0\n114,000\n89.5\n147,000\n60.5\n104,000\n96.4\n124,000\n76.5\n101,000\n113.5\n91,300\n103.4\n89,400\n104.6\n"]], ["block_13", ["13\n7.59\n"]], ["block_14", ["12\n5.48\n"]], ["block_15", ["5\n3.40\n"]], ["block_16", ["6\n2.24\n"]], ["block_17", ["8\n6.50\n"]]], "page_127": [["block_0", [{"image_0": "127_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "127_1.png", "coords": [33, 88, 334, 243], "fig_type": "figure"}]], ["block_2", ["1\u2018 M. Amett, J. Am. Chem. 306., 74, 2027 (1952).\n1MS. Matheson, E.E. Auer, E.B. Bevilacqua, and J.E. Hart, J. Am. Chem. 306., 73, 1700 (1951).\n\u201c\u2018 M. May Jr. and WE. Smith, J. Phys. Chem, 72, 216 (1968).\n"]], ["block_3", ["Propose an explanation for the variation observed.\n6.\nDerive Equation 3.7.27 and Equation 3.7.28.\n7.\nThe equations derived in Section 3.7 are based on the assumption that termination occurs\nexclusively by either disproportionation or combination. This is usually not the case; some\npr0portion of each is more common. If as equals the fraction of chains for which termination\noccurs by disproportionation, it can be shown that\n"]], ["block_4", [{"image_2": "127_2.png", "coords": [35, 482, 193, 525], "fig_type": "molecule"}]], ["block_5", ["Use these data to evaluate the cluster of constants (fkd/kt)\u201d2kp at this temperature. Evaluate\nftp/kt\u201d2 using Arnett\u2019s finding thatf2 1.0 and assuming the kd value determined in Example 3.1\nfor AIBN at 77\u00b0C in xylene also applies in benzene.\n5.\nThe lifetime of polystyrene radicals at 50\u00b0C was measuredI as a function of the extent of\nconversion to polymer. The following results were obtained:\n"]], ["block_6", ["4.\nArnettJr initiated the polymerization of methyl methacrylate in benzene at 77\u00b0C with AIBN and\nmeasured the initial rates of polymerization for the concentrations listed:\n"]], ["block_7", ["1 12\nChain-Growth Polymerization\n"]], ["block_8", [{"image_3": "127_3.png", "coords": [44, 313, 169, 393], "fig_type": "figure"}]], ["block_9", ["1\u2014\n2\n2*\u2014\na+(\na)_\na\nN:\n_\nn\nl\u2014p\nl\u2014p\nl\u2014p\n"]], ["block_10", ["Nw_4\u20143a\u2014ap+2p\n"]], ["block_11", ["From measurements of NJ\u201c and NW/Ml it is possible in principle to evaluate a and p. May\nand Smith)k have done this for a number of polystyrene samples. A selection of their data for\nwhich this approach seems feasible is presented below. Since p is very close to unity, it is\n"]], ["block_12", ["0\n2.29\n32.7\n1.80\n36.3\n9.1\n39.5\n13.1\n43.8\n18.8\n"]], ["block_13", ["9.04\n2.35\n1 1.61\n8.63\n2.06\n10.20\n7.19\n2.55\n9.92\n6.13\n2.28\n7.75\n4.96\n3.13\n7.31\n4.75\n1.92\n5.62\n4.22\n2.30\n5.20\n4.17\n5.81\n7.81\n3.26\n2.45\n4.29\n2.07\n2.1 l\n2.49\n"]], ["block_14", ["and\n"]], ["block_15", ["N11\n(2 a?\n"]], ["block_16", ["Percent conversion\n1\u201c- (s)\n"]], ["block_17", ["[M] (mol L\u201c)\n[I]0 x 104 (mol L\u201c)\nRP x 103 (mol L-1 min\u2014l)\n"]]], "page_128": [["block_0", [{"image_0": "128_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["1S.R. Palit and S.K. Das, Proc. Roy. Soc. London, 226A, 82 (1954).\n"]], ["block_2", ["Problems\n1 1 3\n"]], ["block_3", ["10.\n"]], ["block_4", ["11.\n"]], ["block_5", [{"image_1": "128_1.png", "coords": [46, 226, 246, 335], "fig_type": "figure"}]], ["block_6", [". In the research described in Problem 7, the authors determined the following distribution of\nmolecular weights by a chromatographic procedure (w,- is the weight fraction of i-mer):\n"]], ["block_7", [". Derive the two equations given in the previous problem. It may be helpful to recognize that\nfor any distribution taken as a whole, w. = ixi/Nn.\n"]], ["block_8", [{"image_2": "128_2.png", "coords": [53, 84, 120, 164], "fig_type": "figure"}]], ["block_9", ["Na\nNw/Nn\n"]], ["block_10", ["with a 0.65 and p 0.99754. Calculate some representative points for this function and\nplot the theoretical and experimental points on the same graph. From the expression given\nextract the weight fraction i-mer resulting from termination by combination.\nIn fact, the expression in the previous problem is slightly incorrect. Derive the correct\nexpression, and see if the implied values of a and p are signi\ufb01cantly different. The solution\nto Problem 8 provides part of the answer.\nPalit and Das\u2019r measured 17\u201c at 60\u00b0C for different values of the ratio [SX]/[M] and evaluated\nCsx and 17 for vinyl acetate undergoing chain transfer with various solvents. Some of their\nmeasured and derived results are tabulated below (the same concentrations of AIBN and\nmonomer were used in each run). Assuming that no other transfer reactions occur, calculate\nthe values missing from the table. Criticize or defend the following proposition: The 17 values\nobtained from the limit [SX]/[M] \u2014> 0 show that the AIBN initiates polymerization identi-\ncally in all solvents.\n"]], ["block_11", ["100\n3.25\n800\n6.88\n200\n5.50\n900\n6.10\n300\n6.80\n1200\n4.20\n400\n7.45\n1500\n2.90\n500\n7.91\n2000\n1.20\n600\n7.82\n2500\n0.50\n700\n7.18\n3000\n0.20\n"]], ["block_12", ["t-Butyl alcohol\n6580\n3709\n\u2014\u2014\n0.46\nMethyl isobutyl ketone\n6670\n510\n0.492\n\u2014\u2014\nDiethyl ketone\n6670\n\u2014\n0.583\n114.4\nChloroform\n\u2014\n93\n0.772\n125.2\n"]], ["block_13", ["i\nw. x 104\ns\nw; x 104\n"]], ["block_14", ["adequate to assume this value and evaluate a from NW/Nn and then use the value of at so\nobtained to evaluate a better value of p from N\u201c.\n"]], ["block_15", ["They asserted that the points are described by the expression\n"]], ["block_16", ["Solvent\n:7\n17..\n[SX]/[M]\nCSX x 104\n"]], ["block_17", ["w,- = ai(1 p)2pi_1 + 0.5(1 a)i(i 1)(1 p)3p*'*2\n"]], ["block_18", ["1 129\n1.60\n924\n1.67\n674\n1.73\n609\n1 .74\n"]]], "page_129": [["block_0", [{"image_0": "129_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "129_1.png", "coords": [33, 119, 319, 211], "fig_type": "figure"}]], ["block_2", ["1.\nBrandrup, J. and Immergut, E.H., Eds., Polymer Handbook, 3rd ed., Wiley, New York, 1989.\n2.\nTrommsdorff, E., Kohle, H., and Lagally, P., Makromol. Chem, 1, 169 (1948).\n"]], ["block_3", [{"image_2": "129_2.png", "coords": [34, 78, 376, 233], "fig_type": "figure"}]], ["block_4", ["*R.A. Gregg and ER. Mayo, J. Am. Chem. Soc., 70, 2373 (1943).\n"]], ["block_5", ["1 14\nChain-Growth Polymerization\n"]], ["block_6", ["References\n"]], ["block_7", ["12.\nGregg and Mayor studied the chain transfer between styrene and carbon tetrachloride at 60\u00b0C\nand 100\u00b0C. A sample of their data is given below for each of the temperatures.\n"]], ["block_8", ["Evaluate the chain transfer constant (assuming that no other transfer reactions occur) at each\ntemperature. By means of an Arrhenius analysis, estimate Et\"; Eff for this reaction. Are the\nvalues of T2 in the limit of no transfer in the order expected for thermal polymerization?\nExplain.\n13.\nMany olefins can be readily polymerized by a free-radical route. On the other hand,\nisobutylene is usually polymerized by a cationic mechanism. Explain.\n14.\nDraw the mechanisms for the following processes in the radical polymerization of styrene in\ntoluene: (a) initiation by cumyl peroxide; (b) propagation; (c) termination by disproportiona-\ntion; and (d) transfer to solvent.\n15.\nShow the mechanisms of addition of a butadiene monomer to a poly(butadienyl) radical, to\ngive each of the three possible geometric isomers.\n16.\nConsider the polymerization of styrene in toluene at 60\u00b0C initiated by di-t\u2014butylperoxide for\na solution containing 0.04 mol of initiator and 2 mol of monomer per liter. The initial rates of\ninitiation, Ri, and propagation, RP, are found to be 1.6 x 10'10 M s\u20181 and 6.4 x 10\u20187 M 3\u20181,\nrespectively, at 60\u00b0C.\n"]], ["block_9", ["0.00614\n16.1\n0.00582\n36.3\n0.0267\n35.9\n0.0222\n68.4\n0.0393\n49.8\n0.0416\n109\n0.0704\n74.8\n0.0496\n124\n0.1000\n106\n0.0892\n217\n0.1643\n156\n0.2595\n242\n0.3045\n289\n"]], ["block_10", ["((1)\nThe molecular weight of this polymer is too high. The desired molecular weight of this\npolymer is 40,000 g mol\u201d. How much CCl4 (in g L\u2018I) should be added to the reaction\nmedium to attain the desired molecular weight? CT of CC14 is 9 x 10\u20143.\n(e)\nUnder the conditions stated, the polymerization is too slow. What is the initial rate of\npolymerization if the temperature is raised to 100\u00b0C?\n(f)\nCalculate the conversion attained after the reaction has gone for 5 h at 100\u00b0C. Assume\nvolume expansion does not change concentration significantly and that the initiator\nconcentration is constant throughout the entire reaction.\n"]], ["block_11", ["(a)\nCalculate fkd and kp/ktm.\n(b)\nAssuming no chain transfer, calculate the initial kinetic chain length.\n(c)\nAssuming only disproportionation and under the conditions stated, the transfer constant\nof styrene, CM, is 0.85 x 104. How much does this transfer affect the molecular weight\nof the polymer?\n"]], ["block_12", ["[CC14]/[Styrene]\n17;] x 105\n[CCI4]/[Styrene]\n17;! X 105\n"]], ["block_13", ["At 60\u00b0C\nAt 100\u00b0C\n"]]], "page_130": [["block_0", [{"image_0": "130_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["North, \n"]], ["block_2", ["Odian, \nRempp, P. and Merrill, E.W., Polymer Synthesis, 2nd ed., Hiithig & Wepf, Basel, 1991.\n"]], ["block_3", ["Allcock, \n"]], ["block_4", ["Further \n115\n"]], ["block_5", ["Further \n"]], ["block_6", ["NJ, 1990.\n"]]], "page_131": [["block_0", [{"image_0": "131_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["In the preceding chapters we have examined the two main classes of polymerization, namely step\u2014\ngrowth and chain\u2014growth polymerizations, with the latter exemplified by the free-radical mech-\nanism. These are the workhorses of the polymer industry, permitting rapid and facile production of\nlarge quantities of useful materials. One common feature that emerged from the discussion of these\nmechanisms is the statistical nature of the polymerization process, which led directly to rather broad\ndistributions of molecular weight. In particular, even in the simplest case (assuming the principle of\nequal reactivity, no transfer steps or side reactions, etc.), the product polymers of either a poly-\ncondensation or of a free\u2014radical polymerization with termination by disproportionation would\nfollow the most probable distribution, which has a polydispersity index (MW/Mn) approaching 2.\nIn commercial practice, the inevitable violation of most of the simplifying assumptions leads to even\nbroader distributions, with polydispersity indices often falling between 2 and 10. In many cases the\npolymers have further degrees of heterogeneity, such as distributions of composition (e.g., copoly-\nmers), branching, tacticity, or microstructure (e.g., cis 1,4\u2014, trans 1,4\u2014, and 1,2-configurations in\npolybutadiene).\nThis state of affairs is rather unsatisfying, especially from the chemist\u2019s point of view. Chemists\nare used to the idea that every molecule of, say, ascorbic acid (vitamin C) is the same as every\nother one. Now we are confronted with the fact that a tank car full of the material called\npolybutadiene is unlikely to contain any two molecules with exactly the same chemical structure\n(recall Example 1.4). As polymers have found such widespread applications, we have obviously\nlearned to live with this situation. However, if we could exert more control over the distribution of\nproducts, perhaps many more applications would be realized. In this chapter we describe several\napproaches designed to exert more control over the products of a polymerization. The major one is\ntermed living polymerization, and can lead to much narrower molecular weight distributions.\nFurthermore, in addition to molecular weight control, living polymerization also enables the\nlarge-scale production of block copolymers, branched polymers of controlled architecture, and\nend-functionalized polymers.\nA comparison between synthetic and biological macromolecules may be helpful at this stage. If\ncondensation and free-radical polymerization represent the nadir of structural control, proteins and\nDNA represent the zenith. Proteins are \u201ccopolymers\u201d that draw on 20 different amino acid\nmonomers, yet each particular protein is synthesized within a cell with the identical degree of\npolymerization, composition, sequence, and stereochemistry. Similarly, DNAs with degrees\nof polymerization far in excess of those realized in commercial polymers can be faithfully\nreplicated, with precise sequences of the four monomer units. One long-standing goal of polymer\nchemistry is to imitate nature\u2019s ability to exert complete control over polymerization. There are\ntwo ways to approach this. One is to begin with nature, and try to adapt its machinery to our\npurpose. This is exemplified by \u201ctraining\u201d cells into growing polymers that we want, for example,\nvia recombinant DNA technology. The other approach, and the one described in this chapter, is to\nstart with the polymerizations we already have, and try to improve them. Both approaches have\nmerit, and we select the latter because it is currently much more established, and plays a central\n"]], ["block_2", ["4.1\nIntroduction\n"]], ["block_3", ["Controlled Polymerization\n"]], ["block_4", ["4\n"]], ["block_5", ["117\n"]]], "page_132": [["block_0", [{"image_0": "132_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["be done in a chain-growth mechanism, through the concentration of initiators. The number of\ninitiators will be equal to the number of polymers, assuming 100% initiation efficiency and\nassuming no transfer reactions that lead to new polymers. The second requirement is to distribute\nthe total number of monomers as uniformly as possible among the fixed number of growing chains.\nIf the polymerization then proceeds to completion, we could predict Nn precisely: it would simply\nbe the ratio of the number of monomers to the number of initiators. To allow the reaction to\nproceed to completion, we would need to prevent termination steps, or at least defer them until we\nwere ready. Now, suppose further that the reaction proceeds statistically, meaning that any\nmonomer is equally likely to add to any growing chain at any time. If N,1| was reasonably large,\nwe could expect a rather narrow distribution of the number of monomers in each chain, just by\nprobability. (This argument also assumes no transfer reactions, so that growing polymers are not\nterminated prematurely.) As an illustration, imagine placing an array of empty cups out in a steady\nrain; an empty cup is an \u201cinitiator\u201d and a raindrop is a \u201cmonomer.\u201d As time goes on, the raindrops\nare distributed statistically among the cups, but after a lot of drops have fallen, the water level will\nbe pretty much equal among the various cups. If a cup fell over, or a leaf fell and covered its top,\nthat \u201cpolymer\u201d would be \u201cterminated,\u201d and its volume of water would not keep up with the others.\nSimilarly, if you placed a cup outside a few minutes after the others, the delayed initiation would mean\nthat it would never catch up with its neighbors. What we have just described is, in fact, the essence of a\ncontrolled polymerization: start with a fixed number of initiators, choose chemistry and conditions to\neliminate transfer and termination reactions, and let the reaction start at a certain time and then go to\ncompletion. In order to control the local structural details, such as microstructure and stereochemistry,\nwe have to in\ufb02uence the relative rates of various propagation steps. This can be achieved to some\nextent by manipulating the conditions at the active site at the growing end of the chain.\nThe remainder of this chapter is organized as follows. First we demonstrate how the kinetics of\nan ideal living polymerization leads to a narrow, Poisson distribution of chain lengths. Then, we\nconsider chain\u2014growth polymerization via an anionic propagating center; this has historically\nbeen the most commonly used controlled polymerization mechanism, and it can be conducted in\nsuch a way as to approach the ideal case very closely. In Section 4.4 we explore how the anionic\nmechanism can be extended to the preparation of block copolymers, end-functional polymers, and\nregular branched polymers of various architectures. We then turn our attention to other mechan-\nisms that are capable of controlled polymerization, including cationic (Section 4.5), ring-opening\n(Section 4.8), and, especially, controlled radical polymerizations (Section 4.6). The concluding\nsections also address the concept of equilibrium polymerization, and a special class of controlled\npolymers called dendrimers.\n"]], ["block_2", ["4.2\nPoisson Distribution for an Ideal Living Polymerization\n"]], ["block_3", ["In this section we lay out the kinetic scheme that describes a living polymerization, and thereby\nderive the resulting distribution of chain lengths. This scenario is most closely approached in the\nanionic case, but because it is not limited to anionic polymerizations, we will designate an active\n"]], ["block_4", ["role in much of polymer research. It is worth noting that nature also makes use of many other\nmacromolecular materials that are not so well\u2014controlled as proteins and DNA; examples include\npolysaccharides such as cellulose, chitin, and starch. So in nature, as with commercial polymers,\nuseful properties can still result from materials that are very heterogeneous at the molecular level.\nThe lack of control over molecular weight in polymerization arises directly from the random\ncharacter of each step in the reaction. In a polycondensation, any molecule can react with any other\nat any time; the number of molecules is steadily decreasing, but the mole fraction of monomer is\nalways larger than the mole fraction of any other species. In a free-radical polymerization, chains\nmay be initiated at any time. Growing chains may also add monomer, or undergo a transfer or\ntermination reaction at any time. The first requirement in controlling molecular weight is to fix the\ntotal number of polymers. This cannot be done in an unconstrained step-growth process, but it can\n"]], ["block_5", ["118\nControlled Polymerization\n"]]], "page_133": [["block_0", [{"image_0": "133_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["When the reaction has gone to completion, [M] will be 0, and the kinetic chain length will be\nthe number average degree of polymerization of the resulting polymer. It will also be helpful in the\nfollowing development to differentiate Equation 4.2.5 with respect to time, and then incorporate\nEquation 4.2.3:\n 9?- \n1 d[M]\n"]], ["block_2", ["This is a linear, first-order differential equation for [M], which has the solution\n"]], ["block_3", ["Therefore the concentration of monomer decreases exponentially to zero as time progresses. (Note\nthat we are also assuming that propagation is irreversible, that is, there is no \u201cback arrow\u201d in\nEquation 4.2.2. The possibility of depolymerization reaction steps will be taken up in Section 4.7.)\nAt this stage it is very helpful to introduce a kinetic chain length, a, analogous to the one we\ndefined in Equation 3.5.10, as the ratio of the number of monomers incorporated into polymers to\nthe number of polymers. The former is given by [M]0 [M], and the latter by [I]0, so we write\n"]], ["block_4", ["Note that in using a single propagation rate constant, kp, we are once again invoking the principle\nof equal reactivity.\nWe will now assume that initiation is effectively instantaneous relative to propagation (ki >> kp),\nso that at time t= 0, [Pf] [I]0, and we will not worry about Equation 4.2.1 any further. Note that\nthis criterion is not necessary to have a living polymerization, but it is necessary to achieve a\nnarrow distribution of molecular weights. The concentration of unreacted monomer, [M], will\ndecrease in time as propagation takes over. The overall rate of polymerization, RP, is the sum of the\nrates of consumption of monomer by all growing chains Pf. However, we know that, in the absence\nof termination or transfer reactions, the total concentration of P? is always [I]0: we have fixed the\nnumber of polymers. Therefore we can write\n"]], ["block_5", ["The concentration ofmonomer at time t will be denoted [M]. The initial concentrations\nof monomer and initiator are [M]0 and [I]0, respectively. The reaction steps can be represented as\nfollows:\n"]], ["block_6", ["4.2.1\nKinetic Scheme\n"]], ["block_7", ["polymer of degree of polymerization i by P?\u2018, and its concentration by [Pf], where represents the\nreactive end. is defined as a chain-growth processfor which there are no\nirreversible \nabout but \nwill not \n"]], ["block_8", ["Poisson Distribution Polymerization\n119\n"]], ["block_9", ["[M10 [M]\n_ = \u2014\u2014\u2014-\u2014-\u2014\n4.2.5\nV\n[Ho\n(\n)\n"]], ["block_10", ["(1 M\nRp \u2014\u2014[d7\u2014] MM]2 [Pi] kl][ I 10\n(4.2.3)\n"]], ["block_11", ["k\nPropagation:\nPi\u201c + M \u2014\">\n193\u2018\n"]], ["block_12", ["Initiation:\nI+M ii Pi\"\n(42.1)\n"]], ["block_13", ["[M] [Mloe\u2018kpllb\u2018\n(4.2.4)\n"]], ["block_14", ["k\nPf-\u201c+ M \u2014\"i\nP511\n(4.22)\n"]]], "page_134": [["block_0", [{"image_0": "134_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "134_1.png", "coords": [26, 340, 267, 411], "fig_type": "figure"}]], ["block_2", [{"image_2": "134_2.png", "coords": [31, 415, 152, 454], "fig_type": "molecule"}]], ["block_3", [{"image_3": "134_3.png", "coords": [31, 159, 194, 195], "fig_type": "molecule"}]], ["block_4", ["We could insert Equation 4.2.4 into Equation 4.2.7 to replace [M], and thereby obtain an equation\nthat can be solved. However, a simpler approach turns out to be to invoke the chain rule, as\nfollows:\n"]], ["block_5", ["Now we repeat this process for [PER], beginning with the rate law. This is slightly more compli-\ncated, because [Pfrf] grows by the reaction of Pi\" with monomer, but decreases by the reaction of Pi\"\nwith monomer:\n"]], ["block_6", ["In order to obtain the distribution of chain lengths, we need to do a bit more work. We begin by\nwriting an explicit equation for the rate of consumption of [PT]:\n"]], ["block_7", ["and comparing with Equation 4.2.11 we obtain\n"]], ["block_8", ["This equation has the solution\n"]], ["block_9", ["We can go through this sequence of steps once more, considering the concentration of trimer [133\u2018]:\n"]], ["block_10", ["By invoking the chain rule once more\n"]], ["block_11", ["If we now compare Equation 4.2.7 and Equation 4.2.8 we can see that\n"]], ["block_12", ["leading to\n"]], ["block_13", ["and this equation is readily solved:\n"]], ["block_14", ["120\nControlled Polymerization\n"]], ["block_15", [{"image_4": "134_4.png", "coords": [36, 563, 263, 631], "fig_type": "molecule"}]], ["block_16", ["d 13*\n_\nL114. [133\u2018] : [Pg\u2018] : \ufb02uoe-V\n(4.2.16)\ndii\n"]], ["block_17", ["d 13*\n_\n[1;] + [133\u2018] [133\u2018] [Hoe\u2014V\n(4.2.13)\n"]], ["block_18", ["d[133\u2018]\nd[133\u2018] an\n"]], ["block_19", ["d[131*\u2018] : d[P3\u2018] (1'17 __ d[P1\"]\ndt\ndv E-\nd}? \n(4.2.8)\n"]], ["block_20", ["d P*\n*\n_ _%__\ufb01ll : [131]\n(4.2.9)\n"]], ["block_21", ["[133\u2018] : E[I]0e\u201d'_\u2019\n(4.2.14)\n"]], ["block_22", ["(1 [P3]\ndt \n"]], ["block_23", ["d P*\n{.31 41341114 4 1341 [M]\n"]], ["block_24", ["[133\u2018] [13\u20181\"]04rE [nae\u20145\n(4.2.10)\n"]], ["block_25", ["dr \nd?\nd;\n(4.2.12)\n"]], ["block_26", ["d [PT]\ndt\n2 kp [133\u2018] [M]\n(4.2.7)\n"]], ["block_27", ["d?\n:\np[M]([P\u00a7] [Path 51112:] \n[13311)\n(42.15)\n"]], ["block_28", ["d?\n=41M1<1P\ufb02\u20141P41>=~a;<1311311341)\n(4.2.11)\n"]]], "page_135": [["block_0", [{"image_0": "135_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "135_1.png", "coords": [20, 384, 266, 526], "fig_type": "figure"}]], ["block_2", ["a small number of boxes or polymers). Once the polymerization reaction has gone to\ncompletion, and the polymers by introduction of some appropriate reagent, the resulting\nmolecular weight distribution should obey Equation 4.2.19, with i equal to [M]0/[I]0.\nThe following example illustrates some aspects of the kinetics of a living polymerization.\n"]], ["block_3", ["From this the mole fraction \npolymers, x,, by dividing 4.2.18 by the total number of polymers, [I]0:\nvine\u2014a\nx, \n(i 1)!\n(4.2.19)\n"]], ["block_4", ["The following data were reported for the living anionic polymerization of styrene:f The initial\nmonomer concentration was 0.29 mol LTI, and the initiator concentration was 0.00048 mol L\u2018l.\nThe reactor was sampled at the indicated times, and the resulting polymer was terminated and\nanalyzed for molecular weight and polydispersity. Use these data and Equation 4.2.4 and Equation\n4.2.5 to answer the following questions: Does conversion of monomer to polymer follow the\nexpected time dependence? What is the propagation rate constant under these conditions?\n"]], ["block_5", ["This particular is called Poisson distribution. \nobtained itfrom consideringfact it will describe \nwhenever larger objects or are distributed randomly \n"]], ["block_6", ["This pattern population of i-mer is\n"]], ["block_7", ["Using Equation 4.2.4 we see how p should evolve in time:\n"]], ["block_8", ["Therefore a plot of 1n(1 \u2014p) versus t should give a straight line with slope equal to \u2014kp[I]0. The data\nprovided do not include [M] explicitly, but we can infer [M] and p from Mn. From Equation 4.2.5,\n"]], ["block_9", ["which has \n"]], ["block_10", ["t (s)\nMn (g/mol)\nNn\nPDI\nl\u2014p\n"]], ["block_11", ["We can equate the conversion of monomer to polymer with the familiar extent of reaction, p, as in\nChapter 2 and Chapter 3:\n:[Mlo\u2014[M] ____1_ [M1\n[M10\n[M10\n"]], ["block_12", ["Poisson \n121\n"]], ["block_13", ["Solution\n"]], ["block_14", ["TW. Lee, H. Lee, 1. Cha, T. Chang, K]. Hanley, and T.P. Lodge, Macromolecules, 33, 5111 (2000).\n"]], ["block_15", ["Example 4.1\n"]], ["block_16", [{"image_2": "135_2.png", "coords": [36, 161, 111, 206], "fig_type": "molecule"}]], ["block_17", ["238\n3,770\n36.3\n1.06\n0.940\n888\n20,600\n198\n1.02\n0.672\n1,626\n33,700\n324\n1.02\n0.463\n2,296\n43,000\n413\n1.01\n0.316\n3,098\n49,800\n479\n1.008\n0.207\n4,220\n54,900\n528\n1.006\n0.127\n14,345\n61,700\n593\n1.005\n0.018\n"]], ["block_18", ["p \n1 \n"]], ["block_19", ["1\n_\n[Pi] gazilloe\u2018\u201d\n(4.2.17)\n"]], ["block_20", ["1\n.\n_\n:1:\n_\n-\u20141-1\n\u2014V\n[P,-] \n(F1)!\n:2\n[Hoe\n(4.2.18)\n"]]], "page_136": [["block_0", [{"image_0": "136_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["The suggested plot is shown below and the resulting slope from linear regression implies that\nkl, a:\n1 mol L\u20181 5\u20181. (This is actually a rather low value, and in fact only an apparent value, due to\n"]], ["block_2", ["4.2.2\nBreadth of the Poisson Distribution\n"]], ["block_3", ["Figure 4.1 illustrates the Poisson distribution for values of ? equal to 100, 500, and 1000. For\npolystyrene with M0 104, these would correspond to polymers with number average molecular\nweights of about 104, 5 x 104, and 105, respectively, which are moderate. The width of the distribu-\ntions, although narrow, increases with r, but as we shall see in a moment, the relative width (i.e., the\nwidth divided by 17) decreases steadily. It should be clear that these distributions are very narrow\ncompared to the step-growth or free\u2014radical polymerizations shown in Figure 2.5 and Figure 3.5,\nrespectively. To underscore this, Figure 4.2 compares the theoretical distributions for free-radical\npolymerization with termination by combination (Equation 3.7.26) and for living polymerization, both\nwith E 100. The difference is dramatic, and is made even more so when we recall that termination by\ncombination leads to a relatively narrow distribution with Mw/Mn approaching 1.5 rather than 2.\nFor the Poisson distribution, the polydispersity index, Mw/Mn, in fact approaches unity as 3\nincreases indefinitely. The explicit relation for the Poisson distribution is\n"]], ["block_4", ["the table was obtained as\n"]], ["block_5", ["a phenomenon to be described in Section 4.3 [also see Problem 3]. Also note that the last data point\nhas been omitted from the fit, as it corresponds to essentially complete conversion, and thus is\nindependent of t once the reaction is finished.)\n"]], ["block_6", ["the kinetic chain length is equal to p[I]0/[M]O, and it is also equal to NH \n"]], ["block_7", ["where the approximation applies for large ?. For 17: 1000 Equation 4.2.20 indicates that the\npolydispersity index will be 1.001, which is a far cry from 2.\n"]], ["block_8", ["122\nControlled Polymerization\n"]], ["block_9", [{"image_1": "136_1.png", "coords": [38, 595, 220, 628], "fig_type": "molecule"}]], ["block_10", [{"image_2": "136_2.png", "coords": [44, 190, 284, 416], "fig_type": "figure"}]], ["block_11", ["MW\nNW\n1\nE\n__:_____\n____~1\n4.2.20\nMn\nN.\n+(1+r)2\n(\n)\nElli-d\n"]], ["block_12", ["1_ #1_ N \np \n[M]0 \n(0.29 molL\u201c)\n\u201c\n"]], ["block_13", ["__2-5_I_II_I_III_III_I_IIIIIIII|\nIll\u2014\n0\n1 000\n2000\n3000\n4000\n5000\n"]], ["block_14", ["\u20140.5 \nSlope = \u20140.00051 3\u20181\n-\n"]], ["block_15", ["O\nI\nI\nr'\nI\nI\nI\nI\nI_l\u2014I\nI\nl\u2014l\u2014W\nI\u2014l\u2014I\nITI\nI\nl\u2014\n"]], ["block_16", ["\u2018l\u2014I'\n"]], ["block_17", ["I\n"]], ["block_18", ["lI\ufb01\u2014II\ufb01\u2014Ilj\u2014I\n"]], ["block_19", ["{,5\n"]], ["block_20", ["lrr\n"]]], "page_137": [["block_0", [{"image_0": "137_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "137_1.png", "coords": [19, 400, 298, 655], "fig_type": "figure"}]], ["block_2", [{"image_2": "137_2.png", "coords": [21, 25, 297, 303], "fig_type": "figure"}]], ["block_3", [{"image_3": "137_3.png", "coords": [25, 364, 173, 410], "fig_type": "molecule"}]], ["block_4", ["Figure 4.2\nComparison of Poisson distribution and distribution for free-radical polymerization with\ntennination by combination.\n"]], ["block_5", ["The derivation of Equation 4.2.20 is not too complicated, but it involves a couple of useful\nnicks, as we will show now. From Equation 1.7.2, we recall the definition of N\u201c, and insert\nEquation 4.2.19 to obtain\n"]], ["block_6", ["Poisson \n123\n"]], ["block_7", ["Figure 4.1\nMole fraction of i\u2014mer for the Poisson distribution with the indicated kinetic chain lengths.\n"]], ["block_8", ["s:\n-\n'.\n\u2018\n_\n-.\n-l\n0.02 \n-.\n_\n'\ng\n9.\nV = 500\n-\n"]], ["block_9", ["s;- 0.02 i\n-\nJ_\ni\n.\n\u00b0\n_;\n"]], ["block_10", ["0'0\ni\u2014i\u2014 \nNn:;ixi=:L\n(1\u2014 1)!\n(4.2.21)\n"]], ["block_11", ["0'04 .\u2014\n5 17: 100\ni\n"]], ["block_12", ["g\n' \nIdeal living polymerization, v = 100\n:\n0.03 i:\n. \n1\n"]], ["block_13", ["0.01 E\n:\nZ\n.\n_\n'\n:\n'\n_\n.\nIdeal free-radical polymerization,\n:\n"]], ["block_14", ["0,04\nL'I_l_\nI\nI\nI T I\nI\nI ; I\nl\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI \"l_l\nI\nI\nIT I\nI\nI\nl\n"]], ["block_15", ["0.01 :2\nf\n"]], ["block_16", ["0.03\n\u2014\n-'\n\u2014\n_\n_.\ni\n"]], ["block_17", ["0.05\n"]], ["block_18", ["0\n:\nl\n1\nI 1\u20191\nl _I_Ll \\4\ni\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nl\nI\nI\nI\nI\nI\nl\nI\nI\nI\nI\n"]], ["block_19", ["0\nI}! i\nL\nl_L\n] l_l\nLL_I\nI\nI\nI\nI\n\u2019\n0\n200\n400\n600\n800\n1 000\n1 200\n"]], ["block_20", [{"image_4": "137_4.png", "coords": [62, 132, 293, 284], "fig_type": "figure"}]], ["block_21", ["0\n100\n200\n300\n400\n"]], ["block_22", ["E\n,\n.\ntermination by recombination,\n-\n"]], ["block_23", ["__\n.\n'_\ni7\n= 100\n.I\n"]], ["block_24", ["t\n0 \n1\n"]], ["block_25", ["i\n- j\ni\n"]], ["block_26", ["k\u2014I\u2014\n"]], ["block_27", ["_\nd:\n_{\n"]], ["block_28", ["\u2014\n. _\n17: 1000\n\u2014\n"]], ["block_29", ["E\n'5\n5\u2018\n"]], ["block_30", ["l\nI \u2018l\u2014I\u2014 I _I\u2014|'\nI \u2014I\u2014I\u2014\ni\nI\nI\nI\ni\nl\nI\u2014I\u2014\nI\nI\nI\nI\n"]], ["block_31", ["-\n"]]], "page_138": [["block_0", [{"image_0": "138_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "138_1.png", "coords": [26, 81, 161, 122], "fig_type": "molecule"}]], ["block_2", ["This relationship establishes that Nn = 1 + 3. (You may be wondering where the \u201c1\u201d came from.\nA glance at Equation 4.2.5 reveals the answer: before the reaction begins, when [M] [M]0, then\n3 0 when the degree of polymerization is actually 1. Of course, for any reasonable value of NH,\nthe difference between Nn and M, + 1 is inconsequential.)\nThe development to obtain an expression for NW follows a similar approach, beginning with the\ndefinition from Equation 1.7.4:\n"]], ["block_3", ["We already know that the denominator on the right-hand side of Equation 4.2.25 is equal to 1 + a,\nso we just need to sort out the numerator\n"]], ["block_4", ["This differentiation is straightforward, recalling the rule for differentiating the product of two\nfunctions, and that d(e\")/dx\u2014\n- ex:\n"]], ["block_5", ["which leaves us with some more derivatives to take:\n"]], ["block_6", ["To progress further with this, it is helpful to recall the infinite series expansion of ex (see the\nAppendix if this is unfamiliar):\n"]], ["block_7", ["We will use this expansion to get rid of the factorials. Returning to Equation 4.2.21, we perform a\nseries of manipulations, recognizing that e\u2018\u201d does not depend on i and can be factored out of the\nsum, and that 1'74 can be written as d(v')/dv:\n"]], ["block_8", ["124\nControlled Polymerization\n"]], ["block_9", [{"image_2": "138_2.png", "coords": [35, 298, 226, 329], "fig_type": "molecule"}]], ["block_10", [{"image_3": "138_3.png", "coords": [36, 163, 216, 202], "fig_type": "molecule"}]], ["block_11", [{"image_4": "138_4.png", "coords": [41, 400, 147, 458], "fig_type": "molecule"}]], ["block_12", [{"image_5": "138_5.png", "coords": [45, 486, 246, 517], "fig_type": "molecule"}]], ["block_13", ["00\n00\n2 \ufb01x,\nN, = Ziw, = '3\n(4.2.25)\ni=1\n2 ix,-\ni=1\n"]], ["block_14", ["..;7\nd\n_ d\n__\n3\n_\n_;\nd\n_\n3\n_\n'17\ne\nE(va;{ve })\n\u2014 e\nd? (v{e +ve })\n"]], ["block_15", ["00\nx\u2018\n00\n=\nex; g)\u201c\n(i 1)!\n(4.222)\n"]], ["block_16", ["ivl\u2018\n1 6\u2014D\n_\n00\n1311\nN\n::\n\u2014\u2014\u2014-\u2014\u2014\u2014-\u2014\u2014\u2014\u2014-:\nFV\nn\nIElla\u20141)!\ne\n;(_i\u2014\u2014l)!\n"]], ["block_17", ["oo\n00\n-i-\u2014 1\n00\n\u2014i\n2\n2\u201d\ne\n_\n\u2014?\n.2\nV\nx\u2019= 2\u2019 (5\u20141)!\u2018e\n1230\u20141)!\ni=1\ni=1\n"]], ["block_18", ["\u2019Vd\u2014\ufb01\n\u2014{ve }\u2014e \u201d{e + ve\"\u2014\u2014} 1 +1)\n(4.2.24)\n"]], ["block_19", ["_;d\n00\n_\n31\u20141\n\u201417\nd\n__\n00\n\ufb01h-l\n--l_/ d\n_ 3\n_ e\nE;V(i\u20141)!_e\n3-;{1/1);???\n\u2014\u2014-\u2014-\u2014)\u20141!}\u2014e\nd\u2014?{ve}\n(4.2.23)\n"]], ["block_20", ["_ e_,\n0\u00b0\nd\nv\"\n_e_,\u2014,d\n\u00b0\u00b0\ni7\"\n\u201d\ni=1 \ndam (xx\u20141)!\n"]], ["block_21", [{"image_6": "138_6.png", "coords": [74, 505, 318, 585], "fig_type": "molecule"}]], ["block_22", [{"image_7": "138_7.png", "coords": [79, 519, 308, 553], "fig_type": "molecule"}]], ["block_23", ["\u2014 e\u2018F\u2014d\u2014vfl\u2014 i\n1,:-\n\u2014 e\u2014\ufb01i\ufb01fl\u2014 {reB\n\u2018\ndv\ncw\n.\n(i .\u2014 1)!\n_\nd\ufb01 dr\n(4.2.26)-\n"]], ["block_24", ["= e\u20187 {\ufb01eF + e3 + 2%; + 32 e;}\n= 1 + 33 + v?\n(4.2.27)\n"]]], "page_139": [["block_0", [{"image_0": "139_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "139_1.png", "coords": [30, 426, 267, 610], "fig_type": "figure"}]], ["block_2", ["using \n"]], ["block_3", ["The polydispersity \n(Equation consistently \ntion, but \n"]], ["block_4", ["seem to be approaching result; the implications of this observation are considered in\nProblem \nonly recently a polydispersity index \nwould require an accuracy much in the determination of M and Mn, and this is not\nyet possible using standard techniques in Chapter 7, Chapter 8, and\nChapter 9. In one sample obtained \nlaser desorption/ionization (MALDI)spectrometry shown \nwith the Poisson distribution mean; the the \ntal distribution being only slightly broader than the theoretical one.\nWe conclude this section with a summary of the requirements to achieve a narrow molecular\nweight distribution, and thereby draw an important distinction between \u201clivingness\u201d and the\nPoisson distribution. To recall the basic definition, a living polymerization is one that proceeds\nin the absence of transfer and termination reactions. Satisfying these two criteria is not sufficient to\n"]], ["block_5", ["It is \n"]], ["block_6", ["Finally, \n"]], ["block_7", ["Poisson \n125\n"]], ["block_8", ["Figure 4.3\n(a) Experimental polydispersities versus molecular weight for anionically polymerized polysty-\nrenes, from the data in Example 4.1. (b) The distribution obtained by MALDI mass spectrometry for one\nparticular sample. The smooth curves represent the results for the Poisson distribution, Equation 4.2.29 in (a)\nand Equation 4.2.19 in (b).\n"]], ["block_9", ["Mn,g/mol\nM\n(a)\n(b)\n"]], ["block_10", ["1.06 2-0\n-.\n_\n:\n:\n.5 0'6?\n1.05:-\n\u2014j\n<3\n,\n.\n.\n3\n:\n2\nQ-\n'\ng 1.04 ;\n__\nDC: 0.4_\n_\n\\3\nE\ng\n-\nE\n1.03\n_\u2014\n\u2014_\n.\n"]], ["block_11", [{"image_2": "139_2.png", "coords": [44, 400, 411, 578], "fig_type": "figure"}]], ["block_12", ["1 + 3? + 32\nNW _ _1\u2014+\u00a7\u2014\n(4.2.28)\n"]], ["block_13", ["_N_.,,_:1_\u2014I\u20143>Z;I\u2014TT\u00bb2=_(1\u2014I\u2014P\u2014)i-l2\u20143: +L_2\n(4.2.29)\nN\u201c\n(1 +12)\n(1+V)\n(1 +12)\n"]], ["block_14", ["1.07\n:I\n"]], ["block_15", ["r\n3\n'\n1.01 \no\n,\n\u2014j\n-\n"]], ["block_16", ["102:-\n.\nO\n-:\n02-\n_\n"]], ["block_17", ["Q\ng\nz\n-\n1IIIIIIIllllIIIIIIIIIIIIIIIII_I\u2014l\u2014IIII.\nOllgrllrrrllrllll\n.\n"]], ["block_18", ["0\n1x104 2x104 3x104 4x104 5x104 6x104 7x104\n3.5x104 4x104 4.5>-<1o4 5x104 5.5><1o\u2018l sxro4 6.5x104\n"]], ["block_19", ["l\nI\nI\n\u2018I\nl_l_l_l\nI\nI\nI\nl\nI\nl\nI\nI\nI\nI\nI\nl\nI\nI\nI\nII\nI\nl\nI\nI\nI\nl\nl\nl_\n:\n"]], ["block_20", [{"image_3": "139_3.png", "coords": [199, 390, 441, 591], "fig_type": "figure"}]], ["block_21", ["0.3-\n_\n"]], ["block_22", ["1\nI\nI\n-\n"]]], "page_140": [["block_0", [{"image_0": "140_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Anionic polymerization has been the most important mechanism for living polymerization, since\nits first realization in the 1950s [2]. Both modes of ionic polymerization (i.e., anionic and cationic)\nare described by the same vocabulary as the corresponding steps in the free\u2014radical mechanism for\nchain-growth polymerization. However, initiation, propagation, transfer, and termination are quite\ndifferent in ionic polymerization than in the free-radical case and, in fact, different in many ways\nbetween anionic and cationic mechanisms. In particular, termination by recombination is clearly\nnot an option in ionic polymerization, a simple fact that underpins the development of living\npolymerization. In this section we will discuss some of the factors that contribute to a successful\nliving anionic polymerization, and in the following section we will illustrate the extension of these\ntechniques to block copolymers and controlled architecture branched polymers.\nMonomers that are amenable to anionic polymerization include those with double bonds (vinyl,\ndiene, and carbonyl functionality), and heterocyclic rings (see also Table 4.3). In the case of vinyl\nmonomers CH2 CHX, the X group needs to have some electron withdrawing character, in order\nto stabilize the resulting carbanion. Examples include styrenes and substituted styrenes, vinyl\naromatics, vinyl pyridines, alkyl methacrylates and acrylates, and conjugated dienes. The relative\nstabilities of these carbanions can be assessed by considering the pKa of the corresponding\nconjugate acid. For example, the polystyryl carbanion is roughly equivalent to the conjugate\nbase of toluene. The smaller the pKa of the corresponding acid, the more stable the resulting\ncarbanion. The more stable the carbanion, the more reactive the monomer in anionic polymeriza-\ntion. In the case of anionic ring-opening polymerization (ROP), the ring must be amenable to\nnucleophilic attack, as well as present a stable anion. Examples include epoxides, cyclic siloxanes,\nlactones, and carbonates. At the same time, there are many functionalities that will interfere with\nan anionic mechanism, especially those with an acidic proton (e.g., \u2014OH, \u2014NH3, \u2014COOH) or an\nelectrophilic functional group (e.g., 02, \u2014C(O)\u2014, C02). Anionic polymerization of monomers that\ninclude such functionalities can generally only be achieved if the functional group can be\nprotected. As a corollary, the polymerization medium must be rigorously free of protic impurities\nsuch as water, as well as oxygen and carbon dioxide.\nA wide variety of initiating systems have been developed for anionic polymerization. The \ufb01rst\nconsideration is to choose an initiator that has a comparable or slightly higher reactivity than the\nintended carbanion. If the initiator is less reactive, the reaction will not proceed. If, on the other\nhand, it is too reactive, unwanted side reactions may result. As the pKas of the conjugate acids for\nthe many possible monomers span a wide range, so too must the pKas of the conjugate acids of the\ninitiators. Second, the initiator must be soluble in the same solvent as the monomer and resulting\npolymer. Common classes of initiators include radical anions, alkali metals, and especially\nalkyllithium compounds. We will illustrate two particular initiator systems: sodium naphthalenide,\nas an example of a radical anion, for the polymerization of styrene, and sec\u2014butyllithium, as an\nalkyllithium, in the polymerization of isoprene.\n"]], ["block_2", ["4.3\nAnionic Polymerization\n"]], ["block_3", ["guarantee a narrow distribution, however. The additional requirements for approaching the Poisson\ndistribution are:\n"]], ["block_4", ["1.\nAll active chain ends must be equally likely to react with a monomer throughout the\npolymerization. This requires both the principle of equal reactivity, and good mixing of\nreagents at all times.\n2.\nAll active chain ends must be introduced at the same time. In practice, this means that the rate\nof initiation needs to be much more rapid than the rate of propagation, if all the monomer is\nadded to the reaction mixture at the outset.\n3.\nPropagation must be essentially irreversible, that is, the reverse \u201cdepolymerization\u201d reaction\ndoes not occur to a significant extent. There are, in fact, cases where the propagation step is\nreversible, leading to the concept of an equilibrium polymerization, which we will take up in\nSection 4.8.\n"]], ["block_5", ["126\nControlled Polymerization\n"]]], "page_141": [["block_0", [{"image_0": "141_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "141_1.png", "coords": [31, 383, 272, 437], "fig_type": "figure"}]], ["block_2", ["4.\n"]], ["block_3", ["Now we consider the polymerization of isoprene by sec\u2014butyllithium, in benzene at room\ntemperature. In the first step, one monomer is added, but immediately there are many possibilities,\n"]], ["block_4", [{"image_2": "141_2.png", "coords": [35, 558, 336, 601], "fig_type": "molecule"}]], ["block_5", ["Anionic Polymerization\n127\n"]], ["block_6", ["as indicated:\n"]], ["block_7", ["3.\n"]], ["block_8", ["tetrahydrofuran \n"]], ["block_9", ["1.\n"]], ["block_10", [{"image_3": "141_3.png", "coords": [36, 379, 247, 447], "fig_type": "molecule"}]], ["block_11", ["The first living polymer studied in detail was polystyrene initiated with sodium naphthalenide in\n"]], ["block_12", ["Me\nMe\n1 4\nMe\n/\nMe\n/0\\\n"]], ["block_13", ["Me+Li\nMe\n1 2 3 4\nMe\n+Li\n"]], ["block_14", [{"image_4": "141_4.png", "coords": [49, 284, 128, 342], "fig_type": "figure"}]], ["block_15", [{"image_5": "141_5.png", "coords": [50, 274, 241, 351], "fig_type": "figure"}]], ["block_16", [{"image_6": "141_6.png", "coords": [51, 381, 150, 442], "fig_type": "molecule"}]], ["block_17", ["The latter undergoes radical combination to form the dianion, which subsequently initiates the\npolymerization:\n"]], ["block_18", ["In this case, the degree of polymerization is 2? because the initiator is difunctional; further-\nmore, there will be a single tail-to\u2014tail linkage somewhere near the middle of each chain.\nThe propagation step at either end of the chain can be written as follows:\n"]], ["block_19", ["Of course the structure of the radical anion shown is just one of the several possible resonance\nforms.\nThese green radical ions react with styrene to produce the red styryl radical ion:\n"]], ["block_20", ["The precursor to the initiator is prepared by the reaction of sodium metal with naphthalene and\nresults in the formation of a radical ion:\n"]], ["block_21", ["H\nCH2\nH..c.:H2 \ni\u201d +6\n\u201dJ\u201d\n(4.3)\n"]], ["block_22", ["The carbanion attacks the more electropositive and less sterically hindered carbon to regen\u2014\nerate the more stable benzylic carbanion. Thus, the addition is essentially all head\u2014to-tail in\nthis case. Note also that the sodium counterions have not been written explicitly in Reactions\n(4.B)\ufb02(4.D), although of course they are present. As we will see below, the counterion can\nactually play a crucial role in the polymerization itself.\n"]], ["block_23", ["Na+HNa*+[i_\n(4A)\n"]], ["block_24", ["1,44\n_H_\nH\nCH\n"]], ["block_25", [{"image_7": "141_7.png", "coords": [61, 116, 216, 150], "fig_type": "molecule"}]], ["block_26", ["H .. o\n"]], ["block_27", [{"image_8": "141_8.png", "coords": [81, 192, 311, 258], "fig_type": "molecule"}]], ["block_28", [{"image_9": "141_9.png", "coords": [110, 387, 247, 437], "fig_type": "molecule"}]], ["block_29", [{"image_10": "141_10.png", "coords": [110, 279, 222, 339], "fig_type": "molecule"}]], ["block_30", [{"image_11": "141_11.png", "coords": [124, 551, 386, 616], "fig_type": "molecule"}]], ["block_31", [{"image_12": "141_12.png", "coords": [153, 626, 360, 687], "fig_type": "molecule"}]], ["block_32", [{"image_13": "141_13.png", "coords": [157, 619, 374, 686], "fig_type": "figure"}]], ["block_33", ["+LI\n1,2\nMe\n\u2018Me\n3,4\n\u201c\" JAE\nMel/I:\nMe\n\\CHQ\nMe\nMe\n(413)\n"]], ["block_34", [{"image_14": "141_14.png", "coords": [218, 635, 343, 680], "fig_type": "molecule"}]], ["block_35", [{"image_15": "141_15.png", "coords": [271, 549, 378, 605], "fig_type": "molecule"}]], ["block_36", ["(4D)\n"]], ["block_37", ["(4.C)\n"]]], "page_142": [["block_0", [{"image_0": "142_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "142_1.png", "coords": [24, 527, 454, 667], "fig_type": "figure"}]], ["block_2", [{"image_2": "142_2.png", "coords": [33, 306, 432, 447], "fig_type": "figure"}]], ["block_3", ["THF\nLi\n30\n12 combined\n29\n59\nDioxane\nLi\n15\n3\n1 1\n18\n68\nHeptanea\nLi\n\u2014\u2014 10\n74\n18\n\u2014\n8\nHeptaneb\nLi\n\u201410\n97\n\u2014\n\u2014\n\u2014-\n"]], ["block_4", ["Table 4.1\nPolymerization of Polyisoprene under Various Conditions,\nand the Resulting Microstructure in %\n"]], ["block_5", ["aInitiator concentration 6 x 10\u20193 M.\nbInitiator concentration 8 x 10\"6 M.\n"]], ["block_6", ["Source: From Hsieh, H.L. and Quirk, R.P., Anionic Polymerization, Principles and Practical Applications, Marcel Dekker,\nInc., New York, NY, 1996.\n"]], ["block_7", ["Addition steps occur primarily when the living chain end is not associated. This leads to an\ninteresting dependence of the rate of polymerization, RP, on the living chain concentration, as can\nreadily be understood as follows (recall Equation 4.2.3):\n"]], ["block_8", ["None\nLi\n25\n94\n\u2014\n\u2014\n6\nNone\nNa\n25\n\u2014\n45\n7\n48\nNone\nCs\n25\n4\n51\n8\n37\n"]], ["block_9", ["Which happens, and why? What happens when the next monomer adds? Is it the same configuration,\nor not? What does it all depend on? There is no simple answer to these questions, but we can gain a\nlittle insight into how to control the microstructure of a polydiene by looking at some data.\nTable 4.1 gives the results of chemical analysis of the microstructure of polyisoprene after\npolymerization under the stated conditions. In the first two cases, there is a strong preference for\n3,4 addition, with significant amounts of 1,2; relatively little 1,4 addition is found. The key feature\nhere turns out to be the solvent polarity, as will be discussed below. When switching to heptane, a\nnonpolar solvent, the situation is reversed; now 1,4 cis is heavily favored. Interestingly, decreasing\nthe initiator concentration by a factor of a thousand exerts a significant in\ufb02uence on the 1,4 cis/\ntrans ratio. At first glance this seems strange; the details of an addition step should not depend on\nthe number of initiators. However, the answer lies in kinetics, as the propagation step is not as\nsimple as one might naively expect. Finally, the last three entries show isoprene polymerized in\nbulk, which also corresponds to a nonpolar medium. In this case, we see that changing the\ncounterion has a huge effect. Simply replacing lithium with sodium switches the product from\nalmost all cis 1,4 to a mixture of trans 1,4 and 3,4.\nThe key factor that comes into play in nonpolar solvents is ion pairing or clustering of the living\nends. Ionic species tend to be sparingly soluble in hydrocarbons, as the dielectric constant of the\nmedium is too low. Consequently, the counterion is rather tightly associated with the carbanion,\nforming a dipole; these dipoles have a strong tendency to associate into a small cluster, with\nperhaps it 2, 4, or 6 chains effectively connected as a star molecule. This equilibrium is\nillustrated in the cartoon below for the case n 4:\nair-\n41\u20144\n"]], ["block_10", ["Solvent\nCounterion\nT, \u00b0C\n1,4 cis\n1,4 trans\n1,2\n3,4\n"]], ["block_11", ["128\nControlled Polymerization\n"]], ["block_12", [{"image_3": "142_3.png", "coords": [38, 309, 200, 446], "fig_type": "figure"}]], ["block_13", [{"image_4": "142_4.png", "coords": [189, 355, 374, 407], "fig_type": "molecule"}]], ["block_14", [{"image_5": "142_5.png", "coords": [195, 336, 364, 415], "fig_type": "figure"}]], ["block_15", [{"image_6": "142_6.png", "coords": [210, 529, 439, 646], "fig_type": "figure"}]]], "page_143": [["block_0", [{"image_0": "143_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["a suitable acidic proton source, such as methanol, will cap the growing chain and produce the\ncorresponding salt, for example, Li +OCH3\u2014 . Care must be taken that the termination is conducted\nunder the same conditions of purity as the reaction itself, however. For example, introduction of\noxygen along with the terminating agent can induce coupling of two living chains. However, in\nmany cases it is desirable to introduce a particular chemical functionality at the end of the growing\nChain. One prime example is to switch to a second monomer, which is capable of continued\npolymerization to form a block copolymer. The second example is to use particular multifunctional\nterminating agents to prepare star\u2014branched polymers. These cases, and other uses of end\u2014functional\nchains, are the next subject we take up.\n"]], ["block_2", ["where we recognize [(P*)4] m (1/4)[P*], as most of the chains are in aggregates, and that the\napparent rate constant kappzkp(KdiS/4)U4. The rate of polymerization is therefore first order in\nmonomer concentration, as one should expect, but has a (l/n) fractional dependence on initiator\nconcentration, is the average size. Accurate experimental determination of n is\ntricky, but a large body of data exists. It should also be noted that there is in all likelihood a\ndistribution of of association or ion clustering, so that the actual situation is considerably\nmore complicated than implied by Reaction (4F).\nIncreasing the size of the counterion increases the separation between charges at the end of the\ngrowing chain, thereby facilitating the insertion of the next monomer. The concentration of\ninitiator can also in\ufb02uence:2, presumably by the law of mass action. The dependence of the cis\nisomer concentration in heptane indicated in Table 4.1 is actually thought to be the result of a more\nsubtle effect than however. Itis generally accepted that the cis configuration is preferred\nimmediately after addition of a monomer, but that isomerization to trans is possible, within an\naggregate, given time. The rate of isomerization is proportional to the concentration of chains in\naggregates and therefore proportional to [P*], whereas the rate of addition is proportional to a\nfractional power of [P*]. Increasing the initiator concentration increases both rates, but favors\nisomerization relative to propagation.\nTermination of an anionic polymerization is a relatively straightforward process; introduction of\n"]], ["block_3", ["Before addressing the preparation of block copolymers by anionic polymerization, it is appropriate to\nconsider some of the reasons why block copolymers are such an interesting class of macromolecules.\n"]], ["block_4", ["4.4\nBlock Copolymers, End-Functional Polymers, and Branched\nPolymers by Anionic Polymerization\n"]], ["block_5", ["The central importance of living anionic polymerization to current understanding of polymer\nbehavior cannot be overstated. For example, throughout Chapter 6 through Chapter 13 we will\nderive a host of relationships between observable physical properties of polymers and their\nmolecular weight. These relationships have been largely confirmed or established experimentally\nby measurements on narrow molecular weight distribution polymers, which were prepared by\nliving anionic methods. However, it can be argued that even more important and interesting\napplications of living polymerization arise in the production of elaborate, controlled architectures;\nthis section touches on some of these possibilities.\n"]], ["block_6", ["4.4.1\nBlock Copolymers\n"]], ["block_7", ["Inserting \nR, :kp\u2014\nBr\n2\nn\nm\n+ \ufb01Br --\u2014-* (styrene)n-\u2014\u2014\u2014(methyl methacrylate)2m\u2014(styrene)n\nO\nO O\nO\n|\n|\n9\nMe\n+ 2 LiBr\nM\n(4.G)\n"]], ["block_8", [{"image_4": "146_4.png", "coords": [48, 620, 266, 684], "fig_type": "figure"}]], ["block_9", ["H\nOH\n' _\n+\nO\nH+\nn+1\n[.1 \n3]\nSwo-\nn\n\u201c+Oj\u2014\u2014F\u2014)-\nn+1\n"]], ["block_10", [{"image_5": "146_5.png", "coords": [173, 617, 435, 681], "fig_type": "molecule"}]], ["block_11", [{"image_6": "146_6.png", "coords": [284, 622, 410, 679], "fig_type": "molecule"}]], ["block_12", ["(4.H)\n"]], ["block_13", ["+\n"]]], "page_147": [["block_0", [{"image_0": "147_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["end\u2014functional polymer with\nfunctionality at one end, or at both ends. Such polymers are also referred to as telechelic. It should\nbe apparent that most condensation polymers have reactive groups at each end, and thus fall in\nthis class. here with polymers that have narrow molecular weight\ndistributions In essence, an end-functional is\nmacromolecular characterized and then stored on the shelf until needed\nfor particular application. is a list of a few of the many examples of possible uses\nfor end-functional polymers:\n"]], ["block_2", ["The previous illustration of the macroinitiator approach is an excellent example of the utility of an\n"]], ["block_3", ["4.4.2\nEnd-Functional Polymers\n"]], ["block_4", ["1.\nMacroinitiators. As illustrated in the previous section, a macroinitiator is an end-functional\npolymer in which the functional group can be used to initiate polymerization of a second\nmonomer. In this way, block copolymers can be prepared that are not readily accessible by\nsequential monomer addition. Indeed, the second block could be polymerized by an entirely\ndifferent mechanism than the first; other living polymerization schemes will be discussed in\nsubsequent sections.\n2.\nLabeled polymers. It is sometimes desired to attach a \u201clabel\u201d to a particular polymer, such as a\n\ufb02uorescent dye or radioactive group, which will permit subsequent tracking of the location of\nthe polymer in some process. By attaching the label to the end of the chain, the number of\nlabels is well-defined, and labeled chains can be dispersed in otherwise equivalent unlabeled\nchains in any desired proportion.\n3.\nChain coupling. Both block copolymers and regular branched architectures can be accessed by\ncoupling reactions between complementary functionalities on different chains.\n4.\nMacromonomers. If the terminal functional group is actually polymerizable, such as a carbon\u2014\ncarbon double bond, polymerization through the double bond can produce densely branched\ncomb or \u201cbottlebrush\u201d copolymers.\n5.\nGrafting to surfaces. As mentioned in the context of copolymer adsorption to a surface, a densely\npacked layer of polymer chains emanating from a surface forms a brush. Such brushes can also\nbe prepared by the grafting of end-functional chains, where the functionality is tailored to react with\n"]], ["block_5", ["7.\nNetwork precursors. Telechelic polymers can serve as precursors to network formation, when\ncombined with suitable multifunctional linkers or catalysts. For example, some silicone\nadhesives contain poly(dimethylsiloxane) chains with vinyl groups at each end. In the\nunreacted form, these polymers form a low-viscosity \ufb02uid that can easily be mixed with\ncatalyst and spread on the surfaces to be joined; the subsequent reaction produces an adhesive,\nthree-dimensional network in situ.\n8.\nReactive compatibilization. As noted previously, block copolymers can act as macromolecular\nsurfactants to stabilize dispersions of immiscible homopolymers. However, direct mixing of\nblock copolymers during polymer processing is not always successful, as the copolymers have\n"]], ["block_6", ["the surface. High grafting densities are hard to achieve by this strategy, however, due to steric\ncrowding; the first chains anchored to the surface make it progressively harder for further chain\nends to react.\n6.\nControlled-branched and cyclic architectures. Examples of branched structures will be given\nin the following section. Cyclic polymers can be prepared by intramolecular reaction of an\n\u201cor,m-heterotelechelic\u201d linear precursor, where the two distinct end groups can react. Such\nring-closing reactions have to be run at extreme dilution, to suppress interchain end linking.\n"]], ["block_7", ["Block Copolymers. End-Functional Polymers, and Branched Polymers\n133\n"]], ["block_8", ["a tendency to aggregate into micelles and never reach the interface between the two polymers.\nOne effective way to overcome this is to form the block copolymer at the targeted interface, by\nin situ reaction of suitable functional chains. Note that in this case it is not absolutely\nnecessary that the reactive groups be at the chain ends.\n"]]], "page_148": [["block_0", [{"image_0": "148_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "148_1.png", "coords": [15, 436, 312, 607], "fig_type": "figure"}]], ["block_2", ["Table 4.2\nExamples of Protection Strategies for Preparing End\u2014Functional\nPolymers by Living Anionic Polymerization of Styrenes and Dienes\n"]], ["block_3", ["Termination by short alkanes with a halide at one end and the protected functionality at the other.\n"]], ["block_4", ["Source: From Hirao, A. and Hoyashi, M., Acta Polymerica, 50, 219, 1999.\n"]], ["block_5", ["There are two general routes to end-functional chains: use a functional initiator or use a\nfunctional terminating group. The use of a functional terminating agent proves to be the more\n\ufb02exible strategy for a rather straightforward reason. Any functional group present in the initiator\nmust be inert to the polymerization, which can be problematic in the case of anionic polymeriza-\ntion. Thus, the functional group in the initiator must be protected in some way. In contrast, for the\nterminating agent all that is required are two functionalities: the desired one and another electro-\nphilic one to terminate the polymerization. However, the functionality that is designed to terminate\npolymerization must be substantially more reactive to carbanions than the other functionality, or\nmore than one chain end structure will result. Consequently, in most cases a protection strategy is\nalso employed for the terminating agent. Nevertheless, in the termination case the demands on the\nprotecting group are much reduced relative to initiation; in the former, the protecting group only\nneeds to be significantly less reactive than the electrophile, whereas in the latter the protecting\ngroup must be substantially less reactive than the monomer.\nFor the living anionic polymerization of styrene, butadiene, and isoprene, an effective termin-\nating strategy is to use alkanes that have bromo functionality at one end and the protecting group at\nthe other. The halide is very reactive to the carbanion, readily eliminating the LiBr salt as the chain\nis terminated. Of course, the protecting group must then be removed in a separate step. Examples\nof protecting groups and the desired functionalities are given in Table 4.2. Some of the same\nprotecting groups illustrated in Table 4.2 can also be used in functional initiators. For example, the\ntert-butyl dimethylsilyl moiety used to protect the thiol group can also be used to protect a\nhydroxyl group in the initiator, as in (3-(tert-buty1 dimethylsilyloxy-l-propyllithium)).\nAnother powerful strategy for preparing end-functional polymers by anionic polymerization\nwas implicitly suggested in the previous section, where addition of a nominally polymerizable\nmonomer (ethylene oxide in that instance) to a growing polystyryl anion resulted in the addition of\nonly one new monomer. It turns out that 1,1-dipheny1ethylene and derivatives thereof will only\nreact with organolithium salts to form the associated relatively stable carbanion; no further\npropagation occurs:\n"]], ["block_6", ["Si(CH3)3\n-NH2\n[:1/ \\Si(CH3)3\n"]], ["block_7", ["-COOH\n/C(OCH3)3\n"]], ["block_8", ["\"CECH\n\u2014CEC\"Si(CH3)3\n"]], ["block_9", ["Functional Group\nProtected functionality\n"]], ["block_10", ["._\nO\nOH\n/ \\Si(CH3)3\n"]], ["block_11", ["134\nControlled Polymerization\n"]], ["block_12", [{"image_2": "148_2.png", "coords": [228, 449, 292, 475], "fig_type": "molecule"}]], ["block_13", ["Sr((CH3)2t-Bu)\n"]]], "page_149": [["block_0", [{"image_0": "149_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "149_1.png", "coords": [31, 538, 287, 633], "fig_type": "figure"}]], ["block_2", ["This multifunctional terminating agent is then introduced directly into the reaction vessel\ncontaining the living polystyryl chains. The chains should be in stoichiometric excess to minimize\n"]], ["block_3", ["The kinds of synthetic methodology suggested in the previous section have been adapted to the\npreparation of a wide range of polymer structures with controlled branching [4]. The first\narchitecture to consider is that of the regular star, in which a predetermined number of equal\nlength arms are connected to a central core. There are two general strategies to prepare such a\npolymer by living anionic polymerization: use a multifunctional initiator, and grow the arms\noutwards simultaneously, or use a multifunctional terminating agent to link together premade\narms. The first route is an example of an approach known as grafting from, whereas the second is\ntermed grafting r0. Or, in anticipation of the discussion of dendrimers in Section 4.9, grafting from\nand grafting to are analogous to divergent and convergent synthetic strategies. Although both have\nbeen used extensively, grafting to is more generally applicable to anionic polymerization due to the\ndifficulty in preparing and dissolving small molecules with multiple alkyllithium functionalities.\nFurthermore, in order to achieve uniform arm lengths, it is essential that each initiation site be\nequally reactive and equally accessible to monomers in the reaction medium. If it is desired to\nterminate each star arm with a functional group, however, then grafting from may be preferred.\nShould the anionic polymerization be initiated by a potassium alkoxide group, as for example with\nthe polymerization of ethylene oxide suggested in the context of Reaction (4.H), then preparation of\ninitiators with multiple hydroxyl groups is quite feasible (see Reaction 4.EE for a specific example).\nSimilarly, if other living polymerization routes are employed, such as controlled radical polymer-\nization to be discussed in Section 4.6, then grafting from is more convenient than in the anionic case.\nThe preparation of an eight-arm polystyrene star by grafting to is illustrated in the following\nscheme. The most popular terminating functionality in this context is a chlorosilane, which reacts\nrapidly and cleanly with many polymeric carbanions, and which can be prepared with function-\nalities up to at least 32 without extraordinary effort. An octafunctional chlorosilane can be prepared\nstarting with tetravinylsilane and dichloromethylsilane, using platinum as a catalyst:\n"]], ["block_4", ["In this structure R\u2019 and R\u201d could be any of a variety of protected or even unprotected functional-\nities. Even more interesting is the fact that this carbanion can be used to initiate anionic polymer\u2014\nization of a new monomer (such as methyl methacrylate, dienes, etc.) or even to reinitiate the\npolymerization of styrene. In this way, diphenylethylene derivatives can be used to place particular\nfunctional groups at desired locations along a homopolymer or copolymer, not just at the terminus.\n"]], ["block_5", ["4.4.3\nRegular Branched Architectures\n"]], ["block_6", [{"image_2": "149_2.png", "coords": [35, 42, 282, 154], "fig_type": "figure"}]], ["block_7", ["Block \n135\n"]], ["block_8", [{"image_3": "149_3.png", "coords": [40, 556, 274, 627], "fig_type": "molecule"}]], ["block_9", ["R!\nR\u201d\n,.\nH\nO\nO\n\"\n+Li\n+\nH0\n\u2014+\n\u2014\n+-\n(4.1)\n2\nLI\n"]], ["block_10", ["~\u2014\\-\u2014/S\u20187\n"]], ["block_11", [{"image_4": "149_4.png", "coords": [49, 69, 179, 145], "fig_type": "molecule"}]], ["block_12", ["Me Cl\n\\ /\nCl\u2014Si\nl/\n2\n9'\n"]], ["block_13", ["+ CH3SIC|2H \u2014\u2014-\u2014 CIWSIA\u2019SK CI\n(4.1)\n/\nM9 CI\n8\nSi-Cl\n/ \\\nCI Me\n"]], ["block_14", [{"image_5": "149_5.png", "coords": [144, 537, 275, 640], "fig_type": "molecule"}]], ["block_15", [{"image_6": "149_6.png", "coords": [144, 44, 269, 159], "fig_type": "molecule"}]], ["block_16", ["R!\nR.\u201d\n"]]], "page_150": [["block_0", [{"image_0": "150_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "150_1.png", "coords": [14, 278, 462, 376], "fig_type": "figure"}]], ["block_2", ["the formation of a mixture of stars with different numbers of arms. This will necessitate separation\nof the unattached arms from the reaction mixture, but this is feasible. Moreover, an additional\nadvantage of the grafting to approach is thus exposed: the unattached arms can be characterized\n(for molecular weight, polydispersity, etc.) independently of the stars themselves, a desirable step\nthat is not possible when grafting from.\nThe scheme just outlined is not quite as straightforward as it might appear. The key issue is to make\nall eight terminating sites accessible to the polystyryl chains. As the number of attached arms grows, it\nbecomes harder and harder for new chain ends to find their way into the reactive core. In order to\nreduce these steric effects, more methylene groups can be inserted into the terminating agent to spread\nout the chlorosilanes. In some cases, polystyryl chains have been capped with a few butadienyl units to\nreduce the steric bulk of the chain end. Clearly, all of these issues grow in importance as the number of\narms increases. Note, however, that it is not necessary that all the chlorosilanes be equally reactive in\norder to preserve a narrow molecular weight distribution; it is only necessary that the attachment of the\nnarrowly distributed arms be driven to completion (which may take some time).\nAs the desired number of arms increases, it is practical to surrender some control over the exact\nnumber of arms in favor of a simpler method for termination. A scheme that has been refined to a\nconsiderable extent is to introduce a difunctional monomer, such as divinylbenzene, as a poly-\nmerizable linking agent. The idea is illustrated in the following reaction:\n"]], ["block_3", ["One divinylbenzene molecule can thus couple two polystyryl chains and leave two anions for\nfurther reaction. Each anion might add one more divinylbenzene, each of which could then add one\nmore polystyryl chain. At that point, the growing star molecule would have four arms, emanating\nfrom a core containing three divinylbenzene moieties and four anions. This process can continue\nuntil the divinylbenzene is consumed and the anions terminated. Clearly, there is potential for\n"]], ["block_4", ["a great deal of variation in the resulting structures, both in the size of the core and in the number\nof arms. However, by carefully controlling the reaction conditions, and especially the ratio of\ndivinylbenzene to living chains, reasonably narrow distributions of functionality can be obtained,\nwith average numbers of arms even exceeding 100.\nThe preceding strategy can actually be classified as grafting through, a third approach that is\nparticularly useful for the preparation of comb polymers. A comb polymer consists of a backbone to\nwhich a number of polymeric arms are attached; combs can be prepared by grafting from, grafting to,\nand grafting through. In the first case, the backbone must contain reactive sites that can used to initiate\npolymerization. The backbone can be characterized independently of the arms, but the arms them-\nselves cannot. In grafting to, the backbone must contain reactive sites such as chlorosilanes that can\nact to terminate the polymerization of the arms. Clearly in this case, as with stars, the arms and the\nbackbone can be characterized independently. The grafting through strategy takes advantage of what\nwe previously termed macromonomers: the arms are polymers terminated with a polymerizable group.\nThese groups can be copolymerized with the analogous monomers to generate the backbone. By\nvarying the ratio of macromonomer to comonomer, the spacing of the \u201cteeth\u201d of the comb can be\ntuned. Note that this process is not necessarily straightforward. In Chapter 5 we will consider\ncopolymerization in great detail, but a key concept is that of reactivity ratio. This refers to the relative\nprobability of adding one monomer to a growing chain, depending on the identity of the previous\nmonomer that attached. It is generally the case that there are significant preferences (i.e., the\nreactivity ratios of the two monomers are not unity), which means that the two monomers will not\n"]], ["block_5", ["136\nControlled Polymerization\n"]], ["block_6", [{"image_2": "150_2.png", "coords": [37, 303, 244, 370], "fig_type": "molecule"}]], ["block_7", ["H\nH\nx\n\\\n2\nH\nn\n_\n+\nCH2\n{1\nl\n(4K)\n0 O\n/\nH\n\u2014\nn\n"]], ["block_8", [{"image_3": "150_3.png", "coords": [45, 300, 246, 368], "fig_type": "figure"}]], ["block_9", [{"image_4": "150_4.png", "coords": [228, 270, 416, 377], "fig_type": "figure"}]], ["block_10", [{"image_5": "150_5.png", "coords": [233, 289, 435, 379], "fig_type": "molecule"}]]], "page_151": [["block_0", [{"image_0": "151_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "151_1.png", "coords": [31, 206, 327, 259], "fig_type": "molecule"}]], ["block_2", ["This macromonomer can then be copolymerized with methyl methacrylate, to produce a comb or\ngraft copolymer, witha poly(methyl methacrylate) backbone and polystyrene arms:\n"]], ["block_3", ["1.\nA single initiator species is often not sufficient in cationic polymerizations; frequently a\nsecond ingredient (or cocatalyst) is required.\n2.\nTotal dissociation of the cationic initiator is rather rare, which has implications for the ability\nto start all the chains growing at the same time.\n3.\nAlthough both ionic mechanisms clearly eliminate termination by direct recombination of\ngrowing chains, cationic species are much more prone to transfer reactions than their anionic\ncounterparts. Consequently, living cationic polymerization is much less prevalent than living\nanionic polymerization.\n4.\nMost monomers that can be readily polymerized by anionic mechanisms are also amenable to\nfree-radical polymerization. Thus, in commercial practice the rather more demanding anionic\nroute is only employed when a higher degree of control is required, for example, in the\npreparation of styrene\u2014diene block copolymers.\n5.\nIn contrast, although most monomers that can be polymerized by cationic mechanisms are also\namenable to free-radical polymerization, there are important exceptions. The most significant\nfrom a total production point of view is polyisobutylene (butyl rubber), which is produced\ncommercially by (both living and nonliving) cationic polymerization.\n"]], ["block_4", ["This last example reminds us that the variety of possible controlled branched architectures is\ngreatly enhanced when different chemistries are used for different parts of the molecule. If we\nconfine ourselves to the case of stars, a molecule in which any two arms differ in a deliberate and\nsigni\ufb01cant way has been termed a miktoarm star, from the Greek word for mixed [4]. A whole host of\ndifferent structures have been prepared in this manner. For example, an AZB miktoarm star contains\ntwo equal length arms of polymer A and one arm of polymer B. Among the structures that have been\nreported are AZB, A3B, A2B2, A4B4, and a variety of ABC miktoarm terpolymers. It is even possible\nto produce asymmetric stars, in which the arms consist of the same polymer but differ in length.\n"]], ["block_5", ["add completely to be understood before regular comb molecules with\nvariable branching density can be prepared by grafting through.\nThe grafting through approach can be illustrated through the following sequence [5]. Polystyryl\nchains 4.H) followed \nmethacryloyl \n"]], ["block_6", ["4.5\nCationic Polymerization\n"]], ["block_7", ["Just as anionic polymerization is a chain-growth mechanism that shares important parallels with the\nfree-radical route, so too cationic polymerizations can be discussed within the same framework:\ninitiation, propagation, termination, and transfer. However, there are important differences between\nanionic and cationic polymerizations that have direct impact on the suitability of the latter for living\npolymerization. The principal differences between the two ionic routes are the following:\n"]], ["block_8", ["Cationic Polymerization\n137\n"]], ["block_9", ["H\n.\na\u2014m\n.\nW\u2018\n(W\nCH2\n0\n'\n"]], ["block_10", ["CH2\n0\nCH\n_ _\nH\nn\nomiMe\n+\nMew/\u201c\\O/\n3 PMMA 9 PS\n(4 M)\n0\nCH2\n\u2018\n"]], ["block_11", [{"image_2": "151_2.png", "coords": [73, 115, 331, 177], "fig_type": "molecule"}]], ["block_12", ["CH2\n0\n"]]], "page_152": [["block_0", [{"image_0": "152_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "152_1.png", "coords": [13, 77, 366, 243], "fig_type": "figure"}]], ["block_2", [{"image_2": "152_2.png", "coords": [32, 490, 245, 558], "fig_type": "figure"}]], ["block_3", ["Note: Parentheses indicate not readily polymerized by this route.\n"]], ["block_4", ["Table 4.3\nGeneral Summary of Polymerizability of Various Monomer Types by\nthe Indicated Chain-Growth Modes\n"]], ["block_5", ["Ethylene\n\\/\n(\\/)\n><\na-Olefins\n><\n\\/\n(\\/)\n1,1-Dialkyl alkenes\nx\nx\n\\/\nHalogenated alkenes\n\\/\nx\nx\n1,3-Dienes\n\\/\n\\/\n\\/\n"]], ["block_6", ["Source: Adapted from Odian, G., Principles of Polymerization, 4th ed, Wiley-Interscience,\nHoboken, NJ, 2004.\n"]], ["block_7", ["4.5.1\nAspects of Cationic Polymerization\n"]], ["block_8", ["With insufficient cocatalyst these equilibria lie too far to the left, while excess cocatalyst can terminate\nthe chain or destroy the catalyst. Thus, the optimum proportion of catalyst and cocatalyst varies with\nthe speci\ufb01c monomer and polymerization solvent. In the case of protonic acids, the concentration of\nprotons depends on the position of the standard acid\u2014base equilibria, but in the chosen organic solvent:\n"]], ["block_9", ["Styrenes\n\\/\n\\/\n\\/\nAcrylates, methacrylates\n\\/\n\\/\n><sub>><></sub>sub><sub><sub>><></sub></sub><sub>><></sub>/sub>\nAcrylonitrile\n\\/\n\\/\n><sub>><></sub>sub><sub><sub>><></sub></sub><sub>><></sub>/sub>\nAcrylamide, methacrylamide\n\\/\n\\/\n><sub>><></sub>sub><sub><sub>><></sub></sub><sub>><></sub>/sub>\nVinyl esters\n\\/\n><sub>><></sub>sub><sub><sub>><></sub></sub><sub>><></sub>/sub>\n\\/\nVinyl ethers\nx\nx\n\\/\nAldehydes, ketones\n><sub>><></sub>sub><sub><sub>><></sub></sub><sub>><></sub>/sub>\n\\/\n\\/\n"]], ["block_10", ["Monomer\nRadical\nAnionic\nCationic\n"]], ["block_11", ["A brief summary of the applicability of the three chain\u2014growth mechanisms\u2014radical, anionic,\ncationic\u2014to various monomer classes is presented in Table 4.3. In the remainder of this section we\ndescribe general aspects of cationic polymerization and introduce some of the transfer reactions\nthat inhibit living polymerization. Then, we conclude by discussing the strategies that have been\nused to approach a living cationic polymerization.\n"]], ["block_12", ["In cationic polymerization, the active species is the ion formed by the addition of a proton\nfrom the initiator system to a monomer (partly for this reason the initiator species is often called a\ncatalyst, because it is not incorporated into the chain). For vinyl monomers the substituents which\npromote this type of polymerization are electron donating, to stabilize the propagating carboca-\ntion; examples include alkyl, 1,1-dialky1, aryl, and alkoxy. Isobutylene, a-methylstyrene, and vinyl\nalkyl ethers are examples of monomers commonly polymerized via cationic intermediates.\nThe initiator systems are generally Lewis acids, such as BF3, AlCl3, and TiCl4, or protonic\nacids, such as H2804, HClO4, and HI. In the case of the Lewis acids, 3 proton-donating coinitiator\n(often called a cocatalyst) such as water or methanol is typically used:\n"]], ["block_13", ["138\nControlled Polymerization\n"]], ["block_14", [{"image_3": "152_3.png", "coords": [36, 599, 161, 682], "fig_type": "molecule"}]], ["block_15", ["CH30H + TiCI4 \nH+ + C|4TiOCHg\n"]], ["block_16", ["H2804\n\u2014__=\u2018\u2014-_\nH+ + H804\"\n"]], ["block_17", ["H20 A'C'a\n\u2014\u2018-\u2014.\u2014\nH+ + CI3AIOH\u2018\n(4.N)\n"]], ["block_18", ["H20 + BF3\n=\nH+ + FsBOH'\n"]], ["block_19", ["Helo4\n_..__ H+ + CIO4\u2018\n(4.0)\n"]], ["block_20", ["H|\nH+ + l\u2014\n"]], ["block_21", [{"image_4": "152_4.png", "coords": [137, 90, 334, 228], "fig_type": "figure"}]]], "page_153": [["block_0", [{"image_0": "153_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "153_1.png", "coords": [28, 97, 219, 129], "fig_type": "molecule"}]], ["block_2", [{"image_2": "153_2.png", "coords": [29, 584, 200, 651], "fig_type": "molecule"}]], ["block_3", [{"image_3": "153_3.png", "coords": [33, 443, 90, 505], "fig_type": "molecule"}]], ["block_4", ["The structures expected and found are sketched here:\n"]], ["block_5", ["One of the side reactions that can cationic polymerization is the possibility of the ionic\nrepeat unit undergoing rearrangement during the polymerization. The following example illus\u2014\ntrates this situation.\n"]], ["block_6", ["The electron\u2014donating group helps to stabilize this \npolymerization, the separation of the ions and the possibility of ion pairing play important roles in\n"]], ["block_7", ["If we write formula the initiator system as H+B_, then the initiation and\npropagation steps monomer CH2 =CHR can be written as follows. The proton adds\nto the more electronegative carbon atom in the olefin to initiate chain growth:\n"]], ["block_8", ["Expected\n"]], ["block_9", ["Found\n"]], ["block_10", ["The conversion of the cationic intermediate of the monomer to the cation of the product occurs by\n"]], ["block_11", ["It has been observed that poly(1,1\u2014dimethyl propane) is the product when 3-methylbutene-1\n(CH2:CH\u2014CH(CH3)2) is polymerized with Al in ethyl chloride at \u2014130\u00b0C.T Draw structural\nformulas for the expected and observed repeat units, and propose an explanation.\n"]], ["block_12", ["Aldehydes can also be polymerized in this fashion, with the corresponding reactions for formal-\ndehyde being\n"]], ["block_13", ["the ease proceeds \n"]], ["block_14", ["Cationic \n139\n"]], ["block_15", ["a hydride shift between adjacent carbons:\n.\n"]], ["block_16", ["\u201dP. Kennedy and RM. Thomas, Makromol. Chem, 53, 28 (1962).\n"]], ["block_17", ["Solution\n"]], ["block_18", ["Example 4.2\n"]], ["block_19", [{"image_4": "153_4.png", "coords": [38, 512, 98, 545], "fig_type": "molecule"}]], ["block_20", [{"image_5": "153_5.png", "coords": [43, 177, 236, 208], "fig_type": "molecule"}]], ["block_21", [{"image_6": "153_6.png", "coords": [43, 251, 337, 303], "fig_type": "molecule"}]], ["block_22", [{"image_7": "153_7.png", "coords": [46, 260, 223, 299], "fig_type": "molecule"}]], ["block_23", ["R\nR\nR\nR\nMe\u2014<+B\u2018 +H2CZ< \u2014\u00bbMe>t/\\r+\n-\n(4Q)\nH\nH\nH\nH\n'3\n"]], ["block_24", ["H\nO=<\n+\nH\nH\nH\nH+\n(4R)\n\u2018\n\u2014\u2014\u2014+ H\n+\n-\nH\n-\nH\nB\n+0= Me\u2014(+\nB-\n(4.13)\n"]], ["block_26", ["Me\nMe\n"]], ["block_27", [{"image_8": "153_8.png", "coords": [55, 596, 195, 635], "fig_type": "molecule"}]], ["block_28", ["n\nMe Me\n"]], ["block_29", ["H\nH\n"]], ["block_30", ["Fl\n"]], ["block_31", ["Me\n"]], ["block_32", [{"image_9": "153_9.png", "coords": [127, 180, 230, 212], "fig_type": "molecule"}]], ["block_33", [{"image_10": "153_10.png", "coords": [200, 255, 323, 298], "fig_type": "molecule"}]]], "page_154": [["block_0", [{"image_0": "154_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "154_1.png", "coords": [24, 205, 385, 245], "fig_type": "molecule"}]], ["block_2", ["1.\nClearly, the reaction must be conducted in the absence of nucleophilic species that are capable\nof irreversible termination of the growing chain.\n2.\nSimilarly, the reaction should be conducted in the absence of bases that can participate in\nB\u2014proton transfer. As discussed above, the monomer itself is such as base, and therefore\n"]], ["block_3", ["The preceding discussion provides some insight into the obstacles to achieve a living cationic\npolymerization. Nevertheless, living cationic polymerization is by now a relatively common tool,\nand many of the controlled architectures (block copolymers, end\u2014functional chains, regular branched\nmolecules) that we discussed in the context of anionic polymerization have been accessed [7]. In this\nsection, we brie\ufb02y describe the general strategy behind living cationic polymerization; recall that the\nessential elements are the absence of termination or transfer reactions:\n"]], ["block_4", ["4.5.2\nLiving Cationic Polymerization\n"]], ["block_5", ["4.\nSpontaneous termination. This process, also known as chain transfer to counterion, is essen-\ntially a reversal of the initiation step, as a B-proton is transferred back to the anion (e.g., as in\nReaction 4.P, but with a growing chain rather than the first monomer).\n"]], ["block_6", ["This is a well-known reaction that is favored by the greater stability of the tertiary compared to the\nsecondary carbocation.\n"]], ["block_7", ["In the particular case of isobutylene the resulting primary carbocation is less stable than the\ntertiary one on the chain, so Reaction (4.T) is less of an issue than Reaction (4.8).\n3.\nIntermolecular hydride transfer. This is an example of transfer to polymer, and can be written\ngenerally as\n"]], ["block_8", ["The preceding example illustrates one of the potential complications encountered in cationic\npolymerization, but it is not in itself an impediment to living polymerization. There are several\nother potential transfer reactions, however, that collectively do impede a living cationic polymeri-\nzation. Four of these are the following:\n"]], ["block_9", ["The activated monomer can now participate in propagation reactions, whereas the previous\nchain is terminated. Note that in isobutylene there are two distinct B\u2014protons, and thus two\npossible structures for the terminal unsaturation of the chain. There is also a possibility that\nthese double bonds can react subsequently.\n2.\nHydride transfer from monomer. In this case, the transfer proceeds in the opposite direc-\ntion, but has the same detrimental net effect from the point of view of achieving a living\npolymerization:\n"]], ["block_10", ["14o\nControlled Polymerization\n"]], ["block_11", ["1.\nB\u2014Proton transfer. This is exemplified by the case of polyisobutylene. Protons on carbons\nadjacent (B) to the carbocation are electropositive, due to a phenomenon known as hypercon-\njagation; we can view this as partial electron delocalization through 0 bonds, in contrast\nto resonance, which is delocalization through 17 bonds. Consequently there is a tendency for\nB-protons to react with any base present, such as a vinyl monomer:\n"]], ["block_12", ["Me\nMe\nMe\nMe\nMe\n\u201d\\\\ufb02\n3\u2014\n+ H20=< \u2014+ H30\u2014<+\nB\u2014 +\u201d\\n\u2019\n+ \u201dY\n(4.8)\nMe\nMe\nMe\nCH2\nMe\n"]], ["block_13", ["+ Wyn/i9\nMe\n(4.T)\nMe\nMe\nMe\nN\u2019J\\]/+\nB\u2014\n+ HZC=<\n\u2014-\u2014+\nHZC=<\n"]], ["block_14", [{"image_2": "154_2.png", "coords": [54, 210, 267, 238], "fig_type": "molecule"}]], ["block_15", ["R\nFl\n"]], ["block_16", [{"image_3": "154_3.png", "coords": [114, 430, 279, 471], "fig_type": "molecule"}]], ["block_17", [{"image_4": "154_4.png", "coords": [127, 336, 295, 377], "fig_type": "molecule"}]], ["block_18", [{"image_5": "154_5.png", "coords": [170, 207, 362, 242], "fig_type": "molecule"}]]], "page_155": [["block_0", [{"image_0": "155_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["The carbon\u2014iodide bond is then activated by the relatively weak Lewis acid az to allow insertion\nof the next monomer. The transition state for pr0pagation may be represented schematically as\n.01\nto\n"]], ["block_2", ["In this sequence, the first reaction generates the initiating proton, and the second and third reactions\ncorrespond to standard irreversible initiation and propagation steps involving monomer M. The fourth\nreaction is the key. The growing cationic i-mer P? is converted to a dormant, covalent species BC] by a\nreversible reaction. While the growing chain is in this form, it does not undergo transfer or propagation\nreactions, thereby extending its lifetime. The reversible activation/deactivation reaction must be\nsufficiently rapid to allow each chain to have many opportunities to add monomer during the\npolymerization, and the relative length of time spent in the active and dormant states can be controlled\nby the position ofthe associated equilibrium. This, in turn, offers many opportunities to tune a particular\nchemical system. For example, decreasing the polarity ofthe solvent or adding an inert salt that contains\n"]], ["block_3", ["Experimentally, the system exhibits many of the characteristics associated with a living polymeri-\nzation: polydispersities consistently below 1.1; Mn increasing linearly with conversion; the ability\n"]], ["block_4", ["a common ion (chloride in this case) both push the equilibrium toward the dormant state.\nWe will revisit this idea of a dormant reactive species in the next section on controlled radical\npolymerization, where it plays the central role. We conclude this section with a specific example of\na successful living cationic polymerization scheme. Isobutyl vinyl ether (and other vinyl ethers)\ncan be polymerized by a combination of HI and az [8]. The hydrogen iodide \u201cinitiator\u201d adds\nacross the double bond, but forms an essentially unreactive species:\n"]], ["block_5", ["4.\nCationic polymerization, like most chain-growth polymerization, is often highly exothermic.\nWith the additional feature of very rapid reaction, it becomes important to reduce the rate of\npolymerization in order to remove the excess heat. Low temperature is the first option in this\nrespect, followed by lower concentrations of growing chains.\n5.\nAnother way to view control in this context is to aim to extend the lifetime ofthe growing chain. As\na point ofreference, a living polystyryl carbanion can persist for years in a sealed reaction vessel; a\npolyisobutyl carbocation will probably not last for an hour under equivalently pristine conditions.\nWhile low temperature certainly aids in increasing the lifetime, another useful strategy is to make\nthe growing center inactive or dormant for a significant fraction ofthe elapsed reaction time. This is\ndone via the process of reversible termination, as illustrated by the sequence in Reaction (4.V):\n"]], ["block_6", ["3.\nGenerally, both propagation and transfer are very rapid reactions, with transfer having the\nhigher activation energy. Lower temperatures therefore favor propagation relative to transfer,\nas well as have the advantage of bringing both reactions under better control.\n"]], ["block_7", ["Cationic \n141\n"]], ["block_8", [{"image_1": "155_1.png", "coords": [39, 502, 195, 539], "fig_type": "molecule"}]], ["block_9", [{"image_2": "155_2.png", "coords": [47, 575, 190, 627], "fig_type": "molecule"}]], ["block_10", ["HCI + Till; I: TiCl5' + H+\n"]], ["block_11", ["H++ M \nP14-\n"]], ["block_12", ["on\nonl\n+ HI\n4.W\nc=Pi++ M \u00e9\nPi+1+\n"]], ["block_14", ["Pi++TiCI5' :: PiCI +TiC|4\n"]], ["block_15", ["cationic polymerization always has a \u201cbuilt-in\u201d transfer reaction. The key step, therefore, is to\nchoose reaction conditions to maximize the rate of propagation relative to transfer, given that\ntransfer probably cannot be completely eliminated.\n"]], ["block_16", ["\\[2\u2019\nOF!\nOR\n"]], ["block_17", ["I;\n|--Zn|2\n\u2014-\u2014\u2014h-\nreznlz\n"]], ["block_18", ["(4.V)\n"]], ["block_19", ["(4.X)\n"]]], "page_156": [["block_0", [{"image_0": "156_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["In this section we take up the topic of controlled radical polymerization, which represents one of\nthe most active \ufb01elds in polymer synthesis in recent years. The combination of the general\nadvantages of radical polymerization (a wide range of suitable monomers, tolerance to many\nfunctional groups, characteristically rapid reactions, relatively relaxed polymerization conditions)\nwith the unique features of a living polymerization (narrow molecular weight distributions,\ncontrolled molecular weights, end functionality, block copolymers, and other complex architec-\ntures) has tremendous appeal in many different areas of polymer science. In this section we outline\nfirst in general terms how this combination is achieved, and then give some specific examples of\nthe mechanistic details. We choose the term \u201ccontrolled\u201d rather than \u201cliving\u201d in this section,\nbecause irreversible termination reactions cannot be rigorously excluded.\n"]], ["block_2", ["to resume polymerization after addition of a new charge of monomer. These aspects are illustrated\nin Figure 4.5. The mechanism implied by Reaction (4.W) and Reaction (4.X) is consistent with the\nexperimental observation that Mn is inversely proportional to the concentration of HI, but inde-\npendent of the concentration of ZnIz. On the other hand, the polymerization rate increases with\nadded 21112. az is apparently a sufficiently mild activator that the polymerization is still living at\nroom temperature when conducted in toluene, whereas in the more polar solvent methylene\nchloride lower temperatures are required.\n"]], ["block_3", ["4.6\nControlled Radical Polymerization\n"]], ["block_4", ["Figure 4.5\nLiving cationic polymerization of isobutyl vinyl ether in methylene chloride at \u201440\u00b0C. The\ncalculated curve indicates the expected molecular weight assuming 100% initiator (HI) efficiency. After 100%\nconversion a new charge of monomer was added, demonstrating the ability of the chain ends to resume\npr0pagati0n. (Reproduced from Sawamoto, M., Okamoto, C., and Higashimura, T., Macromolecules, 20,\n2693, 1987. With permission.)\n"]], ["block_5", ["4.6.1\nGeneral Principles of Controlled Radical Polymerization\n"]], ["block_6", ["The first task is to resolve the apparent paradox: given that radicals can always combine to undergo\ntermination reactions, how do we approximate a living polymerization? To develop the answer, it\nis helpful to start by summarizing once again the essence of a chain-growth polymerization in\nterms of initiation, propagation, and termination rates:\n"]], ["block_7", ["142\nControlled Polymerization\n"]], ["block_8", [{"image_1": "156_1.png", "coords": [36, 47, 275, 247], "fig_type": "figure"}]], ["block_9", ["X\n0\nIE:\nIEC\nX\n.\n:\n[EC\n"]], ["block_10", ["\u20185;\n10\n-\nl\n"]], ["block_11", ["15 \nCalcd\n"]], ["block_12", ["5\n-\n5\n\u2014\n"]], ["block_13", ["0\nr\n\u2018\nI\nI\nI\n1.0\n"]], ["block_14", ["0\n100\n200\n"]], ["block_15", ["A\n. no?\n2 A\n\u2014 1 1\n"]], ["block_16", ["E\n\u2014- 1 2\nO\nI\nA\nA\ufb01.\n"]], ["block_17", ["Conversion, %\n"]], ["block_18", ["Monomer\n0\naddition\n"]]], "page_157": [["block_0", [{"image_0": "157_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["The process of controlled radical polymerization can now be seen to take place as follows. The\ndormant species PX spontaneously dissociates into the active radical and the inert partner X.\nThe exposed radical may then undergo propagation steps, or simply recombine with X so that no net\nreaction takes place. If each radical spends most of its time in the dormant state, the instantaneous\nconcentration of radicals is small, and termination is very unlikely (but never impossible). During\nan average active period a given radical may add many new monomers, about one new monomer,\nor essentially no new monomers. It is actually the last situation that is most desirable, because it\nmeans that over time all radicals are equally likely to propagate, one monomer at a time. We can\nunderstand this concept in the following way. After the polymerization has proceeded for a\nreasonable time, so that each chain on average has experienced many active periods, the number\nof active periods per chain will follow the Poisson distribution (Equation 4.2.19). That is because\nwe are randomly distributing a large number of items (active periods) into a smaller number of\nboxes (growing chains). In the limit where the likelihood of adding a monomer per active period is\nsmall, the average number of monomers added per chain will be directly proportional to the\nnumber of active periods, and thus follow the Poisson distribution as well. Of course, we are\nneglecting any termination and transfer reactions.\nIn contrast, if radicals tend to add monomers in a burst during each active period, the molecular\nweight distribution will not be as narrow unless the total degree of polymerization involves many\nsuch bursts. In fact, the length distribution of the \u201cbursts\u201d will be the most probable distribution,\nwhich (recall Equation 2.4.10 and Equation 3.7.19) has a polydispersity approaching 2. We can\nactually rationalize an approximate expression for the polydispersity of the resulting polymers,\n"]], ["block_2", ["where species, [M] is the concentration of unreacted\nmonomer, and [19.] concentration of radicals of any length. The key to a living\npolymerization is that Rt 0, or equivalently in practice, that Rp >> Rt. From Table 3.3 and\nTable 3.4, we of kt are about four orders of magnitude larger than kp.\nTherefore, thanl\u2018i\u2019t we will need [M] to be 108 times larger\nthan [Po]. Given that [M] could be on the order of 1\u201410 mol L\"1 (i.e., in bulk or concentrated\nsolution), and therefore growing chains, will have to be\n104\u2014 10\u20148 mol L4. This is quite small, but from the calculations in Section 3.4.3 we know that it\nis quite feasible.\nWe can do even better than this, however, by a nifty trick already suggested in the context of\nliving cationic polymerization. that the absolute concentration of radical forming species\nis not that the vast majority of its time in an unreactive,\ndormant form. This is illustrated schematically below:\n"]], ["block_3", ["where PX is the dormant species and X is a group (atom or molecular fragment) that can leave and\nreattach to the radical rapidly. (Note that radical species are apparently not conserved in the way\nReaction (4.Y) is expressed, but as we will see subsequently this is not actually the case. Usually X\nis also a radical species, but one that is not capable of propagating.) If the equilibrium constant for\nthe activation process, Kw, is small, then the instantaneous [P-] will be small even if [PX] is\nreasonably large:\n"]], ["block_4", ["Controlled Radical Polymerization\n143\n"]], ["block_5", ["R. kit]\n(4.6.1)\nR, kprPoitM]\n(45-2)\nRt _\u2014_ kt[P\u00b0][P-]\n(4.6.3)\n"]], ["block_6", ["PX = P- + X\n(4.Y)\n"]], ["block_7", ["[PX]\n[P ]\nKact\n[X]\n(4.6.4)\n"]]], "page_158": [["block_0", [{"image_0": "158_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Equation 4.6.5 suggests that even if q is 10, a polydispersity of 1.1 is achievable if the total degree\nof polymerization exceeds 100. A more detailed analysis yields equations similar to Equation\n4.6.5, when the average degree of polymerization is sufficiently large [9]. It is worth noting that\nthere are several complications to this analysis, such as the fact that the value of q will actually\nchange during the polymerization, as [M] decreases.\nIt should be evident from the preceding discussion that termination processes are not rigor-\nously excluded in controlled radical polymerization, only significantly suppressed. Nevertheless,\npolydispersities MW/Mn < 1.1\u20141.2 are routinely obtained by this methodology. It should also be\nevident that the higher the average chain length, the more likely termination steps become. This\ncan be seen directly from Equation 4.6.2 and Equation 4.6.3; as time progresses, l\u2019i\u2019t remains\nessentially constant, whereas Rp decreases because [M] decreases with time. Consequently, the\nrelative likelihood of a termination event increases steadily as the reaction progresses. In fact,\nthere is really a three-way competition in designing a controlled radical polymerization scheme,\namong average molecular weight, polydispersity, and \u201cefficiency,\u201d where we use ef\ufb01ciency to\ndenote a combination of practical issues. For example, the higher the desired molecular weight,\nthe broader the distribution will become, due to termination reactions. This could be mitigated\nto some extent by running at even higher dilution, but this costs time and generates large volumes\nof solvent waste. Or, the reaction vessel could be replenished with monomer, to keep [M]\nhigh even as the reaction progresses, but this wastes monomer, or at least necessitates a\nrecovery process.\n"]], ["block_2", ["A rich variety of systems that fall under the umbrella of Reaction (4.Y) have been reported. Three\ngeneral schemes have so far emerged as the most prevalent, although there is no a priori reason\nwhy others may not become more pOpular in the years ahead. Each has particular advantages and\ndisadvantages relative to the others, but for the purposes of this discussion we are really only\ninterested in their evident success. All three have been the subject of extensive review articles; see\nfor example Refs. [10,11,12].\n"]], ["block_3", ["4.6.2\nParticular Realizations of Controlled Radical Polymerization\n"]], ["block_4", ["4.6.2.1\nAtom Transfer Radical Polymerization (ATRP)\n"]], ["block_5", ["based on what we already know. Suppose the mean number of monomers added per active period\nis q 2 1. The distribution of active periods from chain to chain still follows the Poisson distribution,\nso it is almost as though we were adding one q-length block per active period. Thus, the\npolydispersity index becomes that for the Poisson distribution (recall Equation 4.2.20) with a\nnew \u201ceffective monomer\u201d of molecular weight qMO:\n"]], ["block_6", ["In this approach the leaving group X in Reaction (4.Y) is a halide, such as a chloride or bromide,\nand it is extracted by a suitable metal, such as copper, nickel, iron, or ruthenium. The metal is\nchelated by ligands such as bipyridines, amines, and trialklyphosphines that can stabilize the metal\nin different oxidation states. A particular example of the activation/deactivation equilibrium using\ncopper bromide/2,2\u2019-bipyridine (bipy) can thus be written:\n"]], ["block_7", ["where the copper atom is oxidized from Cu(I)Br to Cu(II)Br2. Reaction (4.2) suggests that the\npolymerization could be initiated by the appropriate halide of the monomer in question, such as\nl-phenylethyl bromide when styrene is the monomer. Alternatively, a standard free-radical\n"]], ["block_8", ["144\nControlled Polymerization\n"]], ["block_9", ["MW\nm1+ \nMn\nMn\n(4.6.5)\n"]], ["block_10", ["PgBI\u2019 + CuBr(bipy)2 = B0 + CuBr2(bipy)2\n(4.2)\n"]]], "page_159": [["block_0", [{"image_0": "159_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["From itbeen reported that apparent propagation rate\nconstant on the order of 10\u20144 s\", where the apparent\nrate constant is What is the order of magnitude of the concentration\nof active \n"]], ["block_2", ["4.6.2.2\nStable Free-Radical Polymerization (SFRP)\n"]], ["block_3", ["In this variant, the leaving group X in Reaction (4.Y) is a free-radical, but a sufficiently stable one\nthat it does not initiate polymerization. The prime example of this class is the nitroxide radical,\nusually embedded in\nthe\n(2,2,6,6\u2014tetramethylpiperidinyloxy)\n\u201cTEMPO\u201d\ngroup\n[13]. When\nattached to a monomer analog or a growing polymer chain terminus through the alkoxyamine\n"]], ["block_4", ["Controlled Radical \n145\n"]], ["block_5", ["initiator of\nATRP is are amenable to this approach: styrene and substituted\nstyrenes, acrylates and methacrylates,and other vinyl monomers. Dienes and amine or carboxylic\n"]], ["block_6", ["acid\u2014containing \nThe following example illustrates some of the quantitative aspects of ATRP of styrene.\n"]], ["block_7", ["The adduct of styrene and TEMPO on the left-hand side of Reaction (4AA) can be prepared rather\nreadily, puri\ufb01ed, and stored inde\ufb01nitely. In contrast to other controlled radical polymerization\nschemes, this approach is based on a single initiating species; no cocatalyst or transfer agent is needed.\nEven in the presence of a large excess of styrene monomer, it is not until the system is brought to an\nelevated temperature such as 125\u00b0C that polymerization proceeds directly. The reaction can be run\nunder nitrogen, and the rigorous purification necessary for living ionic polymerizations is not required.\nMolecular weights well in excess of 105, with polydispersities in the range of 1.1\u20141.2, have been\nachieved. The range of accessible monomers is so far more restricted than with ATRP or reversible\naddition-fragmentation transfer (RAFT), with styrene, acrylate, and methacrylate derivatives being the\n"]], ["block_8", ["monomers of choice. However, the polymerization is relatively tolerant of functional groups, and\nmany functionalized initiators with TEMPO adducts have been designed. This makes SFRP an\nappealing alternative to living ionic polymerization for the production of end-functional polymers\n(recall Section 4.4), and by extension block copolymers and branched architectures, once the initiator\nis available.\n"]], ["block_9", ["From Equation 4.6.2, we can see that kgpp thus defined is actually equal to kp [Po ]. From Table 3.4 in\nChapter 3 we know that a typical value for kp for free-radical polymerization is 102\u2014103 L mol\u2018l s- l,\nand on this basis direct substitution tells us that [Po] is about 104 3\u20141/102'3 L 11101\u201c s\u20141 10\"5\u201410\u20147\nmol L\u2019 '. This is in line with the estimate given in the previous section, of the target concentration of\nactive radicals needed to make the rate of termination small with respect to the rate of propagation.\n"]], ["block_10", ["C\u2014ON bond, homolytic cleavage of the C\u20140 bond produces the stable TEMPO radical and an\nactive radical species. This reaction is illustrated below for the case of styrene:\n"]], ["block_11", ["Solution\n"]], ["block_12", ["lK. Matyjaszewski, T.E. Patten, and J. Xia, J. Am. Chem. 306., 119, 674 (1997).\n"]], ["block_13", ["Example \n"]], ["block_14", [{"image_1": "159_1.png", "coords": [44, 395, 245, 465], "fig_type": "molecule"}]], ["block_15", [{"image_2": "159_2.png", "coords": [45, 397, 251, 461], "fig_type": "figure"}]], ["block_16", ["Me\nOH.\u2018\nMe\n.\nO\u2018M.\nN\n'\nN\n_._\n+\n(4AA)\nMe\nMe\nMe\nMe\n"]], ["block_17", [{"image_3": "159_3.png", "coords": [125, 396, 244, 454], "fig_type": "molecule"}]]], "page_160": [["block_0", [{"image_0": "160_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "160_1.png", "coords": [27, 606, 258, 659], "fig_type": "molecule"}]], ["block_2", ["4.6.2.3\nReversible Addition-Fragmentation Transfer (RAFT) Polymerization\n"]], ["block_3", ["The principal distinction between RAFT polymerization on the one hand and ATRP or SFRP on\nthe other is that RAFT polymerization involves a reversible chain transfer, whereas the other two\ninvolve reversible chain termination. The key player in the RAFT process is the chain transfer\nagent itself; the radicals are generally provided by conventional free-radical initiators such as\nAIBN. Dithioesters (RCSSR\u2019) such as cumyl dithiobenzoate are often used; in this instance R is a\nphenyl ring and R\u2019r is a cumyl group. The growing radical chain P,\u00bb reacts with the transfer agent,\nand the cumyl group departs with the radical:\n"]], ["block_4", ["A different growing radical PJ-o can also react in the analogous manner:\n"]], ["block_5", ["1\u2018CJ. Hawker, G.G. Barclay, and J. Dao, J. Am. Chem. Soc, 118, 11467 (1996).\n"]], ["block_6", ["A 1:] mixture of the two initiators was added to styrene monomer and heated to the polymer-\nization temperature. At various times, the reaction mixture could be cooled, and analyzed. If the\nexchange of TEMPO groups was rapid, then one would expect four distinct chain populations,\nwith roughly equal proportions: one with no hydroxyls, one with a hydroxyl at each end, one\nwith a hydroxyl at the terminus, and one with a hydroxyl at the initial monomer. On the other\nhand, if there was little exchange, there should be just two populations: one with no hydroxyls\nand one with two. Liquid chromatographic analysis gave results that were fully consistent with\nthe former scenario.\n"]], ["block_7", ["The unimolecular nature of the TEMPO-based initiator, plus its susceptibility to functionalization,\noffers a convenient solution, as has been demonstrated.)r These authors prepared the styrene\u2014-\nTEMPO adduct shown in Reaction (4.AA), plus a dihydroxy-functionalized variant:\n"]], ["block_8", ["An interesting question arises upon examination of Reaction (4.AA): does each TEMPO radical\nremain associated with the same chain during the polymerization, or does it migrate freely through\nthe reaction medium? In the case of anionic polymerization in a nonpolar solvent, the counterion is\ncertainly closely associated with the chain end, due to the requirement of electrical neutrality. In\nthe case, of conventional free-radical polymerization, we considered the \u201ccaging effect\u201d that can\nseverely limit the efficiency of an initiator (see Section 3.3). In this case, the relatively high\ntemperature should enhance both the mobility of the individual species and the ability to escape\nfrom whatever attractive interaction would hold the two radical species in proximity. How could\none test this intuition experimentally?\n"]], ["block_9", ["Solution\n"]], ["block_10", ["146\nControlled Polymerization\n"]], ["block_11", ["Example 4.4\n"]], ["block_12", [{"image_2": "160_2.png", "coords": [36, 550, 259, 582], "fig_type": "molecule"}]], ["block_13", [{"image_3": "160_3.png", "coords": [42, 255, 157, 315], "fig_type": "molecule"}]], ["block_14", ["P\" RASH' \n3\nFl\nFl\n"]], ["block_15", ["..\n\u2014-\u2014\n\u00b0\n..\n4.CC\nP! HAS\nHASP,\nP' \u201dHa/\u201c\\sp,\n(\n)\n"]], ["block_16", ["OKN\nHO\n"]], ["block_17", [{"image_4": "160_4.png", "coords": [59, 616, 250, 646], "fig_type": "molecule"}]], ["block_18", ["SP,-\nSP,-\n.\n_.._\n.\n4.BB\n13R.__R.. x\n<\n)\ns\n"]], ["block_19", ["Me\nMe\nOH\n"]], ["block_20", ["Me\nMe\n"]]], "page_161": [["block_0", [{"image_0": "161_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "161_1.png", "coords": [30, 427, 194, 485], "fig_type": "molecule"}]], ["block_2", [{"image_2": "161_2.png", "coords": [32, 567, 182, 597], "fig_type": "molecule"}]], ["block_3", ["In this from chain to chain. of\nthis reaction narrow molecular weight \nthat the independent of the length of the \nchain, or between group.\nThere is an important feature of this scheme that is different from the other two controlled\nradical number of chains is not determined by the number of\ninitiators, initiators (e.g., AIBN) and those from \nRAFT agent, e.g., cumyl radicals (R\u2019\u00b0 in Reaction 4.BB). In fact, given that the decomposi-\ntion of AIBNto use an excess of the RAFT agent,\nthereby dictatingof initiated via R\u201c, which in turn is proportional to the\nconcentration of RCSSR\u2019. This is facilitated by the fact that such dithioesters have\nvery large chain transfer constants (recall Section 3.8), and thus a chain initiated by AIBN\nor by R\" is rapidly transformed into a dormant form, before achieving a significant degree\nof polymerization. The RAFT approach has been successful with a very wide variety of\ndifferent monomers.\n"]], ["block_4", ["The last term indicates that the equilibrium constant is the inverse of the equilibrium monomer\nconcentration, because the concentrations of i-mer and (i + l)-mer must be nearly equal (recall\nEquation 3.7.3). The reason we have not emphasized the possibility of equilibrium so far is that\nalmost all polymerization reactions are run under conditions where the equilibrium lies far to the\nright, in favor of products; the residual monomer concentration is very small. This is not always the\n"]], ["block_5", ["where the reaction quotient, (21,015,, is the same ratio of product and reactant concentrations as K,\nbut not necessarily at equilibrium. The free energy of polymerization is the difference between the\nfree energies of the products and the reactants, in kJ/mol, where for polymeric species we consider\nmoles of repeat units. The superscript 0 indicates the standard quantity, where all species are at\nsome specified standard state (e.g., pure monomer and repeat unit, or perhaps at\n1 mol L\u20141\nin solution). For the polymerization reaction to proceed spontaneously, AGpoly <0. When the\n"]], ["block_6", ["Up to this point we have tended to write chain-growth propagation steps as one-way reactions, with\na single arrow pointing to the product:\n"]], ["block_7", ["case, however, as we shall now discuss.\nThe state of equilibrium is directly related to the Gibbs free energy of polymerization:\n"]], ["block_8", ["In fact, as a chemical reaction, there must be a reverse depropagation or depolymert'zation step, and\nthe possibility of chemical equilibrium:\n"]], ["block_9", ["This equilibrium constant for polymerization, Kpoly, can be written as the ratio of the forward and\nreverse rate constants, and as the appropriate ratio of species concentrations at equilibrium:\n"]], ["block_10", ["4.7\nPolymerization Equilibrium\n"]], ["block_11", ["Polymerization Equilibrium\n147\n"]], ["block_12", ["K\n: =\n[PEI-(+1]\ng\n1\nW\u201d\nk...\n[Pr][M1\n[M].q\n(4.7.1)\n"]], ["block_13", ["It\nPik+M\u2014p>P?c+l\n"]], ["block_14", ["now], A0301), + RT 1n QM\n(4.7.2)\n"]], ["block_15", ["Kpolr\nPik\u2018l'Mpl-l\n"]]], "page_162": [["block_0", [{"image_0": "162_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "162_1.png", "coords": [20, 497, 412, 650], "fig_type": "figure"}]], ["block_2", ["Note: The enthalpy corresponds to the conversion of liquid monomer (gas1n the case of ethylene) to amorphous\n(or slightly crystalline) polymer. The entropy corresponds to conversion of a 1 mol L\u2018I solution of monomer to\npolymer.\n"]], ["block_3", ["reaction is allowed to come to equilibrium, Q1301), \npoly \nknown relation\n"]], ["block_4", ["The free energy change per repeat unit upon polymerization may be further resolved into enthalpic\n(H) and entropic (S) contributions:\n"]], ["block_5", ["Ethylene\n\u2014\u201493\n\u2014 155\n\u201447\nPropylene\n\u201484\n\u20141 16\n\u201449\nIsobutylene\n\u201448\n\u2014121\n\u201412\n1,3-Butadiene\n\u201473\n\u201489\n\u201446\nIsoprene\n\u201475\n\u20141 01\n\u201445\nStyrene\n\u201473\n\u2014104\n\u201442\na-Methylstyrene\n\u201435\n\u20141 10\n\u20142\nTetrafluoroethylene\n-\u2014163\n\u20141 12\n\u2014130\nVinyl acetate\n\u201488\n-\u20141 10\n\u201455\nMethyl methacrylate\n\u201456\n\u20141 17\n\u201421\n"]], ["block_6", ["From Equation 47.3, we can see that the statement that Kpoly is large, favoring products, is\nequivalent to saying that AGpoly18 large and negative. From Equation 47.4, we can see that facile\npolymerization requires that either A1130\u201d13 large and negative, thatIS, the reaction is exothermic,\nor that ASpol), is large and positive. In fact, A5130]y is usually negative, the monomers lose\ntranslational entropy when bonded together1n a polymer. However,\npoly \nthe extra energy of a carbon\u2014carbon double bond relative to a single bond is released. In fact, we\nshould have anticipated this conclusion from the outset: polymers could not be made inexpensively\nin large quantities if we had to put in energy for each propagation step.\nTable 4.4 provides examples of the standard enthalpy and entropy of polymerization for a few\ncommon vinyl monomers. In all cases both the enthalpy and the entropy changes are negative, as\nexpected; furthermore, AGED\u201d, is negative at room temperature (300 K). Starting with ethylene as\nthe reference, the relative enthalpies of polymerization can be understood in terms of two general\neffects. The first is the possibility of resonance stabilization of the double bond in the monomer\nthat is lost upon polymerization. This results in lower exothermicity for butadiene, isoprene,\nstyrene, and a-methylstyrene, for example. The second is steric hindrance in the resulting polymer.\nFor example, disubstituted carbons in the polymer can lead to signi\ufb01cant interactions between\nsubstituents on every other carbon, which therefore destabilize the polymer, as in the case of\nisobutylene, or-methylstyrene, and methyl methacrylate. Tetra\ufb02uoroethylene, with its unusually\nlarge exothennicity, is included in this short table in part to remind us that there are examples\nwhere we may not have a simple explanation. Further discussion of these issues is provided in\nSection 5.4.\nEquation 4.7.4 indicates that as the polymerization temperature increases, the relative import-\nance of entropy increases as well. As AS favors depolymerization, it is possible to reach a\n"]], ["block_7", ["Table 4.4\nValues of the Standard Enthalpy and Entropy of Polymerization, as Reported in\nOdian [6]\n"]], ["block_8", ["Monomer\nAHgoly (kJ mol-l)\nA5130;y (J K Hmol\u2018 )\nAGpoly at 300 K (kJ mol\")\n"]], ["block_9", ["148\nControlled Polymerization\n"]], ["block_10", ["AGO\nRTln K1301),\n(4.7.3)\npoly:\n"]], ["block_11", ["AGO\n= M301, T453013,\n(4.7.4)\nPoly\n"]]], "page_163": [["block_0", [{"image_0": "163_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "163_1.png", "coords": [30, 85, 189, 114], "fig_type": "molecule"}]], ["block_2", ["Using this relation, the data in Table 4.4, and assuming [M] 1 mol L\u2018l, the ceiling temperature is\n45\u00b0C for poly(0t-methylstyrene) and 206\u00b0C for poly(methyl methacrylate). Note the important fact\nthat according to Equation willdepend on the monomer concentration and will therefore\nbe different for a polymerization in dilute solution compared to one in bulk monomer.\nFigure 4.6 illustrates the application of Equation 4.7.6 to poly(a\u2014methylstyrene). The equilib-\nrium monomer concentration is plotted against temperature according to Equation 4.7.6 and\nthe indicated values of AH0 and A30. The smooth curve corresponds to the ceiling temperature\nand the horizontal line indicates a 1 mol L\u20181 solution. Any solution of monomer that falls to the\nright of, or below, the curve (i.e., any combination of [M] and T) will simply not polymerize. Any\nsolution to the left of, or above, the curve can polymerize, but only until the equilibrium monomer\nconcentration is reached. For example, a 1 mol L\u20141 solution at 0\u00b0C could polymerize, but as\n"]], ["block_3", ["the equilibrium monomer concentration is about 0.1 mol LT], the maximum conversion would\nonly be 90%.\nInterestingly, there are a few instances in which polymerization is driven by an increase in\nentropy, and where the enthalpy gain is almost negligible. Examples include the polymerization of\ncyclic oligomers of dimethylsiloxane, such as the cyclic trimer and tetramer, which we will discuss\nin the next section. In this case, the bonds that are broken and reformed are essentially the same,\n"]], ["block_4", ["and combining 4.7.1 through Equation 4.7.3 \ufb01nd\n"]], ["block_5", ["temperature \nspecial temperature is to the ceiling temperature, TC. From have\n"]], ["block_6", ["Figure 4.6\nIllustration of the relation between equilibrium monomer concentration and ceiling temperature\nfor poly(a-methylstyrene).\n"]], ["block_7", ["[\n]\n0.1-\n-:\n_ a\u2014methylstyrene\n"]], ["block_8", ["polymerization Equilibrium\n149\n"]], ["block_9", [{"image_2": "163_2.png", "coords": [41, 476, 284, 608], "fig_type": "figure"}]], ["block_10", ["0.001 \n-50\n0\n50\n100\n150\n"]], ["block_11", ["AHOpoly\n, \n4.7.6\nT\nA8301),+1\u2019i\u2018ln[M],,,l\n(\n)\n"]], ["block_12", ["AGO\npoly \npoly TCASO\n(4-7-5)\nPoly\n"]], ["block_13", ["001:\nA802\u2014110Jmol_1K_1\n2\n"]], ["block_14", [{"image_3": "163_3.png", "coords": [59, 367, 278, 634], "fig_type": "figure"}]], ["block_15", [{"image_4": "163_4.png", "coords": [129, 509, 263, 567], "fig_type": "figure"}]], ["block_16", ["T,\u00b0C\n"]], ["block_17", ["AH\u201d \u201435,000 J mol\"1\n"]]], "page_164": [["block_0", [{"image_0": "164_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["The following values of We and A58; per methylene unit have been estimated at room tempera-\nture (as reported in Ref. [10]) for the process\n"]], ["block_2", ["The subscript \u201c1c\u201d denotes the liquid-to\u2014crystal aspect of the process; as we will see in Chapter 13,\nhigh molecular weight linear polyethylene, the product of the hypothetical polymerization\n"]], ["block_3", ["In this instance the ring contains four atoms, and the A\u2014B bond is the one that is preferentially\ncleaved. The propagating chain is shown with B containing the active center; it is often the case\nthat ROP proceeds by an ionic mechanism. Comparison of monomer and repeat unit structures in\nReaction (4.DD) reveals that the bonding sequence is the same in both cases, in marked contrast to\neither a chain-growth polymerization through, for example, a carbon\u2014carbon double bond or a\nstep-growth polymerization through, for example, condensation of acid and alcohol groups. In\nlight of the previous section, where we considered the thermodynamics of polymerization, a basic\nquestion immediately arises: if the bonding is the same in monomer and polymer, what is the\nprimary driving force for polymerization? The answer is ring strain. The linkage of the atoms into\na ring generally enforces distortion of the preferred bond angles and even bond lengths, effects that\nare grouped together under the title ring strain. The amount of ring strain is a strong function of the\nnumber of atoms in the ring, r. For example, ethylene oxide, with r 3, is quite a explosive gas at\nroom temperature. On the other hand, cyclohexane, with r=6, is almost inert. In fact, r=6\nrepresents a special case, at least for all carbon rings, as the natural sp3 bond angles and lengths\ncan be almost perfectly matched. The following example illustrates the effect of r on the\nthermodynamics of polymerization, for cyclic alkanes.\n"]], ["block_4", ["4.8\nRing-Opening Polymerization (ROP)\n"]], ["block_5", ["4.8.1\nGeneral Aspects\n"]], ["block_6", ["Cyclic molecules, in which the ring contains a modest number of atoms, say 3\u20148, can often be\npolymerized by a ring-opening reaction, in which a particular bond in the cycle is ruptured, and\nthen reformed between two different monomers in a linear sequence. This process is illustrated in\nthe following schematic reaction:\n"]], ["block_7", ["hence M301), :3 0. On the other hand, possibly because of the greater conformational freedom in\nthe linear polymer versus the small cycles, A5301), is positive. In such a case when M301), is slightly\npositive, it is possible in principle to have a\ufb02oor temperature, below which polymerization cannot\noccur at equilibrium. Furthermore, for. the living polymerization of \u201cD3,\u201d the cyclic trimer of\ndimethylsiloxane, for example using an anionic initiator, 111G130]y is never particularly favorable\ncompared to the typical vinyl monomer case. Consequently, one has to be aware of the law of mass\naction, just as in a polycondensation. In other words, the reaction cannot be allowed to go to\ncompletion, because rather than achieving essentially 100% conversion to polymer, the reaction\nmixture stabilizes at the equilibrium monomer concentration. Then, random propagation and\ndepropagation steps will degrade the narrow molecular weight distribution that was initially\nsought. This problem can be circumvented by adding more monomer than is necessary to achieve\nthe target molecular weight, and using trial and error to determine the time (and fractional\nconversion) at which the desired average molecular weight has been achieved.\n"]], ["block_8", ["Example 4.5\n"]], ["block_9", ["150\nControlled Polymerization\n"]], ["block_10", [{"image_1": "164_1.png", "coords": [39, 326, 184, 358], "fig_type": "molecule"}]], ["block_11", ["MB*+pl:||3\u2014\u2014\u2014p\u2014\nMB\u2014A\n3*\n(4.DD)\n"]], ["block_12", ["\u2014(CH2), (liquid ring)\n41>\n\u2014\u2014(CH2),, (crystalline linear polymer)\n"]]], "page_165": [["block_0", [{"image_0": "165_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "165_1.png", "coords": [19, 93, 204, 190], "fig_type": "figure"}]], ["block_2", [{"image_2": "165_2.png", "coords": [26, 239, 175, 346], "fig_type": "figure"}]], ["block_3", ["The preceding example nicely illustrates the importance of ring strain, but the fact is that the\nprimary utility of ROP is not to produce polyethylene from cyclic alkanes. Rather, it is to produce\ninteresting polymer structures from readily accessible cyclic monomers, structures that cannot be\nprepared more conveniently by classical step\u2014growth or chain-growth polymerization. Examples of\nseven different classes of cyclic monomers and the resulting polymer structures are given in Table\n4.5. In all cases the ring contains one or more heteroatoms, such as O, N, and Si. These participate\nin the bond-breaking process that is essential to ROP; in contrast, it is actually rather difficult in\npractice to polymerize cyclic alkanes, even when free energy considerations favor it. In Chapter 1\nwe suggested that the presence of a heteroatom in the backbone was often characteristic of a step\u2014\ngrowth polymerization. One of the beauties of ROP is that it is a chain-growth mechanism,\nenabling the ready preparation of high molecular weight materials. For example, Entry 5 (a\npolyamide, poly(s\u2014caprolactam)) and Entry 6 (a polyester, polylactide) are polymers that could\nbe prepared by condensation of the appropriate AB monomer. However, by using the cyclic\nmonomer, the condensation step has already taken place, and the small molecule byproduct is\n"]], ["block_4", ["7\u2018, while remaining consistently negative. However, we need to recognize that these values\nincorporate the relief of the ring strain, the incorporation of monomer into polymer, and the\nchanges associated with crystallization of the liquid polymer. The opening of the ring affords\nmore degrees of freedom to the molecule, increasing the entropy, but both subsequent polymer-\nization and crystallization reduce it. Consequently, it is dangerous to overinterpret particular\nvalues of A80.\n"]], ["block_5", ["The results indicate that for all but r: 6, polymerization is favored, consistent with the known\nstability of six\u2014membered carbon rings. From the point of view of polymerization, the driving\nforce should be ranked according to r:3, 4 >r= 8 >r=5, 7. These are general trends, and\ndifferent substituents or heteroatoms within the ring can change the numerical values signifi-\ncantly. Finally, while AHO shows a distinct maximum at r 6, A50 decreases monotonically with\n"]], ["block_6", ["We use the relationships AG\u00b0=AH\u00b0\u2014TASO (Equation 4.7.4) and AGO: \u2014RT In K (Equation\n4.7.3), with T:298 K and R 8.314 J moi\u20141 K\u2018l, to obtain the following table:\n"]], ["block_7", ["3\n\u201492\n2 x 1016\n4\n\u201489\n3 x 1015\n5\n\u20148.5\n30\n6\n+ 6.0\n0.09\n7\n\u201417\n103\n8\n\u201434\n8 x 105\n"]], ["block_8", ["_._._\u2014\u2014\u2014\nr\nA119, (kJ mol\u201c)\nASE, (J nior1 K\u201c)\n"]], ["block_9", ["3\n\u20141 13\n\u201469.1\n4\n4105\n\u201455.3\n5\n\u201421.2\n\u201442.7\n6\n+ 2.9\n\u201410.5\n7\n\u201421.8\n\u201415.9\ng\n\u201c34.8\n\u20143.3\n"]], ["block_10", ["reaction, is crystalline at equilibrium at room temperature. Evaluate AGPC and K for this process\nand interpret \n"]], ["block_11", ["Solution\n"]], ["block_12", ["r\nnot: (H moi\u20141)\nK\n"]], ["block_13", ["Ring-Opening \n151\n"]]], "page_166": [["block_0", [{"image_0": "166_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "166_1.png", "coords": [32, 76, 267, 327], "fig_type": "figure"}]], ["block_2", ["4.8.2\nSpecific Examples of Living Ring-Opening Polymerizations\n"]], ["block_3", ["4.8.2.1\nP01y(ethy1ene oxide)\n"]], ["block_4", ["removed. Thus, the law of mass action that typically limits the molecular weight of a step-growth\npolymerization is overcome. (This is not to say that polymerization equilibrium will not be an\nissue. In fact, Entry 6 and Entry 7 both indicate six-membered rings, and the correspondingly lower\nring-strain does bring equilibration into play.)\nFrom the perspective of this chapter the main point of ROP is not just its chain-growth\ncharacter, but the fact that in many cases living ROP systems have been designed. As indicated\nabove, most ROPs proceed via an ionic mechanism, which certainly invites attempts to achieve a\nliving polymerization. We will brie\ufb02y present specific examples of such systems, for three\ndisparate but rather interesting and important polymers in Table 4.5: poly(ethy1ene oxide) (Entry\n1), polylactide (Entry 6), and poly(dimethylsiloxane) (Entry 7). We will also consider a class of\nROP that can produce all carbon backbones, via olefin metathesis.\n"]], ["block_5", ["P01y(ethy1ene oxide) represents one of the most versatile polymer structures for both fundamental\nstudies and in commercial applications. It is readily prepared by living anionic polymerization,\nwith molecular weights ranging all the way up to several millions. It is water soluble, a highly\ndesirable yet relatively unusual characteristic of nonionic, controlled molecular weight polymers.\nFurthermore, it appears to be more or less benign in humans, thereby allowing its use in many\n"]], ["block_6", ["Table 4.5\nExamples of Monomers Amenable\nto Ring-Opening Polymerization\n"]], ["block_7", ["6. Lactones (cyclic esters)\n\u2018z\n"]], ["block_8", ["/ /\n,O-Sk\nMe\nMe\n7. Siloxanes\n:Si\n0\n()Si: ,)\nb\u2014Si:\n0\n"]], ["block_9", ["1 52\nControlled Polymerization\n"]], ["block_10", [".\n.\n.\nT7\nH\n4. Immes (cyclic amines)\nN\nN\nH\nW \\l\n"]], ["block_11", ["2. Cyclic ethers\n[:0\nM04\n"]], ["block_12", ["5. Lactams (cyclic amides)\nGH\n(\u201895\nE4\n"]], ["block_13", ["Monomer class\nExample\nRepeat unit\n"]], ["block_14", ["/\\\n.\n0\n0\n3. Cyclic acetals\nOMS\n(/O\\/ p?\n"]], ["block_15", ["W0\u00bb)\n[:0\n1. Epoxides\n"]], ["block_16", [{"image_2": "166_2.png", "coords": [134, 354, 190, 404], "fig_type": "molecule"}]], ["block_17", [{"image_3": "166_3.png", "coords": [143, 298, 265, 351], "fig_type": "molecule"}]], ["block_18", ["0\no\no\n0\n"]], ["block_19", ["(x w\n0\n0\nMe\n"]], ["block_20", [{"image_4": "166_4.png", "coords": [221, 301, 263, 346], "fig_type": "molecule"}]]], "page_167": [["block_0", [{"image_0": "167_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["is not an effective counter ion in this case. (As noted in the context of Reaction 4.H, this feature is\nactually convenient when it is desired to use ethylene oxide to end-functionalize such polymers.)\nThe following scheme represents an example of a three-arm star prepared by grafting from; the\ninitiator is trimethylol propane, which has three equivalent primary alcohols that can be activated\nby diphenylmethyl potassium. of ethylene oxide monomer is straightforward, and the\nresulting polymer is terminated with acidic methanol to yield a terminal hydroxyl-functionality on\neach arm.\n"]], ["block_2", ["4.8.2.2\nPolylactide\n"]], ["block_3", ["Polylactide is biodegradable and can be derived from biorenewable feedstocks such as corn. The\nformer feature enables a long-standing application as resorbable sutures; after a period of days to\nweeks, the suture degrades and is metabolized by the body. The latter property underscores recent\ninterest in the large-scale commercial production of polylactide for a wide variety of thermoplastic\napplications.\nThe structure of lactic acid is\n"]], ["block_4", ["Clearly, as it contains both a hydroxyl and a carboxylic acid, it could be polymerized directly to the\ncorresponding polyester via condensation. However, if the starting material is lactide, the cyclic\n"]], ["block_5", ["ene oxide that example in proposing the Poisson distribution for chain-growth\npolymers prepared from a \ufb01xed number of initiators with rapid propagation [14].\nIn practice, the living anionic polymerization of ethylene oxide has been achieved by a variety\nof initiator systems, following the general principles laid out in Section 4.3. Examples include\nmetal hydroxides, alkoxides, alkyls, and aryls. In contrast to styrenes and dienes, however, lithium\n"]], ["block_6", ["consumer products, biomedical formulations, etc. In fact, a grafted layer of short chain poly(ethyl-\nene oxide)s (see confer long-term stability against protein adsorption or deposition\nof other biochemical arena short chain poly(ethylene oxide)s (with\nhydroxyl groups at both ends) are more commonly referred to as poly(ethylene glycol)s or PEGs.\nThe grafting of PEG molecules onto a biomacromolecule or other substrate has become such a\nuseful name: PEGylation. Poly(ethylene oxide) crystallizes\nrather readily, witha typical melting temperature near 65\u00b0C, which has led to its use in many\nfundamental crystallization (see Chapter 13). The first block copolymers\nto become the so-called polyoxamers, diblocks and triblocks of\npoly(ethylene oxide) and poly(propylene oxide). Historically, it was the polymerization of ethyl-\n"]], ["block_7", ["a cationic ring-opening mechanism; the high ring-strain of ethylene oxide makes it the exception to\nthis rule.\n"]], ["block_8", ["An interesting feature of this reaction is that it can be carried out in THF, which like ethylene oxide\nis a cyclic ether. This again illustrates the importance of ring-strain in facilitating, or, in this case\nsuppressing, polymerization. In fact, cyclic ethers including THF are usually polymerized only by\n"]], ["block_9", ["Ring-Opening Polymerization (ROP)\n153\n"]], ["block_10", [{"image_1": "167_1.png", "coords": [43, 336, 282, 398], "fig_type": "figure"}]], ["block_11", [{"image_2": "167_2.png", "coords": [44, 335, 293, 401], "fig_type": "molecule"}]], ["block_12", ["Me\nH\nHO\nO\n"]], ["block_13", ["Me/q(\\OH\n"]], ["block_14", [{"image_3": "167_3.png", "coords": [48, 592, 98, 634], "fig_type": "molecule"}]], ["block_15", [{"image_4": "167_4.png", "coords": [49, 340, 205, 393], "fig_type": "molecule"}]], ["block_16", ["K+\nO\nK+ 01\nMeOH\nH0\nK+ _\n"]], ["block_17", ["0\n"]], ["block_18", ["OH\nK+ \n(4.EE)\n"]], ["block_19", ["+\n3\n\u2014p-\nMe\nO\u2014 \n-\u2014-\u2014-\u2014\u2014\u2014\u2014-i-- Fl((CH2CH20)nH)3\n"]], ["block_20", [{"image_5": "167_5.png", "coords": [197, 338, 428, 395], "fig_type": "molecule"}]], ["block_21", ["\u2014\n"]]], "page_168": [["block_0", [{"image_0": "168_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "168_1.png", "coords": [32, 377, 280, 457], "fig_type": "figure"}]], ["block_2", ["4.8.2.3\nPoly(dimethylsiloxane)\n"]], ["block_3", ["Note that the central carbon in lactic acid is chiral, and therefore the corresponding two carbons in\nboth lactide and the polylactide repeat unit are stereocenters. Consequently, there are three possi-\nbilities for the lactide monomer, according to the absolute configuration of these carbons: R, R; S, S;\nand R,S. Polymerization of either of the first two leads to the corresponding stereochemically pure\nPDLA and PLLA, which are crystallizable; the meso dyad leads to an atactic polymer. In this case,\ntherefore, the responsibility of producing a particular tacticity is largely transferred from the catalyst\n(see Chapter 5) to the purification of the starting material. Of course, the catalyst has to guide the\npolymerization in such a way that the stereochemistry is not scrambled or epimerized.\nFrom the point of view of designing a living polymerization of polylactide, there are two general\nissues to confront. First, as polylactide is a polyester, it is susceptible to transesterification reactions.\nThis constraint favors lower reaction temperatures, conditions that are neither too basic nor acidic,\nand acts against the normal desire to make the catalyst more \u201cactive.\u201d Ironically, the advantageous\ndegradability of polylactide through the hydrolysis of the ester linkage is thus a disadvantage from\nthe point of view of molecular weight control. The second problem is equilibration. Recall from\nExample 4.5 that for cyclic alkanes, the six-membered ring has no ring~strain to speak of, and is\ntherefore not polymerizable. Although lactide is a six-membered ring, it does possess sufficient ring-\nstrain, but not a lot. Consequently, narrow molecular weight distributions are usually obtained only\nby terminating well before the reaction has approached completion.\nA typical catalyst is based on a metal alkoxide, such as LMOR\u2019, where L is a \u201cspectator\u201d ligand\nand R\u2019 is a small alkyl group. The initiation step can be written as\n"]], ["block_4", ["where the lactide ring is cleaved at the bond between the oxygen and the carbonyl carbon. The\nsubsequent propagation steps involve the same bond cleavage, with addition of the new monomer\ninto the oxygen\u2014metal bond at the growing chain end. In fact, this kind of polymerization has been\nclassified as \u201canionic coordination,\u201d in distinction to anionic polymerization, as the crucial step is\ncoordination of the metal with the carbonyl oxygen, followed by insertion of the alkoxide into the\npolarized C\u2014O bond. The most commonly employed catalyst for polylactide is tin ethylhexanoate,\nbut more success in terms of achieving living conditions has been realized with aluminum\nalkoxides. Interestingly, these aluminum species have a tendency to aggregate in solution, with\nthe result that the reaction kinetics can become rather complicated; different aggregation states can\nexhibit very different pr0pagation rates. This situation is reminiscent of the aggregation of\ncarbanions in anionic polymerization in nonpolar solvents discussed in Section 4.3.\n"]], ["block_5", ["Poly(dimethylsiloxane) has one of the lowest glass transition temperatures (Tg, see Chapter 12) of\nall common polymers. This is due in part to the great \ufb02exibility of the backbone structure (see\nChapter 6), which re\ufb02ects the longer Si\u2014O bond compared to the C\u2014C bond, the larger bond angle, and\nthe absence of substituents on every other backbone atom. It is also chemically quite robust. It is used if.\n"]], ["block_6", ["dimer of the corresponding ester, then ROP produces the same polymer structure but by a Chain-\ngrowth process:\n"]], ["block_7", ["154\nControlled Polymerization\n"]], ["block_8", [{"image_2": "168_2.png", "coords": [36, 75, 226, 141], "fig_type": "figure"}]], ["block_9", [{"image_3": "168_3.png", "coords": [38, 383, 245, 450], "fig_type": "molecule"}]], ["block_10", [{"image_4": "168_4.png", "coords": [43, 83, 222, 127], "fig_type": "molecule"}]], ["block_11", ["I'_M\nMe\nO\n0\nMe\nOM\nI I\n+LMOR'\nl\ne\no\n(4-GG)\no\n0\nMe\n0\nOJY\non'\n"]], ["block_12", ["Me\n0\no\n0\nMe\nI f\n..\n0%0\n(4.1212)\n0\n0\nMe\nMe\n0\nn\n"]]], "page_169": [["block_0", [{"image_0": "169_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "169_1.png", "coords": [28, 188, 265, 243], "fig_type": "molecule"}]], ["block_2", ["Although Reaction (4.11) illustrates the net outcome, it says nothing about the mechanism.\nMetathesis reactions are catalyzed by transition metal centers, with the associated ligand package\nproviding tunability of reaction characteristics such as rate, selectivity, and stereochemistry of\naddition. In the case of ROMP, the metal forms a double bond with a carbon at one end of the\nchain; thus in Reaction (4.11) the RICH group would be replaced by the metal and its ligands\n(MLn). For a propagation step, R2 would denote a previously polymerized chain, P,. In the\nmonomer to be inserted into the chain, between the metal and the end of the previously polymer-\nized chain, R3 and R4 are covalently linked to form a ring. Thus, the ring plays two key roles:\nthe ring-strain provides the driving force for polymerization, and the ring structure provides the\npermanent connectivity between the two carbon atoms whose double bond is broken. The ROMP\nanalog to Reaction (4.11) can thus be described schematically as\n"]], ["block_3", ["any number of lubrication and adhesive applications (\u201csilicones\u201d), as well as a variety of rubber\nmaterials. of the cyclic hexamethylcyclotrisiloxane, D3, has been\nachieved that this monomer is a six\u2014membered ring, we can anticipate that\nthe polymerization is not strongly favored by thermodynamics. Consequently, narrow molecular weight\ndistributions are achieved by terminating the reaction well before the consumption of all the monomer.\nAn example of a successful protocol is the following. A modest amount of cyclic trimer is\ninitiated solution. Under these conditions initiation is rather\nslow, but propagation is almost thereby allowing for complete initiation. Presumably,\nthe lack of propagation is due to ion clustering as discussed in Section 4.3. The addition of THF, as a\npolar modifier, plus more monomer allows propagation to proceed for an empirically determined\ntime interval. Termination achieved with trimethylchlorosilane (TMSCl):\n/\nO\u2014Si\nO\nO\nO\u2014\nCHX\n+\n:SI/\n\\O+FlOK\u2014I-\u2014 Ro/>SiR1 \n--.\u2014\u2014\n(4.11)\n"]], ["block_12", [{"image_4": "169_4.png", "coords": [48, 391, 213, 463], "fig_type": "figure"}]], ["block_13", ["H\nH\n"]], ["block_14", [{"image_5": "169_5.png", "coords": [81, 248, 286, 293], "fig_type": "molecule"}]], ["block_15", ["THF\nD3\nTMSCI\nO\n"]], ["block_16", ["\\\nn\n[\n"]]], "page_170": [["block_0", [{"image_0": "170_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "170_1.png", "coords": [25, 495, 217, 562], "fig_type": "figure"}]], ["block_2", [{"image_2": "170_2.png", "coords": [33, 490, 374, 571], "fig_type": "figure"}]], ["block_3", ["Dendrimers are an interesting, unique class of polymers with controlled structures. For example,\nthey can have precisely defined molecular weights, even though the elementary addition steps are\nusually of the condensation variety. From an application point of view it is the structure of the\ndendrimer, rather than its molecular weight per se that is the source of its appeal. A cartoon\nexample of a dendrimer is provided in Figure 1.2. The term comes from the Greek word dendron,\nor tree, and indeed a dendrimer is a highly branched polymer molecule. In particular, a dendrimer\nis usually an approximately spherical molecule with a radius of a few nanometers. Thus, a\n"]], ["block_4", ["Note that the substituent on the metal\u2014carbon double bond will become attached to the nonpropa-\ngating terminus of the chain. Collectively, catalysts in these two families have proven capable of\nachieving controlled polymerization of a wide variety of cyclic olefins, including those containing\nfunctional groups. In particular, while the Schrock catalysts tend to be more active, the Grubbs\ncatalysts are more tolerant of functional groups, oxygen, and protic solvents. Reaction conditions\nare often mild, that is, near room temperature, and in some cases the polymerization can be\nconducted in water. Overall control is often quite good, and many block copolymers have\nbeen prepared by ROMP. Some ROMP systems have even been commercialized, including the\npolymerization of norbornene:\n"]], ["block_5", ["4.9\nDendrimers\n"]], ["block_6", ["In this case, the monomer is cyclooctene. After the monomer insertion or propagation step, an\nactive metal carbene remains at the chain terminus, and one carbon\u2014carbon double bond remains in\nthe backbone for each repeat unit.\nThere is a large literature on metathesis catalysts and associated mechanisms, many of\nwhich incorporate multiple components beyond the active metal. However, for the purposes of\ncontrolled ROMP, there are currently two families of single species catalysts that are highly\nsuccessful. One, based on tungsten or molybdenum, is known as a Schrock catalyst [I], and the\nother, based on ruthenium, is a Grubbs catalyst [II]. These investigators were corecipients of the\n2005 Nobel Prize in chemistry (with Y. Chauvin) for their work on the metathesis reaction.\nThe structures of representative examples are given below, where the symbol Cy denotes a\ncyclohexyl ring:\n"]], ["block_7", ["1 56\nControlled Polymerization\n"]], ["block_8", [{"image_3": "170_3.png", "coords": [41, 189, 136, 267], "fig_type": "molecule"}]], ["block_9", [{"image_4": "170_4.png", "coords": [42, 498, 213, 563], "fig_type": "molecule"}]], ["block_10", [{"image_5": "170_5.png", "coords": [44, 289, 127, 364], "fig_type": "molecule"}]], ["block_11", ["N\\\n0\nemail\n"]], ["block_12", ["CI\" I?\u201d\nPCY3\n"]], ["block_13", ["Grubbs catalyst\n"]], ["block_14", ["Ins _\n[II]\n"]], ["block_15", ["\u2014...W\n(4.KK)\n"]]], "page_171": [["block_0", [{"image_0": "171_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "171_1.png", "coords": [15, 465, 235, 540], "fig_type": "molecule"}]], ["block_2", [{"image_2": "171_2.png", "coords": [30, 551, 248, 632], "fig_type": "molecule"}]], ["block_3", [{"image_3": "171_3.png", "coords": [32, 409, 211, 443], "fig_type": "molecule"}]], ["block_4", ["dendrimer and also a well-defined nanoparticle. The outer\nsurface density of functional groups that govern the\ninteractions These exterior groups have the advan-\ntages of being numerous, and readily accessible for chemical transformation. The interior of the\ndendrimer can distinct functionality that can endow the molecule with desirable\nproperties. For example, the dendrimer might incorporate a highly absorbing group, for \u201clight\nharvesting,\u201d or \ufb02uor0phore, for ef\ufb01cient emission. Other possibilities include catalytic centers or\nelectrochemically active groups. By being housed within the dendrimer, this functional unit can be\nprotected from unwanted interactions with the environment. The functional unit may be covalently\nbound within the or it may simply be encapsulated. The possibility of controlling\nuptake and release of specific agents by the dendrimer core also makes them appealing as possible\ndelivery vehicles for pharmaceuticals or other therapeutic agents. As nanoparticles, dendrimers\nshare certain attributes with other objects of similar size, such as globular proteins, surfactant and\nblock copolymer micelles, hyperbranched polymers, and colloidal nanoparticles. Although beyond\nthe scope of this chapter, it is interesting to speculate on the possible advantages and disadvantages\nof these various structures (see Problem 16).\nThere are two distinct, primary synthetic routes to prepare a dendrimer, termed divergent and\nconvergent. In a divergent approach, the dendrimer is built-up by successive additions of mono-\nmers to a central, branched core unit, whereas in the convergent approach branched structures\ncalled dendrons are built-up separately, and then ultimately linked together to form the dendrimer\nin a final step. The divergent approach was conceived first, and is the more easily visualized. The\nprocess is illustrated schematically in Figure 4.7. The core molecule in this case has three\nfunctional groups denoted by the open circles. These are reacted with three equivalents of another\nthree-functional monomer, but in this case two of the functional groups are protected (filled\ncircles). After this reaction is complete, the growing molecule has six functional groups that are\n"]], ["block_5", ["Figure 4.7\nSchematic illustration of the divergent synthesis of a third-generation dendrimer from a\ntrifunctional core. The open circles denote reactive groups and the \ufb01lled circles protected groups.\n"]], ["block_6", ["Dendrimers\n1 57\n"]], ["block_7", ["G2\n+12\n"]], ["block_8", [{"image_4": "171_4.png", "coords": [47, 389, 230, 462], "fig_type": "figure"}]], ["block_9", [{"image_5": "171_5.png", "coords": [52, 395, 208, 458], "fig_type": "molecule"}]], ["block_10", ["A. J/ \n"]], ["block_11", ["J, Deprotect\n"]], ["block_12", [{"image_6": "171_6.png", "coords": [95, 473, 229, 530], "fig_type": "molecule"}]], ["block_13", ["\\k\ufb01%\\<\n"]], ["block_14", ["Deprotect\n\\\n"]]], "page_172": [["block_0", [{"image_0": "172_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Thus, a perfect G5, G6, and G7 dendrimer would have 96, 192, and 384 functional groups,\nrespectively. Note that Equation 4.9.1 would need to be modified in the case of, for example, a\ntetrafunctional core. We have introduced the term \u201cperfect\u201d here to emphasize that it is certainly\npossible for a dendrimer molecule to have defects, or missing functional groups, which will\npropagate through all subsequent generations. For the first few generations it is usually not too\ndifficult to approach perfection, but for G5 and above, the functional groups become rather\ncongested, which makes complete addition of the next generation difficult. It also becomes harder\nto separate out defective structures. It is typically not practical to go beyond G8.\nFurther consideration of the divergent approach in Figure 4.7 reveals that, compared with most\npolymerization reactions, it is rather labor intensive. For example, the addition of each generation\nrequires both an addition step and a deprotection step. The addition will typically be conducted\nin the presence of a substantial excess of protected monomer, to drive the completion of the\nnew layer. The resulting products will need to be separated, in order to isolate the perfect\ndendrimer structure from all other reaction products and reagents. Similarly, the deprotection\nstep needs to be driven to completion, and the pure product isolated. Thus, in the end there are two\nreaction steps and two purification steps required for each generation. This requires a significant\namount of time, and it is challenging to prepare commercial-scale quantities of perfect, high\ngeneration products.\nAs a specific example of a divergent synthesis, we will consider the formation of the poly-\namidoamine (PAMAM) system. In this case, there are two monomers to be added sequentially in\neach generation, rather than one addition and one deprotection step. The core molecule and one of\nthe monomers is typically ethylene diamine and the other monomer is methyl acrylate. The first\nstep is addition of four methyl acrylate molecules to ethylene diamine in a solvent such\nas methanol. The Michael addition\u2014type mechanism involves nucleophilic attack of the electron\npair\non\nthe\nnitrogen\nto\nthe\ndouble\nbond\nof\nthe\nacrylate,\nwhich\nis\nactivated\nby\nthe\nelectron withdrawing character of the ester group:\n"]], ["block_2", ["then deprotected. At this stage the molecule is termed afirst generation (G1) dendrimer. Another\naddition reaction is then performed, but now six equivalents of the protected monomer are required\nto complete the next generation. After deprotection the resulting G2 dendrimer has 12 functional\ngroups. It is straightforward to see that the number of functional groups on the surface grows\ngeometrically with the number of generations, 3:\n"]], ["block_3", ["The next second step involves amidation of each ester group by nucleophilic attack of the nitrogen\non the electropositive carbonyl carbon, with release of methanol:\n"]], ["block_4", ["158\nControlled Polymerization\n"]], ["block_5", [{"image_1": "172_1.png", "coords": [43, 492, 193, 530], "fig_type": "molecule"}]], ["block_6", [{"image_2": "172_2.png", "coords": [44, 461, 360, 564], "fig_type": "figure"}]], ["block_7", ["Number of functional groups 3 X 23\n(4.9.1)\n"]], ["block_8", ["H\nO\nN\nH2N \ufb02\u201d\\n/\n\\Me\n\u2014\u2014\u2014\u2014\u2014\u20141-- Ir\u201d\\\\u201d/ \\/\\NH2 + MeOH\n(4.MM)\n"]], ["block_9", [{"image_3": "172_3.png", "coords": [66, 607, 370, 662], "fig_type": "molecule"}]], ["block_10", ["O\n"]], ["block_11", ["O\nO\n"]], ["block_12", [{"image_4": "172_4.png", "coords": [179, 461, 322, 523], "fig_type": "molecule"}]], ["block_13", [{"image_5": "172_5.png", "coords": [184, 450, 319, 574], "fig_type": "molecule"}]], ["block_14", ["(DE/Q1219\nMe\\0J1\n"]], ["block_15", [{"image_6": "172_6.png", "coords": [190, 453, 327, 567], "fig_type": "figure"}]], ["block_16", [{"image_7": "172_7.png", "coords": [192, 492, 304, 565], "fig_type": "molecule"}]], ["block_17", ["\u2018i\nM\n"]], ["block_18", ["O\n"]], ["block_19", ["9\n"]], ["block_20", ["f\n1Y0\n"]], ["block_21", ["O\\\n0\nMe\n"]], ["block_22", ["(4.LL)\n"]]], "page_173": [["block_0", [{"image_0": "173_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "173_1.png", "coords": [16, 423, 250, 648], "fig_type": "figure"}]], ["block_2", [{"image_2": "173_2.png", "coords": [21, 432, 186, 471], "fig_type": "molecule"}]], ["block_3", [{"image_3": "173_3.png", "coords": [23, 536, 276, 628], "fig_type": "figure"}]], ["block_4", [{"image_4": "173_4.png", "coords": [26, 473, 221, 534], "fig_type": "molecule"}]], ["block_5", ["An alternative general strategy for preparing dendrimers and dendritic fragments, or dendrons,\nis the so-called convergent approach. This is illustrated schematically in Figure 4.8. As the name\nimplies, these molecules are made \u201cfrom the outside in,\u201d that is, the eventual surface group,\ndenoted by \u201cx\u201d in Figure 4.8, is present in the initial reactants. The first reaction produces a\nmolecule with two surface groups and one protected reactive group. The second step, after\ndeprotection, doubles the number of surface groups, and so on. At any stage, a suitable multifunc-\ntional core molecule can be used, to stitch the appropriate number of dendrons (usually 3 or 4)\ntogether. Each growing wedge-shaped dendron possesses only one reactive group, which pre-\nsents a significant advantage in terms of purification. At each growth step, a dendron either\nreacts or it does not, but the product and reactant are significantly different in molecular weight.\nBy contrast, in the divergent approach, the surface of the dendrimer has many reactive groups,\nand it may not be easy to separate a G3 dendrimer with 24 newly added monomers from one with\nonly 23 monomers. Furthermore, because there are so many more reactive groups on the\ndendrimer than on the added monomers, the monomers must be present in huge molar excess\nto drive each reaction to completion. In the convergent approach shown, there are only twice as\nmany dendrons as new coupling molecules at stoichiometric equivalence, so a large excess of\ndendrons is not necessary.\n"]], ["block_6", ["The structure of the resulting Gl PAMAM dendrimer is therefore the following:\n"]], ["block_7", ["X\u2014O\n2\nX\u2014-O\n+ P \u2014\u2014-\u2014>-\n>\u2014.\nX\u2014o\n"]], ["block_8", ["X\u2014o\n2\n>\u20140\n+ P\u2014p\u2014x\nX\u2014o\nX\u2014o\n"]], ["block_9", ["Figure 4.8\nIllustration of the convergent approach to dendrimer synthesis. Each dendron is built-up by\nsuccessive 2:1 reactions, before the final coupling step.\n"]], ["block_10", ["Dendrimers\n159\n"]], ["block_11", [{"image_5": "173_5.png", "coords": [35, 536, 235, 610], "fig_type": "molecule"}]], ["block_12", [{"image_6": "173_6.png", "coords": [36, 542, 193, 586], "fig_type": "molecule"}]], ["block_13", ["2 \n:>\u2014c\n"]], ["block_14", [{"image_7": "173_7.png", "coords": [37, 68, 238, 205], "fig_type": "figure"}]], ["block_15", ["NH2\nHKEHE\n1\nO\n"]], ["block_16", ["+ P \u2014\"\nX\u2014o\n"]], ["block_17", ["X\u2014\u2014O\n"]], ["block_18", ["X\u2014\u2014o\n"]], ["block_19", [{"image_8": "173_8.png", "coords": [56, 73, 196, 202], "fig_type": "molecule"}]], ["block_20", ["HN\n0\n HM) \nK/NH 2\n"]], ["block_21", ["N/\\/N\nf\n\\kfo\nNH2\n"]], ["block_22", [{"image_9": "173_9.png", "coords": [142, 528, 302, 642], "fig_type": "figure"}]], ["block_23", [")(\u2014\u2014-\u2014-\u00abo\n"]], ["block_24", ["X\n"]], ["block_25", [{"image_10": "173_10.png", "coords": [155, 534, 277, 641], "fig_type": "molecule"}]], ["block_26", ["f?\n1111\n"]]], "page_174": [["block_0", [{"image_0": "174_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "174_1.png", "coords": [33, 141, 430, 254], "fig_type": "figure"}]], ["block_2", ["1.\nWhen a living polymerization is conducted such that the rate of initiation is effectively\ninstantaneous compared to propagation, it is possible to approach a Poisson distribution of\nmolecular weights, where the polydispersity is l +(1/Nn).\n2.\nAnionic polymerization is the most established method for approaching the ideal living\npolymerization, and effective protocols for a variety of monomers have been established.\n3.\nCationic polymerization can also be living, although it is generally harder to do so than for the\nanionic case, in large part due to the prevalence of transfer reactions, including transfer to\nmonomer.\n4.\nUsing the concept of a reversibly dormant or inactive species, free-radical polymerizations\nhave also been brought under much greater control. Three general \ufb02avors of controlled\nradical polymerization, known as ATRP, SFRP, and RAFT, are currently undergoing rapid\ndevelopment.\n5.\nLiving polymerization in general, and anionic polymerization in particular, can be used to\nproduce block copolymers, end-functional polymers, and well-defined star and graft polymers,\nfor a variety of possible uses.\n6.\nThrough basic thermodynamic considerations the concepts of equilibrium polymerization,\nceiling temperature, and \ufb02oor temperature have been explored.\n7.\nThe utility of ROPs has been established, where the thermodynamic driving force for chain\ngrowth relies on ring-strain. Specific systems of nearly living ring-opening polymerizations\nhave been introduced, including important metal-catalyzed routes such as ROMP.\n"]], ["block_3", ["In this chapter, we have considered a wide variety of synthetic strategies to exert greater control\nover the products of a polymerization, compared to the standard step-growth and chain-growth\napproaches. Although access to much narrower molecular weight distributions has been the\nprimary focus, production of block copolymers, end-functional polymers, and controlled-branched\narchitectures has also been explored. The central concept of the chapter is that of a living\npolymerization, de\ufb01ned as a chain-growth process that proceeds in the absence of irreversible\nchain termination or chain transfer:\n"]], ["block_4", ["4.10\nChapter Summary\n"]], ["block_5", ["The initial demonstration of this approach was based on the following scheme [15]. The\nbuilding blocks were 3,5-dihydroxybenzyl alcohol and a benzylic bromide. The first reaction,\nconducted in acetone, coupled two of the bromides with one alcohol. The surviving benzylic\nalcohol was then transformed back to a bromide functionality with carbon tetrabromide in the\npresence of triphenyl phosphine. Introduction of more 3,5-dihydroxybenzyl alcohol began the\nformation of the next generation dendron, and the process continued.\n"]], ["block_6", ["160\nControlled Polymerization\n"]], ["block_7", [{"image_2": "174_2.png", "coords": [156, 150, 360, 245], "fig_type": "figure"}]], ["block_8", [{"image_3": "174_3.png", "coords": [276, 128, 431, 254], "fig_type": "figure"}]], ["block_9", [{"image_4": "174_4.png", "coords": [294, 124, 429, 263], "fig_type": "molecule"}]], ["block_10", ["(4.NN)\n"]]], "page_175": [["block_0", [{"image_0": "175_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Problems\n161\n"]], ["block_2", ["8.\nA particular class of highly branched, precisely controlled polymers called dendrimers can be\nprepared by either of two step-growth routes, referred to as convergent and divergent,\nrespectively.\n"]], ["block_3", ["Problems\n"]], ["block_4", ["TA. Soum and M. Fontanille, in Anionic Polymerization, J.E. McGrath, Ed., ACS Symposium Series, Vol1981.\ninn. Plesch, Ed, Cationic Polymerization, Macmillan, New York, 1963.\n"]], ["block_5", ["1.\nExperimental data cited in Example 4.1 for the anionic polymerization of styrene do not really\ntest the relationship between conversion, p and time; why not? What additional experimental\ninformation should have been obtained if that were the object?\nAlthough the polydispersities described in Example 4.1 are very low, they consistently exceed\nthe theoretical Poisson limit. List four assumptions that are necessary for the Poisson distri-\nbution to apply, and then identify which one is most likely not satisfied. Justify your answer,\nbased on the data provided.\nFor the living anionic polymerization of styrene discussed in Example 4.1, the solvent used\nwas cyclohexane, and the kinetics are known to be 0.5 order with respect to initiator. What is\nthe predominant species in terms of ion pairing, and what is the approximate dissociation\nconstant for this cluster if kp is actually 1000 mol L\u20181 s\u2018l?\nOne often-cited criterion for judging whether a polymerization is living is that 114,, should\nincrease linearly with conversion. Why is this not, in fact, a robust criterion?\nA living polymerization of 2\u2014vinyl pyridine was conducted using benzyl picolyl magnesium as\nthe initiator.)r Values of Mn were determined for polymers prepared with different initiator\nconcentrations and different initial concentrations of monomer, as shown below. Calculate the\nexpected Mn assuming complete conversion and 100% initiator efficiency; how well do the\ntheoretical and experimental values agree?\n"]], ["block_6", [{"image_1": "175_1.png", "coords": [41, 359, 270, 454], "fig_type": "figure"}]], ["block_7", ["The following table shows values of AH0 at 298 K for the gas phase reactions X(g) + H+(g)\n"]], ["block_8", ["\u2014> HX+ (g), where X is an olefin: Use these data to comment quantitatively of the\nfollowing points:\n"]], ["block_9", ["0.48\n82\n20\n0.37\n85\n25\n0.17\n71\n46\n0.48\n71\n17\n0.58\n73\n14\n0.15\n150\n115\n"]], ["block_10", ["\u20145 .8 k] mol_l, respectively, evaluate the AH for the rearrangement sec-butyl\nions and compare with the corresponding isomerization for the propyl cation.\n4.\nOf the monomers shown, only isobutene undergoes cationic polymerization to any\nsignificant extent. Criticize or defend the following proposition: the data explain this\nfact by showing that this is the only monomer listed that combines a sufficiently negative\nAH for protonation, with the freedom from interfering isomerization reactions.\n"]], ["block_11", ["1.\nThe cation is stabilized by electron-donating alkyl substituents.\n2.\nThe carbonium ion rearrangement of n-propyl ions to i-propyl ions is energetically\nfavored.\n3.\nWith the supplementary information that AH? of l\u2014butene and cis-Z-butene are +1.6 and\n"]], ["block_12", ["[I](mmolL_1)\n[M]0(mmolL\"l)\nMn (kg 11101\u20141)\n"]]], "page_176": [["block_0", [{"image_0": "176_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["1\u2018A. Duda and S. Penczek Macromolecules, 23, 1636 (1990).\n"]], ["block_2", ["162\n"]], ["block_3", ["13.\n"]], ["block_4", ["14.\n"]], ["block_5", ["10.\n"]], ["block_6", ["11.\n"]], ["block_7", ["12.\n"]], ["block_8", [{"image_1": "176_1.png", "coords": [41, 61, 331, 147], "fig_type": "figure"}]], ["block_9", [{"image_2": "176_2.png", "coords": [51, 539, 280, 612], "fig_type": "figure"}]], ["block_10", ["In the study discussed in Example 4.3, a solution ATRP of styrene gave an apparent\npropagation rate constant of 3.9 x 10\u20145 s\u2018]. Given that the initial monomer concentration\nwas 4.3 mol L\u2018 1, and that the initial concentrations of initiator and CuBr were 0.045 mol L\u201d,\nestimate the equilibrium constant K for activation of the chain end radical.\nFor the solution polymerization of lactide with [M] =1 mol L\u201d, Duda and Penczek\ndetermined AHO \u201422.9 kJ mol\u20181 and A50: 411 J mol\" K\u2018\u201d. What is the associated\nceiling temperature for an equilibrium monomer concentration of\n1 mol L\u2018l? Does the\nvalue you obtain suggest that equilibration is an issue in controlled polymerization of\nlactide?\nCompare\nthese thermodynamic quantities with\nthose for the cyclic alkanes\nin Table 4.4; how do you account for the differences between the six-membered alkane\nand lactide?\nFor the polymerization system in Problem 8, calculate the equilibrium monomer con-\ncentration that would actually be obtained, and the conversion to polymer, at 80\u00b0C\nand 120\u00b0C.\nA typical propagation rate constant, kp, for the anionic ROP of hexamethylcyclotrisiloxane\n(D3) is 0.1 L mol\u20181 5\". Design a polymerization system (initial monomer and initiator\nconcentrations) to obtain a narrow distribution poly(dimethylsiloxane) with Mn=50,000,\nassuming that the reaction will be terminated at 50% conversion. At what time should the\npolymerization be terminated?\nGiven that ROP is often conducted under conditions in which reverse reactions are\npossible, do we need to worry about cyclization of the entire growing polymer? Why or\nwhy not?\nSuggest a scheme to test the hypothesis that in lactide polymerization it is the acyl carbon\u2014\noxygen bond that is cleaved, rather than the alkyl carbon\u2014oxygen bond.\nBoth ethyleneimine and ethylene sul\ufb01de are amenable to ROP. The former proceeds in the\npresence of acid, whereas the latter can follow either anionic or cationic routes. Propose\nstructures for the three propagating chain ends and the resulting polymers.\n"]], ["block_11", ["Draw repeat unit structures for polymers made by ROMP of the following three monomers:\n"]], ["block_12", ["H\nN\n8\nAA\nA\n"]], ["block_13", ["CH3CH2CH CH2\nCH3CH2CH2CH2+\n\u2014682\nCH3CH CHCH3\nCH3CH2C+HCH3\n\u2014782\n(CH3)2C CH2\n(CH3)2CHCH2+\n\u2014695\n"]], ["block_14", ["X\nHX+\nAHO at 298 K (kJ mol\u201c)\n"]], ["block_15", ["CH2 CH2\nCH3CH;\n\u2014640\nCHgCH CH2\nCH3CH2CH2+\n\u201469O\nCH3CH CH2\nCH3C+HCH3\n\u2014757\n"]], ["block_16", ["Me\n"]], ["block_17", ["Me\nMW\nOH\n"]], ["block_18", [{"image_3": "176_3.png", "coords": [201, 546, 276, 593], "fig_type": "molecule"}]], ["block_19", ["Controlled Polymerization\n"]]], "page_177": [["block_0", [{"image_0": "177_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Further Readings\n163\n"]], ["block_2", ["N. Hadjichristidis, S. PiSpas, and G. Floudas, Block Copolymers: Synthetic Strategies, Physical Pr0perties,\nand Applications, John Wiley & Sons, Inc., New York, 2003.\n"]], ["block_3", ["References\n"]], ["block_4", ["15.\n"]], ["block_5", ["16.\n"]], ["block_6", ["18.\n"]], ["block_7", ["Further Readings\n"]], ["block_8", ["17.\n"]], ["block_9", ["12.\n"]], ["block_10", ["13.\n14.\n15.\n"]], ["block_11", ["11.\n"]], ["block_12", ["9\u00b0\n"]], ["block_13", ["1.\n"]], ["block_14", ["249).\u201d?\n"]], ["block_15", ["G. Odian, Principles of Polymerization, 4th ed., Wiley\u2014Interscience, Hoboken, NJ, 2004.\nJ.E. Puskas, P. Antony, Y. Kwon, C. Paulo, M. Kovar, P.R. Norton, G. Kaszas, and V. Altst'adt,\nMacromol. Mater. Eng, 286, 565 (2001).\nM. Sawamoto, C. Okamoto, and T. Higashimura, Macromolecules, 20, 2693 (1987).\n"]], ["block_16", ["Rev., 101, 3661 (2001).\nG. Moad, E. Rizzardo, S.H. Thang, Aust. J. Chem, 58, 379 (2005).\nCJ. Hawker, Acc. Chem. Res, 30, 373 (1997).\nR]. Flory, Nature (London), 62, 1561 (1940).\nC]. Hawker and J.M.J. Fr\u00e9chet, J. Am. Chem. Soc., 112, 7639 (1990).\n"]], ["block_17", ["TR. Darling, T.P. Davis, M. Fryd, A.A. Gridnev, D.M. Haddleton, S.D. Ittel, R.R. Matheson, Jr.,\nG. Moad, and E. Rizzardo, J. Polym. Sci., Polym. Chem. Ed., 38, 1706 (2000).\nM. Szwarc, Nature, 178, 1168 (1956).\nH.L. Hsieh and RP. Quirk, Anionic Polymerization, Principles and Practical Applications, Marcel\nDekker, Inc., New York, NY, 1996.\nN. Hadjichristidis, M. Pitsikalis, S. Pispas, and H. Iatrou, Chem. Rev., 101, 3747 (2001).\nY. Tsukahara, K. Tsutsumi, Y. Yamashita, and S. Shimada, Macromolecules, 23, 5201 (1990).\n"]], ["block_18", ["J .E. Puskas, G. Kaszas, and M. Litt, Macromolecules, 24, 5278 (1991).\nK. Matyjaszewski, (Ed.), Controlled/Living Polymerization: Progress in ATRP, ACS\nSymposium Series 768, Oxford University Press (2000).\nK. Matyjaszewski, Chem. Rev., 101, 2921 (2001); C]. Hawker, A.W. Bosman, and E. Harth, Chem.\n"]], ["block_19", ["Compare and contrast dendrimers with block copolymer micelles, globular proteins, inor-\nganic nanoparticles in terms of attributes and likely utility in the following applications: (a)\ndrug delivery; (b) homogenous catalysis; and (c) solubilization.\nIn an ideal living polymerization, how should Mn and MW/Mn vary with conversion\nof monomer to polymer? How should Mn of the formed polymer vary with time?\nCompare these to a radical polymerization with termination by disproportionation, and no\ntransfer.\nThe following criteria have all been suggested and/or utilized as diagnostics for whether a\npolymerization is living or not. For each one, explain why it might be useful, and then decide\nwhether or not it is a robust criterion, that is, can you think of a situation in which the\ncriterion is satis\ufb01ed but the polymerization is not living? (See also Problem 4.)\nl.\nPolymerization proceeds until all monomer is consumed. Polymerization continues if\nmore monomer is then added.\n2.\nThe number of polymer molecules is constant, and independent of conversion.\n3.\nNarrow molecular weight distributions are produced.\n4.\nThe concentration of monomer decreases to zero, exponentially with time.\n"]], ["block_20", ["Suggest monomer structures that will lead to the following repeat unit structures following\nROMP:\n/\nN\nW\nONO\n"]], ["block_21", [{"image_1": "177_1.png", "coords": [89, 79, 334, 157], "fig_type": "figure"}]], ["block_22", ["I\nMe\n"]]], "page_178": [["block_0", [{"image_0": "178_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["P. Rempp and E.W. Merrill, Polymer Synthesis, 2nd ed., Hijthig & Wepf, Basel, 1991.\nM. Szwarc and M. van Beylen, Ionic Polymerization and Living Polymers, Chapman and Hall, New York, 1993.\n"]], ["block_2", ["H.L. Hsieh and RP. Quirk, Anionic Polymerization, Principles and Practical Applications, Marcel Dekker,\nInc., New York, NY, 1996.\nG. Odian, Principles of Polymerization, 4th ed., Wiley\u2014Interscience, Hoboken, NJ, 2004.\n"]], ["block_3", ["164\nControlled Polymerization\n"]]], "page_179": [["block_0", [{"image_0": "179_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["All polymer molecules have unique features of one sort or another at the level of the individual\nrepeat units. Occasional head-to-head or tail-to-tail orientations, random branching, and the\ndistinctiveness of chain ends are all examples of such details. In this chapter, we shall focus\nattention on two other situations that introduce structural variation at the level of the repeat unit:\nthe presence of two different monomers, and the regulation of configuration of successive repeat\nunits. In the former case copolymers are produced, and in the latter case polymers with differences\nin tacticity. In the discussion of these combined topics, we use statistics extensively because the\ndescription of microstructure requires this kind of approach. This is the basis for merging a\ndiscussion of copolymers and stereoregular polymers into a single chapter. In other respects\nthese two classes of materials and the processes that produce them are very different, and their\ndescription leads us into some rather diverse areas. Copolymerization offers a facile means to tune\nmaterial properties, as the average composition of the resulting polymers can often be varied across\nthe complete composition range. Similarly, control of stereoregularity plays an essential role in\ndictating the crystallinity of the resulting material, which in turn can exert a profound in\ufb02uence on\nthe resulting physical properties.\nThe formation of copolymers involves the reaction of (at least) two kinds of monomers. This\nmeans that each must be capable of undergoing the same propagation reaction, but it is apparent\nthat quite a range of reactivities are compatible with this broad requirement. We shall examine\nsuch things as the polarity of monomers, the degree of resonance stabilization they possess, and the\nsteric hindrance they experience in an attempt to understand these differences in reactivity. There\nare few types of reactions for which chemists are successful in explaining all examples with\ngeneral concepts such as these, and polymerization reactions are no exception. Even for the\nspeci\ufb01c case of free\u2014radical copolymerization, we shall see that reactivity involves the interplay\nof all these considerations.\nTo achieve any sort of pattern in configuration among successive repeat units in a polymer\nchain, the tendency toward random addition must be overcome. Although temperature effects are\npertinent here\u2014remember that high temperature is the great randomizer\u2014real success in regulat-\ning the pattern of successive addition involves the use of catalysts that \u201cpin down\u201d both the\nmonomer and the growing chain so that their reaction is biased in favor of one mode of addition or\nanother. We shall discuss the Ziegler\u2014Natta catalysts that accomplish this, and shall discover these\nto be complicated systems for which no single mechanism is entirely satisfactory. We shall also\ncompare these to the more recently developed \u201csingle\u2014site\u201d catalysts, which offer great potential\nfor controlling multiple aspects of polymer structure.\nFor both copolymers and stereoregular polymers, experimental methods for characterizing the\nproducts often involve spectroscopy. We shall see that nuclear magnetic resonance (NMR)\nspectroscopy is particularly well suited for the study of tacticity. This method is also used for\nthe analysis of copolymers.\n"]], ["block_2", ["5.1\nIntroduction\n"]], ["block_3", ["Copolymers, Microstructure,\nand Stereoregularity\n"]], ["block_4", ["5\n"]], ["block_5", ["165\n"]]], "page_180": [["block_0", [{"image_0": "180_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["In writing Equation 5.2.1 through Equation 5.2.4 we make the customary assumption that the\nkinetic constants are independent of the size of the radical, and we indicate the concentration of all\nradicals ending with the M1 repeat unit, whatever their chain length, by the notation [M10]. This\nformalism therefore assumes that only the nature of the radical chain end in\ufb02uences the rate\nconstant for propagation. We refer to this as the terminal control mechanism. If we wished to\nconsider the effect of the next-to-last repeat unit in the radical, each of these reactions and the\nassociated rate laws would be replaced by two alternatives. Thus Reaction (5.A) becomes\n"]], ["block_2", ["and Equation 5.2.1 becomes\n"]], ["block_3", ["Each of these reactions is characterized by a propagation constant, which is labeled by a two-digit\nsubscript: the first number identifies the terminal repeat unit in the growing radical and the second\nidentifies the added monomer. The rate laws governing these four reactions are:\n"]], ["block_4", ["The polymerization mechanism continues to include initiation, termination, and propagation steps,\nand we ignore transfer reactions for simplicity. This time, however, there are four distinctly\ndifferent propagation reactions:\n"]], ["block_5", ["We begin our discussion of copolymers by considering the free-radical polymerization of a mixture\nof two monomers, M1 and M2. This is already a narrow view of the entire \ufb01eld of copolymers, since\nmore than two repeat units can be present in copolymers and, in addition, mechanisms other than free-\nradical chain growth can be responsible for copolymer formation. The essential features ofthe problem\nare introduced by this simpler special case, and so we shall restrict our attention to this system.\n"]], ["block_6", ["In spite of the assortment of things discussed in this chapter, there are also related topics that\ncould be included but which are not owing to space limitations. We do not discuss copolymers\nformed by the step\u2014growth mechanism, for example, or the use of Ziegler\u2014Natta catalysts to\nregulate geometrical isomerism in, say, butadiene polymerization. Some other important omissions\nare noted in passing in the body of the chapter.\n"]], ["block_7", ["5.2.1\nRate Laws\n"]], ["block_8", ["5.2\nC0polymer Composition\n"]], ["block_9", ["166\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_10", ["12,2, knmzonmz]\n(5.2.4)\n"]], ["block_11", ["Ran \n(5-2-1)\n"]], ["block_12", ["Rp.12 k12[M1\u00b0][M2]\n(5.2.2)\n"]], ["block_13", ["R1321 k21iM2\u00b0liM1l\n(52.3)\n"]], ["block_14", ["\u2014M2\u00b0 + M2L \u2014-M2M2o\n(5.D)\n"]], ["block_15", ["\u2014M1- + M1 L \u2014M1Mlo\n(5A)\n_M,. + M2 A. \u2014M1M2-\n(5.3)\n"]], ["block_16", ["\u2014M2. + M1 L) \u2014M2M1o\n(5.C)\n"]], ["block_17", ["Rp.211 k211[M2M1\u00b0][M1]\n(5.2.6)\n"]], ["block_18", ["Rp,111 kllliMlMl\u2018HMll\n(5.2.5)\n"]], ["block_19", ["km\n*MlMl\u2019 + M] -\u2014-\u2014-\u2014> \u2014M1M1M1\u00b0\n(5.13)\n"]], ["block_20", ["\u2014M2M1- + M1 J\ufb02\u2014v \u2014M2M1M1-\n(5F)\n"]]], "page_181": [["block_0", [{"image_0": "181_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["where the is considered. \nourselves to the only the terminal unit determines behavior, \nin which the penultimate effect is important are well known.\nThe magnitudes of the various k values in Equation 5.2.1 through Equation 5.2.4 describe the\nintrinsic of addition, and the ks \nthe different species determine the rates at which the four kinds of additions occur. It is the\nproportion of different steps that determines the composition of the copolymer produced.\nMonomer M1 is converted to polymer by Reaction (5A) and Reaction (5C); therefore the rate\nat which this occurs is the sum of Rl and R1921:\n"]], ["block_2", ["Likewise, Reaction (5B) convert M2 to polymer, and the rate at \noccurs is the sum of RP,\u201d and R922:\n"]], ["block_3", ["We saw in Chapter 3 that the stationary-state approximation is applicable to free-radical\nhomOpolymerizations, and the same is true of copolymerizations. Of course, it takes a brief time\nfor the stationary-state radical concentration to be reached, but this period is insignificant com-\npared to the total duration of a polymerization reaction. If the total concentration of radicals is\nconstant, this means that the rate of crossover between the different types of terminal units is also\nequal, or that Rp.l2 =Rp.213\n"]], ["block_4", ["Copolymer Composition\n167\n"]], ["block_5", ["The ratio of Equation 5.2.7 and Equation 5.2.8 gives the relative rates of the two monomer\nadditions and, hence, the ratio of the two kinds of repeat units in the copolymer:\n"]], ["block_6", ["or\n"]], ["block_7", ["Although there are a total of four different rate constants for propagation, Equation \nthat the relationship between the relative amounts of the two monomers incorporated \npolymer and the composition of the monomer feedstock involves only two \nof these constants. Accordingly, we simplify the notation by defining reactivity ratios:\n"]], ["block_8", ["Combining Equation 5.2.9 and Equation 5.2.11 yields one form of the important \ncomposition equation or copolymerization equation:\n"]], ["block_9", ["and\n"]], ["block_10", ["diMli/dl\u2018 _ k11[M1\u00b0][M1]+ k21[M2\u00b0][M1]\nd[M2]/d[\n\u2014\nk12[M1\u00b0][M2] + k22[M2\u00b0][M2]\n(5.2.9)\n"]], ["block_11", ["k12[M1\u00b0][M2] k21[M2'][M1]\n(5.2.10)\n"]], ["block_12", ["r1 = ki\n(5.2.13)\nklz\n"]], ["block_13", ["\u2014i:\u2019:l\u2014] k11[M1\u00b0][M1]+ k21[M2\u00b0][M1]\n(5.2.7)\n"]], ["block_14", ["\u2014 \ndt\n= k12[M1'][M2] + k22[M2\u00b0][M2]\n(5.2.8)\n"]], ["block_15", ["d[M1]/dr_ [M1] (kll/k12)[M1] + [M2]\ndil/df\n_\n[M2] (kzz/k21)[M2] + [M1]\n(5.2.12)\n"]], ["block_16", ["r, \n(5.2.14)\n"]], ["block_17", ["[My] _ kzliMll\n__\n5.2.11\n[My]\nklziMz]\n(\n)\n"]]], "page_182": [["block_0", [{"image_0": "182_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["2.\nIf r1 r2, the copolymer and the feed mixture have the same composition at f1 0.5. In this\ncase Equation 5.2.18 becomes F1 :(r + 1)/2(r + 1) 0.5.\n3.\nIf r1 =r2, with both values less than unity, the copolymer is richer in component 1 than the\nfeed mixture for f1 < 0.5, and richer in component 2 than the feed mixture for f1 > 0.5.\n4.\nIf r1 r2, with both values greater than unity, an S-shaped curve passing through the point\n(0.5, 0.5) would also result, but in this case re\ufb02ected across the 45\u00b0 line compared to item (3).\n5.\nIf r1 74 r2, with both values less than unity, the copolymer starts out richer in monomer 1 than\nthe feed mixture and then crosses the 45\u00b0 line, and is richer in component 2 beyond this\n"]], ["block_2", ["This equation relates the composition of the copolymer formed to the instantaneous composition of\nthe feedstock and to the reactivity ratios r1 and r2 that characterize the specific system.\nFigure 5.1 shows a plot of F1 versusf1\u2014\u2014the mole fractions of monomer 1 in the copolymer and\nin the mixture, respectively\u2014for several values of the reactivity ratios. Inspection of Figure 5.1\nbrings out the following points:\n"]], ["block_3", ["Combining Equation 5.2.15 and Equation 5.2.16 into Equation 5.2.17 yields another form of the\ncopolymer composition equation\n"]], ["block_4", ["As an alternative to Equation 5.2.15, it is convenient to describe the composition of both\nthe polymer and the feedstock in terms of the mole fraction of each monomer. Defining F,- as the\nmole fraction of the ith component in the polymer and f,- as the mole fraction of component i in\nthe monomer solution, we observe that\n"]], ["block_5", ["and\n"]], ["block_6", ["With these substitutions, Equation 5.2.12 becomes\n"]], ["block_7", ["The ratio (d[M1]/dt)/(d[M2]/dt) is the same as the ratio of the numbers of each kind of repeat\nunit in the polymer formed from the solution containing M1 and M2 at concentrations [M1] and\n[M2], respectively. Since the composition of the monomer solution changes as the reaction\nprogresses, Equation 5.2.15 applies to the feedstock as prepared only during the initial stages of\nthe polymerization. Subsequently, the instantaneous concentrations in the prevailing mixture apply\nunless monomer is added continuously to replace that which has reacted and maintain the original\ncomposition of the feedstock. We shall assume that it is the initial product formed that we describe\nwhen we use Equation 5.2.15 so as to remove uncertainty as to the monomer concentrations.\n"]], ["block_8", ["5.2.2\nComposition versus Feedstock\n"]], ["block_9", ["168\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_10", ["1.\nIf r1 r2 1, the copolymer and the feed mixture have the same composition at all times. In\nthis case Equation 5.2.18 becomes\n"]], ["block_11", ["[M1]\nf1 \n\u2014_[M1]+[M2]\n(5.2.17)\n"]], ["block_12", ["_\n_\n(\u201cMd/(If\nF\u2018 F2 d[M1]/dt+d[M2]/dt\n(5.2.16)\n"]], ["block_13", ["d[M]]/dt = [M1] \u201c[111] + [M2] = 1+ r([M1]/[M2]\n(5 15)\nd[M2]/dl\u2018\n[M2] r2[M2] + [M1]\nI + r2[M2]/[M1]\n\u2018\n.\n"]], ["block_14", ["F1 M =fl\n(5.2.19)\n(f1 +12)2\n"]], ["block_15", ["1:\nr1f12 +f1f2\nI'1f12'l'2f1f2+r2f22\n"]], ["block_16", ["(5.2.18)\n"]]], "page_183": [["block_0", [{"image_0": "183_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "183_1.png", "coords": [26, 38, 294, 314], "fig_type": "figure"}]], ["block_2", ["Figure 5.1\nMole fraction of component\n1 in the copolymer as a function of feedstock composition for\nvarious reactivity ratios.\n"]], ["block_3", ["There is a parallel between the composition of a copolymer produced from a certain feed and\nthe composition of a vapor in equilibrium with a two-component liquid mixture. The following\nexample illustrates this parallel when the liquid mixture is an ideal solution and the vapor is an\nideal gas.\n"]], ["block_4", ["An ideal gas obeys Dalton\u2019s law; that is, the total pressure is the sum of the partial pressures of the\ncomponents. An ideal solution obeys Raoult\u2019s law; that is, the partial pressure of the ith component\nabove a solution is equal to the mole fraction of that component in the solution times the vapor\npressure of pure component 1'. Use these relationships to relate the mole fraction of component 1 in\nthe equilibrium vapor to its mole fraction in a two\u2014component solution and relate the result to the\nideal case of the copolymer composition equation.\n"]], ["block_5", ["Copolymer Composition\n169\n"]], ["block_6", ["Example 5.1\n"]], ["block_7", ["0.6 \n_\n"]], ["block_8", ["0.8\nr-\nr1: r2 :1\n\u2014\n"]], ["block_9", ["0.4\n\u2014\nF1 0.33. f2 0.57\n_\n"]], ["block_10", ["0.2 \n\u2014\n"]], ["block_11", ["For the case of r1 0.33 and r2 0.67 shown in Figure 5.1, [Md/[M2] equals 0.5 and\nf1 0.33. This mathematical analysis shows that a comparable result is possible with both r1\nand r2 greater than unity, but is not possible for r1 > 1 and r2 < 1.\nWhen r1: l/rz, the copolymer composition curve will be either convex or concave when\nviewed from the F1 axis, depending on whether r1 is greater or less than unity. The further\nremoved from unity r1 is, the farther the composition curve will be displaced from the 45\u00b0 line.\nThis situation where rlrz l is called an idea! copolymerizarion. The example below explores\nthe origin of this terminology.\n"]], ["block_12", ["0\n_|\n1\nl\nI\nI\nl\n1\n[\ni\nI\n1\nI\nl\n1\n1\n1\n"]], ["block_13", ["\u20181\nI\n\u2014l\nI\nI\nI\n1\n1'\n1\nI\nI\nI\nl\nl\nl\n1\n"]], ["block_14", ["crossover point. At the crossover point the copolymer and feed mixture have the same\ncomposition. The monomer ratio at this point is conveniently solved by Equation 5.2.15:\n"]], ["block_15", ["(W)cross_ r1\n(5.2.20)\n"]], ["block_16", ["0\n02\n0.4\n06\n08\n1\n"]], ["block_17", ["[M1]\n_ \n"]], ["block_18", ["._\nr1=10,f2=0.1\n_\n"]], ["block_19", ["\u2014\nr1=r2=0.15\n-\n"]], ["block_20", ["r1=0.1, {2:10\n"]]], "page_184": [["block_0", [{"image_0": "184_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["The parameters r1 and r2 are the vehicles by which the nature of the reactants enter the copolymer\ncomposition equation. We shall call these radical reactivity ratios simply reactivity ratios, although\nsimilarly defined ratios also describe copolymerizations that involve ionic intermediates. There are\nseveral important things to note about reactivity ratios:\n"]], ["block_2", ["This is identical to the ideal liquid\u2014vapor equilibrium if r1 is identified with p?/pg.\nThe vapor pressure ratio measures the intrinsic tendency of component 1 to enter the vapor phase\nrelative to component 2. Likewise r1 measures the tendency of M1 to add to M10 relative to M1\nadding to My. In this sense there is a certain parallel, but it is based on My as a reference radical and\nhence appears to be less general than the vapor pressure ratio. Note, however, that n Hr; means\n"]], ["block_3", ["k1 1/1612 kzl/kgg. In this case the ratio of rate constants for monomer 1 relative to monomer 2 is the\nsame regardless of the reference radical examined. This shows the parallelism to be exact.\n"]], ["block_4", ["Because of the analogy with liquid\u2014vapor equilibrium, copolymers for which n 1/5 are said\nto be ideal. For those nonideal cases in which the copolymer and feedstock happen to have the\nsame composition, the reaction is called an azeotropic polymerization. Just as in the case of\nazeotropic distillation, the composition of the reaction mixture does not change as copolymer is\nformed if the composition corresponds to the azeotrope. The proportion of the two monomers at\nthis point is given by Equation 5.2.20.\nIn this section we have seen that the copolymer composition depends to a large extent on the\nfour propagation constants, although it is sufficient to consider these in terms of the two\nreactivity ratios\n1', and r2. In the next section we shall examine these ratios in somewhat\ngreater detail.\n"]], ["block_5", ["We define F1 to be the mole fraction of component\n1 in the vapor phase and f1 to be its mole\nfraction in the liquid solution. Here p1 and p2 are the vapor pressures of components\n1 and 2 in\nequilibrium with an ideal solution, and p? and p3 are the vapor pressures of the two pure liquids. By\nDalton\u2019s law, Ptot=P1+P2 and F1 =p1/pt0t, since these are ideal gases and p is proportional to\nthe number of moles. By Raoult\u2019s law, p1 =f1p\u20181), p2 fn, and pm, =f1p? +fn. Combining the\ntwo gives\n"]], ["block_6", ["Now examine Equation 5.2.18 for the case of r1 1/r2:\n"]], ["block_7", ["1.\nThe single subscript used to label r is the index of the radical.\n2.\nr1 is the ratio of two propagation constants involving radical 1: The ratio always compares the\npropagation constant for the same monomer adding to the radical relative to the propagation\nconstant for the addition of the other monomer. Thus, if r1 > 1, My adds M1 in preference to\nM2; if r1 < 1, M10 adds M2 in preference to M1.\n3.\nAlthough r1 is descriptive of radical Mr, it also depends on the identity of monomer 2; the\npair of parameters r, and r2 are both required to characterize a particular system, and the\nproduct rlrz is used to quantify this by a single parameter.\n4.\nThe reciprocal of a radical reactivity ratio can be used to quantify the reactivity of monomer\nM2 by comparing its rate of addition to radical M10 relative to the rate of M1 adding M10.\n"]], ["block_8", ["5.3\nReactivity Ratios\n"]], ["block_9", ["Solution\n"]], ["block_10", ["170\nCapolymers, Microstructure, and Stereoregularity\n"]], ["block_11", [{"image_1": "184_1.png", "coords": [39, 145, 194, 188], "fig_type": "molecule"}]], ["block_12", ["F1\n"]], ["block_13", ["F \nfir)?\n:\nfn(P?/p3)\n1\nflPti'l'fZPg\nf1(P?/Pg)+f2\n"]], ["block_14", ["=\nwe +1315\n=r1f1(r1f1+f2):\nma\nr1f12+2flf2+(l/rl)f22\n(r1f1+f22)\nr1f1+f2\n"]]], "page_185": [["block_0", [{"image_0": "185_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Table 5.1\nValues of Reactivity Ratios r1 and r2 and the Product rlrz for a Few Copolymers at 60\u00b0C\n"]], ["block_2", ["The reactivity copolymerization system are the fundamental parameters in terms of\nwhich the system is described. Since the copolymer composition equation relates the compositions\nof the product itis clear that values of r can be evaluated from experimental data\nin which the corresponding compositions are measured. We shall consider this evaluation procedure\nin Section be this approach is not as free of ambiguity as might be\ndesired. For now we shall simply assume that we know the desired r values for a system; in fact,\nextensive tabulations of such values exist. An especially convenient source of this information is\nthe Polymer Handbook [1]. Table 5.] lists some typical r values at 60\u00b0C.\nAlthough Table 5.1 is rather arbitrarily assembled, note that it contains no system for which r]\nand r2 are both greater than unity. Indeed, such systems are very rare. We can understand this by\nrecognizing that, at least in the extreme case of very large r\u2019s, these monomers would tend to\nsimultaneously homopolymerize. Because of this preference toward homopolymerization, any\ncopolymer that does form in systems with r1 and r2 both greater than unity will be a block-type\npolymer with very long sequences of a single repeat unit. Since such systems are only infrequently\nencountered, we shall not consider them further.\nTable 5.1 also lists the product rlrg for the systems included. These products typically lie in the\nrange between zero and unity, and it is instructive to consider the character of the copolymer\nproduced toward each of these extremes.\nIn the extreme case where r1r2:0 because both r1 and r2 equal zero, the copolymer adds\nmonomers with perfect alternation. This is apparent from the definition of r, which compares the\naddition of the same monomer to the other monomer for a particular radical. If both r\u2019s are zero,\nthere is no tendency for a radical to add a monomer of the same kind as the growing end,\n"]], ["block_3", ["Source: Data from Young, L]. in Polymer Handbook, 3rd ed., Brandrup, J. and Immergut, E.H. (Eds), Wiley, New York,\n1989.\n"]], ["block_4", ["Acrylonitrile\nMethyl vinyl ketone\n0.61\n1.78\n1.09\nMethyl methacrylate\n0.13\n1.16\n0.15\nor-Methyl styrene\n0.04\n0.20\n0.008\nVinyl acetate\n4.05\n0.061\n0.25\nMethyl methacrylate\nStyrene\n0.46\n0.52\n0.24\nMethacrylic acid\n1.18\n0.63\n0.74\nVinyl acetate\n20\n0.015\n0.30\nVinylidene chloride\n2.53\n0.24\n0.61\nStyrene\nVinyl acetate\n55\n0.01\n0.55\nVinyl chloride\n17\n0.02\n0.34\nVinylidene chloride\n1.85\n0.085\n0.16\n2-Vinyl pyridine\n0.55\n1.14\n0.63\nVinyl acetate\nl-Butene\n2.0\n0.34\n0.68\nIsobutylene\n2.15\n0.31\n0.67\nVinyl chloride\n0.23\n1.68\n0.39\nVinylidene chloride\n0.05\n6.7\n0.34\n"]], ["block_5", ["Rd]\nDd;\nr1\nr2\nrlrz\n"]], ["block_6", ["5.\nAs the ratio of two rate constants, a radical reactivity ratio follows the Arrhenius equation with\nan apparent activation energy equal to the difference in the activation energies for the\nindividual constants. Thus for r1, Efpp E1351 El\u201d; 12. Since the activation energies for pI\u2018Opa-\ngation are not large to begin with, their difference is even smaller. Accordingly, the tempera-\nture dependence of r is relatively small.\n"]], ["block_7", ["5.3.1\nEffects of r Values\n"]], ["block_8", ["Reactivity \n171\n"]], ["block_9", [{"image_1": "185_1.png", "coords": [121, 451, 456, 640], "fig_type": "figure"}]], ["block_10", [{"image_2": "185_2.png", "coords": [136, 457, 418, 635], "fig_type": "table"}]]], "page_186": [["block_0", [{"image_0": "186_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Returning to the data of Table 5.1, it is apparent that there is a good deal of variability among\nthe r values displayed by various systems. We have already seen the effect this produces on the overall\ncopolymer composition; we shall return to this matter ofmicrostructures in Section 5 .5. First, however,\nlet us consider the obvious question. What factors in the molecular structure of the two monomers\ngovern the kinetics of the different addition steps? This question is considered in the following\nsections; for now we look for a way to systematize the data as the first step toward an answer.\n"]], ["block_2", ["whichever species is the terminal unit. When only one of the r\u2019s is zero, say r1, then alternation\noccurs whenever the radical ends with an M1. unit. There is thus a tendency toward alternation in\nthis case, although it is less pronounced than in the case where both r\u2019s are zero. Accordingly, we\n\ufb01nd increasing tendency toward alternation as r1 \u2014> 0 and r2 \u2014> 0, or, more succinctly, as the\nproduct rlrz \u2014> 0.\nOn the other end of the commonly encountered range we find the product rlrz \u2014> 1. As noted\nabove, this limit corresponds to ideal copolymerization and means the two monomers have the\nsame relative tendency to add to both radicals. Thus if r1 \u2014> 10, monomer 1 is 10 times more likely\nto add to M10 than monomer 2. At the same time r2 0.1, which also means that monomer 1 is 10\ntimes more likely to add to My. In this case the radicals exert the same in\ufb02uence, so the monomers\nadd at random in the proportion governed by the specific values of the r\u2019s.\nRecognition of these differences in behavior points out an important limitation on the copolymer\ncomposition equation. The equation describes the overall composition of the copolymer, but gives\nno information whatsoever about the distribution of the different kinds of repeat units within the\npolymer. While the overall composition is an important property of the copolymer, the detailed\nmicrostructural arrangement is also a significant feature of the molecule. It is possible for\ncopolymers with the same overall composition to have very different properties because of the\ndifferences in microstructure. Reviewing the three categories presented in Chapter 1, we see\nthe following:\n"]], ["block_3", ["2.\nRandom structures are promoted by rlrz \u2014\u2014> 1:\n"]], ["block_4", ["4.\nAlternating copolymers, while relatively rare, are characterized by combining the properties of\nthe two monomers along with structural regularity. We will see in Chapter 13 that a very high\ndegree of regularity\u2014extending all the way to stereoregularity in the configuration of the\nrepeat units\u2014is required for crystallinity to develop in polymers.\n5.\nRandom copolymers tend to average the properties of the constituent monomers in proportion\nto the relative abundance of the two comonomers.\n6.\nBlock copolymers are closer to blends of homopolymers in properties, but without the latter\u2019s\ntendency to undergo phase separation. Diblock copolymers can be used as surfactants to bind\nimmiscible homopolymer blends together and thus improve their mechanical properties. Block\ncopolymers are generally prepared by sequential addition of monomers to living polymers,\nrather than by depending on the improbable rlrg > 1 criterion in monomers, as was discussed\nin Chapter 4.\n"]], ["block_5", ["3.\n\u201cBlocky\u201d structures are promoted by rlrz > 1:\n"]], ["block_6", ["172\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_7", ["1.\nAlternating structures are promoted by r;\n\u2014\u2014> O and r2 \u2014> 0:\n"]], ["block_8", ["Each of these polymers has a 50:50 proportion of the two components, but the products\nprobably differ in properties. As examples of such differences, we note the following:\n"]], ["block_9", ["M1MZMIMZMIMZMIMZMIMZMIMZMIMZMIMZMIMQMIMZ\n"]], ["block_10", ["MlMgMgMlMlMnM1M2M1M2M1M2M1M2M2M1M1M2\n"]], ["block_11", ["M1MiMiMiMiMiMIMlMlM1M2M2M2M2M2M2M2M2M2M2\n"]]], "page_187": [["block_0", [{"image_0": "187_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["We noted above that the rlrzcan used locatea copolymer along \nalternating and random Itmeans of product that some \n"]], ["block_2", ["has been \n"]], ["block_3", ["3.\nThe spacing of the lines is such that each monomer along the base serves as a label for a row of\ndiamonds.\n4.\nEach diamond marks the intersection of two such rows and therefore corresponds to two\ncomonomers.\n"]], ["block_4", ["5.3.2\nRelation of Reactivity Ratios to Chemical Structure\n"]], ["block_5", ["supplemented \n"]], ["block_6", ["Reactivity Ratios\n173\n"]], ["block_7", ["6.\nIndividual monomers have been arranged in such a way as to achieve to the greatest extent\npossible the values of rlrz that approach zero toward the apex of the triangle and values of rlrz\nthat approach unity toward the base of the triangle.\n"]], ["block_8", ["5.\nThe rlrz product for the various systems is the number entered in each diamond.\n"]], ["block_9", ["1,\nVarious monomers are listed along the base of the triangle.\nThe triangle is subdivided into an array of diamonds by lines drawn parallel to the two sides of\nthe triangle.\n"]], ["block_10", ["Figure 5.2\nThe product r1r2 for copolymers whose components define the \nappear. The value marked* is determined in Example 5.4; other values are from Ref. [1].\n"]], ["block_11", ["Increasing\ntendency\n0-00017\n.\ntoward\nalternation\n0.00240ee\n\u2019:\u00b03\u00b0:\u00b0e\n\u201d\u00b0Ai%\u00a3\nALLA\n"]], ["block_12", ["<1)\nas\n(:5\nE\nH\non\na:\n=\na:\n2\u2018\n\u2014\nCD\n03\nE\na\nE\ng\n\u00a32\nCC\u201d\n0\n\u20181\u2019\n91\n3\nS\n'0\n_L\nc\n7\u201c\n0\n_cu\n(DE\n_EU\n.9\n0\nCD\na:\n.9>~\n>~\n0\n>335\nrag.\nE5\n>5\n5\nSP,\n0.\n\u2014c\n+-\nE\u2018\n+-.._\n#0\n>~.:\n+-\n3\u2019\u201d:\nH\nCD\n(Una\n.93\n(.3\nC13\";\n0q\n.Eo\nCUE\n.20\nCD\na\nE\nC]\n<\nE\nE\n>\nE\n>\n"]], ["block_13", [{"image_1": "187_1.png", "coords": [49, 279, 373, 586], "fig_type": "figure"}]], ["block_14", ["0.006\n"]], ["block_15", ["a.)\n_L\n"]], ["block_16", ["a?\n9n\n"]], ["block_17", ["Increasing\ntendency\ntoward\nrandomness\n"]]], "page_188": [["block_0", [{"image_0": "188_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "188_1.png", "coords": [25, 422, 287, 476], "fig_type": "molecule"}]], ["block_2", [{"image_2": "188_2.png", "coords": [31, 423, 275, 474], "fig_type": "figure"}]], ["block_3", ["Before proceeding with a discussion of this display, it is important to acknowledge that the\ncriteria for monomer placement can be met only in part. For one thing, there are combinations for\nwhich data are not readily available. Incidentally, not all of the rlrz values in Figure 5.2 were\nmeasured at the same temperature, but, as noted above, temperature effects are expected to be\nrelatively unimportant. Also, there are outright exceptions to the pattern sought: Generalizations\nabout chemical reactions always seem to be plagued by these. In spite of some reversals of ranking,\nthe predominant trend moving upward from the base along any row of diamonds is a decrease in\nrlrz values.\nFrom the geometry of this triangular display, it follows immediately\u2014if one overlooks\nthe exceptions\u2014that the more widely separated a pair of comonomers are in Figure 5.2, the\ngreater is their tendency toward randomness. We recognize a parallel here to the notion\nthat widely separated elements in the periodic table will produce more polar bonds than those\nwhich are closer together, and vice versa. This is a purely empirical and qualitative trend. The next\norder of business is to seek an explanation for its origin in terms of molecular structure. If we focus\nattention on the electron-withdrawing or electron-donating attributes of the substituents on the\ndouble bond, we find that the substituents of monomers that are located toward the right-hand\ncorner of the triangle in Figure 5.2 are recognized as electron donors. Likewise, the substituents\nin monomers located toward the left-hand corner of the triangle are electron acceptors. The\ndemarcation between the two regions of behavior is indicated in Figure 5.2 by reversing the direction\nof the lettering at this point. Pushing this point of view somewhat further, we conclude that the\nsequence acetoxy < phenyl < vinyl is the order of increase in electron-donating tendency. Chlor-\no < carbonyl < nitrile is the order of increase in electron-withdrawing tendency. The positions of\ndiethyl fumarate and vinylidene chloride relative to their monosubstituted analogs indicates\nthat \u201cmore is better\u201d with respect to these substituent effects. The location of methyl methacrylate\nrelative to methyl acrylate also indicates additivity, this time with partial compensation of\nopposing effects.\nThe reactivity ratios are kinetic in origin, and therefore re\ufb02ect the mechanism or, more\nspeci\ufb01cally, the transition state of a reaction. The transition state for the addition of a vinyl\nmonomer to a growing radical involves the formation of a partial bond between the two species,\nwith a corresponding reduction of the double-bond character of the vinyl group in the monomer:\n"]], ["block_4", ["The contribution of this polar structure to the bonding lowers the energy of the transition state. This\nmay be viewed as a lower activation energy for the addition step and thus a factor that promotes\nthis particular reaction. The effect is clearly larger the greater the difference in the donor\u2014acceptor\nproperties of X and Y. The transition state for the successive addition of the same monomer\n(whether X or Y substituted) is Structure (5.11). This involves a more uniform distribution of\n"]], ["block_5", ["If substituent X is an electron donor and Y an electron acceptor, then the partial bond in the\ntransition state is stabilized by a resonance form (5.1), which attributes a certain polarity to the\nemerging bond:\n"]], ["block_6", ["174\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_7", [{"image_3": "188_3.png", "coords": [38, 427, 223, 464], "fig_type": "molecule"}]], ["block_8", [{"image_4": "188_4.png", "coords": [40, 508, 201, 595], "fig_type": "figure"}]], ["block_9", ["1\n\u201dN.\n+ ///\\Y \u2014..\nWE\n"]], ["block_10", ["5\u2018 3\"\ni\ni\nas\n\u201dMS\nx\nv\nx\nx\n"]], ["block_11", [{"image_5": "188_5.png", "coords": [64, 527, 197, 576], "fig_type": "figure"}]], ["block_12", ["\u2014--\nProduct\n(5G)\nx \nY\nX\n"]], ["block_13", ["[5.I]\n[5-H]\n"]]], "page_189": [["block_0", [{"image_0": "189_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Table 5.2\nValues of the Cross-Propagation Constants kn (L mol'1 3*) for Four Monomer\u2014Radical\nCombinations\n"]], ["block_2", ["Styrene\n145\n49,000\n14,000\n230,000\nAcrylonitrile\n435\n1,960\n2,510\n46,000\nMethyl acrylate\n203\n1,310\n2,090\n23,000\nVinyl acetate\n2.9\n230\n230\n2,300\n"]], ["block_3", ["Source: From Brandrup, J. and Immergut, EH. (Eds), Polymer Handbook, 3rd ed., Wiley, New York, 1989.\n"]], ["block_4", ["a substituent, whether it is in a monomer or a radical. Conjugation allows greater electron\ndelocalization, which, in turn, lowers the energy of the system that possesses this feature.\nComparison of the range of km along rows and columns in Table 5.2 suggests that resonance\nstabilization produces a bigger effect in the radical than in the monomer. After all, the right- and\nleft\u2014hand columns in Table 5.2 (various radicals) differ by factors of 100\u20141000, whereas the top\n"]], ["block_5", ["Monomer\nStyrene\nAcrylonitrile\nMethyl acrylate\nVinyl acetate\n"]], ["block_6", ["Table 5.2 lists a few cross-propagation constants calculated by Equation 5.4.1. Far more extensive\ntabulations than this have been prepared by correlating copolymerization and homopolymerization\ndata for additional systems.\nExamination of Table 5.2 shows that the general order of increasing radical activity is\nstyrene < acrylonitrile < methyl acrylate < vinyl acetate. An additional observation is that any\none of these species shows the reverse order of reactivity for the corresponding monomers. As\nmonomers, the order of reactivity in Table 5.2 is styrene > acrylonitrile > methyl acrylate > vinyl\nacetate. These and similar rankings based on more extensive comparisons are summarized in terms\nof substituents in Table 5.3.\nAn important pattern to recognize among the substituents listed in Table 5.3 is this: Those that\nhave a double bond conjugated with the double bond in the olefin are the species that are more\nstable as radicals and more reactive as monomers. The inverse relationship between the stability of\nmonomers and radicals arises precisely because monomers gain (or lose) stability by converting to\nthe radical: The greater the gain (or loss), the greater (or less) the incentive for the monomer\nto react. It is important to realize that the ability to form conjugated structures is associated with\n"]], ["block_7", ["The tendency toward alternation is not the only pattern in terms of which copolymerization can be\ndiscussed. of radicals and monomers may also be examined as a source of insight\ninto copolymer formation. The reactivity of radical 1 copolymerizing with monomer 2 is measured\nby the rate constant km. The absolute value of this constant can be determined from copolymer-\nization data (r1) and studies yielding absolute homopolymerization constants (kn):\n"]], ["block_8", ["charge and thus lacks the stabilizing effect of the polar\nresonance of addition is that for alternation,\nat least X Y sufficiently different.\nAlthough we use the term resonance in describing the effect of polarity in stabilizing the\ntransition state in alternating copolymers, the emphasis of the foregoing is definitely on polarity\nrather that resonance plays an important role in free-\nradical polarity effects are ignored. In Section 5.4 we examine some\nevidence for this and consider the origin of this behavior.\n"]], ["block_9", ["5.4\nResonance and Reactivity\n"]], ["block_10", ["Resonance and Reactivity\n175\n"]], ["block_11", ["k12 \n(5.4.1)\nr1\n"]], ["block_12", ["Radical\n"]]], "page_190": [["block_0", [{"image_0": "190_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "190_1.png", "coords": [25, 70, 97, 343], "fig_type": "figure"}]], ["block_2", [{"image_2": "190_2.png", "coords": [27, 79, 314, 343], "fig_type": "figure"}]], ["block_3", ["Table 5.3\nList of Substituents Ranked in Terms of Their Effects on\nMonomer and Radical Reactivity\n"]], ["block_4", ["and bottom rows (various monomers) differ only by the factors of 50-100. In order to examine this\neffect in more detail, consider the addition reaction of monomer M to a reactant radical R0 to form\na product radical Po. What distinguishes these species is the presence or absence of resonance\nstabilization (subscript rs). If the latter is operative, we must also consider which species benefit\nfrom its presence. There are four possibilities:\n"]], ["block_5", ["O\n0\nIncreasing\nJLOH\n)LO/R\nIncreasing\nmonomer\nradical\nactivity\nactivity\n"]], ["block_6", ["176\nCopolymers, Microstructure. and Stereoregularity\n"]], ["block_7", ["I.\nUnstabilized monomer converts stabilized radical to unstabilized radical:\n"]], ["block_8", ["This reaction suffers none of the reduction in resonance stabilization that is present in Reaction\n(SH) and Reaction (5.1). It is energetically more favored than both of these, but not as much as\nthe reaction in which. . ..\nStabilized monomer converts unstabilized radical to stabilized radical:\n"]], ["block_9", ["There is an overall loss of resonance stabilization in this reaction. Since it is a radical which\nsuffers the loss, the effect is larger than in the reaction in which. . ..\nStabilized monomer converts stabilized radical to another stabilized radical:\n"]], ["block_10", ["Here too there is an overall loss of resonance stabilization, but it is monomer stabilization\nwhich is lost, and this is energetically less costly than Reaction (5.H).\nUnstabilized monomer converts unstabilized radical to another unstabilized radical:\nR+MeR\nan\n"]], ["block_11", ["Rrs' + Mrs Prs\u2018\n(SI)\n"]], ["block_12", ["R\u00b0 + Mrs Prs'\n(5K)\n"]], ["block_13", ["Rrs- + M \u2014> P.\n(S.H)\n"]], ["block_14", ["it o\nm\n"]], ["block_15", [",0\nFt\nY\n-R\nO\n"]], ["block_16", ["\u2014 CEN\nAO/R\n"]], ["block_17", ["/O\\R\n\u2014H\n"]], ["block_18", ["- Cl\n"]], ["block_19", ["ll\n"]]], "page_191": [["block_0", [{"image_0": "191_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "191_1.png", "coords": [28, 444, 250, 521], "fig_type": "figure"}]], ["block_2", ["Note that this inquiry into copolymer propagation rates also increases our understanding of the\ndifferences in free-radical homopolymerization rates. Recall that in Chapter 3 a discussion of this\naspect of homopolymerization was deferred until copolymerization was introduced. The trends\nunder consideration enable us to make some sense out of the rate constants for propagation in free-\nradical homopolymerization as well. For example, in Table 3.4 we see that kp values at 60\u00b0C for\nvinyl acetate and styrene are 2300 and 165 L mol'1 3\u2014], respectively. The relative magnitude of\nthese constants can be understood in terms of the sequence above.\nResonance stabilization energies are generally assessed from thermodynamic data. If we define\n"]], ["block_3", ["Radical\nStyrene\nStyrene\nVinyl acetate\nVinyl acetate\n+\n<\n+\n<\n+\n<\n+\nMonomer\nVinyl acetate\nStyrene\nVinyl acetate\nStyrene\n"]], ["block_4", ["31 to be the resonance stabilization energy of species 1', then the heat of formation of that species\nwill be less by an amount 31 than for an otherwise equivalent molecule without resonance.\nLikewise, the change in enthalpy AH for a reaction that is in\ufb02uenced by resonance effects is\nless by an amount As (A is the usual difference: products minus reactants) that the AH for a\nreaction which is otherwise identical except for resonance effects:\n"]], ["block_5", ["Thus if we consider the homopolymerization of ethylene (no resonance possibilities),\n"]], ["block_6", [{"image_2": "191_2.png", "coords": [34, 632, 278, 680], "fig_type": "molecule"}]], ["block_7", ["Systems from Table 5.2 which correspond to these situations are the following:\n"]], ["block_8", ["Resonance and Reactivity\n177\n"]], ["block_9", ["we find a value of A3 19 kJ mol\u2018l, according to Equation 5.4.2. Reaction (5.M) is a specific\nexample of the general Reaction (5.1), and the negative value of A3 in this example indicates the\noverall loss of resonance stabilization, which is characteristic of Reaction (5.1).\nAlthough it is not universally true that the activation energies of reactions parallel their heats of\nreaction, this is approximately true for the kind of addition reaction we are discussing. Accord-\ningly, we can estimate E* aAH, with a an appropriate proportionality constant. If we consider\nthe difference between two activation energies by combining this idea with Equation 5.4.2, the\ncontribution of the nonstabilized reference reaction drops out of Equation 5.4.2 and we obtain\n"]], ["block_10", ["as a reference reaction, and compare it with the homopolymerization of styrene (resonance effects\n"]], ["block_11", ["present),\n"]], ["block_12", ["AHrs AHnO ,8 A3\n(5.4.2)\n"]], ["block_13", ["Eii\u2014 Biz a[\u2014A311 0236312)]\n= \u2018(SPF 8R1. _SMI) + (8P2. _BRI. 8M2)\n(5.4.3)\n"]], ["block_14", ["AHno ,, \u2014\u201488.7 kJ mol-1\n"]], ["block_15", ["AH,, \u201469.9 kJ mol\u201d1\n"]], ["block_16", ["\u2014CH2\u2014CH2\u00b0 + CH2=CH2 -+ \u201cCH2CH2CH2CH2\u2018\n(5.L)\n"]], ["block_17", ["This reaction converts the less effective resonance stabilization of a monomer to a more effective\nform of radical stabilization. This is the most favorable of the four reaction possibilities.\n"]], ["block_18", ["Rrso +M< Rrso +Mrs < R- +M< Ro +Mrs\n"]], ["block_19", [{"image_3": "191_3.png", "coords": [58, 141, 315, 194], "fig_type": "molecule"}]], ["block_20", ["In summary, we can rank these reactions in terms of their propagation constants as follows:\n"]], ["block_21", [".H\n+\n\\\n\u2014+\n.H\n"]]], "page_192": [["block_0", [{"image_0": "192_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "192_1.png", "coords": [34, 122, 233, 154], "fig_type": "molecule"}]], ["block_2", ["1.\nDefine styrene to be monomer 1 and vinyl acetate to be monomer 2.\nThe difference in resonance stabilization energy 3p;\n\u2014\u2014 3M. >1, since styrene is resonance\nstabilized and the effect is larger for the radical than the monomer.\n3.\nThe difference 8132\u2018 3M2 \u00a7 0, since neither the radical nor the monomer of vinyl acetate\nshows appreciable stabilization.\n4.\nTherefore, according to Equation 5.4.4 and Equation 5.4.5, n > 1 while r2 < 1.\n5.\nExperimental values for this system are r1 55 and r2: 0.01.\n"]], ["block_3", ["We might be hard-pressed to estimate the individual resonance stabilization energies in\nEquation 5.4.4 and Equation 5.4.5, but the quantitative application of these ideas is not difficult.\nConsider once again the styrene-vinyl acetate system:\n"]], ["block_4", ["In writing the second version of this relation, the proportionality constant has been set equal to\nunity as a simpli\ufb01cation. Note that the resonance stabilization energy of the reference radical Rlo\nalso cancels out of this expression.\nThe temperature dependence of the reactivity ratio r1 also involves the E\u2019\ufb01\u2014 E\u2019f\u2018z difference\nthrough the Arrhenius equation; hence\n"]], ["block_5", ["1.\nThe reactivity ratios are proportional to the product of two exponentials.\n2.\nEach exponential involves the difference between the resonance stabilization energy of the\nradical and monomer of a particular species.\n3.\nA positive exponent is associated with the same species\nas identifies the r (i.e., for\n"]], ["block_6", ["An analogous expression can be written for r2:\nM)\n:.(_8PI'_\u2018\n8W\n5 4 5\nr2 oc exp(\nRT\nexp(\nRT\n)\n(\n.\n. )\n"]], ["block_7", ["1.\nIf resonance effects alone are considered, it is possible to make some sense of the ranking of\nvarious propagation constants.\n2.\nIn this case only random microstructure is predicted.\n3.\nIf polarity effects alone are considered, it is possible to make some sense out of the tendency\ntoward alternation.\n4.\nIn this case homOpolymerization is unexplained.\n"]], ["block_8", ["Although this approach does correctly rank the parameters r1 and r2 for the styrene\u2014vinyl acetate\nsystem, this conclusion was already reached qualitatively above using the same concepts and\nwithout any mathematical manipulations. One point that the quantitative derivation makes clear is\nthat explanations of copolymer behavior based exclusively on resonance concepts fail to describe\nthe full picture. All that we need to do is examine the product rlrg as given by Equation 5.4.4 and\nEquation 5.4.5, and the shortcoming becomes apparent. According to these relationships, the\nproduct rlrz always equals unity, yet we saw in the last section that experimental rlrg values\ngenerally lie between zero and unity. We also saw that polarity effects could be invoked to\nrationalize the rlrz product.\nThe situation may be summarized as follows:\n"]], ["block_9", ["178\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_10", ["According to this formalism, the following applies:\n"]], ["block_11", ["6P: \n3,4,)\n\u2014(8P2\u2018 3M2)\n5 4 4\nr1 oc exp(\u2014\u2014\u2014\u2014\u2014RT\nexp(\nRT\n)\n(\n.\n. )\n"]], ["block_12", ["11, M1\n\u2014-> Pp), whereas the negative exponent is associated with the other species (for\nr], M2 \u2014-> 132' ).\n"]]], "page_193": [["block_0", [{"image_0": "193_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "193_1.png", "coords": [33, 295, 238, 445], "fig_type": "figure"}]], ["block_2", ["It is apparent from the size of the conjugated system here that numerous resonance possibilities\nexist in this species in both the radical and the molecular form. Styrene also has resonance\nstructures in both forms. On the principle that these effects are larger for radicals than monomers,\nwe conclude that the difference 8}\u201d 8M > 0 for both hemin and styrene. On the principle that\ngreater resonance effects result from greater delocalization, we expect the difference to be larger\nfor hemin than for styrene. According to Equation 5.4.4, r1 oc el\u2018t\u2018rg\"\u2018\u201de\u20183\"\u201c\u00a3lller > 1. According to\nEquation 5.4.5, r2 oc esm\u2018i\u2018lh\u2018ire\u2014l\u2018t\u2018rgcr < 1. Experimentally, the values for these parameters turn out to\nbe r1: 65 and r2 0.18.\n"]], ["block_3", ["as these and resonance are both operative, and, if these still fall short of\nexplaining all observations, there is another old standby to fall back on: steric effects. Resonance,\npolarity, and considerations are all believed to play an important role in copolymerization\nchemistry of organic chemistry. Things are obviously simplified if only\none of these is considered, but it must be remembered that doing this necessarily reveals only one\nfacet of are times, particularly before launching an experimental\ninvestigation ofwhen some guidelines are very useful. The following example\nillustrates this point.\n"]], ["block_4", ["It is proposed to polymerize the vinyl group of the hemin molecule with other vinyl comonomers to\nprepare model compounds to be used in hemoglobin research. Considering hemin and styrene to be\nspecies 1 and 2, respectively, use the resonance concept to rank the reactivity ratios of r, and r2.\n"]], ["block_5", ["In Section 5.3 we noted that variations in the product rlrg led to differences in the polymer\nmicrostructure, even when the overall compositions of two systems are the same. In this section we\nshall take a closer look at this variation, using the approach best suited for this kind of detail,\nstatistics.\n"]], ["block_6", ["Hemin is the complex between protoporphyrin and iron in the +3 oxidation state. Iron is in the +2\nstate in the heme of hemoglobin. The molecule has the following structure:\n"]], ["block_7", ["A Closer Look at Microstructure\n179\n"]], ["block_8", ["The way out of this apparent dilemma is easily stated, although not easily acted upon. It is not\nadequate these approaches for the explanation of something as complicated\n"]], ["block_9", ["Solution\n"]], ["block_10", ["5.5\nA Closer Look at Microstructure\n"]], ["block_11", ["Example 5.2\n"]], ["block_12", [{"image_2": "193_2.png", "coords": [36, 300, 208, 411], "fig_type": "molecule"}]], ["block_13", [{"image_3": "193_3.png", "coords": [46, 333, 194, 393], "fig_type": "molecule"}]], ["block_14", ["OH\n"]], ["block_15", ["CH2\n"]]], "page_194": [["block_0", [{"image_0": "194_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Note that p11 + p12 p22 \u2014|\u2014 p21 1. In writing these expressions we make the assumption that only\nthe terminal unit of the radical in\ufb02uences the addition of the next monomer. This same assumption\nwas made in deriving the copolymer composition equation. We shall have more to say below about\nthis particular assumption.\nNext let us consider the probability of \ufb01nding a sequence of repeat units in a copolymer, which\nis exactly 12 units of M1 in length. This may be represented as M2(M1),,M2. Working from left to\nright in this sequence, we note the following:\n"]], ["block_2", ["For the various possible combinations in a copolymer, Equation 5.5.2 becomes\n"]], ["block_3", ["This equation can also be written in terms of the propagation rates of the different types of addition\nsteps which generate the sequences:\n"]], ["block_4", ["1.\nIf the addition of monomer M1 to a radical ending with M2 occurs L times in a sample, then\nthere will be a total of L sequences, of unspecified length, of M1 units in the sample.\n2.\nIf v 1 consecutive M1 monomers add to radicals capped by M1 units, the total number of\nsuch sequences is expressed in terms of p11 to be Lp'ff\u2018.\n3.\nIf the sequence contains exactly 12 units of type M], then the next step must be the addition of\nan M2 unit. The probability of such an addition is given by p12, and the number of sequences is\nLpiilplz-\n"]], ["block_5", ["The similarity of this derivation to those in Section 2.4 and Section 3.7 should be apparent.\nSubstitution of the probabilities given by Equation 5.5.3 and Equation 5.5.4 leads to\n"]], ["block_6", ["A similar result can be written for d)\u201c, where n denotes the length of a sequence of M2 units. These\nexpressions give the fraction of sequences of specified length in terms of the reactivity ratios of the\n"]], ["block_7", ["Suppose we define as p,3,- the probability that a unit of type i is followed in the polymer by a unit of\ntype j, where both i and j can be either 1 or 2. Since an 1' unit must be followed by either an i or aj,\nthe fraction of ij sequences (pairs) out of all possible sequences (pairs) de\ufb01nes pg:\n"]], ["block_8", ["Since L equals the total number of M1 sequences of any length, the fraction of sequences of length\n"]], ["block_9", ["180\nCopolymers, Microstructure. and Stereoregularity\n"]], ["block_10", ["5.5.1\nSequence Distributions\n"]], ["block_11", ["I), (by, is given by\n"]], ["block_12", ["k\nM \nM\nM\nP11=k\n\u201d[\n1H\n1]\n=\n\u201cl\n1]\n(5.5.3)\n11[M1'][M1] + k12[M1'][M2]\nr1[M1] + [M2]\n[M2]\n: \n(5.5.4\n\u201c2\nr1 [M1] + [M2]\n)\n"]], ["block_13", ["k\nM \nM\nM\np22 k\n22[\n2 H\n2]\n2\nrd\n2]\n(5.5.5)\n22[M2'][M2] + k21iM2'HM1]\nVziMzi -|- [M1]\nM\n[)2]\n[\n'1\n(5.5.6)\n: r2[M2] + [M1]\n"]], ["block_14", ["Ra\u201c\n_\nke\u2018li'l [Mr]\n1.. \n_\n(5.5.2\np}\nRij + RH\nkili'] [M1] + kiz\u2018lMi'HMil\n)\n"]], ["block_15", ["Number of if sequences\ni\u201c \n..\n..\n5.5.1\np 1\nNumber of 1] sequences + Number of :1 sequences\n(\n)\n"]], ["block_16", ["9b,, pi\u2019flpnz\n(5.5.7)\n"]], ["block_17", ["\u201cb\" ([MriihiliMzii\ufb02 (\ufb01lls/IT)\n(5'58)\n"]]], "page_195": [["block_0", [{"image_0": "195_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "195_1.png", "coords": [13, 365, 277, 636], "fig_type": "figure"}]], ["block_2", [{"image_2": "195_2.png", "coords": [30, 300, 188, 347], "fig_type": "molecule"}]], ["block_3", ["copolymer system and the composition of the feedstock. Figure 5.3 illustrates by means of a bar\ngraph how (1),, varies with v for two polymer systems prepared from equimolar solutions of\nmonomers. The shaded bars in Figure 5.3 describe the system for which \n"]], ["block_4", ["A Closer Look at Microstructure\n181\n"]], ["block_5", ["unshaded \nTable 5.4 shows the effect of variations in the composition of the feedstock for the system\nrlrz :1. The following observations can be made concerning Figure 5.3 and Table 5.4:\n"]], ["block_6", ["4.\nTable 5.4 also shows that increasing the percentage of M 1 in the monomer solution \ufb02attens\nand broadens the distribution of sequence lengths. Similar results are observed for lower\nvalues of rlrg, but the broadening is less pronounced when the tendency toward alternation\nis high.\n"]], ["block_7", ["2.\nFigure 5.3 shows that for rlrg 0.03, about 85% of the M1 units are sandwiched between two\nMg\u2019s. We have already concluded that low values of the rlrz product indicate a tendency\ntoward alternation.\n"]], ["block_8", ["3.\nFigure 5.3 also shows that the proportion of alternating M1 units decreases, and the fraction of\nlonger sequences increases, as rlrg increases. The 50 mol% entry in Table 5.4 shows that the\ndistribution of sequence lengths gets \ufb02atter and broader for 7'e 1, the ideal case.\n"]], ["block_9", ["Next we consider the average value of a sequence length of M1, 17. Combining Equation 1.7.7\nand Equation 5.5.7 gives\n"]], ["block_10", ["Simplifying this result involves the same infinite series that we examined in connection with\nEquation 2.4.5; therefore we can write immediately\n"]], ["block_11", [{"image_3": "195_3.png", "coords": [35, 458, 272, 601], "fig_type": "figure"}]], ["block_12", ["0/0\n_\n"]], ["block_13", ["451, 1968.)\n"]], ["block_14", ["1.\nIn all situations, the fraction (by decreases with increasing :2.\n"]], ["block_15", ["feedstocks with r1r2:0.03 (shaded) and r1r2:0.30 (unshaded). (Data from Tosi, C., \n"]], ["block_16", ["Figure 5.3\nFraction of sequences of the indicated length for copolymers \n"]], ["block_17", ["100\n\u2018I\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nl\nl\nI\n"]], ["block_18", ["80\ne\n_\n"]], ["block_19", ["60\n\u2014\n..\n"]], ["block_20", ["40\n\u2014\nJ\n"]], ["block_21", ["20\n\u2014\n\u2014\n"]], ["block_22", ["0\n"]], ["block_23", ["17:\n(5.5.9)\n"]], ["block_24", ["\"\nD\n['11'2 0.30\n\"\n"]], ["block_25", ["..\nI\nr1r2 0.03\n"]], ["block_26", ["_\n"]], ["block_27", ["2:1 12(1),, : 2:,\n"]], ["block_28", ["2:195\u201d\n2:1Pif1l\u201912\n"]], ["block_29", ["1\n2\n3\n4\n5\n"]], ["block_30", [{"image_4": "195_4.png", "coords": [144, 416, 235, 462], "fig_type": "figure"}]], ["block_31", ["V1\n"]], ["block_32", ["I\n1\n1\n1\nI\nl W\nI\nI\n"]]], "page_196": [["block_0", [{"image_0": "196_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "196_1.png", "coords": [17, 77, 451, 225], "fig_type": "figure"}]], ["block_2", ["1\n90\n30\n70\n60\n50\n40\n30\n20\n10\n2\n9\n16\n21\n24\n25\n24\n21\n16\n9\n3\n0.9\n3.2\n6.3\n9.6\n12.5\n14.4\n14.7\n12.3\n3.1\n4\n0.09\n0.64\n1.39\n3.34\n6.25\n3.64\n10.3\n10.2\n7.29\n5\n0.13\n0.57\n1.54\n3.13\n5.13\n7.20\n3.19\n6.56\n6\n0.17\n0.62\n1.56\n3.11\n5.04\n6.55\n5.90\n7\n0.05\n0.25\n0.73\n1.37\n3.53\n5.24\n5.31\n3\n0.10\n0.39\n1.12\n2.47\n4.19\n4.73\n9\n0.04\n0.20\n0.67\n1.73\n3.36\n4.30\n10\n0.10\n0.40\n1.21\n2.63\n3.37\n11\n0.05\n0.24\n0.35\n2.15\n3.59\n12\n0.14\n0.59\n1.72\n3.23\n"]], ["block_3", ["The following example demonstrates the use of some of these relationships pertaining to\nmicrostructure.\n"]], ["block_4", ["The hemoglobin molecule contains four heme units. It is proposed to synthesize a hemin (molecule\n1)\u2014styrene (molecule 2) copolymer such that 17 = 4 in an attempt to test some theory concerning\nhemoglobin. As noted in Example 5.2, r1: 65 and r2 0.18 for this system. What should be the\nproportion of monomers to obtain this average hemin sequence length? What is the average styrene\nsequence length at this composition? Does this system seem like a suitable model if the four hemin\nclusters are to be treated as isolated from one another in the theory being tested? Also evaluate (1),.\nfor several v bracketing 17 to get an idea of the distribution of these values.\n"]], ["block_5", ["Table 5.4\nPercentage of Sequences of Length v for Copolymers Prepared from Different Feedstocks f1\nwith rlrz 1\n"]], ["block_6", ["Use Equation 5.5.11 to evaluate [Md/[M2] for r1 65 and 17 4:\n"]], ["block_7", ["A value of 11 is obtained by similar operations:\n"]], ["block_8", ["By combining Equation 5.5.4 and Equation 5.5.10, we obtain\n"]], ["block_9", ["Solution\n"]], ["block_10", ["Example 5.3\n"]], ["block_11", ["182\nCopolymers. Microstructure. and Stereoregularity\n"]], ["block_12", ["14f,\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.3\n0.9\n"]], ["block_13", ["it =1+r2%\n(5.5.12)\n"]], ["block_14", [",\u2014, =\n1\n= _1_\n(5.5.10)\n1 P11\nP12\n"]], ["block_15", ["17=1+r1-\u2014\u2014-\u2014\u2014\n(5.5.11)\n"]], ["block_16", ["[M1]\n17 1\n4 1\n[M2]\n=\n=\n= 0.046\nd\n\u2014 = 21.7\n[Mg]\nr1\n65\nan\n[M1]\n"]], ["block_17", [{"image_2": "196_2.png", "coords": [80, 91, 404, 223], "fig_type": "table"}]]], "page_197": [["block_0", [{"image_0": "197_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "197_1.png", "coords": [21, 233, 284, 287], "fig_type": "molecule"}]], ["block_2", [{"image_2": "197_2.png", "coords": [22, 145, 235, 226], "fig_type": "molecule"}]], ["block_3", ["A Closer Look at Microstructure\n183\n"]], ["block_4", ["Use this ratio 5.5.12 to i1:\n"]], ["block_5", ["The number of styrene units in an average sequence is a little larger than the length of the average\nhemin sequence. It is not unreasonable to describe the hemin clusters as isolated, on the average, in\nthis molecule. The product rlrz 11.7 in this system, which also indicates a tendency toward block\nformation. Use Equation with [Md/[M2] 0.046 and the r1 and r2 values to evaluate\n"]], ["block_6", ["For the systems represented in Figure 5.3 and the equimolar case in Table 5.4, the average\nlengths are 17 1.173 for r1r2= 0.03, 17 = 1.548 for rlrg :0.30, and 17 2.000 for rlrg = 1.0.\nEquation 5.5.1 1 and Equation 5.5.12 suggest a second method for the experimental determination\nof reactivity ratios, in addition to the copolymer composition equation. If the average sequence\nlength can be determined for a feedstock of known composition, then r1 and r2 can be evaluated. We\nshall return to this possibility in the next section. In anticipation of applying this idea, let us review\nthe assumptions and limitation to which Equation 5.5.11 and Equation 5.5.12 are subject:\n"]], ["block_7", ["We have suggested earlier that both the copolymer composition equation and the average sequence\nlength offer possibilities for experimental evaluation of the reactivity ratios. Note that in so doing\nwe are finding parameters which fit experimental results to the predictions of a model. Nothing\nabout this tests the model itself. It could be argued that obtaining the same values of r1 and r2 from\nthe fitting of composition and microstructure data would validate the model. It is not likely,\nhowever, that both types of data would be available and of sufficient quality to make this\nunambiguous. We shall examine the experimental side of this in the next section.\nStatistical considerations make it possible to test the assumption of independent Let\n"]], ["block_8", ["as the conversion of monomers to polymer progresses. As in Section 5.2, we assume that either\nthe initial conditions apply (little change has taken place) or that monomers are continuously\nbeing added (replacement of reacted monomer).\n2.\nThe kinetic analysis described by Equation 5.5.3 and Equation 5.5.4 assumes that no repeat\nunit in the radical other than the terminal unit in\ufb02uences the addition. The penultimate unit in\nthe radical as well as those still further from the growing end are assumed to have no effect.\n"]], ["block_9", ["Solving for several values of V, we conclude that the distribution of sequence length is quite broad:\n"]], ["block_10", ["(1),,\n0.251\n0.188\n0.140\n0.105\n0.079\n0.059\n"]], ["block_11", ["us approach this topic by considering an easier problem: coin tossing. Under conditions where two\nevents are purely random\u2014as in tossing a fair coin\u2014the probability of a specific sequence of\n"]], ["block_12", ["3.\nItem (2) requires that each event in the addition process be independent of all others. We have\nconsistently assumed this throughout this chapter, beginning with the copolymer composition\nequation. Until now we have said nothing about testing this assumption. Consideration of\ncopolymer sequence length offers this possibility.\n"]], ["block_13", [":2\n1\n2\n3\n4\n5\n6\n"]], ["block_14", ["5.5.2\nTerminal and Penultimate Models\n"]], ["block_15", ["1.\nThe instantaneous monomer concentration must be used. Except at the azeotrope, this changes\n"]], ["block_16", ["_\n[M2l\n=1+r\u2014=1+0.1821.7\n=49\n"]], ["block_17", ["= (%)V\u20141 <\ufb01) : (0.0749)\u201d\"1(0.251)\n"]]], "page_198": [["block_0", [{"image_0": "198_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["We recall that the fraction of times a particular outcome occurs is used to estimate probabilities.\nTherefore we could evaluate pm; by counting the number of times NH the first toss yielded a head\nand the number of times NHH two tosses yielded a head followed by a head and write\n"]], ["block_2", ["If the events are not independent, provision must be made for this, so we de\ufb01ne a quantity called\nthe conditional probability. For the probability of a head given the prior event of a head, this is\nwritten pH/H, where the first quantity in the subscript is the event under consideration and that\nfollowing the slash mark is the prior condition. Thus pry/H is the probability of a tail following a\nhead. If the events are independent, ppm, :pH; if not, then pm; must be evaluated as a separate\nquantity. If the coin being tossed were biased, that is, if successive events are not independent,\nEquation 5.5.13 would become\n"]], ["block_3", ["This procedure is readily extended to three tosses. For a fair coin the probability of three heads is\nthe cube of the probability of tossing a single head:\n"]], ["block_4", ["If we were testing whether a coin were biased or not, we would use ideas like these as the basis\nfor a test. We could count, for example, HHH and HH sequences and divide them according to\nEquation 5.5.19. If pmHH 75 pH, we would be suspicious.\nA similar logic can be applied to copolymers. The story is a bit more complicated to tell, so we\nonly outline the method. If penultimate effects operate, then the probabilities p11, p12, etc., defined\nin Equation 5.5.3 through Equation 5.5.6 should be replaced by conditional probabilities. As a\nmatter of fact, the kind of conditional probabilities needed must be based on the two preceding\nevents. Thus Reaction (5.E) and Reaction (5.E) are two of the appropriate reactions, and the\ncorresponding probabilities are p1,\u201d and plm. Rather than work out all of the probabilities in\ndetail, we summarize the penultimate model as follows:\n"]], ["block_5", ["outcomes is given by the product of the probabilities of the individual events. The probability of\ntossing a head followed by a head~indicated HH\u2014is given by\n"]], ["block_6", ["If the coin is biased, conditional probabilities must be introduced:\n"]], ["block_7", ["Using Equation 5.5.15 to eliminate mm from the last result gives\n"]], ["block_8", ["01'\n"]], ["block_9", ["1.\nA total of eight different reactions are involved, since each reaction like Reaction (5A) is\nreplaced by a pair of reactions like Reaction (5.E) and Reaction (5.E).\n2.\nThere are eight different rate laws and rate constants associated with these reactions. Equation\n5.2.1, for example, is replaced by Equation 5.2.5 and Equation 5.2.6.\n"]], ["block_10", ["184\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_11", ["PHHH PH/HHpH/HPH\n(5.5.17)\n"]], ["block_12", ["PHH\nPHHH PH/HH (p\u2014H)PH\n(5.5.18)\n"]], ["block_13", ["PHH PHPH\n(5.5.13)\n"]], ["block_14", ["PHH PH/HPH\n(55-14)\n"]], ["block_15", ["_PHH _NHH\nPH/H \nW\nNH\n(55.15)\n"]], ["block_16", ["PHHH PHPHPH\n(5.5.16)\n"]], ["block_17", ["PH/HH \n(5.5.19)\nPHH\nNHH\n"]]], "page_199": [["block_0", [{"image_0": "199_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["those of four units, tetrads; those of five units, pentads; and so on. Next we examine the ratio\nNMIM/NMl and NMIMIMI /NM1M1. If these are the same, then the mechanism is Shown to have\nterminal control; if not, it may be penultimate control. To prove the penultimate model it would\nalso be necessary to count the number of M1 tetrads. If the tetrad\u2014triad ratio were the same as the\ntriad\u2014dyad ratio, the penultimate model is established.\nThis situation can be generalized. If the ratios do not become constant until the ratio of pentads\nto tetrads is considered, then the unit before the next to last\u2014called the antepenultimate unit\u2014\nplays a role in the addition. This situation has been observed, for example, for propylene oxide\u2014\nmaleic anhydride copolymers. The foregoing discussion has been conducted in terms of M]\nsequences. Additional relationships of the sort we have been considering also exist for dyads,\ntriads, and so forth, of different types of specific composition. Thus an ability to investigate\nmicrostructure experimentally allows some rather subtle mechanistic effects to be studied. In the\nnext section we shall see how such information is obtained.\n"]], ["block_2", ["As we have already seen, the reactivity ratios of a particular copolymer system determine both the\ncomposition and microstructure of the polymer. Thus it is important to have reliable values for\nthese parameters. At the same time it suggests that experimental studies of composition and\nmicrostructure can be used to evaluate the various r\u2019s.\n"]], ["block_3", ["Evaluation of reactivity ratios from the copolymer composition equation requires only composition\ndata\u2014~that is, relatively straightforward analytical chemistry\u2014and has been the method most widely\nused to evaluate r1 and r2. As noted in the last section, this method assumes terminal control and\n"]], ["block_4", ["quences, of any definite composition, of two units are called dyads; those of three units, triads;\n"]], ["block_5", ["These observations suggest how the terminal mechanism can be proved to apply to a copoly-\nmerization reaction if experiments exist which permit the number of sequences of a particular\nlength to be determined. If this is possible, we should count the number of Ml\u2019s (this is given by\nthe copolymer composition) and the number of MIMI and MIMIMI sequences. Speci\ufb01ed se-\n"]], ["block_6", ["seeks the best fit of the data to that model. It offers no means for testing the model, and as we shall see,\nis subject to enough uncertainty to make even self-consistency difficult to achieve. Microstructure\n"]], ["block_7", ["6.\nEquation 5.5.4 shows that p11 is constant for a particular copolymer if the terminal model\napplies; therefore the ratio NM.M./NM.\nalso equals this constant. Equation 5.5.20 shows\nthat P1111 is constant for a particular copolymer if the penultimate model applies; therefore\nthe ratio NM]M|M|/NM1M1 also equals this constant, but the ratio NMIMI /NM. does not have the\nsame value.\n"]], ["block_8", ["5.\nThe probability p11 can be written as the ratio NMIMl/NMI using Equation 5.5.15. This is\nreplaced by pm], which is given by the ratio NMM]Ml/NM,Ml according to Equation 5.5.19.\n"]], ["block_9", ["3.\nEight rate constants are clustered in four ratios, which define new reactivity ratios. Thus r1 as\ndefined in Equation 5.2.13 is replaced by rf (cm/km and rf\u2019 \ufb01rm/((212 whereas r; is\nreplaced by r5 k222/k221 and r\ufb01\u2019 km/km.\n4.\nThe probability p11 as given by Equation 5.5.3 is replaced by the conditional probability pl,1 1,\nwhich is defined as\n"]], ["block_10", ["5.6\nCopolymer Composition and Microstructure: Experimental Aspects\n"]], ["block_11", ["5.6.1\nEvaluating Reactivity Ratios from Composition Data\n"]], ["block_12", ["Copolymer Composition and Microstructure: Experimental Aspects\n185\n"]], ["block_13", ["=\nk111[M1M1'][M1]\n_\nr\ufb02Mll/[Mg]\npV1\u2018\nknitMe-HMI] + k112[M1M1o][M2]_1+ r{[M1]/[M2]\n(5.5.20)\n"]], ["block_14", ["There are eight of these conditional probabilities, each associated with the reaction described\nin item (1).\n"]]], "page_200": [["block_0", [{"image_0": "200_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["studies, by contrast, offer both a means to evaluate the reactivity ratios and also to test the\nmodel. The capability to investigate this level of structural detail was virtually nonexistent\nuntil the advent of modem instrumentation, and even now is limited to sequences of rather\nmodest length.\nIn this section we shall use the evaluation of reactivity ratios\nas the unifying theme;\nthe experimental methods constitute the new material introduced. The copolymer composition\nEquation 5.2.18 relates the r\u2019s to the mole fractions of the monomers in the feedstock and in\nthe copolymer. To use the equation to evaluate r1 and r2, the composition of a copolymer\nresulting from a feedstock of known composition must be measured. The composition of the\nfeedstock itself must also be known, but we assume this poses no problems. The copolymer\nspecimen must be obtained by pr0per sampling procedures and puri\ufb01ed of extraneous materials.\nRemember that monomers, initiators, and possibly solvents and soluble catalysts are involved\nin these reactions also, even though we have been focusing attention on the copolymer alone.\nThe proportions of the two kinds of repeat unit in the copolymer are then determined by\neither chemical or physical methods. Elemental analysis is a widely used chemical method, but\nspectroscopic analysis (UV\u2014visible, IR, NMR, and mass spectrometry) for functional groups is\ncommonly employed.\nSince the copolymer equation involves both r1 and r; as unknowns, at least two polymers\nprepared from different feedstocks must be analyzed. It is preferable to use more than this\nminimum number of observations, and it is helpful to rearrange the copolymer composition\nequation into a linear form so that simple graphical methods can be employed to evaluate the\nr\u2019s. Several ways to linearize the equation exist:\n"]], ["block_2", ["Each of these forms weigh the errors in various data points differently, so some may be more\nsuitable than others, depending on the precision of the data. Ideally all should yield the same\nvalues of the reactivity ratios. The following example illustrates the use of Equation 5.6.1 to\nevaluate r1 and r2.\n"]], ["block_3", ["2.\nIn terms of ratios rather than fractions, Equation 5.6.1 may be written as\nW(\ufb02_1)=rlw_r,\n(5.6.2)\nill/\u201d2\n\u201d2\n\u201d1/712\n"]], ["block_4", ["where n1 refers to the number of repeat units in the polymer. This expression is also of\nthe form y=mx+b if x=1\n)_\u2019 2 \u2014r2\u2014 _|_ r1\n(5.6.3)\nx\nx\n"]], ["block_10", ["1\nx:_y+2\nGeo\nr1\nr1\n"]], ["block_11", ["y\nrz)\u2019\nV1\n"]], ["block_12", ["This is the equation of a straight line, so r1 and r; can be evaluated from the slope and intercept\nof an appropriate plot.\n"]], ["block_13", [{"image_2": "200_2.png", "coords": [54, 428, 244, 457], "fig_type": "molecule"}]], ["block_14", ["1\n1\n"]]], "page_201": [["block_0", [{"image_0": "201_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "201_1.png", "coords": [22, 78, 358, 164], "fig_type": "figure"}]], ["block_2", [{"image_2": "201_2.png", "coords": [25, 463, 397, 649], "fig_type": "figure"}]], ["block_3", ["f1\nF1\nf1\nF1\n"]], ["block_4", ["ff(F1\u2014\u20141)/F1(1\u2014f1)2\n\u20140.0083\n20.0143\n\u20140.0316\n\u20140.0486\n\u20140.0832\n\u20140.1315\n\u20140.2792\n\u20140.7349\n"]], ["block_5", ["f1(1\u20142F1)/F1(1-f1)\n0.0217\n\u20140.1036\n\u20140.2087\n\u20140.3832\n\u20140.6127\n\u20140.9668\n\u20142.8102\n46.4061\n"]], ["block_6", ["Table \nValues of F1 as a Function for f1 for the Methyl Acrylate (MO\u2014Vinyl\nChloride \n"]], ["block_7", ["Note: These 5.4.\n"]], ["block_8", ["Source: Data from Chapin, and Fordyce, R., J. Am. Chem. Soc, 70, 538, 1948.\n"]], ["block_9", ["We calculate the variables to be used as ordinate and abscissa for the data in Table 5.5 using\nEquation 5.6.1:\n"]], ["block_10", ["0.075\n0.441\n0.421\n0.864\n0.154\n0.699\n0.521\n0.900\n"]], ["block_11", ["0.237\n0.753\n0.744\n0.968\n0.326\n0.828\n0.867\n0.983\n"]], ["block_12", ["The data in Table 5.5 list the mole fraction of methyl acrylate in the feedstock and in the\ncopolymer for the methyl acrylate (MO\u2014vinyl chloride (M2) system. Use Equation 5.6.1 as the\nbasis for the graphical determination of the reactivity ratios, which describe this system.\n"]], ["block_13", ["Least-square analysis of these values gives a slope r1 8.929 and an intercept r22 0.053.\nFigure 5.4b shows these data plotted according to Equation 5.6.1. The line is drawn with the\nleast\u2014squares slope and intercept. The last point on the left in Figure 5.4b, which this line passes\nthrough, corresponds to F1 0.983 andf1 0.867. Because the functional form plotted involves the\nsmall differences F 1 l and l f1, this point is also subject to the largest error. This illustrates\nthe value of having alternate methods for analyzing the data. The authors of this research carried\n"]], ["block_14", ["Copolymer Composition and Microstructure: Experimental Aspects\n187\n"]], ["block_15", ["the various methods were r1 9.616 i 0.603 and r2 0.0853 i 0.0239. The standard deviations\n"]], ["block_16", ["Figure 5.4\n(a) Mole fraction of methyl acrylate in copolymers with vinyl chloride as of feedstock\ncomposition, and (b) Finemann\u2014Ross plot to extract reactivity ratios, as described in Example 5.4.\n"]], ["block_17", ["Solution\n"]], ["block_18", ["out several different analyses of the same data; the values they obtained for r1 and r2 averaged over\n"]], ["block_19", ["Example 5.4\n"]], ["block_20", ["0'8:\u2014\n.\n_\n.25..\n_\u2018\nF1\n3\n\u00b0\n-\nE\nE\n0.7}\n.\n\u2014_\nZ\u2014\n'\n\u2014;\n"]], ["block_21", ["(a)\n\u20191\n(b)\n"]], ["block_22", ["0.9:\u2014\n.\n\u2014i\n05\n\u20182\n:\n\u00b0\n\u201c\nL\n\u2014;\n"]], ["block_23", [{"image_3": "201_3.png", "coords": [47, 470, 217, 628], "fig_type": "figure"}]], ["block_24", ["E\n\u2018\n\u201441\n\u2014f\n0'67\n\u201c\nE\n:\n"]], ["block_25", [":\nI\n=_\n_i\n0'5:\n:\n-6:-\nr]\n\u2014\no\n\u2014\n:\n2\n04\u2018liilltlillllllllrrr'l\n 'lllilllllllllllll'i \n0\n0.2\n0.4\n0.6\n0.8\n1\n-0-8\n\u20140-6\n\u20140.4\n\u20140.2\n0\n"]], ["block_26", ["171T\n[II\nill\nlrl\nIII\nII\n[II\nIII\nIII\nI_\n:\nI\nI\nI\n.1.\nr\n"]], ["block_27", [{"image_4": "201_4.png", "coords": [166, 485, 383, 633], "fig_type": "figure"}]], ["block_28", ["I\nI\n"]]], "page_202": [["block_0", [{"image_0": "202_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["In the second version of this equation c is the speed of light and )t the wavelength of the radiation.\n3.\nThe more widely separated two states are in energy, the shorter the wavelength of the radiation\nabsorbed. Transitions between electronic states have higher energies, and correspond to UV\u2014\nvisible wavelengths, whereas vibrational quantum states are more closely spaced and are\ninduced by IR radiation.\n4.\nDifferent light-absorbing groups, called chromophores, absorb characteristic wavelengths,\nopening the possibility of qualitative analysis based on the location of an absorption peak.\n5.\nIf there is no band overlap in\na spectrum, the absorbance at a characteristic\nwave\u2014\nlength is proportional to the concentration of chromophores present. This is the basis of\nquantitative analysis using spectra. With band overlap, things are more complicated but\nstill possible.\n6.\nThe proportionality between the concentration of chromophores and the measured absorbance\nis given by Beer\u2019s law (recall the discussion in Section 3.3.4):\nA=wc\n658\n"]], ["block_2", ["The difference is positive for absorbed energy.\n2.\nThe energy absorbed is proportional to the frequency of the radiation via Planck\u2019s constant\n(h=6c3x1044te:\n"]], ["block_3", ["In spite of the compounding of errors to which it is subject, the foregoing method was the\nbest procedure for measuring reactivity ratios until the analysis of microstructure became feasible.\nLet us now consider this development. Most of the experimental information concerning\ncopolymer microstructure has been obtained by modern instrumental methods. Techniques such\nas UV-visible, IR, NMR, and mass spectroscopy have all been used to good advantage in\nthis type of research. Advances in instrumentation have made these physical methods particularly\nsuitable to answer the question we pose: With what frequency do particular sequences of repeat\nunits occur in a c0polymer? The choice of the best method for answering this question is governed\nby the specific nature of the system under investigation. Few general principles exist beyond\nthe importance of analyzing a representative sample of suitable purity. Our approach is to consider\nsome specific examples. In view of the diversity of physical methods available and the\nnumber of copolymer combinations which exist, a few samples barely touch the subject. They\nwill suffice to illustrate the concepts involved, however. The simpler question\u2014What is the mole\nfraction of each repeat unit in the polymer sample?\u2014can usually be answered via the same\ninstrumental techniques.\nSpectroscopic techniques based on the absorption of UV, visible, or IR radiation depend\non the excitation from one quantum state to another. References in physical or analytical chemistry\nshould be consulted for additional details, but a brief summary will be sufficient for our purposes:\n"]], ["block_4", ["of about 6% and 28% in r1 and r2 analyzedfrom the same data indicate the hazards of this method\nfor determining r values.\n"]], ["block_5", ["5.6.2\nSpectroscopic Techniques\n"]], ["block_6", ["1.\nThe excitation energy AE re\ufb02ects the separation between the final (subscript f) and initial\n(subscript i) quantum states:\nAEz\ufb02\u2014Ei\n65$\n"]], ["block_7", ["188\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_8", ["where A is the (dimensionless) absorbance, b is the sample thickness, 0 is the chromophore\nconcentration, and e is the absorptivity. Usually quantitative measurements are facilitated by\n"]], ["block_9", ["AE=hp=h\u00a7\nGan\n"]]], "page_203": [["block_0", [{"image_0": "203_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["1.\nNuclei with an odd number of protons plus neutrons\u2014especially 1H and 13C\u2014possess magnetic\nmoments and show two quantum states (spin up and spin down) in a strong magnetic field.\n2.\nIf energy of the proper frequency is supplied, a transition between these quantum states occurs\nwith the absorption of an amount of energy equal to the separation of the states, just as in UV\u2014-\nvisible and IR absorption. For NMR the frequency of the absorbed radiation lies in the radio\nfrequency range and depends on the local magnetic field at the atom in question.\n3.\nElectrons in a molecule also have magnetic moments and set up secondary magnetic fields,\nwhich partly screen each atom from the applied field. Thus atoms in different chemical\nenvironments display resonance at slightly different magnetic fields.\n4.\nThe displacement 5 of individual resonances from that of a standard is small, and is measured\nin parts per million (ppm) relative to the applied field. These so-called chemical shifts are\ncharacteristic of a proton or carbon in a specific environment.\n5.\nThe interaction between nuclei splits resonances into multiple peaks, the number and relative\nintensity of which also assist in qualitative identification of the proton responsible for the\nabsorption. Proton splitting is most commonly caused by the interaction of protons on adjacent\ncarbons with the proton of interest. If there are m equivalent hydrogens on an adjacent carbon,\nthe proton of interest produces 771 + 1 peaks by this coupling.\n6.\nMore distant coupling is revealed in high magnetic fields. Unresolved fine structures in a field\nof one strength may be resolved at higher field where more subtle long-range influences can be\nprobed. The use of NMR spectroscopy to characterize copolymer microstructure takes advan-\ntage of this last ability to discern environmental effects that extend over the length of several\nrepeat units. This capability is extremely valuable in analyzing the stereoregularity of a\npolymer, and we shall have more to say about it in that context in Section 5.7.\n7.\nIn NMR spectroscopy the absorptivitiesare, in essence, all the same, so that the integrated area\nunder a peak is directly proportional to the number of nuclei of that type in the sample. Thus if\ndifferent repeat units have identifiably different peaks, as is almost always the case, the\nrelative abundance of each type can be extracted by peak integration without any additional\ncalibration.\n"]], ["block_2", ["NMR spectroscopy is especially useful for microstructure studies, because of the sensitivity to\nthe chemical environment of a particular nucleus. We shall consider its application to copolymers\nnow, and to questions of stereoregularity in Section 5.7. NMR has become such an important\ntechnique (actually a family of techniques) in organic chemistry that contemporary textbooks in\nthe subject discuss its principles quite thoroughly, as do texts in physical and analytical chemistry,\nso here also we note only a few pertinent highlights:\n"]], ["block_3", ["7 ,\nFor copolymers, or any other mixture of chromophores, the measured absorbance is given by\nthe sum of individual Beer\u2019s law terms:\n"]], ["block_4", ["Copolymer Composition and Microstructure: Experimental Aspects\n189\n"]], ["block_5", ["140.2) 810.2)i + 820.2)2902\n(5.610)\n"]], ["block_6", ["14011) 81011l1+ 82(A1)b02\n"]], ["block_7", ["A 81bC1+ 82bC2 -|- 831563 -|- \n(5.69)\n"]], ["block_8", ["These relations amount to a system of two equations with two unknowns, c1 and C2, which can\nbe solved in a straightforward manner.\n"]], ["block_9", ["Recalling that .9 depends on the chromophore and on the wavelength, measurements at\ndifferent wavelengths can be used to extract the concentrations of each component. For a\ncopolymer with two monomers, at least two wavelengths would be needed, and ideally they\nshould be chosen to such that if .91 is large at M, then 82 is large at A2.\n"]], ["block_10", ["calibration with standards of known concentrations, so that a, b, and various other instrumental\nparameters need not be determined individually.\n"]]], "page_204": [["block_0", [{"image_0": "204_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "204_1.png", "coords": [26, 419, 318, 623], "fig_type": "figure"}]], ["block_2", ["It is these conjugated double bonds that are the chromophores of interest in this system. What\nmakes this particularly useful is the fact that the absorption maximum for this chromophore is\ndisplaced to longer wavelengths the more conjugated bonds there are in a sequence. Qualita-\ntively, this can be understood in terms of a one-dimensional particle in a box model for which the\nenergy level Spacing is inversely proportional to the square of the length of the box. In this case\nthe latter increases with the length of the conjugated polyene system. This in turn depends on the\nnumber of consecutive butadiene repeat units in the copolymer. For an isolated butadiene\nmolecule dehalogenation produces one pair of conjugated double bonds; two adjacent butadienes,\nfour conjugated double bonds; three adjacent butadienes, six conjugated double bonds; and so on.\nSequences of these increasing lengths are expected to absorb at progressively longer wave-\nlengths.\nFigure 5.5 shows the appropriate portion of the spectrum for a copolymer prepared from a\nfeedstock for which f1 0.153. It turns out that each polyene produces a set of three bands: the\n"]], ["block_3", ["As suggested in the foregoing, the analysis for overall composition in a copolymer sample is by now a\nrelatively straightforward affair. The analysis for sequence distribution, however, is not. The primary\ndifficulty is that the energy of a particular transition, be it electronic, vibrational, or nuclear, is\ndetermined primarily by the immediate chromophore of interest, and only weakly in\ufb02uenced by\nchemically bonded neighbors. NMR offers the most promise in this respect, especially with the advent\nof higher magnetic \ufb01elds; this feature can provide sufficient resolution to detect the in\ufb02uence of repeat\nunits up to about five monomers down the chain. Nevertheless, there are cases where UV\u2014visible\nspectroscopy can help. An elegant example is the copolymer of styrene (molecule 1) and 1-chloro-1,3-\nbutadiene (molecule 2). These molecules quantitatively degrade with the loss of HCl upon heating in\nbase solution. This restores 1,3-unsaturation to the butadiene repeat unit:\n"]], ["block_4", ["Figure 5.5\nUltraviolet\u2014visible spectrum of dehydrohalogenated copolymers of styrene\u2014l-chloro-1,3-buta\u2014\ndiene. (Redrawn from Winston, A. and Wichacheewa, P., Macromolecules, 6, 200, 1973. With permission.)\n"]], ["block_5", ["5.6.3\nSequence Distribution: Experimental Determination\n"]], ["block_6", ["190\nCopolymers. Microstructure, and Stereoregularity\n"]], ["block_7", ["\u00e9\u2019\nTetrad\nTriad\nDyad\n"]], ["block_8", ["6 L-\n8g\nPentad\nEo\n4 -\n"]], ["block_9", [{"image_2": "204_2.png", "coords": [36, 191, 272, 268], "fig_type": "figure"}]], ["block_10", [{"image_3": "204_3.png", "coords": [39, 196, 256, 263], "fig_type": "molecule"}]], ["block_11", ["2 L\n"]], ["block_12", ["0\nL\nl\n_1\n_L\n1\n550\n500\n450\n400\n350\n300\n"]], ["block_13", ["3\n.\u2014\n"]], ["block_14", ["CI /\n\u2014nHC|\n\\ \\\nn\nm\nn\nm\n(5.N)\n"]], ["block_15", [{"image_4": "204_4.png", "coords": [108, 440, 304, 548], "fig_type": "figure"}]], ["block_16", ["Wavelength (nm)\n"]]], "page_205": [["block_0", [{"image_0": "205_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Mole fraction of dyads\nf1\n1 1\n12\n22\n"]], ["block_2", [{"image_1": "205_1.png", "coords": [34, 500, 162, 629], "fig_type": "figure"}]], ["block_3", ["MQMIMZ, MQMZMQ, MzMl, and MIMZMI. These can be divided into two groups of three,\ndepending on the identity ofcentral unit. Thus the center of a triad can be bracketed by two\n"]], ["block_4", ["same resonance feature in the copolymer of methyl methacrylate (M1) and acrylonitrile (M2).\nFigure 5.6 shows that 60 MHz spectrum several of these copolymers in the neighborhood of the\nmethoxy resonance. Three resonance peaks rather than one are observed. Figure 5.6 also lists the\nmethyl methacrylate content of each of these polymers. As the methyl methacrylate content\ndecreases, the peak on the right decreases and the left increases. We therefore identify the peak\non the right-hand peak with the MIMIMI sequence, the left-hand peak with MlMz, and the\npeak in the center with MIMIMQ. The MIMIM, peak occurs at the same location as in the methyl\nmethacrylate homopolymer.\nThe areas under the three peaks give the relative proportions of three sequences. In the\nfollowing example we consider some results on dyad sequences determined by comparable\nprocedures in vinylidene chloride\u2014isobutylene copolymers.\n"]], ["block_5", ["0.584\n0.68\n0.29\n\u2014\n"]], ["block_6", ["observed \ncompounds and the location and relative intensities of the peaks have been shown be \n"]], ["block_7", ["dent of these features have been identified, \ninterpreted \nunits and compared sequences of various lengths. Further consideration \n"]], ["block_8", ["monomers by of each. In \ncentral repeat unit in a differentand a characteristic proton in that repeat unit \n"]], ["block_9", ["different location,that environment. As a specific example, consider\nmethoxy group in poly(methyl methacrylate). The hydrogens in the group are magnetically\nequivalent and hence produce a single resonance at 8 3.74 ppm. Now suppose we look for the\n"]], ["block_10", ["367, and \nbands overlaps that of the triad and is not resolved. Likewise, only one band (at 473 nm) is\n"]], ["block_11", ["dyad is identified with the peaks at )1 298, 312, and 327 nm; the triad with the peaks at )t 347,\n"]], ["block_12", ["The mole fractions of various dyads in the vinylidine chloride (M1)\u2014\u2014isobutylene (M2) system\nwere determined,r by NMR spectroscopy. A selection of the values obtained are listed below, as\nwell as the compositions of the feedstocks from which the copolymers were prepared; assuming\nterminal control, evaluate r1 from each of the first three sets of data, and r2 from each of the last\nthree.\n"]], ["block_13", ["system is left for Problem through Problem 5 at the end of the chapter.\nWe now illustrate the application of NMR to gather 00polymer sequence information. Suppose\nwe consider the various triads of repeat units. There are six possibilities: MlMlMl, MIMIMQ,\n"]], ["block_14", ["0.505\n0.61\n0.36\n-\u2014\u2014\n"]], ["block_15", ["0.471\n0.59\n0.38\n\u2014\u2014\n"]], ["block_16", ["0.130\n\u2014\n0.67\n0.08\n0.121\n\u2014\n0.66\n0.10\n0.083\n\u2014\n0.64\n0.17\n"]], ["block_17", ["Copolymer Composition and Microstructure: Experimental Aspects\n191\n"]], ["block_18", ["tJ.B. Kinsinger, T. Fischer, and CW. Wilson, Polym. Lett., 5, 285 (1967).\n"]], ["block_19", ["Example 5.5\n"]]], "page_206": [["block_0", [{"image_0": "206_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "206_1.png", "coords": [10, 238, 271, 442], "fig_type": "figure"}]], ["block_2", [{"image_2": "206_2.png", "coords": [21, 37, 259, 466], "fig_type": "figure"}]], ["block_3", ["Equation 5.5.3 and Equation 5.5.5 provide the method for evaluating the r\u2019s from the data given.\nWe recognize that a 12 dyad can come about from\n1 adding to 2 as well as from 2 adding to 1;\ntherefore we use half the number of 12 dyads as a measure of the number of additions of monomer\n2 to chain end 1. Accordingly, by Equation 5.5.1,\n"]], ["block_4", ["Figure 5.6\nChemical shift (from hexamethyldisiloxane) for acrylonitrile\u2014methyl methacrylate copolymers\nof the indicated methyl methacrylate (M1) content. Methoxyl resonances are labeled as to the triad source.\n(From Chujo, R., Ubara, H., and Nishioka, A., Polym. J., 3, 670, 1972. With permission.)\n"]], ["block_5", ["Since [M1]/[M2] :f1/(1_f1)a Equation 5.5.2 can be written\n"]], ["block_6", ["Solution\n"]], ["block_7", ["192\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_8", ["N11\n2N11\n21s2\n=\nz\nand\n2\u2014\u2014\nN11+(1/2)N12\n2N11+N12\np22\n2N22 +N12\nP11\n"]], ["block_9", ["2\nrilfl/(l\u2014fill\nand\np\n=\nr2l(1\u201cf1)/f1l\n1+r1[f1/(1\u2014f1)]\n2\u20192\n1+r2l(1*~f1)/f:l\nP11\n"]], ["block_10", ["l\n|\nL\n|\n4.00\n3.90\n3.80\n3.70\n"]], ["block_11", [{"image_3": "206_3.png", "coords": [97, 65, 237, 277], "fig_type": "figure"}]], ["block_12", ["Chemical shift (ppm)\n"]], ["block_13", ["M1M1M2\n"]], ["block_14", ["33%\n"]], ["block_15", ["24%\n"]], ["block_16", ["50%\n"]]], "page_207": [["block_0", [{"image_0": "207_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["f1\nr1\nfl\n\u20192\n"]], ["block_2", [{"image_1": "207_1.png", "coords": [34, 528, 251, 636], "fig_type": "figure"}]], ["block_3", ["From\nFrom\n"]], ["block_4", ["Characterizing Stereoregularity\n193\n"]], ["block_5", ["2N1}\n=\nF|[f1/(1\u2014f1)]\n2s\n=\nr2[(1\u2014f1)/fl]\n2N11+N12\n1+?1lf1/(1-f1)]\n2N22+N12\n1+r2[(1-fI)/f11\n"]], ["block_6", ["0.584\n3.33\n0.130\n0.036\n"]], ["block_7", ["0.471\n3.48\n0.083\n0.048\nAverage\n3.38\nAverage\n0.042\n"]], ["block_8", ["We introduced the concept of Stereoregularity in Section 1.6. Figure 1.3 illustrates \nsyndiotactic, and atactic structures ofvinyl polymer in which \nfully extended chain lie, on the same side, alternating sides, \nto the backbone. It is important to appreciate the fact that these different \ncon\ufb01gurations\u2014have their origin in the bonding of the polymer, and no \nbonds-whanges in conformation-dwill convert one structure into another.\nOur discussion of stereoregularity in this chapter is primarily \nmonosubstituted ethylene repeat units. We shall represent these by\n"]], ["block_9", ["the penultimate model describes this better than the terminal model, \ncomings of the latter are not evident in the example. Problem 6 and Problem 7 at the of\nchapter also refer to this system.\n"]], ["block_10", ["0.505\n3.32\n0.121\n0.042\n"]], ["block_11", ["Particularly when rthis method for evaluating small \ngraphical analysis of composition data (compare Example 5.4 and Figure 5.4).\n"]], ["block_12", ["last example were able of tetrads of different composition \nvinylidene chloride\u2014isobutylene copolymer. Based on the longer sequences, they concluded \n"]], ["block_13", ["arising from still the authors of \n"]], ["block_14", ["these symbols, the isotactic, syndiotactic, and atactic structures \n"]], ["block_15", ["Monosubstituted ethylene\n"]], ["block_16", ["In this representation the X indicates the substituent; other bonds involve \nformalism also applies to 1,1-disubstituted ethylenes in which the substituents \n"]], ["block_17", ["5.7\nCharacterizing Stereoregularity\n"]], ["block_18", ["by Structure (5.111) through Structure (5V), reSpectively:\n"]], ["block_19", [{"image_2": "207_2.png", "coords": [39, 526, 162, 619], "fig_type": "figure"}]], ["block_20", ["+\ufb02xv\n"]], ["block_21", ["By making measurements at higher magnetic \ufb01elds, it is possible to resolve spectral features\n"]], ["block_22", ["l\nI\n"]], ["block_23", ["X\nX\n"]], ["block_24", ["X\nX\nX\nl\nI\n(5.111)\n"]], ["block_25", ["X\nX\n"]], ["block_26", ["1\nl\n"]], ["block_27", ["I\nX\n|\n"]], ["block_28", ["|\n|\n(5.1V)\n"]], ["block_29", ["i\n"]], ["block_30", ["l\nI\n(5V)\n"]]], "page_208": [["block_0", [{"image_0": "208_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["The carbon atoms carrying the substituents are not truly asymmetric, since the two chain\nsections\u2014while generally of different length\u2014are locally the same on either side of any carbon\natom, except near the ends of the chain. As usual, we ignore any uniqueness associated with\nchain ends.\nThere are several topics pertaining to stereoregularity that we shall not cover to simplify the\npresentation:\n"]], ["block_2", ["The successive repeat units in Structure (5.III) through Structure (5.V) are of two different\nkinds. If they were labeled M1 and M2, we would find that, as far as microstructure is concerned,\nisotactic polymers are formally the same as homopolymers, syndiotactic polymers are formally the\nsame as alternating copolymers, and atactic polymers are formally the same as random copoly-\nmers. The analog of block copolymers, stereoblock polymers, also exist. Instead of using M1 and\nM2 to differentiate between the two kinds of repeat units, we shall use the letters D and L as we did\nin Chapter 1.\nThe statistical nature of polymers and polymerization reactions has been illustrated at many\npoints throughout this volume. It continues to be important in the discussion of stereoregularity.\nThus it is generally more accurate to describe a polymer as, say, predominantly isotactic rather\nthan perfectly isotactic. More quantitatively, we need to be able to describe a polymer in terms of\nthe percentage of isotactic, syndiotactic, and atactic sequences.\nCertain bulk properties of polymers also re\ufb02ect differences in stereoregularity. We will see in\nChapter 13 that crystallinity is virtually impossible unless a high degree of stereoregularity is\n"]], ["block_3", ["1.\nStereoregular copolymers. We shall restrict our discussion to stereoregular homopolymers.\nComplications arising from other types of isomerism. Positional and geometrical isomerism,\nalso described in Section 1.6, will be excluded for simplicity. In actual polymers these are not\nalways so easily ignored.\n3.\nPolymerization\nof\n1,2-disubstituted\nethylenes.\nSince\nthese\nintroduce\ntwo\ndifferent\n\u201casymmetric\u201d carbons into the polymer backbone (second substituent Y), they have the\npotential to display ditacticity. Our attention to these is limited to the illustration of some\nterminology, which is derived from carbohydrate nomenclature (Structure (5.VI) through\nStructure (5.IX)).\n"]], ["block_4", ["194\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_5", [{"image_1": "208_1.png", "coords": [42, 201, 196, 510], "fig_type": "figure"}]], ["block_6", [{"image_2": "208_2.png", "coords": [46, 307, 195, 356], "fig_type": "figure"}]], ["block_7", [{"image_3": "208_3.png", "coords": [48, 355, 187, 476], "fig_type": "figure"}]], ["block_8", [{"image_4": "208_4.png", "coords": [52, 369, 189, 411], "fig_type": "figure"}]], ["block_9", ["Y\nX\nY\nX\nJ\nI\nI\nI\nI\n|\nI\n1x\nI I n\nI I I\nI\n6-\n>\n"]], ["block_10", ["Y\nX\nY\nX\nY\nX\nY\nX\n"]], ["block_11", ["X\nX\nX\nl\nI\nI\nI\nl\nI\n|\nI\n|\n|\nI\nI\nI\n|\nI\n|\n(5.VIII)\nY\nY\nY\nY\n"]], ["block_12", ["Y\nX\nY\nX\nI\nI\nI\nI\nI\nI\nI\nI\n"]], ["block_13", ["I\n|\nI\nI\n|\n|\n|\n|\n(5.VII)\nY\nX\nY\nX\n"]], ["block_14", ["I\n|\n|\nI\nl\nl\nI\n|\n(5 \nI\nI\nI\nI\n|\nI\nl\nl\n\u00b0\n)\n"]], ["block_15", ["Th reo-di-syndiotactic\n"]], ["block_16", ["Erythro-di-syndiotactic\n"]], ["block_17", ["Erythro-di-isotactic\n"]], ["block_18", ["Threo-di-isotactic\n"]]], "page_209": [["block_0", [{"image_0": "209_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["The main conclusion we wish to draw from this line of development is that the difference between\nEi* and E3\u2018 could vary widely, depending on the nature of the active center. If the active center in a\npolymerization is a free radical unencumbered by interaction with any surrounding species, we\n"]], ["block_2", ["and, since the concentration dependences are identical, the relative rate of the two processes is\ngiven by the ratio of the rate constants. This same ratio also gives the relative number of dyads\nhaving the same or different configurations:\n"]], ["block_3", ["The Arrhenius equation enables us to expand on this still further:\n"]], ["block_4", ["1.\nThese are addition polymerizations in which chain growth is propagated through an active\ncenter. The latter could be a free radical or an ion; we shall see that a coordinated intermediate\nis the more usual case.\n2.\nThe active\u2014center chain end is open to front or rear attack in general; hence the configuration\nof a repeat unit is not fixed until the next unit attaches to the growing chain.\n3.\nThe reactivity of a growing chain is, as usual, assumed to be independent of chain length. In\nrepresenting this schematically, as either DM* or LM*, the M* indicates the terminal active\ncenter, and the D or L, the penultimate units of \ufb01xed con\ufb01guration. From a kinetic point of\nView, we ignore what lies further back along the chain.\n4.\nAs in Chapter 3 and Chapter 4, the monomer is represented by M.\n"]], ["block_5", ["What is significant about these reactions is that only two possibilities exist: addition with the\nsame configuration (D \u2014> DD or L 6 LL) or addition with the opposite configuration (D ~e> DL or\nL \u2014> LD). We shall designate these isotactic (subscript i) or syndiotactic (subscript s) additions,\nrespectively, and shall define the rate constants for the two steps k, and k5. Therefore the rates of\nisotactic and syndiotactic propagation become\n"]], ["block_6", ["and\n"]], ["block_7", ["gross, bulk properties provide qualitative evidence for differences in stereoregularity, but, as with\ncopolymers, itdetail that quantitatively characterizes the tacticity of a\npolymer. We shall examine the statistics of this situation in the next section, and the application\nof NMR in Section \nThe analogy between stereoregular polymers and copolymers can be extended still further. We\ncan write chemical equations for propagation reactions leading to products that differ in config-\nuration along with the rate laws. We do this without specifying anything\u2014\"at least for\nnow\u2014about the There are several things that need to be defined to do this:\n"]], ["block_8", ["present in a polymer. Since crystallinity plays such an important part in determining the mecha-\nnical properties itself in these other behaviors also. These\n"]], ["block_9", ["Characterizing \n195\n"]], ["block_10", ["RN _ k, _\nNumber dyads with same configuration\n(5 3)\nRP,S \nkS \nNumber dyads with different configurations\n'\n'\n"]], ["block_11", ["Rai k, [M*] [M]\n(5.7.1)\n"]], ["block_12", ["RP. ks[M*l[Ml\n(5.7.2)\n"]], ["block_13", ["ISO dyads\na \ufb02ewwf\u2014E\ufb02/RT\n(5.7.4)\nSyndio dyads\nF\nAS\n"]], ["block_14", ["DDM*\nLLM*\n\u2014\u2014DM* + M/\nor\n\u2014LM* + M\u201d\n(5.0)\n"]], ["block_15", ["With these definitions in mind, we can write\n"]], ["block_16", ["DLM*\n\\ LDMi\"\n"]]], "page_210": [["block_0", [{"image_0": "210_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Since it is unlikely that a polymer will possess perfect stereoregularity, it is desirable to assess\nthis pr0perty quantitatively, both to describe the polymer and to evaluate the effectiveness of\nvarious catalysts in this regard. In discussing tacticity in terms of microstructure, it has\nbecome conventional to designate a dyad as meso if the repeat units have the same configuration,\nand as racemic if the configuration is reversed. The terminology is derived from the stereochem\u2014\nistry of small molecules; its basis is seen by focusing attention on the methylene group in the\nbackbone of the vinyl polymer. This methylene lies in a plane of symmetry in the isotactic\nmolecule [5.X],\n"]], ["block_2", ["ki/k,s 0.63 at 60\u00b0C. The preference for syndiotactic addition is greater than this (i.e., Ef\u2018 E3 is\nlarger) in some systems, apparently because there is less repulsion between substituents when they\nare staggered in the transition state. In all cases, whatever difference in activation energies exists\nmanifests itself in product composition to a greater extent at low temperatures. At high temper-\natures small differences in E* value are leveled out by the high average thermal energy available.\nThe foregoing remarks refer explicitly to free-radical polymerizations. If the active center is\nsome kind of associated species\u2014an ion pair or a coordination complex\u2014then predictions based\non unencumbered intermediates are irrelevant. It turns out that the Ziegler\u2014Natta catalysts\u2014which\nwon their discoverers the Nobel Prize\u2014apparently operate in this way. The active center of the\nchain coordinates with the catalyst in such a way as to block one mode of addition. High levels of\nstereoregularity are achieved in this case. Although these substances also initiate the polymeriza-\ntion, the term catalyst is especially appropriate in the present context, since the activation energy\nfor one mode of addition is dramatically altered relative to the other by these materials. We shall\ndiscuss the chemical makeup of Ziegler\u2014Natta catalysts and some ideas about how they work in\nSection 5.10. For now it is sufficient to recognize that these catalysts introduce a real bias into\nEquation 5.7.4 and thereby favor one pattern of addition.\nIn the next section we take up the statistical description of various possible sequences.\n"]], ["block_3", ["would expect Ef\" E;k to be small. Experiment confirms this expectation; for vinyl chloride it is on\nthe order of 1.3 kJ mol\". Thus at the temperatures usually encountered in free-radical polymer-\nizations (ca. 60\u00b0C), the exponential in Equation 5.7.4 is small and the proportions of isotactic\nand syndiotactic dyads are roughly equal. This is the case for poly(vinyl chloride), for which\n"]], ["block_4", ["and thereby defines a meso (subscript m) structure as far as the dyad is concerned. Considering\nonly the dyad, we see that these two methylene protons are in different environments. Therefore\neach will show a different chemical shift in an NMR spectrum. In addition, each proton splits the\nresonance of the other into a doublet, so a quartet of peaks appears in the spectrum. Still\nconsidering only the dyad, we see that the methylene in a syndiotactic grouping [5.XI] contains\ntwo protons in identical environments:\nX i\n+0~|~\n(5.XI)\n.L x\n"]], ["block_5", ["These protons show a single chemical shift in the NMR spectrum. This is called a racemic\n(subscript r) structure, since it contains equal amounts of D and L character. In the next section\nwe shall discuss the NMR spectra of stereoregular polymers in more detail.\n"]], ["block_6", ["196\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_7", ["5.8\nA Statistical Description of Stereoregularity\n"]], ["block_8", ["X\nIII\nX\n+9+\n(5.X)\n"]], ["block_9", ["H\n"]]], "page_211": [["block_0", [{"image_0": "211_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "211_1.png", "coords": [20, 358, 301, 489], "fig_type": "figure"}]], ["block_2", [{"image_2": "211_2.png", "coords": [33, 320, 284, 575], "fig_type": "figure"}]], ["block_3", ["A Statistical Description of\n197\n"]], ["block_4", ["modes, respectively, then pm + pr possibilities. \nthe analog of pi,- for copolyrners; hence, by analogy with Equation 5.5.1, this equals the fraction of\nisotactic kinetic approach of the last \n"]], ["block_5", ["the rate rates of iso and syndio \n"]], ["block_6", ["This expression is the equivalent of Equation 5.5.2 for copolymers.\nThe system of notation we have defined can readily be extended to sequences of greater length.\nTable 5.6 illustrates how either In or r dyads can be bracketed by two additional repeat units to\nform a tetrad. Each of the outer units is either m or r with respect to the unit it is attached to, so the\nmeso dyad generates three tetrads. Note that the tetrads mmr and rmm are equivalent and are not\ndistinguished. A similar set of tetrads is generated from the r dyad.\nThe same system of notation can be extended further by focusing attention on the backbone\nsubstituents rather than on the methylenes. Consider bracketing a center substituent with a pair of\nmonomers in which the substituents have either the same or opposite configurations as the central\nsubstituent. Thus the probabilities of the resulting triads are obtained from the probabilities of the\nrespective m or r additions. The following possibilities exist:\n"]], ["block_7", ["Table 5.6\nThe Splitting of M680 and Racemic Dyads into Six Tetrads\n"]], ["block_8", ["3\nI\nA\n"]], ["block_9", ["Pm \nki +ks\n(5.8.1)\n"]], ["block_10", ["If we define pm and pr as the probability of addition occurring in the meso and racemic\n"]], ["block_11", ["X\nXX\nX\n"]], ["block_12", ["r\nlmlrlr'\nX\nx\n"]], ["block_13", [{"image_3": "211_3.png", "coords": [72, 492, 323, 688], "fig_type": "figure"}]], ["block_14", [{"image_4": "211_4.png", "coords": [108, 528, 295, 653], "fig_type": "figure"}]], ["block_15", [{"image_5": "211_5.png", "coords": [112, 334, 296, 694], "fig_type": "figure"}]], ["block_16", ["XX \nlrlmlrl\nX\nx\nXX\nr \n|m|r|m|\nXX\n"]], ["block_17", ["3..\u2014\n3\u2014\u2014\n"]], ["block_18", ["X\nX\n"]]], "page_212": [["block_0", [{"image_0": "212_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "212_1.png", "coords": [22, 588, 269, 660], "fig_type": "figure"}]], ["block_2", ["These triads can also be bracketed by two more units to generate 10 different pentads following\nthe pattem established in Table 5.6. It is left for the reader to verify this number by generating the\nvarious structures.\nThe probabilities of the various dyad, triad, and other sequences that we have examined have all\nbeen described by a single probability parameter pm. When we used the same kind of statistics for\ncopolymers, we called the situation one of terminal control. We are considering similar statistics\nhere, but the idea that the stereochemistry is controlled by the terminal unit is inappropriate. The\nactive center of the chain end governs the chemistry of the addition, but not the stereochemistry.\nEquation 5.7.1 and Equation 5.7.2 merely state that an addition must be of one kind or another, but\nthat the rates are not necessarily identical.\nA mechanism in which the stereochemistry of the growing chain does exert an influence on the\naddition might exist, but at least two repeat units in the chain are required to define any such\nstereochemistry. Therefore this possibility is equivalent to the penultimate mechanism in copoly-\nmers. In this case the addition would be described in terms of conditional probabilities, just as\nEquation 5.5.20 does for copolymers. Thus the probability of an isotactic triad controlled by the\nstereochemistry of the growing chain would be represented by the reaction.\n"]], ["block_3", ["and described by the probability\n"]], ["block_4", ["2.\nA syndiotactic triad (5.XIII) is generated by two successive racemic additions:\n"]], ["block_5", ["3.\nA heterotactic triad (5.XIV) is generated by mr and rm sequences of additions:\n"]], ["block_6", ["198\nCopolymers, Microstructure. and Stereoregularity\n"]], ["block_7", ["1.\nAn isotactic triad (5.XII) is generated by two successive meso additions:\n"]], ["block_8", [{"image_2": "212_2.png", "coords": [44, 181, 144, 225], "fig_type": "figure"}]], ["block_9", ["Pcontrol Pu/m\n(So-85)\n"]], ["block_10", ["X\nX\nXIXIX\nq_l_t.MH 'I'I'\n(5.1:)\nm\nImlml\n"]], ["block_11", [{"image_3": "212_3.png", "coords": [51, 73, 149, 119], "fig_type": "figure"}]], ["block_12", ["p, p3,,\n(5.3.2)\n"]], ["block_13", ["[\u20193 =(1_pm)2\n(5.8.3)\n"]], ["block_14", ["Ph 2pm(1\u2014pm)\n(5.8.4)\n"]], ["block_15", ["The probability of the isotactic triad is\n"]], ["block_16", ["The probability of the syndiotactic triad is given by pf, which becomes\n"]], ["block_17", ["The probability of a heterotactic (subscript h) triad is\n"]], ["block_18", ["The factor 2 arises because this particular sequence can be generated in two different orders.\n"]], ["block_19", ["I\nl\n|\nI\nI\n(5.x1n)\n"]], ["block_20", ["X\nJ\n"]], ["block_21", ["i\nl\ni\n|\n1\n(5.xn)\n"]], ["block_22", ["m\nm\n"]], ["block_23", ["X\n"]], ["block_24", ["X\n"]], ["block_25", [{"image_4": "212_4.png", "coords": [97, 594, 253, 635], "fig_type": "figure"}]], ["block_26", ["1\nX\n"]], ["block_27", ["(5 .XIV)\n"]]], "page_213": [["block_0", [{"image_0": "213_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "213_1.png", "coords": [22, 395, 274, 644], "fig_type": "figure"}]], ["block_2", ["Evaluate Equation 5.8.2 through Equation 5.8.4 for pm between zero and unity; these results are\nplotted in Figure 5.7.\n"]], ["block_3", ["Use zero-order Markov statistics to evaluate the probability of isotactic, syndiotactic, and\nheterotactic triads for the series of pm values spaced at intervals of 0.1. Plot and comment on\nthe results.\n"]], ["block_4", ["is exerted such situations are known, we shall limit our discussion to the\nsimple case where the single probability pm is sufficient to describe the various additions. The\nlatter, Markov (or Bernoulli) statistics to avoid the vocabu-\nlary of terminal control. The case where the addition is in\ufb02uenced by whether the last linkage in\nthe chain is m or r is followa first-order Markov process.\nThe number of m or r linkages in an \u201crt-ad\u201d is n\u2014~l. Thus dyads are characterized by a single\nlinkage (either In or r), triads by two linkages (either mm, mr, or rr), and so forth. The m and r\nnotation thus reduces by l order of the description from what is obtained when the repeat units\nthemselves are described. reason the terminal control mechanism for copolymers is a first-\norder Markov process and model is a second\u2014order Markov process. Note that\nthe compound probabilities which describe the probability of an n-ad in terms of pm are also of\norder n\u2014l. In the following example we calculate the probability of various triads on the basis\nof zero\u2014order Markov statistics.\n"]], ["block_5", ["ptriad\n\"\n-\n"]], ["block_6", ["A Statistical \n199\n"]], ["block_7", ["where the conditional is the probability of an m addition, given the fact ofa prior\nm addition. As with copolymers, triads must be considered in order to test whether the simple\nstatistics tobe examined to whether stereochemical \n"]], ["block_8", ["Figure 5.7\nFractions of iso, syndio, and hetero triads as a function of pm, calculated assuming zero\u2014order\nMarkov statistics in Example 5.6.\n"]], ["block_9", ["Example 5.6\n"]], ["block_10", ["Solution\n"]], ["block_11", ["0.8 \nps\nPi\n\u2014\n"]], ["block_12", ["0.5 \n\u2014\n"]], ["block_13", ["l'\n._\n0.4 F\n\u2014\n"]], ["block_14", ["0.2 \n._\n"]], ["block_15", ["0\n"]], ["block_16", ["1\nI\nI\nI\nI\nI\nI\nl\nI\nI\nI\nl\nI\nI\nI\nI\nI\n"]], ["block_17", ["0\n0.2\n0.4\n0 6\n0 8\n1\npm\n"]], ["block_18", ["_\nPh\n-\n"]], ["block_19", ["F\n"]], ["block_20", ["I\nI\nI\nI\nl\nl\nl\n1\nl\ni\nl\nl\nI\nl\nI\n"]], ["block_21", ["'\u2014\n"]]], "page_214": [["block_0", [{"image_0": "214_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "214_1.png", "coords": [26, 35, 196, 214], "fig_type": "figure"}]], ["block_2", ["0\n0\n0\n0\n0.1\n0.01\n0.81\n0.18\n0.2\n0.04\n0.64\n0.32\n0.3\n0.09\n0.49\n0.42\n0.4\n0.16\n0.36\n0.48\n0.5\n0.25\n0.25\n0.50\n0.6\n0.36\n0.16\n0.48\n0.7\n0.49\n0.09\n0.42\n0.8\n0.64\n0.04\n0.32\n0.9\n0.81\n0.01\n0.18\n1.0\n1.0\n0\n0\n"]], ["block_3", ["200\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_4", ["1.\nThe probabilities give the fractions of the three different types of triads in the polymer.\n2.\nIf the fractions of triads could be measured, they either would or would not lie on a single\nvertical line in Figure 5.7. If they did occur at a single value ofpm, this would not only give the\nvalue of pm (which could be obtained from the fraction of one kind of triad), but would also\nprove the statistics assumed. If the fractions were not consistent with a single pm value, higher-\norder Markov statistics are indicated.\n3.\nThe fraction of isotactic sequences increases as pm increases, as required by the definition of\nthese quantities.\n4.\nThe fraction of syndiotactic sequences increases as Pm -\u2014> 0, which corresponds to pr -\u2014> 1.\n5.\nThe fraction of heterotactic triads is a maximum at Pm Pr 0.5 and drops to zero at either\nextreme.\n6.\nFor an atactic polymer the proportions of isotactic, syndiotactic, and heterotactic triads are\n0.25 20.251050.\n"]], ["block_5", ["pm\npi]\n\u201c\u2014m\n2pm(1_pm)\n"]], ["block_6", ["The following observations can be made from these calculations:\n"]], ["block_7", ["To investigate the triads by NMR, the resonances associated with the chain substituent are\nexamined, since Structure (5.XII) through Structure (5.XIV) show that it is these that experience\ndifferent environments in the various triads. If dyad information is sufficient, the resonances of the\nmethylenes in the chain backbone are measured. Structure (5X) and Structure (5.XI) show that\nthese serve as probes of the environment in dyads. In the next section we shall examine in more\ndetail how this type of NMR data is interpreted.\n"]], ["block_8", ["It is not the purpose of this book to discuss in detail the contributions of NMR spectroscopy to\nthe determination of molecular structure. This is a specialized field in itself and a great deal\nhas been written on the subject. In this section we shall consider only the application of NMR\nto the elucidation of stereoregularity in polymers. Numerous other applications of this powerful\ntechnique have also been made in polymer chemistry, including the study of positional and\ngeometrical isomerism (Section\n1.6) and copolymers (Section 5.7). We shall also make no\nattempt to compare the NMR spectra of various different polymers; instead, we shall examine\nprimarily the NMR spectra of different poly(methyl methacrylate) preparations to illustrate the\ncapabilities of the method using the first system that was investigated by this technique as\nthe example.\nFigure 5.8 shows the 60 MHz spectra of poly(methy1 methacrylate) prepared with different\ncatalysts so that predominantly isotactic, syndiotactic, and atactic products are formed. The three\nspectra in Figure 5.8 are identified in terms of this predominant character. It is apparent that the\n"]], ["block_9", ["5.9\nAssessing Stereoregularity by Nuclear Magnetic Resonance\n"]]], "page_215": [["block_0", [{"image_0": "215_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "215_1.png", "coords": [30, 36, 281, 343], "fig_type": "figure"}]], ["block_2", ["1.\nHydrogens of the methylene group in the backbone of the poly(methy1 methacrylate) produce\na single peak in a racemic dyad, as illustrated by Structure (5.XIII).\n2.\nThe same group of hydrogens in a meso dyad (5.X) produces a quartet of peaks: two different\nchemical shifts, each split into two by the two hydrogens in the methylene.\n3.\nThe peaks centered at 5 1.84 ppm\u2014a singlet in the syndiotactic and a quartet in the isotactic\npolymers\u2014are thus identified with these protons. This provides an unambiguous identification\nof the predominant stereoregularity of these samples.\n4.\nThe features that occur near 5 1.0 ppm are associated with the protons of the (rt-methyl\ngroup. The location of this peak depends on the configurations of the nearest neighbors.\n5.\nWorking from the methylene assignments, we see that the peak at 5 1.22 ppm in the isotactic\npolymer arises from the methyl in the center of an isotactic triad, the peak at 5 0.87 ppm\nfrom a syndiotactic triad, and the peak at 5 1.02 ppm from a homotactic triad.\n6.\nThe peak at 5 3.5 ppm is due to the methoxy group.\n"]], ["block_3", ["Figure 5.8\nNuclear magnetic resonance spectra of three poly(methyl methacrylate) samples. Curves are\nlabeled according to the predominant tacticity of samples. (From McCall, D.W. and Slichter, W.P., in Newer\nMethods of Polymer Characterization, Ke, B. (Ed), Interscience, New York, 1964. With permission.)\n"]], ["block_4", ["Once these assignments are made, the areas under the various peaks can be measured to\ndetermine the various fractions:\n"]], ["block_5", ["spectra are quite different, especially in the range of 5 values between\n1 and 2 ppm. Since the\natactic polymer has the least regular structure, we concentrate on the other two to make the\nassignment of the spectral features to the various protons.\nSeveral observations from the last section provide the basis of interpreting these spectra:\n"]], ["block_6", ["Assessing \n201\n"]], ["block_7", ["1.\nThe area under the methylene peaks is proportional to the dyad concentration: The singlet\ngives the racemic dyads and the quartet gives the meso dyads.\n"]], ["block_8", ["1\n1\nl\n5\n4\n3\n2\n5 (ppm)\n"]], ["block_9", [{"image_2": "215_2.png", "coords": [53, 50, 270, 227], "fig_type": "figure"}]], ["block_10", ["Syndiotactic\n"]], ["block_11", ["lsotactic\n"]], ["block_12", ["Atactic\n"]], ["block_13", ["er-\nOl-\n"]]], "page_216": [["block_0", [{"image_0": "216_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Source: Data from McCall, D.W. and Slichter, WP. in Newer Methods of Potymer Characterization, Ke, B. (Ed),\nInterscience, New York, 1964.\n"]], ["block_2", ["Table 5.7\nThe Fractions of Meso and Racemic Dyads and Iso, Syndio, and Hetero Triads\nfor the Data in Figure 5.8\n"]], ["block_3", ["Atactic\n0.22\n0.78\n0.07\n0.55\n0.38\nSyndiotactic\n0.17\n0.83\n0.04\n0.68\n0.28\nIsotactic\n0.87\n0. 13\n0.89\n0.04\n0.07\n"]], ["block_4", ["Sample\nMeso\nRacemic\nIso\nSyndio\nHetero\n"]], ["block_5", ["1.\nThe number of isotactic sequences containing ni iso repeat units is Nni.\n2.\nThe number of syndiotactic sequences containing as syndio repeat units is \ufb01lms.\n"]], ["block_6", ["The spectra shown in Figure 5.8 were early attempts at this kind of experiment, and the\nmeasurement of peak areas in this case was a rather subjective affair. We shall continue with an\nanalysis of these spectra, even though improved instrumentation has resulted in greatly enhanced\nspectra. One development that has produced better resolution is the use of higher magnetic fields.\nAs the magnetic field increases, the chemical shifts for the various features are displaced propor-\ntionately. The splitting caused by spin\u2014spin coupling, on the other hand, is unaffected. This can\nproduce a considerable sharpening of the NMR spectrum. Other procedures such as spin decoup-\nling, isotopic substitution, computerized stripping of superimposed spectra, and 13C-NMR also\noffer methods for identifying and quantifying NMR spectra.\nTable 5.7 lists the estimated fractions of dyads of types m and r and the fractions of triads of\ntypes i, s, and h from Figure 5.8. These fractions represent the area under a specific peak (or four\npeaks in the case of the meso dyads) divided by the total area under all of the peaks in either the\ndyad or triad category. As expected for the sample labeled isotactic, 89% of the triads are of type i\nand 87% of the dyads are of type m. Likewise, in the sample labeled syndiotactic, 68% of the triads\nare s and 83% of the dyads are r.\nThe sample labeled atactic in Figure 5.8 was prepared by a free\u2014radical mechanism and is\nexpected to follow zero-order Markov statistics. As a test for this, we examine Figure 5.7 to see\nwhether the values of 19,, p3, and ph, which are given by the fractions in Table 5.7, agree with a\nsingle set of Pm values. When this is done, it is apparent that these proportions are consistent with\nthis type of statistics within experimental error and that pm \u201c=V 0.25 for poly(methyl methacrylate).\nUnder the conditions of this polymerization, the free-radical mechanism is biased in favor of\nsyndiotactic additions over isotactic additions by about 3:1, according to Equation 5.8.1. Presum-\nably this is due to steric effects involving the two substituents on the oc-carbon.\nWith this kind of information it is not difficult to evaluate the average lengths of isotactic\nand syndiotactic sequences in a polymer. As a step toward this objective, we define the\nfollowing:\n"]], ["block_7", ["3.\nSince isotactic and syndiotactic sequences must alternate, it follows that:\n"]], ["block_8", ["2.\nThe area under one of the methyl peaks is proportional to the concentration of the correspond-\ning triad.\n3.\nIt is apparent that it is not particularly easy to determine the exact areas of these features when\nthe various contributions occur together to any significant extent. This is clear from the atactic\nspectrum, in which slight shoulders on both the methylene and methyl peaks are the only\nevidence of meso methylenes and iso methyls.\n"]], ["block_9", ["202\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_10", [{"image_1": "216_1.png", "coords": [46, 489, 133, 523], "fig_type": "molecule"}]], ["block_11", ["EN!\u201c :a\n(5.9.1)\n"]], ["block_12", ["Dyads\nTriads\n"]]], "page_217": [["block_0", [{"image_0": "217_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["4,\nThe number of iso triads in a sequence of n, iso repeat units is ni\u2014 l, and the number of syndio\ntriads in a sequence of ns syndio repeat units is rig\u20141. We can verify these relationships by\nexamining a specific chain segment:\n"]], ["block_2", ["Assessing \n203\n"]], ["block_3", ["5.\nThe number of racemic dyads in a sequence is the same as the number of syndiotactic units (is.\nThe number of meso dyads in a sequence is the same as the number of iso units In. These can\nalso be verified from structure above.\n"]], ["block_4", ["Use the dyad and triad fractions in Table 5.7 to calculate the average lengths \nsyndiotactic sequences for the polymers of Figure 5.8. Comment on the results.\n"]], ["block_5", ["Example 5.7\n"]], ["block_6", [{"image_1": "217_1.png", "coords": [39, 222, 252, 276], "fig_type": "molecule"}]], ["block_7", [{"image_2": "217_2.png", "coords": [46, 411, 154, 455], "fig_type": "molecule"}]], ["block_8", ["In this example both the iso and syndio sequences consist of eight repeat units, with seven\ntriads in each. The repeat unit marked is counted as part of each type of triad, but is itself the\ncenter of a hetero triad.\n"]], ["block_9", ["In this equation the summations are over all values of n of the specified type. Also remember\nthat the V\u2019s and n\u2019s in this discussion (with subscript i or s) are de\ufb01ned differently from the 12\u2019s\nand n\u2019s defined earlier in the chapter for 00polymers. Using Equation 5.9.1 and remembering\nthe definition of an average provided by Equation 1.7.7, we see that Equation 5.9.2 becomes\n"]], ["block_10", ["where the overbar indicates the average length of the indicated sequence.\n"]], ["block_11", ["With these definitions in mind, we can immediately write expressions for the ratio of the\ntotal number of iso triads, vi, to the total number of syndio triads, vs:\n"]], ["block_12", [" 2 _ \n(5 9 2)\nVs\u2014Za(ns_1)mZN\u201d5(nS)\u2014ZNHS\n.\nI\n"]], ["block_13", ["5 \n(5.9.3)\n"]], ["block_14", ["Equation 5.9.3 and Equation 5.9.4 can be solved simultaneously for E, and rig in terms of the\ntotal number of dyads and triads:\n"]], ["block_15", ["\u2014DDLDLDLDLD*DDDDDDDDL-\n"]], ["block_16", ["A similar result can be written for the ratio of the total number (12) of dyads of the two types\n(m and I), using item (5) above:\n"]], ["block_17", ["and\n"]], ["block_18", ["Use of these relationships is illustrated in the following example.\n"]], ["block_19", ["_ _\n1 Vi/Vs\nHi \n"]], ["block_20", ["a, \n1 \nVi/VS\n(5.9.6)\n(Vim/VI) (Vi/Vs)\n"]], ["block_21", ["VS\n\ufb01s\ufb02l\n"]], ["block_22", ["pm\n2mm # \ufb01r\n(5.9.4)\n:7.\n\u2014 22mm.) \u201c\n(is\n"]], ["block_23", ["1\u2014(Vi/Vs)(Vr/Vm)\n(59.5)\n"]]], "page_218": [["block_0", [{"image_0": "218_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "218_1.png", "coords": [30, 472, 349, 601], "fig_type": "figure"}]], ["block_2", ["Figure 5.9\n13C-NMR assignments for polypropylene. (From Bruce, MD. and Waymouth, R.M., Macro-\nmolecules, 31, 2707, 1998. With permission.)\n"]], ["block_3", ["Thus the fractions in Table 5.7 can be substituted for the 12\u2019s in Equation 5.9.3 and Equation 5.9.4.\nThe values of \ufb01i and fl, so calculated for the three polymers are:\n"]], ["block_4", ["We conclude this section via Figure 5.9, which introduces the use of l3C-NMR obtained at 100\nMHz for the analysis of stereoregularity in polypropylene. This spectrum shows the carbons\non the pendant methyl groups for an atactic polymer. Individual peaks are resolved for all the\npossible pentad sequences. Polypropylene also serves as an excellent starting point for the next\nsection, in which we examine some of the catalysts that are able to control stereoregularity in\nsuch polymers.\n"]], ["block_5", ["This analysis adds nothing new to the picture already presented by the dyad and triad probabilities.\nIt is somewhat easier to visualize an average sequence, however, although it must be remembered\nthat the latter implies nothing about the distribution of sequence lengths.\n"]], ["block_6", ["Atactic\n1 .59\n5.64\nSyndiotactic\n1.32\n6.45\nIsotactic\n9.14\n1.37\n"]], ["block_7", ["Since the total numbers of dyads and triads always occur as ratios in Equation 5.9.3 and Equation\n5.9.4, both the numerators and denominators of these ratios can be divided by the total number of\ndyads or triads to convert these total numbers into fractions, i.e.,\n"]], ["block_8", ["204\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_9", ["Solution\n"]], ["block_10", [{"image_2": "218_2.png", "coords": [39, 440, 189, 607], "fig_type": "figure"}]], ["block_11", ["I\nl\nI\nI\nI\nI\nI\nF\nI\nI\nI\nI\nl\nI\nI\nl\nl\u2014r\nI\u2014l\nI\u2014I\n1\u2014l\n1\nI_|\n22\n21.5\n21\n20.5\n20\n19.5\n"]], ["block_12", [{"image_3": "218_3.png", "coords": [46, 363, 341, 637], "fig_type": "figure"}]], ["block_13", ["Vi/Vs (Vi/Vtot)/(Vs/Vtot) Pi/Ps\n"]], ["block_14", ["\u201di\n\u201ds\n"]], ["block_15", [{"image_4": "218_4.png", "coords": [112, 393, 294, 588], "fig_type": "figure"}]], ["block_16", [{"image_5": "218_5.png", "coords": [118, 379, 251, 418], "fig_type": "molecule"}]]], "page_219": [["block_0", [{"image_0": "219_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Since the coordination almost certainly involves the transition metal atom, there is a resemblance\nhere to anionic polymerization. The coordination is an important aspect of the present picture,\nsince it is this feature that allows the catalyst to serve as a template for stereoregulation.\nThe assortment of combinations of components is not the only variable to consider in describing\nZiegler\u2014Natta catalysts. Some other variables include the following:\n"]], ["block_2", ["Among other possibilities in these reactions, these free radicals can initiate ordinary free-radical\npolymerization. The Ziegler\u2014Natta systems are thus seen to encompass several mechanisms for the\ninitiation of polymerization. Neither ionic nor free-radical mechanisms account for stereoregular-\nity, however, so we must look further for the mechanism whereby the Ziegler\u2014Natta systems\nproduce this interesting effect.\nThe stereoregulating capability of Ziegler\u2014Natta catalysts is believed to depend on a coordin-\nation mechanism in which both the growing polymer chain and the monomer coordinate with the\ncatalyst. The addition then occurs by insertion of the monomer between the growing chain and the\ncatalysts by a concerted mechanism (5.XV):\n"]], ["block_3", ["In this discussion we consider Ziegler\u2014Natta catalysts and their role in achieving stereoregularity.\nThis is a somewhat of the situation, since there are other catalysts\u2014such as phenyl\nmagnesium bromide, a Grignard reagent\u2014which can produce stereoregularity; the Ziegler\u2014Natta\ncatalysts polymers\u2014unbranched polyethylene to name one\u2014which lack\nstereoregularity. catalysts are historically the most widely used and best\u2014\nunderstood stereoregulating systems, so the loss of generality in this approach is not of great\nconsequence.\nThe fundamental Ziegler\u2014Natta recipe consists of two components: a halide or other compound\nof a transition metal from among the group IVB to VIIIB elements, and an organometallic\ncompound of metal from groups IA to IIIA. Some of the transition metal\ncompounds studied include TiCl4, TiCl3, VC14, VC13, ZrCl4, Cl'Cl3, MoCls, and CuCl. Represen-\ntative organometallics include (C2H5)3A1, (C2H5)2Mg, C4H9Li, and (C2H5)22n. These are only a\nfew of the possible compounds, so the number of combinations is very large.\nThe individual components of the Zieglerm-Natta system can separately account for the initiation\nof some forms of polymerization reactions, but not for the fact of stereoregularity. For example,\nbutyl lithium can initiate anionic polymerization (see Section 4.3) and TiCl4 can initiate cationic\npolymerization (see Section 4.5). In combination, still another mechanism for polymerization,\ncoordination polymerization, is indicated. When the two components of the Ziegler\u2014Natta system\nare present together, complicated exchange reactions are possible. Often the catalyst must \u201cage\u201d to\nattain maximum effectiveness; presumably this allows these exchange reactions to occur. Some\npossible exchange equilibria are\n"]], ["block_4", ["The organotitanium halide can then be reduced to TiCl3:\n"]], ["block_5", ["Ziegler-Natta \n205\n"]], ["block_6", ["5.10\nZiegler\u2014Natta Catalysts\n"]], ["block_7", ["H\nH\nX\n_(:3_\\/*\u2018/\\H2\n(5.XV)\n"]], ["block_8", ["2A16 e [A1(C2H5>2]+[A1(C2H5>4]\u2018\nTiCl4 + [A1(C2H5)2]+ 4:) C2H5TiC13 + [A1(C2H5)Cl]+\n(5.Q)\n"]], ["block_9", ["C2H5TiC13 \u2014> TiCl3 + Cs'\n(5.R)\n"]], ["block_10", [{"image_1": "219_1.png", "coords": [50, 549, 105, 595], "fig_type": "molecule"}]], ["block_11", ["X\nCat\n"]]], "page_220": [["block_0", [{"image_0": "220_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["1.\nCatalyst solubility. Polymerization systems may consist of one or two phases. Titanium-based\ncatalysts are the most common of the heterogeneous systems; vanadium-based catalysts are\nthe most common homogeneous systems. Since the catalyst functions as a template for the\nformation of a stereoregular product, it follows that the more extreme orienting effect of\nsolid surface (i.e., heterogeneous catalysts) is required for those monomers that interact\nonly weakly with the catalyst. The latter are nonpolar monomers. Polar monomers interact\nmore strongly with catalysts, and dissolved catalysts are able to exert sufficient control for\nStereoregularity.\n2.\nCrystal structure of solids. The a-crystal form of TiCl3 is an excellent catalyst and has been\ninvestigated extensively. In this particular crystal form of TiCl3, the titanium ions are located\nin an octahedral environment of chloride ions. It is believed that the stereoactive titanium ions\nin this crystal are located at the edges of the crystal, where chloride ion vacancies in the\ncoordination sphere allow coordination with the monomer molecules.\n3.\nTacticity of products. Most solid catalysts produce isotactic products. This is probably because\nof the highly orienting effect of the solid surface, as noted in item (1). The preferred isotactic\nconfiguration produced at these surfaces is largely governed by steric and electrostatic\ninteractions between the monomer and the ligands of the transition metal. Syndiotacticity is\nmostly produced by soluble catalysts. Syndiotactic polymerizations are carried out at low\ntemperatures, and even the catalyst must be prepared at low temperatures; otherwise specifi\u2014\ncity is lost. With polar monomers syndiotacticity is also promoted by polar reaction media.\nApparently the polar solvent molecules compete with monomer for coordination sites, and\nthus indicate more loosely coordinated reactive species.\n4.\nRate of polymerization. The rate of polymerization for homogeneous systems closely resem-\nbles anionic polymerization. For heterogeneous systems the concentration of alkylated tran-\nsition metal sites on the surface appears in the rate law. The latter depends on the particle size\nof the solid catalyst and may be complicated by sites of various degrees of activity. There is\nsometimes an inverse relationship between the degree of Stereoregularity produced by a\ncatalyst and the rate at which polymerization occurs.\n"]], ["block_2", ["The catalysts under consideration both initiate the polymerization and regulate the polymer\nformed. There is general agreement that the mechanism by which these materials exert their\nregulatory role involves coordination of monomer with the transition metal atom, but proposed\ndetails beyond this are almost as numerous and specific as the catalysts themselves. We shall return\nto a description of two specific mechanisms below. The general picture postulates an interaction\nbetween monomer and catalyst such that a complex is formed between the qr electrons of the olefin\nand the d orbitals of the transition metal. Figure 5.10 shows that the overlap between the \ufb01lled\norbitals of the monomer can overlap with vacant dx2_y2 orbitals of the metal. Alternatively, hybrid\norbitals may be involved on the metal. There is a precedent for such bonding in simple model\ncompounds. It is known, for example, that Pt2+ complexes with ethylene by forming a dsp2\nhybridnqr sigma bond and a dp hybrid~rr* pi bond. A crucial consideration in the coordination is\nmaximizing the overlap of the orbitals involved. Titanium(III) ions seem ideally suited for this\n"]], ["block_3", ["206\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_4", ["(a)\n(b)\n"]], ["block_5", [{"image_1": "220_1.png", "coords": [35, 557, 341, 648], "fig_type": "figure"}]], ["block_6", ["Figure 5.10\nPossible orbital overlaps between a transition metal and an olefin.\n"]], ["block_7", [{"image_2": "220_2.png", "coords": [43, 563, 150, 638], "fig_type": "molecule"}]], ["block_8", ["dx2 y2\n"]], ["block_9", ["a\nc\nG>\nl\n1\u201d\n\u20ac09\n050/\"Q M .\\n*\n\\6\nC (3/\n"]], ["block_10", [{"image_3": "220_3.png", "coords": [168, 556, 319, 658], "fig_type": "figure"}]], ["block_11", [{"image_4": "220_4.png", "coords": [180, 593, 339, 650], "fig_type": "molecule"}]]], "page_221": [["block_0", [{"image_0": "221_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["function; higher effective nuclear charge on the metal results in less spatial extension of d orbitals\nand diminished overlap.\nMany mechanisms have been proposed that elaborate on this picture. These are often so specific\nthat they cannot be generalized beyond the systems for which they are proposed. Two schemes that\ndo allow some generalization are presented here. Although they share certain common features,\nthese mechanisms are distinguished by the fact that one\u2014the monometallic model\u2014does not\ninclude any participation by the representative metal in the mechanism. The second\u2014the bimetallic\nmodel\u2014does assume the involvement of both metals in the mechanism.\nThe monometallic mechanism is illustrated by Figure 5.11. It involves the monomer coordin-\nating with an alkylated titanium atom. The insertion of the monomer into the titanium\u2014carbon bond\npropagates the chain. As shown in Figure 5.11 this shifts the vacancy\u2014represented by the\nsquare\u2014in the coordination sphere of the titanium to a different site. Syndiotactic regulation\noccurs if the next addition takes place via this newly created vacancy. In this case the monomer and\nthe growing chain occupy alternating coordination sites in successive steps. For the more common\nisotactic growth the polymer chain must migrate back to its original position.\nThe bimetallic mechanism is illustrated in Figure 5.12; the bimetallic active center is the\ndistinguishing feature of this mechanism. The precise distribution of halides and alkyls is not\nspelled out because of the exchanges described by Reaction (5Q). An alkyl bridge is assumed\nbased on observations of other organometallic compounds. The coordination of the olefin with\nthe titanium is followed by insertion of the monomer into the bridge to propagate the reaction.\nAt present it is not possible to determine which of these mechanisms or their variations most\naccurately represents the behavior of Ziegler\u2014Natta catalysts. In view of the number of variables in\n"]], ["block_2", [",r\u2019\nu\n\u201c1,0\nWCIH\nMe\n\"To\n:,Cl\nMe\nMe\n,0\nol-,Ti--EI \nC'7Tui\"'\u201c=\n\u2014-\u2014--\nCI-,Ti---~CH2 \u2014-\nC|7Ti\n\u2014\u00bb-\nCI-,Ti--n\n0'83.\nC'CI\nH20\n0'63.\n0' ('3.\n0'33\n"]], ["block_3", ["Ziegler\u2014Natta \n207\n"]], ["block_4", ["Polypropylene polymerized with triethyl aluminum and titanium trichloride \ncontain various kinds of chain ends. Both terminal vinylidene unsaturation and aluminum-bound\nchain ends have been identified. Propose two termination reactions to account \ntions. Do the termination reactions allow any discrimination between the monometallic \nbimetallic propagation mechanisms?\n"]], ["block_5", ["these catalyzed polymerizations, both mechanisms may be valid, each for different specific \nIn the following example the termination step of coordination polymerizations is considered.\n"]], ["block_6", ["Figure 5.11\nMonometallic mechanism. The square indicates a vacant ligand site.\n"]], ["block_7", ["HTMe\ng\nH+\nMe\nH\nMe\nMew/g\nI\n\u2019,.C\"'2\n3515+ H202\nH2C/\nEC\"\n9\n/Ti:~\n:\u2018AI/\n\u2014\u20141-\u00bb\n/Ti\\\n:Al/\ni/\n\u2014\u2014\u2014I-\u2014 \\ ., x I\u201c-\nI\n\u201d\u201c- ,\u2019\n\\\nI\n\u2018\n, \n\\\nTl\u201c\n\u2019Al\n/TI\u00ab.\n:AI\nR\nR\n/\n\u201cFl\u2019\n\\\nTR\u201d\n"]], ["block_8", ["Example 5.8\n"]], ["block_9", ["Figure 5.12\nThe bimetallic mechanism.\n"]], ["block_10", [{"image_1": "221_1.png", "coords": [85, 564, 458, 665], "fig_type": "figure"}]], ["block_11", [{"image_2": "221_2.png", "coords": [98, 63, 440, 121], "fig_type": "molecule"}]], ["block_12", [{"image_3": "221_3.png", "coords": [139, 589, 339, 645], "fig_type": "molecule"}]], ["block_13", [{"image_4": "221_4.png", "coords": [205, 577, 448, 650], "fig_type": "molecule"}]], ["block_14", [{"image_5": "221_5.png", "coords": [207, 58, 334, 123], "fig_type": "molecule"}]], ["block_15", ["Me\n"]], ["block_16", [{"image_6": "221_6.png", "coords": [288, 57, 430, 124], "fig_type": "molecule"}]]], "page_222": [["block_0", [{"image_0": "222_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "222_1.png", "coords": [25, 212, 258, 265], "fig_type": "figure"}]], ["block_2", ["These reactions appear equally feasible for titanium in either the monometallic or bimetallic\nintermediate. Thus they account for the different types of end groups in the polymer, but do not\ndifferentiate between propagation intermediates. In the commercial process for the production of\npolypropylene by Ziegler\u2014Natta catalysts, hydrogen is added to terminate the reaction, so neither\nof these reactions is pertinent in this case.\n"]], ["block_3", ["The discussion in the preceding section indicates that Ziegler\u2014Natta catalysts represent a rather\ncomplicated subject. This complexity is often re\ufb02ected in the structure of the polymers produced.\nFor example, the different ways that the two metal centers may or may not interact during an\naddition step suggest that there are, in fact, multiple catalytic sites active in a given polymerization.\nThis can lead to sites with greatly different propagation rates, different stereoselectivity, and\ndifferent propensities to incorporate any comonomers present. The net result is that polymer\nmaterials produced by Ziegler\u2014Natta catalysts, eSpecially under commercial conditions, tend to\nbe highly heterogeneous at the molecular level. A broad strategy to overcome this limitation is\nbased on the concept of a single-site catalyst, i.e., one that has a single, well-defined catalytic\ngeometry that can control the desired aspect of prOpagation. In this section, we brie\ufb02y consider\nsome examples of such catalysts for stereochemical control in the polymerization of a-olefins. We\nbegin with a little more consideration of catalysis in general.\nThe majority of catalysts in commercial use are heterogeneous. In this usage, the term\nheterogeneous means that the phase of the catalyst (e.g., solid) is distinct from that of the reagents\nand products (usually gases and liquids). When the catalyst is a relatively small molecule, it is\nretained in the solid phase by immobilization on some kind of inert, robust support. The reaction of\ninterest therefore takes place at the solid\u2014liquid or solid\u2014gas interface. The fact of immobilization\ncan itself contribute to the multiple site nature of heterogeneous catalysts, for example by exposing\ndifferent faces of the catalytically active metal center, by restricting accessibility of reagents to\ncatalyst particles deep within a porous support, and by presenting a distribution of different cluster\nsizes of catalytic particles. Given these disadvantages, one might ask why heterogeneous catalysis\nis the norm. The answer is simple: It is much easier to separate (and possibly regenerate)\nheterogeneous catalysts from products and unreacted reagents. Note that if the activity of a catalyst\nis sufficiently high (in terms of grams polymer produced per gram catalyst employed), then\n"]], ["block_4", ["The transfer of a tertiary hydrogen between the polymer chain and a monomer can account for the\nvinylidene group in the polymer:\n"]], ["block_5", ["A reaction analogous to the alkylation step of Reaction (4.Q) can account for the association of an\naluminum species with chain ends:\n"]], ["block_6", ["208\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_7", ["5.11\nSingle-Site Catalysts\n"]], ["block_8", ["Solution\n"]], ["block_9", [{"image_2": "222_2.png", "coords": [37, 213, 234, 261], "fig_type": "molecule"}]], ["block_10", ["Ti/W\u2018J\n+ MeVAIVMe \u2014\u2014\u2014\u2014)- Me\\./\nH\nMe\n:W\nMeH\n"]], ["block_11", ["+\n"]], ["block_12", [{"image_3": "222_3.png", "coords": [156, 210, 356, 273], "fig_type": "molecule"}]], ["block_13", [{"image_4": "222_4.png", "coords": [197, 224, 352, 259], "fig_type": "molecule"}]], ["block_14", ["Me\n./\\/Me\n(5.81\u00bb)\nMe\nT'\n+ ACHQ\n"]], ["block_15", ["-->+\n(5.8a)\n"]], ["block_16", ["HMe+\nIVMe \u2014:-- \ufb02AK/AIVMe + TiAMe\n"]]], "page_223": [["block_0", [{"image_0": "223_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["as chloride.\nThe choice of metal, ligands, and design of the overall constraining geometry provide a rich\npalette from which catalysts may be designed. In general, the stereoregulation of monomer\naddition can be achieved through one of two modes. Under chain\u2014end control, the addition of a\nmonomer is influenced mostly by the configuration of the previous repeat unit, which is reminis-\ncent of the terminal model of copolymerization. To appreciate how this can happen, it is important\nto realize that the growing polymer remains bound to the metal center during the addition step.\nAlternatively, under Site control the ligand set may be chosen to provide a chiral confining\nenvironment, which exerts a dominant in\ufb02uence on the stereochemistry of addition. The symmetry\nof the catalyst is often strongly correlated with the mode and effect of stereocontrol. This is\nsummarized in Figure 5.13. Catalysts with a plane of symmetry, or CS, tend to produce syndiotactic\npolymers under site control, but either iso\u2014 or syndiotactic polymers under chain-end control.\nWhen the symmetry is C2, i.e., identical after rotation by 180D about a single axis, the addition is\n"]], ["block_2", ["However, this representation is not complete. Just as Ziegler\u2014Natta catalysts always involve a\nmixture of at least two active ingredients, single\u2014site catalysts involve another component. The\nmost common is a partially hydrolyzed trimethyl aluminum species\u2014methylaluminoxane (MAO).\nThe active center is more properly denoted [LnMP]+[X]_, where the metal site is cationic by virtue\nof being coordinatively unsaturated, and the counterion contains MAO and a displaced ligand, such\n"]], ["block_3", ["Figure 5.13\nRole of catalyst symmetry in stereocontrol. The open square represents the unsaturated site for\nmonomer addition, and the Cp rings are represented by the pendant lines. A catalyst of type (a) is isospecific\nand (b) is syndiospeci\ufb01c, when under site control; (c) and (d) can be either 130- or syndiospecific, under chain-\nend control. (From Coates, G.W., Chem. Rev., 100, 1223, 2000.)\n"]], ["block_4", ["separation and recovery may not be necessary. In contrast, single-site polymeriza-\ntion catalysts are usually they are molecularly dispersed within the reaction\nmedium. This situation leads to better defined products, and is much more amenable to detailed\nstudies of mechanism. strategies for immobilizing such catalysts are available,\nmaking them also of potential commercial interest.\nMost single-site catalysts have the general formula [LnMP], where L,., represents a set of\nligands, M is the active center, and P is the growing polymer. Furthermore, a common\nmotif is for two of the ligands to contain cyclopentadienyl (Cp) rings, which may themselves be\ncovalently linked or bridged. The example shown below (5.XVI) was one of the first such\nmetallocene systems and produces highly isotactic polypropylene.\n"]], ["block_5", ["I\nl\nI\n(a)\nP\n(b)\nP\n(C)\nP\n(d)\nP\n"]], ["block_6", ["Single-Site Catalysts\n209\n"]], ["block_7", ["C \nVol)\nV\nC 0\np\nDEM\nmaul/l\nDEM\np\nElm-M\np\n"]], ["block_8", ["Cp\nCD\nCp\nCp\nCp\nCp\n"]], ["block_9", [{"image_1": "223_1.png", "coords": [115, 533, 360, 603], "fig_type": "molecule"}]], ["block_10", [{"image_2": "223_2.png", "coords": [261, 535, 351, 599], "fig_type": "molecule"}]], ["block_11", ["(5.xv1)\n"]]], "page_224": [["block_0", [{"image_0": "224_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "224_1.png", "coords": [4, 51, 297, 126], "fig_type": "figure"}]], ["block_2", [{"image_2": "224_2.png", "coords": [19, 42, 427, 151], "fig_type": "figure"}]], ["block_3", [{"image_3": "224_3.png", "coords": [28, 62, 203, 116], "fig_type": "figure"}]], ["block_4", ["As indicated by the double arrows, the catalyst actually oscillates between two isomeric structures.\nThe structure on the left is chiral with C2 symmetry, and gives isotactic polypropylene (note that\nthe chloride ligands are not in the plane of the page). The structure on the right, however, is achiral,\n"]], ["block_5", ["iSOSpecific under site control. When a further mirror plane exists, in C2,, symmetry, chain-end\ncontrol leads to either iso- or syndiospecific addition.\nWe now illustrate these phenomena with two particular catalysts and a cartoon sketch of\nthe mechanism of monomer insertion. The monomer in question is polypropylene, the commer-\ncially most important stereogenic polyolefin and the most studied model system. However, it\nshould be noted that the \ufb02exibility of design for single-site catalysts offers the possibility of\nmore tolerance toward monomer polarity or functionality than in the Ziegler\u2014Natta analogs,\nthereby enabling stereocontrol of many different monomers or comonomers. The catalyst\n(5.XVI) has C2 symmetry and is isospecific under site control. The mechanism is illustrated in\nFigure 5.14, where for simplicity the Cp-containing ligands are represented by horizontal lines.\nThe polymer chain is bound to the metal through the unsubstituted backbone carbon and the\norientation of the incoming monomer is in\ufb02uenced by the location of the Cp ligand. In the\ntransition state the unsubstituted carbon of the new monomer coordinates with the metal and will\nbecome the new terminal carbon of the growing chain. A key role is thought to be played by a\nso-called \u201ca-agostic\u201d interaction between the metal and the hydrogen on the terminal carbon of\nthe polymer chain, which stabilizes the particular geometry of the transition state. After the\nincorporation of the monomer, the polymer chain (or a last few repeat units thereof) has\n\u201c\ufb02ipped\u201d to the other side of the metal center, in a process which is often compared to the\naction of a windshield wiper.\nIn contrast, the following zirconocene (5.XVII) is syndiospecific, consistent with its C,\nsymmetry. The mechanism is analogous to that illustrated in Figure 5.14, except that the inversion\nof the position of the bulky ligand inverts the preferred orientation of the incoming monomer.\n"]], ["block_6", ["The range of possibilities afforded by this class of catalysts is vast. As one last example,\nconsider the following zirconocene (5.XVIII), developed by Coates and Waymouth [2]:\nm=6\u201d\nZrCI2\nZrCI2\n(5.XVIII)\n"]], ["block_7", ["210\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_8", ["Figure 5.14\nProposed mechanism of iSOSpecific polymerization of polypropylene. (From Coates, G.W.,\nChem. Rev, 100, 1223, 2000.)\n"]], ["block_9", ["c\nH /\nM~.\nH\nH H\nH\nH\nH\nH/L\u201c/H M\nH\nHH/\u00e9\nW\u201c\nJk\nH\nH\n|\u2014\n\u2019\nMe\nMe\nMe HMe\nH\nMe\u2014H\n"]], ["block_10", [{"image_4": "224_4.png", "coords": [37, 535, 314, 618], "fig_type": "figure"}]], ["block_11", [{"image_5": "224_5.png", "coords": [37, 450, 110, 502], "fig_type": "molecule"}]], ["block_12", [{"image_6": "224_6.png", "coords": [41, 575, 146, 617], "fig_type": "molecule"}]], ["block_13", [{"image_7": "224_7.png", "coords": [170, 56, 289, 120], "fig_type": "figure"}]], ["block_14", [{"image_8": "224_8.png", "coords": [191, 52, 429, 123], "fig_type": "figure"}]], ["block_15", [{"image_9": "224_9.png", "coords": [277, 46, 423, 138], "fig_type": "molecule"}]], ["block_16", [{"image_10": "224_10.png", "coords": [287, 75, 411, 116], "fig_type": "molecule"}]], ["block_17", ["(5.xvn)\n"]]], "page_225": [["block_0", [{"image_0": "225_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["This chapter has covered a broad range of issues relating to the structure of polymer chains at the\nlevel of a few repeat units. The two main topics have been copolymerization and stereoregularity.\nThese topics share many features in common, including (i) the importance of the relative reactivity\nof a growing chain end to addition of a particular monomer, or a monomer in a particular\nconfiguration; (ii) the use of statistics in describing composition, average sequence lengths, and\nsequence length distribution; (iii) the central role of spectroscopic methods, and especially NMR,\nin characterizing structural details.\n"]], ["block_2", ["and actually leads to random stereochemistry, i.e., atactic polypropylene. Now, consider the\ninteresting situation the of monomer insertion is more rapid than the rate of exchange\nbetween factor of 20. suchthe \n\u201cstereoblock copolymer,\u201d with alternating sequences of isotactic and atactic polypropylene, where\nthe average sequence length would be about 20. Such a polymer has some very appealing\nproperties. The isotactic blocks can crystallize, as will be discussed in detail in Chapter 13,\nwhereas the atactic blocks cannot. The result is that for temperatures above the glass transition\nof the atactic block (about \u201410\u00b0C, see Chapter 12) but below the melting temperature of the\nstereoregular block (about 140\u00b0C) the material acts as a crosslinked elastomer (see Chapter 10).\nThe crystallites tie the different molecules together, imparting mechanical strength, but the atactic\nblocks can be stretched appreciably without breaking, like a rubbery material. The mechanical\nresponse is sensitive to the relative lengths of the two blocks, which can be tuned through\nmonomer concentration and polymerization temperature. The result is an appealing situation in\nwhich an inexpensive monomer can be used to produce a variety of different products by\nstraightforward modi\ufb01cation to the reaction conditions.\n"]], ["block_3", ["1.\nThe key parameters in copolymerization are the reactivity ratios, which in\ufb02uence the relative\nrates at which a given radical will add the same monomer versus a comonomer. Thus a given\nreactivity ratio is specific to a particular pair of monomers, and copolymerization of two\nmonomer system requires specification of two reactivity ratios.\n2.\nThe copolymerization equation relates the mole fraction of monomers in polymer to the\ncomposition of the feedstock via the reactivity ratios. Different classes of behavior may be\nassigned based on the product of the reactivity ratios, including an \u201cideal\u201d copolymerization\nwhen the two reactivity ratios are reciprocals of one another.\n3.\nThe relative magnitudes of reactivity ratios can be understood, at least qualitatively, by\nconsidering the contributions of resonance stabilization, polarity differences, and possible\nsteric effects.\n4.\nStatistical considerations give predictions for the average sequence length and sequence \ndistributions in a copolymer on the basis of reactivity ratios and feedstock composition.\nHowever, the probability of adding a given monomer to a growing chain end may be determined\n"]], ["block_4", ["by the last, the last plus next-to-last, or even the last, next-to-last \nmers added. These mechanisms are referred to as terminal, penultimate, and \ncontrol, respectively.\n5.\nStereoregularity may be viewed as a subset of copolymerization, in of\nmonomer with an asymmetric center may follow the same stereochemistry previous\nrepeat unit, thereby forming a meso dyad, or by the opposite stereochemistry, \nracemic dyad. Isotactic, syndiotactic, and atactic polymers thus correspond to \nmeso dyads, predominantly racemic dyads, or random mixtures of the two, respectively.\n"]], ["block_5", ["Chapter Summary\n211\n"]], ["block_6", ["6.\nCopolymer sequence lengths (dyads, triads, tetrads, etc.) can be determined by \nThese in turn may be used to discriminate among terminal, penultimate, and antepenultimate\n"]], ["block_7", ["5.12\nChapter Summary\n"]]], "page_226": [["block_0", [{"image_0": "226_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "226_1.png", "coords": [34, 527, 226, 615], "fig_type": "figure"}]], ["block_2", ["control mechanisms. Similarly NMR gives access to stereochemical information, being\nsensitive to sequences of meso dyads, racemic dyads, and even longer sequences.\n7.\nStereoregularity is obtained by coordination polymerization in the presence of particular\ncatalysts. The most commonly used systems for the polymerization of a-olefins are referred\nto as Ziegler\u2014Natta catalysts, a class which actually spans a large variety of particular\ncompounds. The mechanisms of action of these catalysts are typically rather complicated.\nMore recently there have been rapid advances in the development of single\u2014site catalysts,\nwhich are usually based on metallocenes: a metal center coordinated to one or more\ncyclopentadienyl ligands. The terminology refers to the presence of a well-defined catalytic\nsite throughout the polymerization medium, leading to more homogeneous products. These\nsystems are capable of being fine-tuned to regulate a variety of structural features, including\nstereochemistry and comonomer addition.\n"]], ["block_3", ["212\nCopolymers, Microstructure, and Stereoregularity\n"]], ["block_4", ["1.\nWrite structural formulas for maleic anhydride (M1) and stilbene (M2). Neither of these\nmonomers homopolymerize to any significant extent, presumably owing to steric effects.\nThese monomers form a copolymer, however, with r1=r2=0.03.Jf Criticize or defend the\nfollowing proposition: The strong tendency toward alternation in this copolymer suggests that\npolarity effects offset the steric hindrance and permit copolymerization of these monomers.\n2.\nStyrene and methyl methacrylate have been used as comonomers in many investigations of\ncopolymerization. Use the following list of n values for each of these copolymerizing with the\nmonomers listed below to rank the latter with respect to reactivity. To the extent that the data\nallow, suggest where these substituents might be positioned in Table 5.3.\n"]], ["block_5", ["4.\nAdditional data from the research of the last problem yield the following pairs off], F 1 values\n(remember that styrene is component 1 in the styrene\u20141-chloro\u20141,3-butadiene system). Use the\n"]], ["block_6", ["Problems\n"]], ["block_7", ["3.\nAs part of the research described in Figure 5.5, Winston and Wichacheewa measured\nthe weight percentages of carbon and chlorine in copolymers of styrene (molecule 1) and\n1-chloro-1,3-butadiene (molecule 2) prepared from various feedstocks. A portion of their data\nis given below. Use these data to calculate F 1 , the mole fraction of styrene in these copolymers.\n"]], ["block_8", ["TRM. Lewis and ER. Mayo, J. Am. Chem. 506., 70, 1533 (1943).\n"]], ["block_9", ["f1\nPercent C\nPercent Cl\n"]], ["block_10", [{"image_2": "226_2.png", "coords": [51, 358, 326, 464], "fig_type": "figure"}]], ["block_11", ["M2\nStyrene as M1\nMethyl methacrylate as M1\n"]], ["block_12", ["Acrylonitrile\n0.41\n1.35\nAlly] acetate\n90\n23\n"]], ["block_13", ["1 ,2-Dichloropropene-2\n5\n5.5\nMethacrylonitrile\n0.30\n0.67\nVinyl chloride\n17\n12.5\nVinylidene chloride\n1.85\n2.53\n2-Vinyl pyridine\n0.55\n0.395\n"]], ["block_14", ["0.892\n81.80\n10.88\n0.649\n71.34\n20.14\n0.324\n64.95\n27.92\n0.153\n58.69\n34.79\n"]], ["block_15", [{"image_3": "226_3.png", "coords": [144, 380, 304, 457], "fig_type": "figure"}]]], "page_227": [["block_0", [{"image_0": "227_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "227_1.png", "coords": [34, 278, 279, 380], "fig_type": "figure"}]], ["block_2", ["6.\nUse the values determined in Example 5.5 for the vinylidene chloride (M1)\u2014isobutylene (M2)\nsystem to calculate F 1, for various values off1, according to the terminal mechanism. Prepare a\nplot of the results. On the same graph, plot the following experimentally measured values off1\nand F1. Comment on the quality of the fit.\n"]], ["block_3", ["5.\nThe reactivity ratios for the styrene (M1)\u20141-chloro\u20141,3-butadiene (M2) system were found to\nbe r, 0.26 and r2 1.02 by the authors of the research described in the last two problems,\nusing the results of all their measurements. Use these r values and the feed compositions listed\nbelow to calculate the fraction expected in the copolymer of 1-chlorobutadiene sequences of\nlengths v: 2, 3, or 4. From these calculated results, evaluate the ratios Nag/N22 and N2222/\nN222. Copolymers prepared from these feedstocks were dehydrohalogenated to yield the\npolyenes like that whose spectrum is shown in Figure 5.5. The absorbance at the indicated\nwavelengths was measured for 1% solutions of the products after HCl elimination.\n"]], ["block_4", ["7.\nSome additional dyad fractions from the research cited in the last problem are reported at\nintermediate feedstock concentrations (M1 :vinylidene chloride; Mzzisobutylenefr Still\nassuming terminal control, evaluate r, and r2 from these data. Criticize or defend the following\nproposition: The copolymer composition equation does not provide a very sensitive test for\n"]], ["block_5", ["TJ.B. Kinsinger, T. Fischer, and CW. Wilson, Polym. Left, 5, 285 (1967).\n"]], ["block_6", ["Problems\n213\n"]], ["block_7", [{"image_2": "227_2.png", "coords": [41, 483, 187, 570], "fig_type": "figure"}]], ["block_8", ["f1\nA=312nm\nA2367 nm\nA=412nm\n"]], ["block_9", ["f1\nF1\nf1\nF1\n"]], ["block_10", ["f1\nF1\nf1\nF1\n"]], ["block_11", ["0.947\n0.829\n0.448\n0.362\n0.861\n0.688\n0.247\n0.207\n0.698\n0.515\n0.221\n0.200\n0.602\n0.452\n"]], ["block_12", ["form suggested by Equation 5.6.1 to prepare a graph based on these data and evaluate r1\nand r2.\n"]], ["block_13", ["0.548\n0.83\n0.225\n0.66\n0.471\n0.79\n0.206\n0.64\n0.391\n0.74\n0.159\n0.61\n0.318\n0.71\n0.126\n0.58\n0.288\n0.70\n0.083\n0.52\n"]], ["block_14", ["As noted in Section 5.6, these different wavelengths correspond to absorbance by sequences of\ndifferent lengths. Compare the appropriate absorbance ratios with the theoretical sequence\nlength ratios calculated above and comment brie\ufb02y on the results.\n"]], ["block_15", ["0.551\n154\n77\n20\n0.490\n151\n78\n42\n"]], ["block_16", ["0.734\n71\n19\n\u2014\n"]], ["block_17", ["0.829\n74\n13\n\u2014\n"]], ["block_18", ["Absorbance\n"]]], "page_228": [["block_0", [{"image_0": "228_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["1\u2018J.C. Randall, J. Polym. Sci, Polym. Phys. Ed, 13, 889 (1975).\n"]], ["block_2", ["214\nCopolymers, Microstructure. and Stereoregularity\n"]], ["block_3", ["10.\nRandall,r used 13C-NMR to study the methylene spectrum of polystyrene. In 1,2,4-trichlor-\nobenzene at 120\u00b0C, nine resonances were observed. These were assumed to arise from a\ncombination of tetrads and hexads. Using m and r notation, extend Table 5.6 to include all 20\npossible hexads. Criticize or defend the following proposition: Assuming that none of the\nresonances are obscured by overlap, there is only one way that nine methylene resonances\ncan be produced, namely, by one of the tetrads being split into hexads whereas the remaining\ntetrads remain unsplit.\n"]], ["block_4", ["11.\nIn the research described in the preceding problem, Randall was able to assign the five peaks\nassociated with tetrads in the 13C\u2014NMR spectrum on the basis of their relative intensities,\nassuming zero-order Markov statistics with pm 2: 0.575. The five tetrad intensities and their\nchemical shifts from TMS are as follows:\n"]], ["block_5", ["9.\nA hetero triad occurs at each interface between iso and syndio triads. The total number of\nhetero triads, therefore, equals the total number of sequences of all other types:\n"]], ["block_6", ["8.\nFox and Schnecko carried out the free-radical polymerization of methyl methacrylate\nbetween \u201440\u00b0C and 250\u00b0C. By analysis of the a\u2014methyl peaks in the NMR spectra of the\nproducts, they determined the following values of B, the probability of an isotactic placement\nin the products prepared at different temperatures.\n"]], ["block_7", [{"image_1": "228_1.png", "coords": [47, 88, 223, 205], "fig_type": "figure"}]], ["block_8", [{"image_2": "228_2.png", "coords": [52, 372, 164, 407], "fig_type": "molecule"}]], ["block_9", ["f,\n11\n12\n22\n"]], ["block_10", ["_\nIll-1\n_\n2\np\u201c\nvh+vi+vffzi+\ufb01s\n"]], ["block_11", ["the terminal control mechanism. Dyad fractions are more sensitive, but must be examined\nover a wide range of compositions to provide a valid test.\n"]], ["block_12", ["B\n0.36\n0.33\n0.27\n0.27\n0.24\n0.22\n0.20\n0.18\n0.14\n"]], ["block_13", ["Evaluate Ei\u201c 15;\" by means of an Arrhenius plot of these data using B/(l B) as a measure of\nki/ks. Brie\ufb02y justify this last relationship.\n"]], ["block_14", ["0.418\n0.55\n0.43\n0.03\n0.353\n0.48\n0.49\n0.04\n0.317\n0.44\n0.52\n0.04\n0.247\n0.38\n0.58\n0.04\n0.213\n0.34\n0.62\n0.04\n0. 198\n0.32\n0.64\n0.05\n"]], ["block_15", ["Criticize or defend the following proposition: The sequence DL\u2014 is already two thirds of the\nway to becoming a hetero triad, whereas the sequence DD\u2014 is two thirds of the way toward an\niso triad. This means that the fraction of heterotactic triads is larger when the average length\nof syndio sequences is greater than the average length of iso sequences.\n"]], ["block_16", ["T (\u00b0C)\n250\n150\n100\n95\n60\n30\n0\n\u201420\n\u201440\n"]], ["block_17", ["Use this relationship and Equation 5.9.1 to derive the expression\n"]], ["block_18", ["1\"h :Nni \u20181':a\n"]], ["block_19", ["Mole fraction of dyads\n"]]], "page_229": [["block_0", [{"image_0": "229_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Problems\n21 5\n"]], ["block_2", ["1K. Ito and Y. Yamashita, J. Polym. sci. 3A, 2165 (1965).\n"]], ["block_3", ["13.\nThe following are experimental tacticity fractions of polymers prepared from different\nmonomers and with various catalysts. On the basis of Figure 5.7, decide whether these\npreparations are adequately described by a single parameter pm or whether some other type\nof statistical description is required (remember to make some allowance for experimental\nerror). On the basis of these observations, criticize or defend the following proposition:\nRegardless of the monomer used, zero-order Markov statistics apply to all free-radical,\nanionic, and cationic polymerizations, but not to Ziegler\u2014Natta catalyzed systems.\n"]], ["block_4", ["12.\nThe fraction of sequences of the length indicated below have been measured for a copolymer\nsystem at different feed ratios.)r From appropriate ratios of these sequence lengths, what\nconclusions can be drawn concerning terminal versus penultimate control of addition?\n"]], ["block_5", ["14.\nReplacing one of the alkyl groups in R3Al with a halogen increases the stereospecificity of the\nZiegler\u2014Natta catalyst in the order I > Br > C1 > R. Replacement of a second alkyl by halogen\ndecreases specificity. Criticize or defend the following pr0position on of these\nobservations: The observed result of halogen substitution is consistent the effect on \n"]], ["block_6", [{"image_1": "229_1.png", "coords": [45, 245, 264, 328], "fig_type": "figure"}]], ["block_7", [{"image_2": "229_2.png", "coords": [48, 54, 225, 135], "fig_type": "figure"}]], ["block_8", ["45.38\n0.10\n44.94\n0.28\n44.25\n0.13\n43.77\n0.19\n42.84\n0.09\n"]], ["block_9", ["The remaining 21% of the peak area is distributed among the remaining hexad features. Use\nthe value of Pm given to calculate the probabilities of the unSplit tetrads (see Problem 10) and\non this basis assign the features listed above to the appropriate tetrads. Which of the tetrads\nappears to be split into hexads?\n"]], ["block_10", ["Methyl methacrylatea\nThermal\nToluene\n60\n8\n33\n59\nn-Butyl lithium\nToluene\n\u2014\u201478\n78\n16\n6\nn-Butyl lithium\nMethyl isobutyrate\n\u201478\n21\n31\n48\n(Jr-Methyl styreneb\nTiCl4\nToluene\n\u201478\n\u2014\n19\n8 1\nEt3Al/TiCl4\nBenzene\n25\n3\n35\n62\n"]], ["block_11", ["aMethyl methacrylate data from K. Hatada, K. Ota, and H. Yuki, Polym. Left, 5, 225 (1967).\nl\u2019ut-Methyl styrene data from S. Brownstein, S. Bywater, and OJ. Worsfold, Makromol. Chem,\n48, 127 (1961).\n"]], ["block_12", ["n\u2014Butyl lithium\nCyclohexane\n4\n\u2014\u2014\n31\n69\n"]], ["block_13", ["Catalyst\nSolvent\nT (\u00b0C)\n130\nHetero\nSyndio\n"]], ["block_14", ["[Mll/[l\nP(M1)\nP(M1M1)\nP(M1M1M1)\n"]], ["block_15", [{"image_3": "229_3.png", "coords": [57, 470, 378, 560], "fig_type": "figure"}]], ["block_16", ["l3C STMS (ppm)\nRelative area under peak\n"]], ["block_17", ["3\n0.168\n0.0643\n0.0149\n4\n0.189\n0.0563\n0.0161\n9\n0.388\n0.225\n0.107\n19\n0.592\n0.425\n0.278\n"]], ["block_18", [{"image_4": "229_4.png", "coords": [204, 447, 371, 565], "fig_type": "figure"}]], ["block_19", ["Fraction of polymer\n"]]], "page_230": [["block_0", [{"image_0": "230_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "230_1.png", "coords": [32, 142, 358, 265], "fig_type": "figure"}]], ["block_2", ["216\n"]], ["block_3", ["l.\nBrandrup, J. and Immergut, E.H., Eds., Polymer Handbook, 3rd ed., Wiley, New York, 1989.\n2.\nCoates, G.W. and Waymouth, R.M., Science, 267, 217 (1995).\n"]], ["block_4", ["Allcock, HR. and Lampe, F.W., Contemporary Polymer Chemistry, 2nd ed., Prentice-Hall, Englewood Cliffs,\nNJ, 1990.\nBovey, F.W., High Resolution NMR of Macromolecules, Academic Press, New York, 1972.\nCoates, G.W., Precise control of polyole\ufb01n stereochemistry using single\u2014site metal catalysts, Chem. Rev., 100,\n1223 (2000).\nKoenig, J.L., Chemical Microstructure of Polymer Chains, Wiley, New York, 1980.\nNorth, A.M., The Kinetics of Free Radical Polymerization, Pergamon Press, New York, 1966.\nOdian, G., Principles of Polymerization, 4th ed., Wiley, New York, 2004.\nRempp, P. and Merrill, E.W., Polymer Synthesis, 2nd ed., Hiithig & Wepf, Basel, 1991.\n"]], ["block_5", ["References\n"]], ["block_6", ["lFJ. Karol and W.L. Carrick, J. Am. Chem. Soc, 83, 585 (1960).\n"]], ["block_7", ["Further Readings\n"]], ["block_8", ["15.\n"]], ["block_9", ["16.\n"]], ["block_10", ["ease of alkylation produced by substituents of different electronegativity. This evidence thus\nadds credence to the monometallic mechanism, even though the observation involves the\norganometallic.\n"]], ["block_11", ["21](Cs )2\n4.5\nVOC13\n2.4\nZn(n-Bu)2\n4.5\n"]], ["block_12", ["The weight percent propylene in ethylene\u2014propylene copolymers for different Ziegler\u2014Natta\ncatalysts was measured for the initial polymer produced from identical feedstocksfr The\nfollowing results were obtained. Interpret these results in terms of the relative in\ufb02uence of the\ntwo components of the catalyst on the product found.\n"]], ["block_13", ["VCl4, plus\nAl(i-Bu)3, plus\nAl(i-Bu)3\n4.5\nHfCl4\n0.7\nCH3TiCl3\n4.5\nZrCl4\n0.8\n"]], ["block_14", ["Imagine a given single-site catalyst for polypropylene introduced a stereodefect on average\nonce every 10 monomer additions. Furthermore, assume the catalyst was supposed to be\nhighly isospecific. Explain how measurements of triad populations (e.g., mmm, mmr, etc.)\ncould be used to distinguish between chain-end control and site control. (Hint: consider the\nsequences of D and L in the two cases.)\n"]], ["block_15", ["Catalyst\nWeight percent\nCatalyst\nWeight percent\ncomponents\npropylene\ncomponents\npropylene\n"]], ["block_16", ["Copolymers, Microstructure, and Stereoregularity\n"]]], "page_231": [["block_0", [{"image_0": "231_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["The remarkable properties of polymers derive from their size. As pointed out in Chapter 1, it is not\nthe high molecular weight per se that gives polymers mechanical strength, \ufb02exibility, elasticity,\netc., but rather their large spatial extent. In this chapter, we will learn how to describe the three-\ndimensional shape of polymers in an average sense, and how the average size of the object in space\nwill depend on molecular weight. We will also explore the equilibrium distribution of sizes.\nTo gain an appreciation of the possibilities, consider a polyethylene molecule with M 280,000\ng/mol (this is a reasonable value for a randomly selected molecule in some commercial grades of\npolyethylene). As the monomer (\u2014CHz\u2014CH2\u2014) molecular weight is 28 g/mol, the degree\nof polymerization, N, is 10,000, and there are 20,000 C\u2014C backbone bonds. Assuming a perfectly\nlinear structure (actually not likely for a commercial polyethylene), the contour length L of this\nmolecule would be roughly 20,000 X 1.5 A 30,000 A or about 3 pm, because 1.5 A is approximately\nthe average length of a C\u2014C bond. This is simply huge. If stretched out to its full extent, L would be\nhalf the size of a red blood cell, and possibly visible under a high-power optical microscope. Some\ncommercial polymers are 10 times bigger than this one, and some DNA molecules have molecular\nweights in excess of 109 g/mol. However, as we will see, it is very rare indeed for a chain to be so\nextended, and the contour length is not usually the most useful measure of size. Now consider the\nopposite extreme, where the same polyethylene molecule collapses into a dense ball or glabule. The\ndensity of bulk polyethylene is about 0.9 g/mL. The volume occupied by this 280,000 g/mol molecule\nwould be (280,000/0.9)/(6 x 1023) mL 520,000 A3, and if we assume it is a sphere, the radius would\nbe ((3/411)volume)1/as 50 A. The range from 50 A at the smallest to 3 pm at the largest covers three\norders of magnitude; it is a remarkable fact that such a mundane molecule could adopt conformations\nwith sizes varying over that range. If the dense sphere were a tennis ball, the chain contour would be the\nlength of a football field.\nPolyethylene, and most carbon chain polymers, is not likely to adopt either of these extreme\nconformations. The reason is easy to see. Select a C\u2014C bond anywhere along the chain; we can\nrepresent the structure as R\u2019CHr\u2014CHZR\u201d. There is rotation about this bond, with three energet-\nically preferred relative orientations of R\u2019 and R\u201d called trans (t), gauche plus (g+), and gauche\nminus (g') (see Figure 6.1a). For the chain of 20,000 bonds, there are three possible conformations\nfor each bond, and therefore 320\u2019000 m 1010\u2019000 possible conformations. This number is effectively\nin\ufb01nite. If our molecule were in a high temperature liquid state, and if we assume it takes 1 ps to\nchange one bond conformation, then the molecule would not even approach sampling all possible\nconformations over the history of the universe. Similarly, it would be highly improbable for it to\neven visit any given conformation twice. We can now see why the chance of being fully extended,\nin the all\u2014trans state, is unlikely to say the least; the probability is about 1 in 1010\u2019000. The dense\nsphere state might be marginally more probable, as there are many sequences of t, g+, and g\u2018 that\nmight produce something close to that, but it is still essentially impossible without the action of\nsome external force. What the polymer does instead is form what is called a random coil (Figure\n6.2). The different sequences of t, g+, and g\u2018 cause the chain to wander about haphazardly in\nspace, with a typical size intermediate between the dense sphere and the extended chain. Withso\n"]], ["block_2", ["6.1\nConformations, Bond Rotation, and Polymer Size\n"]], ["block_3", ["Polymer Conformations\n"]], ["block_4", ["6\n"]], ["block_5", ["217\n"]]], "page_232": [["block_0", [{"image_0": "232_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "232_1.png", "coords": [32, 278, 292, 592], "fig_type": "figure"}]], ["block_2", ["218\nPolymer Conformations\n"]], ["block_3", ["Figure 6.1\nBackbone bond conformations for polyethylene. (a) Illustration of trans and gauche arrange-\nments of the backbone bonds. (b) Schematic plot of the potential energy as a function of rotation angle about a\nsingle backbone bond.\n"]], ["block_4", ["Figure 6.2\nIllustration of a random walk in three dimensions. The walk has 4000 steps, and the walk\ntouches each face of the box. (From Lodge, A.S., An Introduction to Elastomer Molecular Network Theory,\nBannatek Press, Madison, 1999. With permission.)\n"]], ["block_5", [{"image_2": "232_2.png", "coords": [40, 122, 220, 217], "fig_type": "figure"}]], ["block_6", [{"image_3": "232_3.png", "coords": [52, 409, 302, 620], "fig_type": "figure"}]], ["block_7", [{"image_4": "232_4.png", "coords": [60, 132, 214, 220], "fig_type": "molecule"}]], ["block_8", [{"image_5": "232_5.png", "coords": [72, 53, 400, 214], "fig_type": "figure"}]], ["block_9", [{"image_6": "232_6.png", "coords": [85, 506, 266, 594], "fig_type": "molecule"}]], ["block_10", [{"image_7": "232_7.png", "coords": [91, 472, 263, 598], "fig_type": "figure"}]], ["block_11", [{"image_8": "232_8.png", "coords": [219, 28, 394, 229], "fig_type": "figure"}]], ["block_12", ["(arbitary\n"]], ["block_13", ["Potential\n"]], ["block_14", ["units)\n"]], ["block_15", ["energy\n"]], ["block_16", ["I\n\u20181\u2019\nI\nI\nI\n\u2014180 \u2014120\n\u201460\n0\n60\n120\n180\n"]], ["block_17", ["Rotation angle (deg)\n(b)\n"]], ["block_18", ["gauche\ngauche\n"]], ["block_19", ["Hans\n"]]], "page_233": [["block_0", [{"image_0": "233_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["many possibilities, cannot predict any instantaneous conformation or size, but we\nwill \nThe preceding argument, although fundamentally sound, neglects a very important aspect of\nchain conformations. ofthe t, g+, and g\u201d states are not equal; for polyethylene t is\nenergetically favorable relative to g+ or g\u2018 by about AB 3 kJ/mol (or 0.7 kcal/mol). Therefore, at\nequilibrium, the of t states will exceed that of g+ or g\u2018 by the appropriate Boltzmann\nfactor, exp(\u2014AE/RT), in thiscase is about 2 at T: 500 K (RT x42 kJ/mol or 1 kcal/mol).\n(We have injected some statistical mechanics here. For a large collection or ensemble of molecules at\nequilibrium, the of any two possible states are given by this Boltzmann factor.)\nThis bias is to put much of a dent into the vast number of possible conformations, but\nit does matter detailed conformational statistics for a given polymer chain.\nThere is a further issue of importance, namely, how high are the energy barriers among the t, g+,\nand g\" states? If these are too high, conformational rearrangements will not occur rapidly. Figure\n6.1b shows schematic plot of the potential energy as a function of rotation about the C\u2014C bond in\npolyethylene. The barrier heights are on the order of 10 kJ/mol (2.5 kcal/mol), which corresponds to\nabout 2.5 times RT, the available thermal energy per mole at 500 K. Thus rotation should be\nrelatively facile for polyethylene. (To put this energy barrier on a Chemist\u2019s scale, it is compar-\nable to the energy of a weak hydrogen bond.) However, for polymers with larger side-groups or\nwith more complicated backbone structures, these barriers can become substantial. For example,\nin poly(n-hexyl isocyanate) (\u2014N(C6H13)\u2014C(O)\u2014) the n-hexyl side chain forces the backbone to\nfavor a helical conformation, and the molecule becomes relatively extended.\nAt this point it might look like a very daunting task to calculate the probable conformation of a\ngiven polymer, and it will require some detailed information about bond rotational potentials, etc.,\nfor each structure. However, it turns out that we can go a long way without any such knowledge.\nWhat we will calculate \ufb01rst is the average distance between the ends of a chain, as a function of\nthe number of steps in a chain. We will show that this is given by a simple formula, and that all the\ndetails about chemical structures, bond potentials, etc., can be grouped into a single parameter.\nWe will also consider the distribution of possible values of this end\u2014to-end distance. The average\ncould, in principle, be taken in two different ways. One would be to follow a single chain as it\nsamples many different conformations\u2014a time average. Another would be to look at a large\ncollection of structurally identical chains at a given instant in time\u2014an ensemble average. In this\nexample, these two averages should be the same; when this occurs, we say the system is ergodic. In\n"]], ["block_2", ["a real polymer sample, a measurement will also average over a distlibution of chain lengths or\nmolecular weights; that is a different average, which we will have to reckon with when we consider\nparticular experimental techniques.\n"]], ["block_3", ["Average \n219\n"]], ["block_4", ["In this section, we calculate the root-mean-square (n'ns) end-to-end distance 022)\u201c2 for an\nimaginary chain, made up of n rigid links, each with length 6'. The model is sketched in\nFigure 6.3. At this stage there is no need to worry about whether the link is meant to represent a\nreal C\u2014C bond or not; we will make the correspondence to real polymers later. If we arbitrarily\nselect one end as the starting point, each link can be represented by a vector,\n13,-, with\ni 1,2,3, . . . n. The instantaneous end\u2014to\u2014end vector, 5, is simply the sum of the link \n"]], ["block_5", ["6.2\nAverage End\u2014to-End Distance for Model Chains\n"]], ["block_6", ["If we have a chain that wiggles around over time, or if we look at an ensemble of similar chains,\nthere13 no reason for h to point in any one direction more than any other, and the average ()1): 0;\nwe say the sample18 isotropic. What we really care about13 the average end-to\u2014end distance, which\n"]], ["block_7", [{"image_1": "233_1.png", "coords": [45, 588, 92, 628], "fig_type": "molecule"}]], ["block_8", ["\u201d2 :15}\n(6.2.1)\n"]], ["block_9", ["i=1\n"]]], "page_234": [["block_0", [{"image_0": "234_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "234_1.png", "coords": [28, 41, 164, 170], "fig_type": "figure"}]], ["block_2", ["In this simplest possible case, the orientation of link 1' is unaffected by link i\u2014l, and is equally\nlikely to point in any direction. It can even lie on top of link i\u2014l by pointing in the opposite\ndirection. (Remember we are dealing with imaginary links, not real chemical bonds, so this is\npermissible.) Mathematically we represent this approximation using the relation\n"]], ["block_3", ["The double sum can be broken into two parts, remembering also that the summations can be taken\noutside the average as follows:\n"]], ["block_4", ["where the first term, 1262, accounts for the \u201cself-terms,\u201d i.e., each of the n link vectors dotted into\nitself, and the second term accounts for the \u201ccross terms,\u201d i.e., each link vector dotted into the n\u2014\u20141\nother link vectors.\nWe now develop explicit results for (I22) ,with three different rules for how the orientation of\na given link, 19,-,IS constrained by the orientation of its predecessor, \u20ac,_ 1.\n"]], ["block_5", ["end-to\u2014end vector it extends from the start of the \ufb01rst link to the end of the last one.\n"]], ["block_6", ["we can calculate remembering that the length of a vector is obtained by taking the dot product of\nthe vector with itself:\n"]], ["block_7", ["Case 6.2.1\nThe Freely Jointed Chain\n"]], ["block_8", ["220\nPolymer Conformations\n"]], ["block_9", ["where 6 is the angle between E; and \u00a3311. For the freely jointed chain, 6 ranges freely from 0\u00b0 to\n180\u00b0. Thus on average\n"]], ["block_10", [{"image_2": "234_2.png", "coords": [35, 247, 223, 285], "fig_type": "molecule"}]], ["block_11", ["Figure 6.3\nModel chain consisting of .12 links of length 6. Each link is represented by a vector, E, and the\n"]], ["block_12", [{"image_3": "234_3.png", "coords": [36, 324, 220, 363], "fig_type": "molecule"}]], ["block_13", ["h2>=<2eza>=22\nI:\nj:\nr:\nj=\n"]], ["block_14", [{"image_4": "234_4.png", "coords": [45, 311, 230, 440], "fig_type": "figure"}]], ["block_15", ["6\", 2;, :22 cos 6\n(6.2.4)\n"]], ["block_16", ["1/2\n(112)1/2: (h- I?)_<:6o :E>\n(6.2.2)\n"]], ["block_17", ["<12} EH) 62(6639) 0\n(6.2.5)\n"]], ["block_18", [{"image_5": "234_5.png", "coords": [51, 360, 211, 401], "fig_type": "molecule"}]], ["block_19", [{"image_6": "234_6.png", "coords": [55, 317, 215, 441], "fig_type": "molecule"}]], ["block_20", ["=2??? E>+ZZ<\u20acW>\ni=1j9\u00e91\n: M + Z 2 <5- - (5)\n(6.2.3)\ni=1 #i\n"]]], "page_235": [["block_0", [{"image_0": "235_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "235_1.png", "coords": [19, 441, 269, 651], "fig_type": "figure"}]], ["block_2", [{"image_2": "235_2.png", "coords": [26, 540, 257, 640], "fig_type": "figure"}]], ["block_3", [{"image_3": "235_3.png", "coords": [28, 164, 196, 189], "fig_type": "molecule"}]], ["block_4", ["Now we make the model a bit more realistic. We will constrain the angle between adjacent links to\nbe a fixed value, 6, but still allow free rotation of the link around the cone defined by 6 (see Figure\n6.4). What happens? Now, 6,0 61-1 62\ncos6 does not average to zero because 6 is fixed. For\nsimplicity, we will define 01\u2014\n\u2014 cos 6 just to avoid writing cos 6 over and over. Returning to Equation\n62.3, what we need to calculateIS the double sum over all possible (6,- 6),i.e., the cross\nterrnzs\nas\nwell as the self-terms. We now know (6, 6-)\u2014\n- 6201 when |1'\u20141|\n\u2014 1, but what about |1' \u20141|\netc.? This1s a little sneaky: 6,- has a component parallel to 6,-_1, with length 601, but it also hzas3a\ncomponent perpendicular to 62-1 (with length 6 sin 6). However, because of the free rotation, over\ntime the perpendicular part will average to zero (see Figure 6.4). So from the point of view of bond\n6,-2, 011 average 6, looks just like a bond of length 601 pointing in the same direction as 6,-_1, and thus\n(6,-2 6-,-\u2014)62012. The same argument can be extended to any pair of bonds 1', j:\n"]], ["block_5", ["\u20141 to +1, with + or values equally probable. If we consider the relative orientations of any two\ndifferent links jointed chain, they must all be uncorrelated, so that\n"]], ["block_6", ["Average End-to-End Distance for Model Chains\n221\n"]], ["block_7", ["When the two linksthen (cos 6) 0, because cos 6 ranges from\n"]], ["block_8", ["because all the cross terms vanish. This is the classic result for the so-called random walk (or\nrandom\ufb02ight): the root-mean\u2014square excursion is given by the step length times the square root of\nthe number of steps. will invokethis result repeatedly in subsequent chapters.\nAt this point you may be thinking \u201cFine, but even if the link is not a C\u2014\u2014\u2014C bond, a real polymer\nchain cannot reverse its direction 180\u00b0 at a joint, so how can this result be relevant?\u201d Good\nquestion. Be patient for a bit.\n"]], ["block_9", ["Case 6.2.2\nThe Freely Rotating Chain\n"]], ["block_10", ["Now we define a new summation index k li\u2014j] and write\nii <1- 1-) 2\": DW\ni=1\n#1\ni=1\n#1\n"]], ["block_11", ["Figure 6.4\nDe\ufb01nition of the angles 6 and\n(15 for the freely rotating and hindered rotation chains; the\ncorrespondence to a polyethylene molecule is suggested by the locations of the carbons.\n"]], ["block_12", ["whenever\n1' ;\u00a3 j. In other words if the orientation of a given link is unaffected by its nearest\nneighbor, it must also be unaffected by more distant neighbors. From Equation 6.2.3 we now\nobtain the simple but important result\n"]], ["block_13", [{"image_4": "235_4.png", "coords": [46, 476, 232, 555], "fig_type": "molecule"}]], ["block_14", [{"image_5": "235_5.png", "coords": [47, 504, 232, 543], "fig_type": "molecule"}]], ["block_15", ["<17; 17,-) 0\n(6.2.6)\n"]], ["block_16", ["(16:11:32\nor (av/am\n(6.2.7)\n"]], ["block_17", ["<6,- 6,) tzali\u2014\ufb02\n(6.2.8)\n"]], ["block_18", ["11-1\n11\u20141\n11\u20141\n2 Z tzatm k) 21112 Z a\" 2112 Z ka\"\n"]], ["block_19", ["C\n/\n"]], ["block_20", ["\\9\n"]], ["block_21", ["60036\n"]], ["block_22", ["(6.2.9)\n"]]], "page_236": [["block_0", [{"image_0": "236_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "236_1.png", "coords": [33, 623, 217, 667], "fig_type": "molecule"}]], ["block_2", ["The factor of 2 comes about because there are two ways to get each value of k (one with i >j and\none with i < j). The term (n k) arises because there are n 1 nearest neighbors (I: 1), n 2 next\nnearest neighbors (k 2), etc. Note also that careful attention has to be paid to the limits\nof the sums.\nThe relevant summations have algebraic answers that are derived in the Appendix, namely:\n"]], ["block_3", ["222\nPolymer Conformations\n"]], ["block_4", ["4!) are possible to some extent (see Figure 6.1b). The derivation of (122) is more complicated for this\ncase, as you might expect, but it is similar in spirit to that for the freely rotating chain; it may be\nfound in Flory\u2019s second book [1]. The large :1 result is\n"]], ["block_5", ["In a real polyethylene chain, the rotation about the cone is not free; there are three preferred\nconformations (t, g+, g\u2018) as discussed in Section 6.1. Furthermore, all values of the rotation angle\n"]], ["block_6", ["forn \u2014> oo, assuming |a| < 1 (i.e., 6 79 90\u00b0), and\n"]], ["block_7", ["where again we let n\u2014> 00 to reach the last expression. Now we insert Equation 6.2.10a and\nEquation 6.2.10b into Equation 6.2.9, and we recall Equation 6.2.3 to obtain\n"]], ["block_8", ["where again we assume n is large in the penultimate step. (We also reinserted cos 6 for a.) We can\nlearn three important things from this result.\n"]], ["block_9", ["1.\n(112) is larger than the freely jointed chain result if 6 < 90\u00b0 (cos 6 > 0). This is very reasonable;\nif each link has some preference for heading in the same direction as the previous link, the\nchain will double back on itself less often. As an example, for C\u2014C single bonds, 6 is close to\n705\u00b0 (the complement to the tetrahedral angle) and for this value (112) x 21262.\n2.\n(/22) is still proportional to n62; the proportionality factor is just a number that depends on the\ndetails of the local constraints placed on link orientation. In particular, therefore, (122) is still\nproportional to (/13.\n3.\nThe previous statement applies strictly only in the large n limit, i.e., when the term propor-\ntional to 1/11 in Equation 6.2.11 is negligible and when of\u201d vanishes. This is a commonly\nencountered caveat in polymer science: we can derive relatively simple expressions, but they\nwill often be valid only in the large n limit. The answer to \u201cHow large is large enough?\u201d will\ndepend on the particular property, but when the correction is proportional to 1/12, as it is in\nEquation 6.2.11, it will drop to the order of 1% when n m 100, which is not a particularly large\nnumber of backbone bonds.\n"]], ["block_10", ["Case 6.2.3\nHindered Rotation Chain\n"]], ["block_11", [{"image_2": "236_2.png", "coords": [37, 122, 180, 158], "fig_type": "molecule"}]], ["block_12", [{"image_3": "236_3.png", "coords": [40, 181, 193, 213], "fig_type": "molecule"}]], ["block_13", [{"image_4": "236_4.png", "coords": [41, 243, 263, 319], "fig_type": "molecule"}]], ["block_14", ["Zkak \n1\u2014 (In\nN\nC!\n"]], ["block_15", ["\"\u20141\nk\n1 Gin\u2014I\na\na a\n%\n(6.2.10a)\nk=l\n1\u20140:\n1\u20140:\n"]], ["block_16", ["2\n_\n2\nl+cos6\n1+(cosqb)\n(h) \u2014n\u20ac{1\u2014cos6}{l\u2014(cosqb)}\n(6.2.12)\n"]], ["block_17", ["2 _\n2\n2\na\n_\n2\n0f\n(h)\u2014\u2014n\u20ac +2121? (1_)\n26(1_a)2\n\u201c,2 m3;\nW92 Li\u201c.\n1\u20140:\nIt\n(1\u2014092\n1\u20140:\nl+cos6\n__\n2\n_ \u201dE {1\u20140036}\n(6.2.11)\n"]], ["block_18", ["=\n(1 a)2\n(1 a)2\n(6.2.10b)\n"]]], "page_237": [["block_0", [{"image_0": "237_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["a theorem:\nIf we take the limit n~>oo, and if we consider \u201cphantom\u201d chains that can double back on\nthemselves, C is a numerical constant that depends local\nconstraints and not on n.\nThe physical content of this theorem can be summarized as follows. If we have a chain of links\nwith any degree of conformational freedom, no matter how limited that freedom may be, and if we\ntrack the conformation over enough links, the orientation of the last link will have lost all memory\nof the orientation of the first. At this point we could replace that entire subset of links with one new\nlink, and it would be freely jointed with respect to the next set and the previous set. In other words,\nfor any chain of n links whose relative orientations are constrained, we can always generate an\nequivalent chain with a new (bigger) link that is freely jointed, so that the original chain and the\nnew chain have the same (h2). We will illustrate this concept in the next section.\nThe issue of how large it needs to be for this theorem to be useful was mentioned before; it of a\nhundred or so is usually more than adequate. The fact that real polymers occupy real volume means\nthat polymer chains cannot double back on themselves, or have two or more links at the same point\nin space. This so-called \u201cexcluded volume\u201d problem is actually very serious, and makes an exact\nsolution for (hz) of a real polymer much more complicated. However, it turns out that there are two\npractical situations in which we can make this problem essentially go away; one is in a molten\npolymer, and the other is in a particular kind of solvent, called a theta solvent. Under these\ncircumstances, a polymer is said to exhibit \u201cunperturbed\u201d dimensions. Thus this theorem is of\ntremendous practical importance. Further discussion of the excluded volume effect will be deferred\nto Section 6.8, and then it will be revisited in more detail in Chapter 7.7.\n"]], ["block_2", ["where (cos (I5) is the average of cos over the appropriate potential energy curve (Figure 6.1b).\nHowever, the important message is that (hz) is still proportional to n32; all that has changed is a\nnumerical prefactor that specific local constraints. This point, in fact, as\n"]], ["block_3", ["We can define a quantity Cn, called the characteristic ratio, which for any polymer structure\ndescribes the effect of local constraints on the chain dimensions:\n"]], ["block_4", ["Characteristic Ratio and Statistical Segment Length\n223\n"]], ["block_5", ["In this equation, (h2)0 is the actual mean-square end-to-end distance of the polymer chain,\nand the subscript O reminds us that we are referring to unperturbed dimensions. In Equation\n6.3.1, n denotes the number of chemical bonds along the polymer backbone, and is actual\nlength of a backbone bond, e.g., 1.5 A for C\u2014C. (For polymers containing different kinds\nof backbone bonds, such as polyisoprene or poly(ethylene oxide), it is appropriate simply to\nadd n16? + ngl\u2019g \u2014l\u2014\n\u00ab - 4 where n,- and E, are the number and length of bonds of type i, respectively.)\n"]], ["block_6", ["6.3\nCharacteristic Ratio and Statistical Segment Length\n"]], ["block_7", ["caused the chain to extend in one direction. As defined in Equation 6.3.1, C\u201d but\nit approaches a constant value at large n; this is often denoted C00. For the freely rotating C00 is\n"]], ["block_8", ["(1 + cos 6)/(1 cos 6) from Equation 6.2.11. The dependence of C,, on n is shown in 6.5 for\nseveral theoretical chains. The values of C00 for several common polymers are listed 6.1.\nFor polymers that have primarily C\u2014C or C\u2014O single bonds along the backbone, C00 from\nabout 4 to about 12. Using these values, or those provided in reference books, it is to\nestimate (h2)0 for any polymer of known structure and molecular weight.\nAlthough calculating (h2)0 for a given polymer is thus a solved problem, this C00\nis not always the most convenient. For example, it requires remembering particular lengths\n"]], ["block_9", ["C,, is a measure of chain flexibility: the larger the value of C,,, the more the local have\n"]], ["block_10", [{"image_1": "237_1.png", "coords": [37, 425, 101, 460], "fig_type": "molecule"}]], ["block_11", ["(hzlo\nCn\nr1132\n(6.3.1)\n"]]], "page_238": [["block_0", [{"image_0": "238_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "238_1.png", "coords": [23, 499, 372, 613], "fig_type": "table"}]], ["block_2", [{"image_2": "238_2.png", "coords": [29, 477, 434, 643], "fig_type": "figure"}]], ["block_3", [{"image_3": "238_3.png", "coords": [30, 31, 232, 242], "fig_type": "figure"}]], ["block_4", ["Table 6.1\nRepresentative Values of the Characteristic Ratio, Statistical Segment Length, and Persistence\nLength for Various Flexible Polymers, Calculated from the Experimental Quantities, (h2)0/M (A2 mol/g),\nvia Equation 6.3.2 and Equation 6.4.5b\n"]], ["block_5", ["The experimental chain dimensions were obtained by small-angle neutron scattering, as compiled in Fetters, L.J., Lohse,\nD.J., Witten, T.A., and Zirkel, A., Macromolecules, 27, 4639, 1994. The uncertainties in (h2)0 and M are typically a few\npercent. The temperature of the measurement is indicated, because the distribution of chain conformations depends on\ntemperature.\n"]], ["block_6", ["Figure 6.5\nCharacteristic ratio as a function of the number of bonds for three model chains. The dotted\ncurve represents the freely rotating chain with 6:68\u00b0. The long dashed curve corresponds to a parti-\ncular hindered rotation chain with the preferred values of (b 120\u00b0 apart, but in which values of qb for\nneighboring bonds are independent. The smooth curve applies to an interdependence among values\nof (b on neighboring bonds. (Reproduced from Flory, P.J., Statistical Mechanics of Chain Molecules,\nWiley-Interscience, New York, 1969. With permission.)\n"]], ["block_7", ["Poly(ethylene oxide)\n5.6\n6.0\n4.1\n0.805\n140\n1,4-Polybutadiene\n5 .3\n6.9\n4.0\n0.876\n140\n1,4\u2014Polyisoprene\n4.8\n6.5\n3.5\n0.625\n140\nPoly(dimethylsiloxane)\n6.6\n5.8\n5.3\n0.457\n140\nPolyethylene\n7.4\n5.9\n5.7\n1.25\n140\nPolyprOpylene\n5.9\n5.3\n4.6\n0.67\n140\nPolyisobutylene\n6.7\n5.6\n5.2\n0.57\n140\nPoly(methyl methacrylate)\n9.0\n6.5\n6.9\n0.425\n140\nPoly(vinyl acetate)\n8.9\n6.5\n6.8\n0.49\n25\nPolystyrene\n9.5\n6.7\n7.3\n0.434\n140\n"]], ["block_8", ["Polymer\nCo,\nb (ti)\n3,, (ti)\n(h2)0/M (.38 mol/g)\nT (\u00b0C)\n"]], ["block_9", ["224\nPolymer Conformations\n"]], ["block_10", ["can replace the number of bonds, n, with the number of monomers or repeat units, N, and subsume\nthe proportionality factor between rt and N into a new effective step length, b. The quantity b,\ndefined by Equation 6.3.2, is called the statistical segment length. The calculation of (h2)0 through\n"]], ["block_11", ["where Eeff fix/E is a new effective bond length with the following meaning: the real chain with\nlocal constraints has an end\u2014to\u2014end distance, which is the same as that of a freely jointed chain with\nthe same number of links rt, but with a different (larger) step length Eeff. Continuing in this vein, we\n"]], ["block_12", ["and the number of bonds per repeat unit. A more popular approach was suggested in the previous\nsection. We could rewrite Equation 6.3.1 in the following way:\n"]], ["block_13", ["6.87\nl\nI\nI\nI\nl\n!\n__l_\n"]], ["block_14", ["6.0\nInterdependent rotations\n"]], ["block_15", [", \n2 0\n\"\nI\n.\u2019\n\u2019\n"]], ["block_16", ["(11% Cam/32 2262.. s\n(6.3.2)\n"]], ["block_17", ["0\nI\n|\nl\n|\nI\n|\n0\n50\n100\n1 50\nNumber of bonds, it\n"]], ["block_18", ["Independent hindered rotations\n"]], ["block_19", [{"image_4": "238_4.png", "coords": [172, 503, 445, 626], "fig_type": "figure"}]]], "page_239": [["block_0", [{"image_0": "239_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["N and b treats the real chain though it were a freely jointed chain with N links of length I). This\nproves to be a useful computational scheme, but it is important to realize that b has no simple\ncorrespondence with the physical chain; it is not a measure of the real size of a real monomer.\n"]], ["block_2", ["Furthermore, it contains new information beyond that embodied in C00. Values of b are also\ngiven in Table 6.1. Note that although b varies monotonically with C00, it is not as simple a measure\nof \ufb02exibility. For example, b for polystyrene (6.7 A) and polyisoprene (6.5 A) differ by only a few\npercent, whereas polyiSOprene is considered to be a relatively \ufb02exible polymer, and polystyrene a\nrelatively stiff one. The resolution of this apparent paradox is left to Problem 6.2. Values of C00 and\n"]], ["block_3", ["It is an interesting fact that bulk polyethylene has a positive coefficient of thermal expansion (about\n2.5 x 10\u20144 per \u00b0C at room temperature), whereas the individual chain dimensions have a negative\ncoefficient (d In (h2)0/dT: $1.2 >< 10\u20143deg\u201c1). In other words, when a piece of polyethylene is\nheated, the volume increases while the individual chain dimensions shrink. How does this\ncome about?\n"]], ["block_4", ["measurements of the chain dimensions; this method will be described in detail in Chapter 8.\nThe approach taken above, and indeed for the remainder of the book, is that C0\u2018, or b are\nstructure\u2014specific parameters that we can look up as needed; the dependence of (h2)0 on molecular\nweight is universal and therefore more important to understand. However, it is of considerable\ninterest to ask whether the techniques of computational statistical mechanics can be used to\ncalculate CDO from first principles, i.e., from knowledge of bond angles, rotational potentials,\netc. A highly successful scheme for doing so, called the rotational isomeric state approach, was\ndeveloped by Flory [1]. It is beyond the scope of this book to describe it, but it is worth mentioning\nthat even today it is not a trivial matter to execute such calculations, and that controversy exists\nabout the correct values of C00 for some relatively simple chemical structures. These controversies\nare also not easily resolved by experiment; combined uncertainties in measured molecular weights\nand chain dimensions often exceed 10%.\n"]], ["block_5", ["Thermal expansion corresponds to a decrease in the density of the material, which re\ufb02ects\nprimarily an increase in the average distance between molecules; the radii of the individual\n"]], ["block_6", ["atoms and the bond lengths also tend to increase, but to a much smaller extent. In contrast, the\nreduction in (h2)0 is primarily of intramolecular origin. From Equation 6.3.1, we can see that as n is\nindependent of T and E, if anything, increases with T, then there must be a decrease in the\ncharacteristic ratio, C00. The origin of this effect can be seen from Figure 6.1. As temperature\nincreases, the Boltzmann factors that dictate the relative equilibrium populations of trans and\ngauche conformations change, and the gauche states become relatively more populated. As the\ntrans conformations favor larger (h2)0, the net result is a reduction in C00. Note that this simple\nrelation between C00 and the relative populations of trans and gauche states does not necessarily\nextend to more complicated backbone structures. For example, d In (h2)0/dT is positive 100%\ncis-l,4\u2014polybutadiene, but negative for the all-trans versions (see Problem 65). This observation is\nnot easy to anticipate based on the molecular structure.\n"]], ["block_7", ["b may be determined experimentally in various ways, but the most direct route is by scattering\n"]], ["block_8", ["For many macromolecules, the backbone does not consist of a string of single bonds with facile\nrotations, but rather some combination of bonds that tend to make the backbone in one\ndirection. Such chains are called semiflexible, and examples (see Figure 6.6) include polymers\nwith mostly aromatic rings along the backbone, such as poly(p\u2014phenylene); polymers with large\n"]], ["block_9", ["Semiflexible Chains and the Persistence Length\n225\n"]], ["block_10", ["6.4\nSemiflexible Chains and the Persistence Length\n"]], ["block_11", ["Example 6.1\n"]], ["block_12", ["Solution\n"]]], "page_240": [["block_0", [{"image_0": "240_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "240_1.png", "coords": [21, 59, 121, 107], "fig_type": "molecule"}]], ["block_2", [{"image_2": "240_2.png", "coords": [29, 497, 281, 581], "fig_type": "figure"}]], ["block_3", ["226\nPolymer Conformations\nsi\n"]], ["block_4", ["side\u2014groups that for steric reasons induce the backbone to adopt a helical conformation, such as\npoly(n-hexyl isocyanate)\nand poly('y\u2014benzyl-L\u2014glutamate); biopolymers\nsuch\nas DNA and\ncollagen that involve intertwined double or triple helices. Short versions of these molecules\nare essentially \u201crigid rods,\u201d but very long versions will wander about enough to be random coils.\nThe description of chain dimensions in terms of either COO or b turns out not to be as useful for\nthis class of macromolecules. Therefore it is desirable to have a method to calculate the\ndimensions of such molecules, and particularly to understand the crossover from rod-like to\ncoil\u2014like behavior. Such a scheme is provided by the so\u2014called worm-like chain of Kratky and\nPorod [2]; the fundamental concept is that of the persistence length, 6p, which is a measure of\nhow far along the backbone one has to go before the orientation changes appreciably. A garden\nhose provides a good everyday analogy to a semi\ufb02exible polymer, with a persistence length on\nthe order of 1 ft. A 2 in. section of hose is relatively stiff or rigid, whereas the full 50 ft hose can\nbe wrapped around and tangled with itself many times like a random coil. We will \ufb01rst define 3p\nfor \ufb02exible chains and see how it is simply related to C00. Then we will develop in terms of 6,, an\nexpression for (/12) that can be used to describe \ufb02exible, semi\ufb02exible, and rigid chains.\nThe persistence length represents the tendency of the chain to continue to point in a\nparticular direction as one moves along the backbone. It can be calculated by taking the projection\nof the end-to-end vector on the direction of the \ufb01rst bond (\u201957: /\u20ac is a unit vector in the direction\n"]], ["block_5", ["For the freely jointed chain, as discussed above, all the terms in the expansion in Equation 6.4.1 are\nzero except the \ufb01rst, and thus El, E. For chains with more and more conformational constraints\nthat encourage the backbone to straighten out, more and more terms in the expansion wili\ncontribute positively, and 6p increases. In the limit that every bond points in the same direction\nthe persistence length tends to infinity. When Ep > L, where L n3 is the contour length of the\nchain, such a molecule is called a rigid rod.\n"]], ["block_6", ["Of 61):\n"]], ["block_7", ["Figure 6.6\nExamples of polymer structures that are semi\ufb02exible or stiff chains.\n"]], ["block_8", ["Poly(n\u2014hexyl isocyanate)\nPoly(y-benzyl-L-glutamate)\n"]], ["block_9", ["Poly(p\u2014phenylene)\nPoly(p\u2014phenylene terephthalamide)\n"]], ["block_10", [{"image_3": "240_3.png", "coords": [51, 125, 110, 195], "fig_type": "molecule"}]], ["block_11", [{"image_4": "240_4.png", "coords": [54, 503, 250, 579], "fig_type": "molecule"}]], ["block_12", ["=\u00a7{<:-1>++...+}\n(64-12\n"]], ["block_13", ["Ci )\nNJ\nn\n"]], ["block_14", ["N\nn\n"]], ["block_15", ["2.\n.\n"]], ["block_16", [{"image_5": "240_5.png", "coords": [149, 52, 298, 97], "fig_type": "molecule"}]], ["block_17", [{"image_6": "240_6.png", "coords": [163, 41, 297, 135], "fig_type": "molecule"}]], ["block_18", [{"image_7": "240_7.png", "coords": [190, 121, 267, 201], "fig_type": "molecule"}]], ["block_19", [{"image_8": "240_8.png", "coords": [201, 122, 267, 163], "fig_type": "molecule"}]], ["block_20", ["O\nH\n"]]], "page_241": [["block_0", [{"image_0": "241_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "241_1.png", "coords": [33, 322, 182, 360], "fig_type": "molecule"}]], ["block_2", ["The difference between Equation 6.4.5a and Equation 6.4.5b is not particularly important, espe-\ncially for stiffer chains where CDO>>1; we will use the latter form below, because it is simpler.\nA related quantity in common use is the Kuhn length [3], 6k, which is de\ufb01ned as twice the\npersistence length:\n"]], ["block_3", ["We thus have three different, but fully equivalent expressions for the mean-square unperturbed\n"]], ["block_4", ["Where did the extra E on the right hand side of Equation 6.4.3 come from? Well, the double\ncounting was not quite complete; there were two terms with |x j | 1, two with |x \u2014j| :2, etc.,\nbut only one with x:j. We need this contribution of E in order to obtain 26p, and therefore the\nmissing \u201cself\u201d term is appended to Equation 6.4.3.\nIn order to remove the arbitrary choice of bond )6, we sum over all possible choices of x, and\nassume that we get the same answer for each it; this approximation neglects the effects of chain\nends, so it is valid only in the limit of large n.\n"]], ["block_5", ["end-to\u2014end distance of a \ufb02exible chain:\n"]], ["block_6", ["In some derivations of this relation, the joint limit it \u2014> 00 and E \u2014> 0 is taken (see following\nsection). In this case, the extra 6 on the right hand side of Equation 6.4.3 would vanish, and\nEquation 6.4.5a would become\n"]], ["block_7", ["All three are useful and frequently employed, so they are worth remembering. Estimates of the\npersistence lengths of \ufb02exible polymers are also listed in Table 6.1.\n"]], ["block_8", ["Here we divided by n in front of the double sum because we added 72 identical terms through the\nsum over x. All of these manipulations finally pay off: we recognize this double sum as exactly\n([12) from Equation 6.2.2, and hence\n"]], ["block_9", ["We now seek a relation between 6,, and COO for long, \ufb02exible chains. We can rewrite Equation\n6.4.1 as\n"]], ["block_10", ["where x is any arbitrary bond in the chain. We can make this substitution because for a \ufb02exible\nchain, only a few terms j with small\n|j x| will contribute. Now we change the limits of the sum\nover j to extend over the complete chain; in other words we look in both directions from bond 16.\nThis amounts to a double counting, so we multiply 6,, by 2:\n"]], ["block_11", ["Semiflexible Chains and the Persistence Length\n227\n"]], ["block_12", ["6.4.1\nPersistence Length of Flexible Chains\n"]], ["block_13", [{"image_2": "241_2.png", "coords": [37, 194, 157, 233], "fig_type": "molecule"}]], ["block_14", [{"image_3": "241_3.png", "coords": [38, 99, 210, 139], "fig_type": "molecule"}]], ["block_15", [{"image_4": "241_4.png", "coords": [45, 482, 211, 512], "fig_type": "molecule"}]], ["block_16", ["(2,. 2 2:2,, :C006\n(6.4.6)\n"]], ["block_17", ["2.6, :%\n12-1:\n<4; 4}) + 6\n(6.4.3)\n"]], ["block_18", ["E\n(7,0662 + E :((3,, + 1) \n(6.4.5a)\n13p\n2\n"]], ["block_19", [">>1>>\n>>1>>\nE\nE :_I22\n= \u2014\u2014C\n42 (300\u2014\n6.4.5b\n>P>\n2n}?<\n>\n2m?\n00\u201d\n2\n(\n)\n"]], ["block_20", ["1\n\u201d\n\"\n4\n4\n(2\nepzm;;<\u00a3,-\u20acj>+\u00a7\n(6.4.4)\n"]], ["block_21", ["gp:%i=%:\n(6.4.2)\n1:1\nJ=x\n"]], ["block_22", ["09),, :(3,0622 :N62 Lfk\n(6.4.7)\n"]], ["block_23", ["I\nE\n1\nE\n__\n2\n_:_____\n_2n!3\u2019<\n>+2\n2m.\u201d\n"]]], "page_242": [["block_0", [{"image_0": "242_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "242_1.png", "coords": [28, 262, 263, 333], "fig_type": "molecule"}]], ["block_2", ["So far, all that the persistence length has given us is a new way to express (h2)0 for \ufb02exible chains.\nTo obtain a useful result for semi\ufb02exible and stiff chains, we will return to the freely rotating chain\nof Section 6.2 and transform it into a continuous worm-like chain. This we do by taking a special\nlimit alluded to earlier; we will let the number of bonds, n, go to infinity, but the length of each\nbond, 6, will go to zero, while maintaining the contour length L n6 constant. We begin by\nrelating E], to a cos 6 of the freely rotating chain, starting from Equation 6.4.1:\nmm.a>++~-+}\n1\n= EM +\u20ac2a+\u20ac2a2 +~-+\u20ac2a\u201d_1}\n"]], ["block_3", ["The last transformation utilized the summation results in Equation 6.2.10a, and the large n limit.\nThus we can write\n"]], ["block_4", ["This expression is the result for the worm-like chain obtained by Kratky and Porod [2]. (Note that\nwe took (12 1).? 9: n6 L in Equation 6.4.10.) It is left as an exercise (Problem 6.6) to show\nthat in the coil limit (L >> 61,) this expression reverts to Equation 6.4.7 and that in the rod limit\n(L < \u20ac13), ([12) =L2, as it should. Examples of experimental persistence lengths for semi\ufb02exible\npolymers are given in Table 6.2.\n"]], ["block_5", ["Returning to Equation 6.4.1, we can actually extract a very appealing physical meaning for El,\nand Bk. The terms in the expansion become progressively smaller as the average orientation of\nbond\n1' becomes less correlated with that of the first bond. In fact, when bond 1' is on average\nperpendicular to the first bond (i.e., is uncorrelated), (\u00a3143) a: 0. All higher-order terms will also\nvanish. Thus the persistence length measures how far we have to travel along the chain before it\nwill, on average, bend 90\u00b0. Similarly, the Kuhn length tells us how far we have to go along the\nchain contour before it will, on average, reverse direction completely. Equation 6.4.6 also provides\na simple interpretation for COO: it is the number of backbone bonds needed for the chain to easily\nbend 180\u00b0.\n"]], ["block_6", ["Now we recall Equation 6.2.11 for the freely rotating chain, but retaining the term in a\u201d:\n"]], ["block_7", ["and thus\n"]], ["block_8", ["6.4.2\nWorm-Like Chains\n"]], ["block_9", ["where we invoke the series expansion (see the Appendix):\n"]], ["block_10", ["228\nPolymer Conformations\n"]], ["block_11", [{"image_2": "242_2.png", "coords": [40, 382, 222, 411], "fig_type": "molecule"}]], ["block_12", [{"image_3": "242_3.png", "coords": [42, 482, 188, 513], "fig_type": "molecule"}]], ["block_13", ["ex=1+x+\u00a32+m\n2!\n"]], ["block_14", ["6\na21\u2014exp(\u2014\u20ac/\u20acp)\nfor\n\u20ac\u2014>0\n(6.4.9)\np\n"]], ["block_15", ["2\n__\n21+a_\n2(1\u2014a\u201d)\n(h ) \u2014n\u00a3\u2019\n\u2014\u2014\u2014\u2014\u20141\n\u2014oz\n213\n(10 \u2014o:)2\n"]], ["block_16", ["(1?) 26,11, 2e\u00a7(1 exp[\u2014L/\u00a3p])\nas\ne \u2014+ 0\n(6.4.11)\n"]], ["block_17", [{"image_4": "242_4.png", "coords": [50, 496, 300, 569], "fig_type": "molecule"}]], ["block_18", ["1_\nn\n=\u20ac{1+a+a2+~-+a\u201d*l}=\u20ac(\na):\n6\n(6.4.8)\n1\u20140:\n1\u20140:\n"]], ["block_19", ["2\u2014_\u20ac/\u20acp)\n_\n2\n_\n1\u2014 exp[\u2014L/\u20acp]\n\u2014n\u20ac (\n3/3p\n26 (1\nE/Ep)\n(3/592\n= L\u20acp(2 E/Ep) 263(1 E/\u00e9\u2019p)(1 \u2014 exp[\u2014L/\u20acp])\n(6.4.10)\n"]], ["block_20", ["_\n2\n"]]], "page_243": [["block_0", [{"image_0": "243_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "243_1.png", "coords": [18, 82, 381, 224], "fig_type": "figure"}]], ["block_2", ["Table \nRepresentative Values of Persistence Lengths for Semiflexible Polymers.\nNote That Values May Depend Either Weakly or Strongly on the Choice of Solventa\n"]], ["block_3", ["_,__\u2014\nPolymer\nSolvent\n1?], (231)\n"]], ["block_4", ["Hydroxypropyl cellulose\nDimethylacetamide\n65\nPoly(p-phenylene)b\nToluene\n130\nPoly(p-phenylene \nMethane sulfonic acid0\n100\n96% Sulfuric acid\n180\nPoly(n-hexyl \nHexane\u2018:l\n420\nDichloromethane\n210\nDNA (double helix)\n0.2 M NaCl\n600\nXanthan (double helix)\n0.1 M NaCl\n1200\nPoly(y-benzyl-L-glutamate)\nDimethylformamide\n1500\nSchizophyllan\nWater\n2000\n"]], ["block_5", ["Semiflexible Chains and the \n229\n"]], ["block_6", ["8\u2018Data are summarized in Sato, T. and Polym. Sci, 126, 85, 1996, except for I\u201d\\l\u2019anhee,\nS., Rulkens, R., Lehmann, U., Rosenauer, M.,Wegner, G., Macromolecules,\n29, 5136, 1996; cChu, S.G., Venkatraman, 8., Y., Macromolecules, 14, 939, 1981;\nand dMuralucami, H., Norisuye, T., and Fujita, 13, 345, 1980.\n"]], ["block_7", ["Figure 6.7 shows a plot of a dimensionless form of Equation 6.4.11, obtained by dividing\nthrough by 6% and plotting against L/\u00e9\u2019p. This independent variable is the number of\nlengths in the chain, i.e., an effective degree of polymerization. The curve illustrates \ncrossover from the rod-like behavior at small L/6p, with (h2)0~M2, to the coil\u2014like \nlarge L/Q, with (h2)0 M. Thus the worm-like chain model is able to describe \nsemiflexible chains with one expression. The double logarithmic format of \nemployed in polymer science, when both the independent variable (such as M) and the \n"]], ["block_8", ["(h2)/\u20ac\u00a7\n"]], ["block_9", ["function of the number of persistence lengths \naccording to the Kratky\u2014Porod worm-like chain. The asymptotic \n"]], ["block_10", ["also shown, as is the location of a chain with length equal to \n"]], ["block_11", ["Figure 6.7\nThe mean-square end-to-end distance, normalized by the \n"]], ["block_12", [{"image_2": "243_2.png", "coords": [40, 356, 114, 615], "fig_type": "figure"}]], ["block_13", [{"image_3": "243_3.png", "coords": [49, 362, 300, 619], "fig_type": "figure"}]], ["block_14", ["1o\u20141\n"]], ["block_15", ["10-3;'\nWorm-like chain\n"]], ["block_16", ["10\u20145-\n||||||III\nn\nnunnnnl\nI\nIlllllll\nI\nIIIIIII'\nr\nllll\n-\n"]], ["block_17", ["103;\n"]], ["block_18", ["101\n"]], ["block_19", ["10-2\n10-1\n1o0\n101\n1o2\n103\n"]], ["block_20", [{"image_4": "243_4.png", "coords": [89, 94, 291, 212], "fig_type": "figure"}]], ["block_21", ["L/Ep\n"]], ["block_22", [{"image_5": "243_5.png", "coords": [192, 88, 363, 215], "fig_type": "figure"}]]], "page_244": [["block_0", [{"image_0": "244_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "244_1.png", "coords": [25, 494, 249, 653], "fig_type": "figure"}]], ["block_2", [{"image_2": "244_2.png", "coords": [25, 308, 284, 383], "fig_type": "figure"}]], ["block_3", ["230\nPolymer Conformations\n"]], ["block_4", ["variable can range over several orders of magnitude. If the functional relation is a power law, then\nin this format the plot will be a straight line and the slope gives the power law exponent.\n"]], ["block_5", ["So far we have considered chain dimensions solely in terms of the average end-to-end distance.\nHowever, there are two severe limitations to this approach. First, the end-to-end distance is\ngenerally very difficult to measure experimentally. Second, for many interesting polymer struc-\ntures (e.g., stars, rings, combs, dendrimers, etc.) it cannot even be defined unambiguously. The\nend-to-end distance assigns particular significance to the first and last monomers, but all mono-\nmers are of importance. A useful way to incorporate this fact is to calculate the average distance of\nall monomers from the center of mass. We denote the instantaneous vector from the center of mass\nto monomer i as 5}, as shown in Figure 6.8. The center of mass at any instant in time for any\npolymer structure is the point in space such that\n"]], ["block_6", ["Here, just as with the end-to-end vector, it is useful to take (5?) (5} 5}) in order to obtain\nan average distance rather than an average vector (which would zero by isotropy). In the second\ntransformation we have assumed equal masses, i.e., a homopolymer, and m,-=m cancels out.\n(Note that the summations run up to N, the number of monomers, and not 11, the number of\nbackbone bonds.) It is worth mentioning that the term radius of gyration is unfortunate, in that it\ninvites confusion with the radius of gyration in mechanics; the latter refers to the mass-weighted,\nroot-mean-square distance from an axis of rotation, not from a single point. However, the term\nradius of gyration in reference to Equation 6.5.2 appears to be firmly entrenched in polymer\n"]], ["block_7", ["6.5\nRadius of Gyration\n"]], ["block_8", ["where m,- is the mass of monomer 5. Note that the center of mass does not need to be actually on\nthe chain (in fact, it is unlikely to be). The root-mean-square, mass-weighted average distance of\nmonomers from the center of mass is called the radius ofgyration, Rg, or (52) 1/2, and is determined by\n"]], ["block_9", ["Figure 6.8\nIllustration of the vectors from the center of mass to monomers i and j, E} and 5}, respectively,\nand the vector from monomer 1' to monomer j, Ff}.\n"]], ["block_10", [{"image_3": "244_3.png", "coords": [40, 228, 106, 265], "fig_type": "molecule"}]], ["block_11", ["N\n1/2\nE\n.-\nJ;\n1/2\nRg:\n(52>l/2 \niii-25:2\n=\nfif\u2014XN:\n(6.52)\n"]], ["block_12", ["N\nEms;- 0\n(6.5.1)\n"]], ["block_13", ["i=1\n"]], ["block_14", [{"image_4": "244_4.png", "coords": [171, 320, 270, 364], "fig_type": "molecule"}]]], "page_245": [["block_0", [{"image_0": "245_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["This equation turns out to be a useful alternative de\ufb01nition of Rg. It expresses Rg in terms of the\naverage distances between all pairs of monomers in the molecule; the location of the center of mass\nis not needed. Furthermore, Equation 6.5.8 is valid for any structure; it need not be a linear chain,\nand it need not have unperturbed dimensions.\nNow we can derive the specific result Equation 6.5.3 for the freely jointed chain by realizing that\n"]], ["block_2", ["This expression equals zero because\n"]], ["block_3", ["from Equation 6.5.1 (assuming all masses are equal). Returning to the second part of Equation\n6.5.5, and utilizing Equation 6.5.2, we obtain\n"]], ["block_4", ["where d) is the angle between E,- and 5}, and r?!- is the square distance between monomers i and j\n(see Figure 6.8). Now we take the average of each term in Equation 6.5.4, and then double sum\nover i and j:\n"]], ["block_5", ["which can be rearranged to\n: \ufb02\nEN;\nEN; (r5)\n(6.5.8)\n"]], ["block_6", ["science. A more correct description will emerge in Section 6.7, namely that (52) is the \u201csecond\nmoment of the monomer distribution about the center of mass,\u201d but this terminology is rather\nunwieldy for daily use.\nIt is clear that Rg can be defined for any polymer structure, and thus avoids the second objection\nto ([12) listed above. It can be measured directly by light scattering techniques, as will be described\nin Chapter 8, and indirectly through various solution dynamics properties, as explained in Chapter\n9, thereby avoiding the first objection. However, we went to some trouble to calculate (h2)0 for\nvarious chains, and to establish the utility of C00, b, and 6p. Is that all out the window? No, it is not.\nWe will now show that is, in fact, very simply related to (h2)0 for an unperturbed linear chain,\nnamely\n"]], ["block_7", ["Consider the dot product of the vectors from the center of mass to any two monomers i andj. By\nthe law of cosines\n"]], ["block_8", ["Radius of Gyration\n231\n"]], ["block_9", [{"image_1": "245_1.png", "coords": [38, 477, 229, 519], "fig_type": "molecule"}]], ["block_10", [{"image_2": "245_2.png", "coords": [39, 177, 154, 214], "fig_type": "molecule"}]], ["block_11", [{"image_3": "245_3.png", "coords": [42, 533, 142, 574], "fig_type": "molecule"}]], ["block_12", ["1 [3,? +512 \u2014r-2]\n(6.5.4)\n\ufb01'\u00e9\u2018IrZIE'z-llilws\u00e9:\n.\n2\nJ\n"]], ["block_13", ["N\nN\n1\nN\n1\nN\n1\nN\nN\n2245'?\u00bb25233536226372Z=0\n(65-5)\n"]], ["block_14", ["ii@ '51) \n(656)\n\u00a321\n"]], ["block_15", ["(r3) : |i \u2014j|b2\n(6.5.9)\n"]], ["block_16", ["g \n= \u2014 = \u2014\n(6.5.3)\n"]], ["block_17", [{"image_4": "245_4.png", "coords": [55, 405, 262, 440], "fig_type": "molecule"}]], ["block_18", ["121\nj=l\n"]], ["block_19", ["N\n"]], ["block_20", [{"image_5": "245_5.png", "coords": [88, 335, 371, 389], "fig_type": "molecule"}]], ["block_21", [" <53>25:: \n(6.5.7)\n"]], ["block_22", ["I:\nj:\n"]]], "page_246": [["block_0", [{"image_0": "246_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "246_1.png", "coords": [26, 76, 248, 130], "fig_type": "molecule"}]], ["block_2", ["which comes from the fact that there are 2(N\u20141) terms where\n|1' \u2014-j| 1; 2(N 2) terms where\nIi \u2014j| 2; 2(N 3) terms where 11' \u2014j| = 3; etc. (If this seems mysterious, draw an N x N matrix,\nwhere the rows are numbers 1': 1 . . .N and the columns are j 1 ... N. For each matrix element,\nenter If \u2014j|. There will be N 0\u2019s along the main diagonal, N 1 1\u20193 immediately adjacent to [and\non both sides of] the main diagonal, N\u20142 2\u2019s in the next place over, etc.). Therefore\n"]], ["block_3", ["232\nPolymer Conformations\n"]], ["block_4", ["or, in other words, (\u20193) represents the end-to-end mean\u2014square distance between any pair of\nmonomers 1' andj separated by k Ii j | links. We can thus write\n"]], ["block_5", ["A useful rule of thumb for polymers15 that Rg is about 100 A when M\u2014\n\u2014 105 g./mol This number\ncan be used to estimate R3 for any other M by recalling the pr0portionality of R3 to M112.Use the\ndata in Table 6.1 to assess the reliability of this rule of thumb.\n"]], ["block_6", ["We can take the values for ghz)O/M directly from the fifth column and multiply each one by 105.\nThe largest will be 125,000 A2 for polyethylene, and the smallest will be 42,500 A2 for poly(methyl\nmethacrylate). Then we need to divide by 6 and take the square root to obtain R.\n"]], ["block_7", ["The straight line fit to the data gives Rg =0.25 M051.To make things convenient, we can choose\nN\u2014\n\u2014 105, for which M\u2014\n\u2014 104 x 105\u2014\n-\u2014 104 X 107 g/mol. From the fitting equation we obtain\nRg =948 A. The number of backbone bonds :1 =2 x 105, and we use a more precise estimate\nof the bond length of 1 5.3 A. From Equation 6.3.2 and Equation 6.5.3, then\n\ufb02_\n6 x (948)\n_\n= 11.5\n462\n2 x 105(153)2\nCOO:\n"]], ["block_8", ["where once again the last formula applies in the high N limit.\n"]], ["block_9", ["All of the other polymers in Table 6.1 will give values between these two. We may conclude that the\nrule of thumb is reliable to at least one significant \ufb01gure, and is better than that for many polymers.\n"]], ["block_10", ["Use the experimental data for Rg for polystyrenes dissolved in cyclohexane in Figure 6.9 to\nestimate Coo, 6p, and b. Note that these data are for remarkably large molecular weights.\n"]], ["block_11", ["Example 6.2\n"]], ["block_12", ["Solution\n"]], ["block_13", ["Solution\n"]], ["block_14", ["Example 6.3\n"]], ["block_15", [{"image_2": "246_2.png", "coords": [42, 635, 209, 672], "fig_type": "molecule"}]], ["block_16", [{"image_3": "246_3.png", "coords": [42, 190, 249, 231], "fig_type": "molecule"}]], ["block_17", ["2\nN-l\nRg= ZNb\u2014622Ii\u2014J\n(6.5.10)\n"]], ["block_18", ["For polyethylene 11g_\u2014(125,\u2014_\u2014000/6)\u2018/2 144 A\n"]], ["block_19", ["For poly(methyl methacrylate), Rg (42,500/6)\u201d84 A\n"]], ["block_20", ["N2(N-\u20141)\nN(N\u20141)(2N\u20141)\n:=;_:{N;1_;kz}=;_:{\n2\n6\n}\n"]], ["block_21", ["\u2014 1) \u2014 (N \u20141)(2N2\n\u2014N)}\n(6.5.11)\nN2\n6\n_N62\nb2\nN62\n\u201cYT\u2014671\u2019\u201d?\n"]], ["block_22", ["_ \n{3N2(N\n"]], ["block_23", [{"image_4": "246_4.png", "coords": [72, 86, 238, 117], "fig_type": "molecule"}]], ["block_24", ["k=1\ni=lj=\n"]]], "page_247": [["block_0", [{"image_0": "247_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "247_1.png", "coords": [24, 529, 256, 578], "fig_type": "molecule"}]], ["block_2", [{"image_2": "247_2.png", "coords": [30, 407, 188, 441], "fig_type": "molecule"}]], ["block_3", ["These values are systematically larger than those given in Table 6.1. Part of this difference may be\nattributed to experimental uncertainty, but most of the difference stems from the fact that the\npolystyrene data in Table 6.1were obtained from molten polystyrenes, whereas the data in Figure\n6.9 are for dilute solutions. Although the chain dimensions in a dilute theta solution and in the bulk\nboth increase as M 2, the prefactor (e.g., C0c,) can be slightly different. In fact, the dimensions of a\ngiven polymer may differ by as much as 10% between two different theta solvents.\n"]], ["block_4", ["Figure 6.9\nRadius of gyration for very high molecular weight polystyrene in cyclohexane at the theta\ntemperature. (Data from Miyake, Y., Einaga, Y., and Fujita, M., Macromolecules, 11, 1180, 1978.)\n"]], ["block_5", ["Also from Equation 6.3.2 and Equation 6.5.3,\nz\ufb01R =\u2014\u2014\u2014\u2018/3\nx/N\ng\n\\/1><105\n"]], ["block_6", ["This result can be derived from the expression for (I12), Equation 6.4.11, by way of the relation\n6.5.8 and a transformation of sums to integrals; this is left as Problem 6.8. An example of the\napplication of the worm-like chain model is shown in Figure 6.10. The material is poly(n-hexyl\nisocyanate) (see Figure 6.6) dissolved in hexane and the coil dimensions were measured by light\nscattering (see Chapter 8). The smooth curve corresponds to Equation 6.5.12 with a persistence\nlength of 42 mn, and the contour length determined as L (nm) =M (g/mol)/715 (g/mol/nm). The\nfactor of 715 therefore re\ufb02ects the molar mass per nanometer of contour length. The correspond\u2014\nence between the data and the model is extremely good, except for the two very highest M samples.\n"]], ["block_7", ["Finally, from Equation 6.4.5b we have\n"]], ["block_8", ["Radius of Gyration\n233\n"]], ["block_9", ["H, (A)\n"]], ["block_10", [{"image_3": "247_3.png", "coords": [37, 356, 205, 394], "fig_type": "molecule"}]], ["block_11", ["4 x 103 \n-\nPolystyrene in cyclohexane at 345\u00b0C\n"]], ["block_12", ["2x103-\n-\n"]], ["block_13", ["3 x 102\n-\nHg = 0.25 lug-51\n"]], ["block_14", ["6x102\u2014\n_\n"]], ["block_15", ["b\nx948=7.3.&\n"]], ["block_16", ["The worm-like chain of Section 6.4.2 also has an expression for R2, which is\n"]], ["block_17", [{"image_4": "247_4.png", "coords": [46, 40, 287, 297], "fig_type": "figure"}]], ["block_18", ["3x103-\n-\n"]], ["block_19", ["1x103\n\u2014\n"]], ["block_20", ["g \n5\np\n\u2014\u2014\np \nE\u2014z\u2014(exp[\u2014\u2014L/\u20acp]\n\u2014 1) +7\n(6.5.12)\n"]], ["block_21", ["8X1051X107\n2x107\n4x107\n6X107\nMW\n"]], ["block_22", ["6\n1053\no\n"]]], "page_248": [["block_0", [{"image_0": "248_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["We have derived results for (h2)0 for coils, rods, and everything in between. The conformation is\nassumed to be determined by the steric constraints induced by connecting the monomers chem-\nically. But we could also think about the molecule more abstractly, as a series of freely jointed\neffective subunits. Now suppose we have the ability to \u201cdial in\u201d a through-space interaction\nbetween these subunits, an interaction that might be either attractive or repulsive. The former\ncould arise naturally through dispersion forces, for example; all molecules attract one another in\nthis way, and if we put our chain in a vacuum, those forces might dominate. If the attractive\ninteraction were sufficiently strong, the chain could collapse into a dense, roughly spherical ball or\nglobule. Repulsion could arise if each subunit bore a charge of the same sign, a so-called\npolyelectrolyte. This is commonly encountered in biological macromolecules, DNA for example.\nIf the repulsive interaction were sufficiently strong, the chain could extend out to be a rod. It is\ninstructive to think of the globule, coil, and rod as the three archetypical possible conformations of\na macromolecule, and for many systems coil\n<\u2014> globule and coil\n<\u2014> rod transitions are experi-\nmentally accessible. For example, proteins in their native state are often globular, but upon\ndenaturing the attractive interactions that cause them to fold are released, and the molecule\nbecomes more coil-like. Similarly, a synthetic, neutral polymer dispersed in a bad solvent will\ncollapse into a globule when it precipitates out of solution. A relatively short DNA double helix is\n"]], ["block_2", ["Figure 6.10\nRadius of gyration versus molecular weight for poly(n-hexyl isocyanate) in hexane. The curve\ncorresponds to the worm-like chain model with a persistence length of 42 nm. (Data from Murakami, H.,\nNorisuye, T., and Fujita, H., Macromolecules, 13, 345, 1980.)\n"]], ["block_3", ["This deviation may be attributed to the onset of excluded volume effects, whereby the coil\nconformations are larger than anticipated by the freely jointed chain (Gaussian) limit.\n"]], ["block_4", ["6.6\nSpheres, Rods, and Coils\n"]], ["block_5", ["234\nPolymer Conformations\n"]], ["block_6", ["\u201cE\n104\n_\u2014\n1\n5\n_\n_\n"]], ["block_7", ["cum\n\"\n'\n0:\nI\nZ\n"]], ["block_8", [{"image_1": "248_1.png", "coords": [48, 40, 333, 309], "fig_type": "figure"}]], ["block_9", ["1\u00b0\n:\nWorm-like chain with 6,, = 42 nm\n3\n"]], ["block_10", ["1C)5\n.\u2014\n\u2014_\n"]], ["block_11", ["-\nJ\n"]], ["block_12", ["-\n"]], ["block_13", ["_\n.\n_\n"]], ["block_14", ["I\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\n1\n.I\nI\nI\nI\nI\nI\nI\nI\n"]], ["block_15", ["Poiy(n-hexy| isocyanate) in hexane\n"]], ["block_16", ["105\n1o6\n107\n"]], ["block_17", ["1\nI\nI\nI\nI\nI\nI\nI\n"]], ["block_18", ["M (g/mol)\n"]], ["block_19", ["-\n"]]], "page_249": [["block_0", [{"image_0": "249_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "249_1.png", "coords": [13, 607, 243, 680], "fig_type": "figure"}]], ["block_2", ["and thus the radius goes as (volume)\u201d3. For the unperturbed coil V: 1/2, as we have seen.\nHowever, we will find out in Chapter 7 that in a good solvent V as 3/5 due to the excluded volume\neffect. For a rod, clearly I\u00bb 1. Equation 6.6.1 is an example of a scaling relation; it expresses the\nmost important aspect of chain dimensions, namely how the size varies with the degree of\npolymerization, but provides no numerical prefactors. The value of the exponent is universal, in\n"]], ["block_3", ["reasonably rod-like (Epw600 A), but if the double helix is denatured or \u201cmelted,\u201d the two\nseparated strands can become coils. A poly(carboxylic acid) such as poly(methacrylic in\nwater will have a different density of charges along the chain as the pH is varied. At high pH,\nvirtually every monomer charge and although the chain willout\ninto a completely rigid, chain, it will show a size that scales linearly\nwith M rather than as \\//1_/I. In short, polymers can adopt conformations varying from spheres\nthrough \ufb02exible coils to rods.\nThe scaling of the size with molecular weight is quite different in each case. We can encompass the\nvarious possibilities by writinga proportionality between the size and the degree of polymerization:\n"]], ["block_4", ["the sense that any particular value of 12 (1/3, 1/2, 3/5, or 1) will apply to all molecules in the\nsame class.\nAs illustrated at the very beginning of this chapter, the size of these various structures (globule,\ncoil, and rod) would be very different for a given polymer. Then we considered a polyethylene\nmolecule, which in the liquid state will always be a coil. Now consider a representative DNA from\nbacteriophage T2. It has a contour length of 60 um or almost 0.1 mm. As a coil, it should have\n# (Lk\n/6)1/2 N 1 mm or 10,000 A. With MN 108 and assuming a density of 1 g/mL, it would\nform a dense sphere with a radius of 340 A. It15 an amazing fact that the bacteriophage actually\npackages the DNA molecule to almost this extent. As shownin Figure 6.11, upon experiencing an\nosmotic shock the bacteriophage releases the DNA, which had been tightly wound up inside its\nhead. The mechanism by which the DNA is packed so tightly remains incompletely understood. It\nis particularly remarkable, given that DNA carries negative charges all along its contour, which\nshould create a strong repulsion between two portions of helix. This example also underscores\nanother important point about chain dimensions: they can be very sensitive to the environment of\nthe molecule, and not only to the intramolecular bonding constraints.\n"]], ["block_5", ["For the globule or dense object,V 1/3. The volume occupied by the molecule is proportional to N\n"]], ["block_6", ["I) will have an end-to\u2014end vector, h, lying between h and h + dh, as illustratedin Figure 6.12a. In\nother words, if the start of the chain defines the origin, we want the probability that end\nfalls in an infinitesimal box with coordinates between .1: and x + dx, y and y + dy, andandz + dz.\nFrom such a function, we will be able to obtain related functions for the probability P(N, h) dh that\nthe same walk has an end-to\u2014end distance, /1 [/1], lying between 11 and h + dh, and the probability\np(N, r) dr that a monomer will be found between a distance r and r + dr from the center of mass. It\nturns out that all of these distributions are approximately Gaussian functions, just like familiar\nnormal distribution for error analysis. In particular, the answer for P(N, It)IS\n"]], ["block_7", ["So far in this chapter we have only considered the average size and conformation of a polymer.\nNow we will \ufb01gure out how to describe the distribution of sizes or conformations for a particular\nchain. We seek an expression for the probability P(N, h) dh that a random walk of N steps of length\n"]], ["block_8", ["6.7\nDistributions for End-to-End Distance and Segment Density\n"]], ["block_9", ["Distributions forSegment \n235\n"]], ["block_10", ["a result that we will now derive.\n"]], ["block_11", ["R, N\u201d\n(6.6.1)\n"]], ["block_12", ["_ \n2N!)2\n(6.7.1)\n#\n3/2\nP(N,h) [ZaZ]\nex\n"]], ["block_13", [{"image_2": "249_2.png", "coords": [56, 614, 209, 654], "fig_type": "figure"}]]], "page_250": [["block_0", [{"image_0": "250_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "250_1.png", "coords": [30, 477, 367, 636], "fig_type": "figure"}]], ["block_2", ["236\nPolymer Conformations\n"]], ["block_3", ["Kleinschmidt, A.K., \n"]], ["block_4", ["Figure 6.11\nThe DNA within a single \n"]], ["block_5", ["We begin with a one-dimensional \n"]], ["block_6", ["6.7.1\nDistribution of the End-to-End \n"]], ["block_7", ["step we go a distance\n"]], ["block_8", [{"image_2": "250_2.png", "coords": [37, 476, 185, 634], "fig_type": "figure"}]], ["block_9", ["Figure 6.12\nA \ufb02exible coil with one \n"]], ["block_10", ["a spherical shell of volume 4nh2\n"]], ["block_11", [{"image_3": "250_3.png", "coords": [202, 479, 321, 625], "fig_type": "figure"}]], ["block_12", ["(b)\n"]], ["block_13", ["m\\\n"]]], "page_251": [["block_0", [{"image_0": "251_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "251_1.png", "coords": [25, 264, 187, 307], "fig_type": "molecule"}]], ["block_2", ["We insert a proportional sign here to allow for the appropriate normalization (see below). Now\nto convert to a three-dimensional N-step random walk, we take N/3 steps along x, N/3 along y, and\nNB along 2', and recognize that the probabilities along the three directions are independent.\n"]], ["block_3", ["or\n"]], ["block_4", ["The expression is simplified by means of Stirling\u2019s large N approximation, namely\n"]], ["block_5", ["and realize that (x/bN) \u2014> 0 as N gets very big. Thus ln(l +x) sex applies\n"]], ["block_6", ["which gets rid of the nasty factorials. (It is worth noting that Stirling\u2019s approximation is excellent\nwhen N is on the order of Avogadro\u2019s number, but it is not quantitatively accurate for N m 100;\nnevertheless these errors largely cancel in deriving the Gaussian distribution.) Utilizing this,\nEquation 6.7.4 can be expanded:\n"]], ["block_7", ["after some algebra. Now we recall the expansion of ln(1 +x) when x << 1, namely\n"]], ["block_8", ["This expression arises as follows. The probability of a sequence of events that are independent is\nequal to the product of the probabilities of each event; this gives the factors of (1/2)\u201d and (1/2)?\nHowever, this would underestimate what we want, namely a net displacement of x. This is because\n(1/2)p(1/2)q assumes we have all p+ steps in succession, followed by (3\u2014 steps, whereas in fact the\norder of the individual steps does not matter, only the total p and q. There are N possible choices\nfor the first step, then N l for the next and so on, which increases the total probability by a factor\nof N!. This, however, now overestimates the answer, because all of the p+ steps are indistinguish-\nable, as are all of the q\u2014 steps. There are p! possible permutations of the+ steps, all of which\nwould give the same answer, and similarly q! permutations of the steps; both of these factors are\ncounted in N!, so we have to divide by them out. Thus we arrive at Equation 6.7.2.\nNow we make the simple substitutions\nFilm! Far!)\n"]], ["block_9", ["to obtain\nr\n1\nN\n2!\n2!\nP(N,x) _ (2) N!\n(ME)! (N g)!\n(67.4)\n"]], ["block_10", ["being 1/2. At the end of the N steps, let p be the total number of + steps and q be the total number\nof steps. Clearly N p + q and the net distance traveled will be x =b( p q). The probability of\nany given outcome for this kind of process is given by the binomial theorem:\n"]], ["block_11", ["Distributions for End-to-End Distance and Segment Density\n237\n"]], ["block_12", [{"image_2": "251_2.png", "coords": [41, 95, 162, 135], "fig_type": "molecule"}]], ["block_13", ["P N\n\"x2\n6 7 9\n(\n,X) exp W\n(\n-\n-\n)\n"]], ["block_14", ["1\nl\nln(l+x):x\u2014\u00a7x2+\u00a7x3~-\n(6.7.7)\n"]], ["block_15", ["1nP(N,x) \u2014% (N + 2) 1%! + 3%) g (1 biN) 1%! {9%)\n(6.7.6)\n"]], ["block_16", [{"image_3": "251_3.png", "coords": [46, 311, 218, 355], "fig_type": "molecule"}]], ["block_17", ["1\nP\n1\nq N!\nP(N,x)= (2) (5)\npl\u2014q!\n(6.7.2)\n"]], ["block_18", ["In N! e N In N N\n(6.7.5)\n"]], ["block_19", ["lnP(N,x) : -%(N+g) (7%) \u00a3073) (Nib)\n_x2\n(6.7.8)\nI 2s\n"]], ["block_20", [{"image_4": "251_4.png", "coords": [147, 444, 324, 475], "fig_type": "molecule"}]]], "page_252": [["block_0", [{"image_0": "252_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "252_1.png", "coords": [10, 59, 291, 117], "fig_type": "molecule"}]], ["block_2", ["and thus from the triple integral of Equation 6.7.11 we find that the result is [3/21TNb2]_3/2.\nTherefore we need to multiply our exponential factor by [3/2'rrN1'32]3\u201d2 to satisfy Equation 6.7.11\nand arrive at the result given in Equation 6.7.1.\nThis Gaussian distribution function for I; is plotted against IE] in Figure 6.13a. Although it is a\ndistribution function for a vector quantity, it only depends on the length of h; this is a natural\nconsequence of assuming that x, y, and 2 steps are equally probable. It is peaked at the origin,\nwhich means that the single most probable outcome is I; 0, Le, the walk returns to the origin.\nHowever, because of the prefactor, the probability of this particular outcome shrinks as N2, even\nthough it is more likely than any other single outcome. Finally, this expression for P(N, I?) was\nobtained by assuming large N (no surprise here). How large does N have to be for the Gaussian\nfunction to be useful? It turns out that even for N a: 10, the real distribution for a random walk\nlooks reasonably Gaussian. Already for very small N it is symmetric and peaked at the origin, but it\n"]], ["block_3", ["where we have inserted |h|2 :x2 + y2 + 22. The final step is the normalization, which accounts for\nthe fact that if we look over all space for the end of our walk, we must find it exactly once. This is\nexpressed by\n"]], ["block_4", ["Therefore\n"]], ["block_5", ["Figure 6.13\nGaussian probability distributions for a chain of N steps of length I), plotted as (a) the\nprobability of an end\u2014to-end vector k versus lit] and (b) the probability of an end-to-end distance 11.\n"]], ["block_6", ["You can find in a table of integrals that\n"]], ["block_7", ["238\nPolymer Conformations\n"]], ["block_8", [{"image_2": "252_2.png", "coords": [38, 218, 155, 257], "fig_type": "molecule"}]], ["block_9", [{"image_3": "252_3.png", "coords": [38, 428, 262, 621], "fig_type": "figure"}]], ["block_10", ["a><10r6\n\u2014\na4::\n113\u2018\n\u00a30.02\u2014\n-l\ng\n.5\n_\nIf?\nD.\n6X10\ng\nL\n_.1\nEL\n*5\n_.\n4X10\n0.01 k\n\u2018\n"]], ["block_11", ["a\nN\nN\nN\nh :P \nP\n\u2014\nP\n\u2014\nm\nPW,\n)\n(3,26)\n(3.y)\n(3.2)\next)\n"]], ["block_12", ["\u201400 \u2014oo \u201400\n"]], ["block_13", ["r-OO\n"]], ["block_14", [{"image_4": "252_4.png", "coords": [46, 71, 258, 109], "fig_type": "molecule"}]], ["block_15", ["1.2)(10\u20145\nl\nl\nl\nr\nl\n_l\u2014\nI\nl\n"]], ["block_16", ["000000\n"]], ["block_17", ["00\n"]], ["block_18", ["J\nJ\nJP(N,1?)dx,dy,dz=1\n(6.7.11)\n"]], ["block_19", ["J exp(\u2014kx2) dx \ufb02\n"]], ["block_20", ["2x10\u201d6\n_\n__\n_\n"]], ["block_21", [{"image_5": "252_5.png", "coords": [53, 416, 431, 627], "fig_type": "figure"}]], ["block_22", ["0.03L\n1><1O\u20185\n-\nN=1000\n7\n"]], ["block_23", ["(a)\n(b)\n"]], ["block_24", ["O\n0\nI\n1\n1\n"]], ["block_25", ["100\n0\n20\n40\n50\n80\n100\n"]], ["block_26", ["\u20183lh12\n2s\n(6.7.10)\n"]], ["block_27", [{"image_6": "252_6.png", "coords": [238, 421, 435, 631], "fig_type": "figure"}]], ["block_28", ["#\nb=l\n_\n"]]], "page_253": [["block_0", [{"image_0": "253_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Note that the normalization integral goes from 0 to 00, as It cannot be negative.\nThis distribution function is plotted in Figure 6.13b and is rather different from P(N,}i). In\nparticular, it vanishes at the origin and has a peak at a finite value of h before decaying to zero as\nN \u2014> 00. The fact that it vanishes at the origin is due to multiplying the exponential decay by hz.\nThere is thus a big difference between finding the most probable vector position (which is the\norigin) and finding the most probable distance (the position of the peak in P(N,h), see Problem\n6.10). You have probably encountered this contrast before, for example, in the radial distribution\nfunction for the s electrons of a hydrogen\u2014like atom, or for the Maxwell distribution of molecular\nvelocities in a gas. In the former, the most probable position of the electron is at the nucleus, but\nthe most probable distance is a finite quantity, the Bohr radius. In the latter, the most probable\nvelocity in the gas is zero, but the most probable speed is finite.\nAs a simple application of this distribution function, we can ask what is the mean square\nvalue of h?\n"]], ["block_2", ["This function is already normalized correctly. It should satisfy the one-dimensional integral\n"]], ["block_3", ["and you can show that it does, armed with the knowledge that\n"]], ["block_4", ["will be \u201cbumpy\u201d because has a discrete value. As N gets larger, the fact that b is discrete\nbecomes less and less although the fact that we are representing a discrete function by a\ncontinuous one never away for finite N. Indeed, we can assert that even though the Gaussian\ndistribution may provide numerical answers that are very accurate, it can never be exactly\ncorrect for a real chain. That is because a real chain has a finite contour length that Hi] can never\nexceed, whereas Equation 6.7.1 provides a finite probability for any value of Iii] all the way out\nto infinity.\n"]], ["block_5", ["We now turn to obtaining P(N, h) for the distance h. The transformation is illustrated in Figure\n6.12b. We consider a spherical shell at a distance 11 from the origin. It has a surface area 4'rrh2 and a\nthickness dh, so its volume is 41't dh. Any walk whose end\u2014to\u2014end vector 1; lies in this shell will\nhave the same length h, so the answer we seek is just\n"]], ["block_6", ["6.7.2\nDistribution of the End-to-End Distance\n"]], ["block_7", ["Distributions for End-to-End Distance and Segment Density\n239\n"]], ["block_8", [{"image_1": "253_1.png", "coords": [39, 572, 261, 677], "fig_type": "figure"}]], ["block_9", ["00\n1\n1T\nsexp(\u2014kx2) dx If I\n0\n"]], ["block_10", ["JP(N,h) dh :1\n(6.7.13)\n"]], ["block_11", ["0\n"]], ["block_12", ["P(N, h) dh :4171121007, 7?) dh\n"]], ["block_13", ["(722) Jh2P(N,h)dh\n"]], ["block_14", [{"image_2": "253_2.png", "coords": [60, 614, 247, 669], "fig_type": "figure"}]], ["block_15", [{"image_3": "253_3.png", "coords": [78, 238, 249, 285], "fig_type": "figure"}]], ["block_16", ["00\na \n(6.7.14)\n"]], ["block_17", ["0\n"]], ["block_18", ["0\n"]], ["block_19", ["\u2018\n(6.7.12)\ndh\n:4 I22 \nTr\n[211's\n"]], ["block_20", ["2\n2\n3\n3/2\n\u20143 h\n"]], ["block_21", ["3/2\n4\u2018}?\n]\nCXP 2s\n"]], ["block_22", ["2\n"]]], "page_254": [["block_0", [{"image_0": "254_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "254_1.png", "coords": [30, 584, 211, 651], "fig_type": "figure"}]], ["block_2", ["The last thing we consider in this section is the related distribution p(N, r) dr, the probability that a\nmonomer is between r and r+dr from the center of mass. It turns out that there is no simple\nanalytical expression for this distribution, even for a Gaussian chain [4], but the resulting\ndistribution is well approximated by a Gaussian:\n"]], ["block_3", ["Thus for a solid sphere\n"]], ["block_4", ["The integral in the denominator provides the necessary normalization. As an example, consider\na solid sphere with radius R0 and uniform density p0. The distribution function for p(f\u2019) is just a\nconstant, p0, for OErSRO, and O for r>R0. (Note that this simple function is p07), not p(r)\nbecause the latter must increase as r2 for r SR0; there are more monomers near the surface of the\nsphere than at the center.) Substituting p07) into Equation 6.7.18 we obtain\n"]], ["block_5", ["where we use the second moment (52) :Nb2/6 explicitly in the expression (compare Equation\n6.7.1, wheres could have been replaced by 012)). One important point is the factor ofN in front.\nThis is simply a new normalization so that\n"]], ["block_6", ["Applying this in Equation 6.7.14 we obtain\n"]], ["block_7", ["240\nPolymer Conformations\n"]], ["block_8", ["6.7.3\nDistribution about the Center of Mass\n"]], ["block_9", ["as expected. The quantity (/12) is also called the second moment of the distribution P(N, h). (You\nmay recall the discussion of moments in the context of molecular weights in Chapter 1.7.)\n"]], ["block_10", ["which re\ufb02ects the fact that when we look for monomers over all space, we must find all N of them.\nThe segment density distribution can also be used for a solid object, where there is no need to\nidentify N separate subunits. In such a case p(r) can be used to find Rg (32)\u201d2, from\n"]], ["block_11", ["where now we need to know\n"]], ["block_12", [{"image_2": "254_2.png", "coords": [35, 449, 192, 529], "fig_type": "molecule"}]], ["block_13", [{"image_3": "254_3.png", "coords": [42, 586, 205, 650], "fig_type": "molecule"}]], ["block_14", ["N,\n: N4\n\u2014\n\u2014\u2014-\n6.7.16\n6\nr)\n\u201cr i2wi\n\u201dp i2i\n(\n)\n"]], ["block_15", ["00\n3\nit\nJx4 exp(\u2014kx2) dx w\nE\n"]], ["block_16", ["<52) 000\n2 30\n(6.7.18)\nI p(r) dr\nI 411731007) dr\n0\n0\n"]], ["block_17", ["R0I 904117\u201d4 dr\nR5 5\n3\n(32) 20\n= R2/3 = 5R3\n(6.7.19)\nI p04'rrr2 dr\n0/\n"]], ["block_18", ["3\n3/23\n3\n\u20182\n2s \n2\n2\n=\n_.\n=\n.7.\n(h)\n4w[2w2]\n8(2Nb2) \ufb01<\n3 \nNb\n(6\n15)\n"]], ["block_19", ["Jp(N, r) dr N\n(6.7.17)\n"]], ["block_20", ["0\n"]], ["block_21", ["T r2p(r) dr\n3? 41174p(F') dr\n"]], ["block_22", ["0\n"]], ["block_23", [{"image_4": "254_4.png", "coords": [96, 140, 276, 174], "fig_type": "molecule"}]]], "page_255": [["block_0", [{"image_0": "255_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "255_1.png", "coords": [0, 60, 373, 304], "fig_type": "figure"}]], ["block_2", [{"image_2": "255_2.png", "coords": [28, 71, 210, 309], "fig_type": "figure"}]], ["block_3", ["This result serves to emphasize an important point that Rg re\ufb02ects an average of the monomer\ndistribution and not the total Spatial extent of the object. There are always monomers further than\nRg from the center of mass as well as monomers closer than R3 to the center of mass. Expressions\nfor Rg for other shapes are listed in Table 6.3.\n"]], ["block_4", ["We mentioned at the end of Section 6.2 that there is a further important issue in chain conforma-\ntions, that of excluded volume. The simple fact is that a real polymer occupies real Space and no\ntwo monomers in the chain can share the same location concurrently. This means that the statistics\nof the conformation are no longer those of a random walk, but rather a self-avoiding walk. This\ndifference might appear subtle, which it is, but the consequences are profound.\nMost importantly, we can no longer write down an analytical expression for any of the desired\ndistribution functions such as P(N,h). Nor can we calculate in any simple yet rigorous way the\ndependence of Rg on N. The mathematical reason for this difficulty is the way the problem\nbecomes more complicated as N increases. In the case of the random walk, the orientation of\nany link or step is dictated by random chance and its position in space is determined only by where\nthe previous link is. For the self\u2014avoiding walk, in contrast, we would need to ask where every\nprevious link is in order to establish whether a particular orientation would be allowed for the link\nin question. It would not be allowed if it intersected any previous link. Consequently, the\ncalculation becomes more and more complicated as N increases. Some very sophisticated mathe-\nmatics has been employed on this problem, but we will not discuss this at all. We can draw an\nimportant qualitative conclusion, however: the excluded volume effect will tend to make the\naverage coil size larger, as the chain seeks conformations without self\u2014intersections. From the\nmost sophisticated analysis, nNO'SBQ instead of NW; in other words the exponent\n12 from\nEquation 6.6.1 is 0.589 (although most people use 0.6 as a reasonable approximation).\n"]], ["block_5", ["1\n1\nCylinder\n5L2 +\n"]], ["block_6", ["Thin rod\n1\u201471?\nRod length L\n"]], ["block_7", ["1\nThin disk\nEr?\nDisk radius r\n"]], ["block_8", ["Nbg\nStatistical segment length b\nGaussian ring\n\u201412\u2014\nDegree of polymerization N\n"]], ["block_9", ["6.8\nSelf-Avoiding Chains: A First Look\n"]], ["block_10", ["Solid ellipsoid\nEllipsoid principal radii R1, R2, R3\n"]], ["block_11", ["Table 6.3\nFormulae for the Radii of Gyration for Various Shapes\n"]], ["block_12", ["3\nStatistical segment length b\nSolid sphere\n3R2\nSphere radius R\n"]], ["block_13", ["Structure\nR:\nParameters\n"]], ["block_14", ["_\n.\nN62\n,\n,\nGauss1an \n.?\nDegree of polymerization N\n"]], ["block_15", ["3f \nb2\nStatistical segment length b\nGaussian \nT\u2014 \nArm degree of polymerization Narm\n"]], ["block_16", ["Self\u2014Avoiding Chains: A First Look\n"]], ["block_17", [{"image_3": "255_3.png", "coords": [39, 318, 103, 356], "fig_type": "molecule"}]], ["block_18", ["3\nR, : \\n0\n(6.7.20)\n"]], ["block_19", ["\u20142\u2014\n"]], ["block_20", ["2\nCylinder radius r, length L\n"]], ["block_21", ["Number of armsf\n"]], ["block_22", ["241\n"]]], "page_256": [["block_0", [{"image_0": "256_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["1.\nExperimental chain dimensions for poly(ethylene terephthalate) (PET) at 275\u00b0C are given by\n(h2)0/M @090 2&2 mol/g. Calculate Coo, the statistical segment length, and the persistence\nlength for this polymer. Based on these numbers, is PET a \ufb02exible polymer, or not? What\nwould you expect based on the molecular structure?\n2.\nResolve the paradox noted in Section 6.3: polystyrene is considered to be relatively stiff,\nand\npolyisOprene\nrelatively\n\ufb02exible,\nyet\ntheir statistical\nsegment\nlengths\nare\nalmost\nidentical.\n3.\nFor freely jointed copolymers with nA steps of length EA and nB steps of length EB \ufb01nd (I12)\n(large n limit) for strictly alternating, random, and diblock architectures. Are the answers the\nsame or different? Why?\n"]], ["block_2", ["As also mentioned in Section 6.2, there are two experimental situations in which this compli-\ncation goes away, and v = 0.5 again. One case is a molten polymer. Here a chain still cannot have\ntwo monomers occupying the same place, but there is no benefit to expanding the coil. The reason\nis that space is full of monomers, and a monomer on one chain cannot tell if its immediate neighbor\nin space belongs to a different chain, or is attached by many bonds to the same chain. Conse-\nquently, it does not gain anything by expanding beyond the Gaussian distribution. The second case\nis for a chain dissolved in a particular kind of solvent called a theta solvent. A theta solvent is\nactually a not-very-good solvent, in the sense that for energetic reasons monomers would much\nprefer to be next to other monomers than next to solvent molecules. This has a tendency to shrink\nthe chain, and a theta solvent refers to a particular solvent at a particular temperature where the\nexpansion due to the self-avoiding nature of the chain is exactly canceled by shrinking due to\nunfavorable polymer\u2014solvent interactions. We will explore this in more detail in the next chapter,\nwhere we consider the thermodynamics of polymer solutions.\n"]], ["block_3", ["1.\nA single chain can adopt an almost infinite number of possible conformations; we must settle\nfor describing the average size.\n2.\nFor any chain with some degree of conformational freedom the average size will grow as the\nsquare root of the degree of polymerization: this is the classic result for a random walk.\nFurthermore, the distribution of chain sizes is approximately Gaussian.\n3.\nThe prefactor that relates size to molecular weight is a measure of local \ufb02exibility; three\ninterchangeable schemes for quantifying the prefactors are the characteristic ratio, the statis-\ntical segment length, and the persistence length.\n4.\nChemical structures for which the chain orientation can reverse direction in about 20 backbone\nbonds or less are called \u201c\ufb02exible\u201d; much stiffer polymers are termed \u201csemi\ufb02exible.\u201d This\nlatter class is best considered through the worm-like chain model using the persistence length\nas the key measure of local \ufb02exibility.\n5.\nThe radius of gyration is the most commonly employed measure of size; it can be de\ufb01ned for\nany chemical structure and it is directly measurable.\n6.\nBecause of excluded volume, real chains dissolved in a good solvent are not random walks, but\nself-avoiding walks. The corresponding size grows with a slightly larger power of molecular\nweight. In theta solvents or in the melt, the excluded volume effect is canceled out and the\nrandom walk result applies.\n"]], ["block_4", ["In this chapter, we have examined the spatial extent of polymer chains as a function of molecular\nweight and chemical structure. The principal results are the following:\n"]], ["block_5", ["242\nPolymer Conformations\n"]], ["block_6", ["6.9\nChapter Summary\n"]], ["block_7", ["Problems\n"]]], "page_257": [["block_0", [{"image_0": "257_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["4.\nAssume a freely rotating, strictly alternating copolymer chain, with alternating bond angles 6A\nand 63 and step lengths EA and EB; find (222).\n5.\nThe unperturbed dimensions of polymer chains depend weakly on temperature. Interestingly,\nthe sign of the temperature coefficient can be either positive or negative. For example,\nfor\n100%\ncis-polybutadiene,\nd(ln (h2))/dTr\u2014e 0.0004 deg\u2018l,\nwhereas\nfor\n100%\ntrans-\npolybutadiene, the same quantity is \u20140.0006 deg\u20141. Provide an explanation for the observa-\ntion that this quantity can, in general, be either positive or negative, and speculate how your\nexplanation might apply in this particular case. Note that it would require a very careful\ncalculation to actually show why these two coefficients have different signs.\n6.\nShow that the expression for the mean-square end-to-end distance of the worm-like chain\ngiven in Equation 6.4.11 reduces to the expected answers for a random coil and a rigid rod,\nwhen the limits L >> 6p and 6p > L are taken, respectively.\n7.\nFrom the following data of KirsteT estimate the persistence length of calf thymus DNA and the\nmass per persistence length Mp:\n"]], ["block_2", ["9.\nMiyaki, Einaga, and Fujita3F reported measurements of R3 for very high M polystyrenes in\nbenzene at 25\u00b0C (a good solvent) and cyclohexane at 345\u00b0C (theta conditions). Find the\nrelation between Rg and MW in the case of benzene (the cyclohexane data were analyzed in\nExample 6.3). How does the exponent compare with expectations? If you extrapolate to lower\n"]], ["block_3", ["tRo. Kirste, Disc. Farad. Soc. 49, 51 (1970).\n"]], ["block_4", ["Problems\n243\n"]], ["block_5", ["1Y. Miyaki, Y. Einaga, and H. Fujita, Macromolecules 11, 1180 (1978).\n"]], ["block_6", ["Equation 6.5.12 is a good place to start. One approach is graphical, i.e., to compare the data\n(plotted logarithmically) against a theoretical plot of the dimensional quantities Rg/Ep versus\nM/Mp. Alternatively, the data can be fit to Equation 6.5.12 using a nonlinear regression\nroutine. However, some care must be taken in weighting the data in the fit. Is there a reason\nwhy the highest molecular weight data should be accorded less significance?\n8.\nDerive the expression for the mean square radius of gyration of the worm\u2014like chain, Equation\n6.5.12, from the corresponding expression for the mean-square end-to-end distance, Equation\n6.4.11. First show that this relation is equivalent to Equation 6.5.8.\n"]], ["block_7", [{"image_1": "257_1.png", "coords": [37, 531, 157, 576], "fig_type": "molecule"}]], ["block_8", [{"image_2": "257_2.png", "coords": [47, 232, 236, 337], "fig_type": "figure"}]], ["block_9", [{"image_3": "257_3.png", "coords": [49, 446, 141, 492], "fig_type": "molecule"}]], ["block_10", ["1\nn\nj\u2014 1\n2 _\n2\n4r6224>\nj=2\ni=1\n"]], ["block_11", ["M,\nRg (.41)\nM...\nR4251)\n"]], ["block_12", ["3.5 x 105\n450\n3.45 x 106\n1700\n4.6 x 105\n480\n4.6 x 106\n2000\n6.9 x 105\n650\n6.3 x 106\n2300\n1.15 x 106\n900\n6.9 x 106\n2500\n1.6 x 106\n1100\n9.2 x 106\n3000\n2.3 x 106\n1390\n1.35 x 107\n3600\n2.8 x 106\n1550\n"]], ["block_13", ["Then make the correspondences L = 1212, x = if, and y \ufb02? when using Equation 6.4.1 1 for (\u00a325.).\nThe final step is to equate the double sum above with the following integrals and carry out the\nintegration:\n"]], ["block_14", ["\u2014d\ne I\n0\n"]], ["block_15", ["1\ny\n"]], ["block_16", ["ml:\u2014\n"]], ["block_17", ["dx\n"]]], "page_258": [["block_0", [{"image_0": "258_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["P9P?\u201c\n"]], ["block_2", ["244\n"]], ["block_3", ["References\n"]], ["block_4", ["10.\n"]], ["block_5", ["14.\n"]], ["block_6", ["11.\n"]], ["block_7", ["12.\n"]], ["block_8", ["13.\n"]], ["block_9", ["15.\n"]], ["block_10", ["16.\n"]], ["block_11", [{"image_1": "258_1.png", "coords": [42, 85, 285, 185], "fig_type": "figure"}]], ["block_12", ["Flory, P.J., Statistical Mechanics of Chain Molecules, Wiley-Interscience, New York, 1969.\nKratky, O. and Porod, G., Rec. Trav. Chim., 68, 1106 (1949).\nKuhn, W., Kollold 2., 68, 2 (1934).\nYamakawa, H., Modern Theory of Polymer Solutions, Harper & Row, New York, 1971.\n"]], ["block_13", ["MW x 10\u20196\nR3 (nrn), Benzene\nRg (nm), Cyclohexane\n"]], ["block_14", ["56.2i 1\n506 _-_|; 10\n228 i 5\n39.5i 1\n32.0i 0.6\n23.5 i 0.5\n15.1 _-|; 0.5\n8.77i 0.3\n"]], ["block_15", ["For the Gaussian distribution function for the end-to-end distance, calculate the most\nprobable distance, the mean distance, and the root\u2014mean-square distance. Generate a good\nplot of this distribution function for some value of N and indicate where these three\ncharacteristic distances fall on the plot.\nFind the mean square radius of gyration for an infinitely thin rod of length L, with mass\ndensity (per unit length) p in the center L/2 section and mass density (per unit length) 2p for\nthe U4 sections at each end.\nFind the radius of gyration of a sphere with an outer shell of different density, where the inner\nspherical core has a radius of R1 and a density of p, and the outer shell (the \u201ccorona\u201d) extends\nto a radius of R2=2R1 and has a density equal to 2p. (This geometry is a reasonable\nrepresentation of a spherical micelle, which can be formed in a solution of block copolymers\nwhen one block is relatively insoluble; see Chapter 4.4.)\nConsider the coil dimensions of an A\u2014B diblock copolymer, with total degree of polyme-\nrization N and the fraction of A monomers given byf. Assuming that the mass of each A and\nB monomers is the same, show that\n"]], ["block_16", ["where (RE), is the mean square radius of gyration for block i and (22)AB is the mean square\nseparation of block centers-of\u2014mass.\nConsider a statistical A\u2014B 00polymer, with total degree of polymerization N and the fraction\nof A monomers given by f. Derive a simple expression for (/12) in terms of N,f, bA, and b3,\nassuming Gaussian statistics. This equation will probably not be exactly correct, in practice,\neven for large N. One reason is that thermodynamic interactions between A and B monomers\n(usually effectively repulsive) will tend to expand the chain. However, there is another\nreason, connected to the nature of the statistical length; what is it?\nA distinguished polymer scientist is reputed to have remarked \u201can infinite steel girder would\nbe a random coil.\u201d Explain the important point that this striking comment is intended to\nillustrate.\nHow does the average polymer concentration inside an individual coil vary with M in good\nand theta solvents? Estimate this concentration in g/cm3 for polystyrene with M = 106 under\nboth conditions, using Miyake et al.\u2019s data given in Problem 9. Explain why this concentra-\ntion is often referred to as the coil overlap concentration, c*.\n"]], ["block_17", [" =fA+ (1 f)AB\n"]], ["block_18", ["M, by what degree of polymerization would excluded volume increase Rg (i) by only 10%?\n(ii) by a factor of 2?\n"]], ["block_19", ["392i8\n353i7\n297_-|;9\n227i7\n164i4\n"]], ["block_20", ["87.9;l\u2014_2\n"]], ["block_21", ["183:4\n167i4\n145i?)\n116i2\n"]], ["block_22", ["Polymer Conformations\n"]]], "page_259": [["block_0", [{"image_0": "259_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Graessley, W.W., Polymeric Liquids and Networks: Structure and Properties, Garland Science, York,\n2003.\nRubinstein, M. and Colby, R.H., Polymer Physics, Oxford University Press, New York, 2003.\nVolkenstein, M.V., Configurational Statistics of Polymeric Chains, Wiley\u2014Interscience, London, 1963.\nYamakawa, H., Modern Theory of Polymer Solutions, Harper & Row, New York, 1971.\n"]], ["block_2", ["Doi, M. and Edwards, S.F., The Theory of Polymer Dynamics, Clarendon Press, Oxford, \nFlory, P.J., Statistical Mechanics of Chain Molecules, Wiley\u2014Interscience, New York, 1969.\n"]], ["block_3", ["Further Readings\n245\n"]], ["block_4", ["Further Readings\n"]]], "page_260": [["block_0", [{"image_0": "260_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["In this chapter we shall consider some thermodynamic properties of solutions in which a polymer\nis the solute and some low-molecular\u2014weight species is the solvent. Our special interest is in the\napplication of solution thermodynamics to problems of phase equilibrium.\nAn important fact to remember about the field of thermodynamics is that it is blind to details\nconcerning the structure of matter. Thermodynamics is concerned with observable quantities and\nthe relationships among them, although there is a danger of losing sight of this fact in the\nsomewhat abstract mathematical formalism of the subject. For example, we will take the position\nthat entropy is often more intelligible from a statistical, atomistic point of view than from purely\nphenomenological perspective. It is the latter that is pure thermodynamics; the former is the\napproach of statistical thermodynamics. In this chapter we shall make extensive use of the\nstatistical point of view to understand the molecular origin of certain phenomena. The treatment\nof heat capacity in physical chemistry provides an excellent example of the relationship between\npure and statistical thermodynamics. Heat capacity is defined experimentally as the heat required\nto change the temperature of a sample in, say, a constant-pressure experiment. Various equations\nrelate the heat capacity to other thermodynamic quantities such as enthalpy, H and entropy, S. The\nalternative approach to heat capacity would be to account for the storage of energy in molecules in\nterms of the various translational, rotational, and vibrational degrees of freedom. Doing thermo-\ndynamics does not even require knowledge that molecules exist, much less how they store energy,\nwhereas understanding thermodynamics benefits considerably from the molecular point of view.\nOne drawback of the statistical approach is that it depends on models, and models are bound to\noversimplify. Nevertheless, we can learn a great deal from the attempt to evaluate thermodynamic\nproperties from molecular models, even if the effort falls short of quantitative success.\nThere is probably no area of science that is as rich in mathematical relationships as thermo-\ndynamics. This makes thermodynamics very powerful, but such an abundance of riches can also be\nintimidating. In this chapter we assume that the reader is familiar with basic chemical and\nstatistical thermodynamics at the level that these topics are treated in undergraduate physical\nchemistry textbooks. This premise notwithstanding, a brief review of some pertinent relationships\nwill be a useful way to get started.\nNotation frequently poses problems in science and this chapter is an example of such a situation.\nOur problem at present is that we have too many things to count: they cannot all be designated n. In\nthermodynamics n is widely used to designate the number of moles and so we will adhere to this\nconvention. Since we deal with (at least) two-component systems in this chapter, any count of the\nnumber of moles will always carry a subscript to indicate the component under consideration. We\nshall use the subscript 1 to designate the solvent and 2 to designate the solute. We have consistently\nused N to designate the degree of polymerization and shall continue with this notation, although the\ndefinition of \u201cmonomer\u201d will be modified slightly.\nTo describe the state of a two-component system at equilibrium, we must specify the number of\nmoles n1 and n2 of each component, as well as the pressure p and the absolute temperature T. It is\n"]], ["block_2", ["7.1\nReview of Thermodynamic and Statistical Thermodynamic Concepts\n"]], ["block_3", ["Thermodynamics of Polymer Solutions\n"]], ["block_4", ["7\n"]], ["block_5", ["247\n"]]], "page_261": [["block_0", [{"image_0": "261_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["the Gibbs free energy, G, that provides the most familiar access to a discussion of equilibrium. The\nincrement of G associated with increments in the independent variables mentioned above is given\nby the equation\n"]], ["block_2", ["where V is the volume, 5 is the entropy, and p.) is the chemical potential of component 1'. An\nimportant aspect of thermodynamics is the fact that the state variables (in the present context,\nespecially H, G, and the internal energy U) can be expanded as partial derivatives of the\nfundamental variables. Hence we can also write\n"]], ["block_3", ["Comparing Equation 7.1.1 and Equation 7.1.2 gives\n"]], ["block_4", ["The chemical potential is an example of a partial molar quantity: p.) is the partial molar Gibbs free\nenergy with respect to component 1'. Other partial molar quantities exist and share the following\nfeatures:\n"]], ["block_5", ["4.\nA useful feature of the partial molar properties is that the property of a mixture (subscript m)\ncan be written as the sum of the mole-weighted contributions of the partial molar properties of\nthe components:\n"]], ["block_6", ["248\nThermodynamics of Polymer Solutions\n"]], ["block_7", ["where Y V, H, or 5, respectively. Except for the partial molar Gibbs free energy, we shall use\nthe notation 7; to signify a partial molar quantity, where Y stands for the symbol of the\nappropriate variable.\n2.\nPartial molar quantities have per mole units, and for Y, this is understood to mean per mole\nof component i. The value of this coefficient depends on the overall composition of the\nmixture. Thus VH20 is not the same for a water\u2014alcohol mixture that is 10% water as for\none that is 90% water.\n3.\nFor a pure component the partial molar quantity is identical to the molar (superscript A) value\nof the pure substance. Thus for pure component i\n"]], ["block_8", ["and\n"]], ["block_9", ["1.\nWe may define, say, partial molar volume, enthalpy, or entropy by analogy with Equation 7.1.5:\n"]], ["block_10", [{"image_1": "261_1.png", "coords": [41, 242, 125, 316], "fig_type": "molecule"}]], ["block_11", [{"image_2": "261_2.png", "coords": [41, 327, 119, 367], "fig_type": "molecule"}]], ["block_12", ["dG=Vdp-\u2014SdT+Zp.,-dm\n(7.1.1)\n"]], ["block_13", ["3G\n3G\n(9G\n36\ndG <\u2014\u2014)\ndp + <\u2014)\ndT \u2014|\u2014 (\u2014)\ndnl + (\u2014)\ndag\n(7.1.2)\nap\nT,n1,n2\n6T\np,m,r12\nanl\np,T,nz\nan;\np,T,n1\n"]], ["block_14", ["1/ z\n(\u201810.)\n(7.1.3)\nap\nT,m,n2\n"]], ["block_15", [{"image_3": "261_3.png", "coords": [47, 93, 180, 131], "fig_type": "molecule"}]], ["block_16", ["3G\nS = _ _\n7.1.4\n(3T) p,nl ,Hz\n(\n)\n"]], ["block_17", ["Y.- (21:)\n(7.1.6)\nam\nplay-5E,-\n"]], ["block_18", ["Ym 111.71 -|- H2372\n(1.18)\n"]], ["block_19", ["ant.\npiranja\ufb01\n"]], ["block_20", ["A\n"]], ["block_21", ["3G\n"]], ["block_22", ["i=1,2\n"]], ["block_23", [{"image_4": "261_4.png", "coords": [157, 184, 374, 222], "fig_type": "molecule"}]], ["block_24", [{"image_5": "261_5.png", "coords": [250, 186, 365, 220], "fig_type": "molecule"}]]], "page_262": [["block_0", [{"image_0": "262_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["The quantity 11.? is called the standard state (superscript 0) value of #4; it is the value of 11.,- when\na; 1. Neither 11.,- nor G can be measured absolutely; we deal with differences in the quantities and\nthe standard state value disappears when differences are taken. Although the standard state is\nde\ufb01ned differently in various situations, we shall generally take the pure component as the\nstandard state and 11.? Gf\u2019.\n"]], ["block_2", ["6.\nRelationships that exist among ordinary thermodynamic variables also apply to the corre-\nsponding partial molar quantities. Two such relationships are\n"]], ["block_3", ["5.\nTo express the value of property Ym on a per mole basis, it is necessary to divide Equation\n7.1.8 by the total number of moles, n1 + 11;. The mole fraction x,- of component i is written\n"]], ["block_4", ["It is convenient to define a thermodynamic concentration called the activity a, in terms of\nwhich the chemical potential is given by the relationship\n"]], ["block_5", ["Regular solution theory illustrates how a relatively simple statistical model can provide a useful\nexpression for the free energy of mixing for a binary solution of two components. Furthermore, the\nFlory\u2014Huggins theory, which we will develop in the next section and which is the starting point for\ndiscussions of both polymer solutions and polymer blends, is really nothing more than regular\nsolution theory extended to polymers. We will derive these results using a lattice approach because\nthe physical picture is particularly clear, but it should be realized that the expression for AG,1n that\nresults could be obtained without such a seemingly artificial assumption.\nWe will designate the number of molecules of the two species ml and m2 and the number of\nmoles 121 and 122. We assume that the two molecules have equal volumes and also that their partial\nmolar volumes are equal (and concentration-independent): V1 V2. The cell size of the lattice is\nchosen to equal the molecular volume, and the lattice has a coordination number 2. The total number\nof molecules m m1 + m2 is also the total number of lattice sites. A section of a two-dimensional\nsquare lattice is illustrated in Figure 7.1.\n"]], ["block_6", ["Regular Solution Theory\n249\n"]], ["block_7", ["The entropy of mixing is obtained from the Boltzmann definition of entropy:\n"]], ["block_8", ["7.2\nRegular Solution Theory\n"]], ["block_9", ["7.2.1 Regular Solution Theory: Entropy of Mixing\n"]], ["block_10", ["521(a\n(7.2.1)\n"]], ["block_11", ["n,-\nxi \n(7.1.9)\n25:12 \"i\n"]], ["block_12", ["In this expression n, and n; are the numbers of moles of components 1 and 2 in the mixture\nunder consideration.\n"]], ["block_13", ["therefore\n"]], ["block_14", ["Ym\n_\n_\n=x1Y1+x2Y2\n(7.1.10)\n111 + n;\n"]], ["block_15", ["p,=\ufb01,\u2014\u2014T\u00a7,\n(7.1.11)\n"]], ["block_16", ["and\n_\n6\n.\nV1: (\u20141\u201c)\n(7.1.12)\n3p\nTan-#1\n"]], ["block_17", ["rt, =u?+RTlna,\n(7.1.13)\n"]], ["block_18", ["As noted above, all of the partial molar quantities are, in general, concentration-dependent.\n"]]], "page_263": [["block_0", [{"image_0": "263_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "263_1.png", "coords": [12, 35, 303, 327], "fig_type": "figure"}]], ["block_2", ["where the terms in the denominator take account of the fact that the m1 molecules of type\n1 are\nindistinguishable from one another, as are the m2 molecules of type 2 (compare this with Equation\n6.7.2). Stirling\u2019s approximation (In N I azN ln N\u2014N for large N) is used to get rid of the factorials,\njust as in Section 6.7:\n"]], ["block_3", ["Now the con\ufb01gurational entropy for each pure component (and we are assuming that configur-\national entropy is the only source of entropy) is zero:\n"]], ["block_4", ["where Q is the total number of possible configurations that the system can adopt. For placing m1\nobjects of type 1 and m; objects of type 2 on a lattice of m sites the total number of configurations is\n"]], ["block_5", ["250\nThermodynamics of Polymer Solutions\n"]], ["block_6", ["Figure 7.1\nSection of a two\u2014dimensional square lattice with each site occupied by either of two species.\n"]], ["block_7", ["I\nQ \nm-\n(7.2.2)\nmllmgl\n"]], ["block_8", ["I\n51 21min] :klnT\u2014E: 0\nm1.\n(7.2.4)\nI\n52 kind2 =k1nm\u20142;= 0\nm2.\n"]], ["block_9", ["Sm :k{lnm!~\u2014\u2014lnm1!\u2014lnm2!}\n= k{mlnm m mllnm1+ m1\u2014 m2 lnmg + m2}\n2 k{(m1+ m2)ln(m1+ m2) m1 ln m1 m2 ln m2}\n= ~\u2014k{m1(ln m1 ln m) + 1722(1a In 172)}\n= \u2014k{m11nx1 + 1722111162}\n(723)\n"]], ["block_10", ["00000000000000\n00000000000000\n00000000000000\n0000000000000.\n00000000000000\n00000000000000\n00000000000000\n00000000000000\n00000000000000\n00000000000000\n00000000000000\n0.000000000000\n00000000000000\n00000000000000\n"]]], "page_264": [["block_0", [{"image_0": "264_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["We now compute the enthalpy of mixing for this model, Mn\u201c, and will assume that the resulting\nenergy term is sufficiently small that it does not matter in A5,\u201c. First, we assume that AH\u201c, = AUm,\nin other words no p V terms contribute to H; this is consistent with the lattice approach and\nemphasizes that we assume there is no volume change on mixing. We introduce the interaction\nenergies in the pure state, w\u201c and W22, which act between two molecules of type 1 and between\ntwo molecules of type 2, respectively; this is shown schematically in Figure 7.2a. All molecules\nattract one another by dispersion forces, so w\u201d and W22 are negative. Under these assumptions the\nenthalpy in the pure state is given by\n"]], ["block_2", ["which means each molecule has 2 neighbors and therefore 2 interactions, but we divide by 2\nbecause there is only w\u201d worth of interaction energy per pair of molecules. We are also assuming\nthat molecules only interact with their nearest neighbors.\nIn the mixture we assume that the placement of\n1 and 2 is completely random, as in the\ncalculation of A5,\u201c. Thus the probability that a neighboring site is occupied by a molecule of type\n"]], ["block_3", ["7.2.2\nRegular Solution Theory: Enthalpy of Mixing\n"]], ["block_4", ["It is important to recognize that Equation 7.2.5a through Equation 7.2.5d are all the same in\nphysical content, but different in units, and one must be careful to use the apprOpriate form of AS\u201c,\nin calculations.\nThere are three important features to this expression for A3,\u201c.\n"]], ["block_5", ["1.\nAs x1 and x2 are always between 0 and 1 in a mixture, the natural logarithm terms are always\nnegative and the overall AS\u201c, > 0. Therefore, configurational entropy always favors spontan-\neous mixing.\n2.\nThe expression is symmetric with respect to exchange of 1 and 2, which is a consequence of\nthe assumption of equal molecular sizes. In real mixtures this condition will be hard to satisfy.\n3.\nThis calculation of the entropy assumed that all configurations on the lattice were equally\nprobable, i.e., there was no energetic benefit or price for having 1 next to 1 and 2 next to 2,\nversus having 1 next to 2. If there were such an energy term, each configuration ought to be\nfurther weighted by the appropriate Boltzmann factor exp[\u2014E/RT], where E is the total energy\nof that con\ufb01guration.\n"]], ["block_6", ["or per mole of lattice sites (and therefore, in this case, per mole of molecules)\n"]], ["block_7", ["where we have factored out Nav. It is often useful to have an intensive expression, i.e., the entropy\nof mixing per lattice site (and therefore per molecule)\n"]], ["block_8", ["This expression applies to the entire mixture, and could also be written\n"]], ["block_9", ["so Equation is actually the change in entropy with mixing\n"]], ["block_10", ["Regular Solution Theory\n251\n"]], ["block_11", ["ASm -\u2014R{x11nx1 +x2 111.152}\n(725d)\n"]], ["block_12", ["ASm 46051111151 +x2 111362)\n(7.2.5c)\n"]], ["block_13", ["ASm \u2014R{n1 lnxl + nglnxg}\n(7.2.5b)\n"]], ["block_14", ["ASm Sm Sl 52 \u2014k(m11nx1 + m2 lnxg)\n(7.2.5a)\n"]], ["block_15", ["H2 =\n\u201dimzZs\n"]], ["block_16", ["H1 \nEmlzwn\n(7.2.6)\n"]]], "page_265": [["block_0", [{"image_0": "265_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "265_1.png", "coords": [26, 45, 226, 144], "fig_type": "figure"}]], ["block_2", ["which is the exchange energy per molecule, normalized by the thermal energy Id\". In other words,\nX is the fraction of H\" you must pay in order to lift one molecule of type 1 out of its beaker, one\n"]], ["block_3", ["We define an exchange energy Aw :(wlrwl 1/2 w22/2), which represents the difference between\nthe attractive cross-interaction of l and 2 and the average self-interaction of 1 with 1 and 2 with 2.\nFor dispersion forces, and for the regular solution theory, Aw 2 0, as we will discuss in Section 7.6;\nthis is a manifestation of \u201clike prefers like,\u201d indicating that the self-interactions are more attractive\nthan cross-interactions. For now, however, we can just take it as an energy parameter that is\nprobably positive.\nWe make a further definition, that of the interaction parameter, x:\n"]], ["block_4", ["252\nThermodynamics of Polymer Solutions\n"]], ["block_5", ["and thus\n"]], ["block_6", ["Figure 7.2\nIllustration of pairwise nearest\u2014neighbor interaction energies, WU, in (a) the pure components and\n(b) a mixture.\n"]], ["block_7", ["g\n4\u2014)-\nO\n"]], ["block_8", ["1 and 2 Wu (wlz is also negative), then (see Figure 7.2b)\n"]], ["block_9", ["1 or 2 is given by x1 or x2, respectively. If we call the interaction energy between molecules\n"]], ["block_10", ["<(-)> \nH\nH\n(a)\n14/20\nw=0\n"]], ["block_11", [".\nO .\n\u00a2W11\nW22\u00a2 \nO . O\n0\n(b)\nW12\n"]], ["block_12", ["__ \n2\u2014\n.2.9\nX\nkT\n(7\n)\n"]], ["block_13", ["1\nHnn \nEm1(zx1w\u201c + zx2w12)+ \u00a7m2(zx1w12 + zx2w22)\n(7.2.7)\n"]], ["block_14", ["AHm =Hm \u2014H1\u2014H2\n"]], ["block_15", [{"image_2": "265_2.png", "coords": [65, 453, 309, 522], "fig_type": "molecule"}]], ["block_16", [{"image_3": "265_3.png", "coords": [65, 187, 192, 272], "fig_type": "molecule"}]], ["block_17", [": \n\u00a7z{w12(m1x2 \n"]], ["block_18", ["mlmz\nml\ufb02\u2019Q\n=\u2014Z\nW12\n\u2018W11(\n)\u2018W22(\n)\n2\nm\nm\nm\n:\n2(w12 _K1_l_323) :zAw\n(7.2.8)\nm.\n2\n2\nm\n"]], ["block_19", ["1\n"]], ["block_20", ["{\n(21711\u201d?2)\n"]], ["block_21", ["0..\n000\nO\n"]], ["block_22", [{"image_4": "265_4.png", "coords": [108, 52, 226, 144], "fig_type": "molecule"}]]], "page_266": [["block_0", [{"image_0": "266_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["For the mixture of 2 mol the total free energy of mixing is therefore (\u2014 1.3 x 2 X 8.3 x 298) m -6.5\nkJ. Although we should not take the exact numbers too seriously, the overall negative sign con\ufb01rms\nour expectation that toluene and cyclohexane should be quite happy to mix, even though X is\npositive.\n"]], ["block_2", ["Assuming that w\u201c,\nW12,\nand\nW22\nare approximately\n\u20141.17 x 10\u201920,\n\u2014 1.08 x 10\u201420,\nand\n\u2014 1.01 x 10\u201420 J for toluene and cyclohexane, respectively, estimate the free energy of mixing\nfor 1 mol of toluene with 1 mol of cyclohexane at room temperature. (We will see in Section 7.6\nhow these interaction energies can be estimated from experimental measurements.)\n"]], ["block_3", ["We will first calculate a value for X, and then substitute into Equation 7.2.11a. We need to assume\na value of the coordination number, 2; it is typically about 10 for small molecule liquids (recall that\nin a close\u2014packed lattice of spheres there are 12 nearest neighbors).\n"]], ["block_4", ["per site. The first two terms represent the entropy, and, as noted above, they favor mixing, whereas\nthe last enthalpic term is assumed to be positive and therefore opposes mixing. The second form of\nA6,\u201c, Equation 7.2.11b, is probably the easier one to remember because of its obvious symmetry.\nHowever, when we subsequently compute chemical potentials, we will need to take partial\nderivatives with respect to 121 and 112, and the form in Equation 7.2.11a will be more appropriate.\nThe implications of regular solution theory, particularly in terms of the predictions for the phase\nbehavior, will be discussed in Section 7.5.\n"]], ["block_5", ["We have r11: r12: 1, and x1 =x2=O.5, so\n"]], ["block_6", ["for the whole system, or\n"]], ["block_7", ["per lattice site or molecule, and\n"]], ["block_8", ["per mole. When X is positive AH\", is positive, and therefore Opposes Spontaneous mixing. We can\nnow express the free energy of mixing, AGm AHm\u2014TASm as\n"]], ["block_9", ["for the system as a whole, or\n"]], ["block_10", ["Solution\n"]], ["block_11", ["molecule ofbeaker, and exchange them. Note that although X is dimensionless,\nits value does depend on the chosen size of the lattice site, through Aw. Using this definition we\ncan write\n"]], ["block_12", ["Example 7.1\n"]], ["block_13", ["Regular Solution Theory\n253\n"]], ["block_14", ["ZAW\n10\n1.17 +1.01\n_,0\n:_g\n\u20141.08 \u201c\n10\n20.24\nX\n H \n(1.4 x 10-23)(298){\n+\nl \n"]], ["block_15", ["AGm\nF \n"]], ["block_16", ["77\u2014,\u2014 \n(7-2-11b)\n"]], ["block_17", ["AGm\nF\n= (r11 lnxl + n; lnX2 + HIJQX)\n(72-113)\n"]], ["block_18", ["AGm\n_\n"]], ["block_19", ["AHm xlxsT\n(7.2.10c)\n"]], ["block_20", ["AHm m1X2XkT nixsT\n(7.2.10a)\n"]], ["block_21", ["AH,\u201c xlk\n(7.2.10b)\n"]]], "page_267": [["block_0", [{"image_0": "267_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "267_1.png", "coords": [24, 408, 259, 640], "fig_type": "figure"}]], ["block_2", ["Before proceeding to polymer solutions, one further comment about this derivation of AG\", is in\norder. In computing AHm we assumed completely random mixing, as in Equation 7.2.7. But, if X is\nnonzero, there will presumably be some finite preference for \u201cclustering,\u201d e.g., 1 with 1 and 2 with\n2 if x> 0. Thus the probability that a lattice site immediately adjacent to a type\n1 molecule is\noccupied by another\n1 may actually be larger than x1 and the same for component 2. This\npossibility is simply not accommodated in the model and represents a fundamental limitation.\nA theory that assumes that the local interactions are determined solely by the bulk average\ncomposition, i.e., x1 and x2 in this case, is called a \u201cmean\u2014field\u201d theory. We will encounter\nother examples of mean\u2014field theories in this book. They are popular, as one might expect, because\nthey are relatively tractable; it also tums out that in many polymer problems, especially in\nundiluted polymers, they are remarkably reliable.\n"]], ["block_3", ["Flory and Huggins independently considered AG\", for polymer solutions, and the essence of their\nmodel is developed here [1,2]. As noted in the previous section, the Flory\u2014Huggins theory is a\nnatural extension of regular solution theory to the case where at least one of the components is\npolymeric. To proceed, we will adopt the same lattice model as before, with one important\ndifference. We choose the lattice site to have the volume of one solvent molecule (subscript 1)\nand each polymer (subscript 2) occupies N lattice sites. This is illustrated in Figure 7.3. Thus N is\nproportional to the degree of polymerization, but the monomer unit is now defined to have the\nsame volume as the solvent. (Equivalently, N is the ratio of the molar volume of the polymer to that\nof the solvent.) We will refer to this subunit of the polymer as a \u201csegment.\u201d We will also switch\nfrom mole fractions to volume fractions, qbl and (152, in describing composition. We do this because\nin order to use moles accurately with polymers, one needs to know the molecular weight precisely,\nand one also has to know the full molecular weight distribution, whereas the volume fraction is\neasily obtained from the measured mass and known densities. (However, we will need to be a little\ncareful in some thermodynamic manipulations in subsequent sections, where for example the\n"]], ["block_4", ["254\nThermodynamics of Polymer Solutions\n"]], ["block_5", ["Figure 7.3\nSection of a two-dimensional square lattice with each site occupied either by a solvent molecule\nor a polymer segment.\n"]], ["block_6", ["7.3\nFlory\u2014Huggins Theory\n"]], ["block_7", ["0\nO\nO\nO\nO\nO\nO\nO\n0000\n"]], ["block_8", ["OO O\n0\n"]], ["block_9", ["00000000000\n"]], ["block_10", ["00000000\n"]]], "page_268": [["block_0", [{"image_0": "268_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["This expression has the same physical content as Equation 7.3.4a, but it brings out the importance\nof N. Because N is usually a large number, the contribution to the entropy of mixing from the\npolymer is very small, often almost negligible. Thus immediately we can predict that for systems\nwith X > 0, where spontaneous mixing is driven only by A5,\u201c, it will be harder to mix a solvent with\na polymer than with another small molecule.\n"]], ["block_2", ["Herewith a more detailed derivation is presented. Let us suppose we have added 5 polymers onto\nthe lattice and we inquire about the number of possibilities for placing the (i + 1)th chain. At this\nmoment, the probability that a lattice site does not contain a monomer isf, =1 (iN/m), where m is\n"]], ["block_3", ["To obtain AS\", for the entire system, we multiply the respective terms by the number of molecules\nof each type, and sum:\n"]], ["block_4", ["If we compare this expression with Equation 7.2.5 for regular solution theory, the only change is the\nswitch to volume fractions. Furthermore, if we let N =1, then (:51 =xl and qbz=x2, and regular\nsolution theory can be recovered as a special case of the Flory\u2014Huggins model. To obtain the\nexpression for AS\", per site, we divide by m 2m1 + Nmz:\n"]], ["block_5", ["The polymer molecule has many internal degrees of freedom, as described in Chapter 6. However,\nwe can assume that the number of conformational degrees of freedom is the same in the solution as\nin the pure polymer; in other words, the polymer is the same random coil with the same segment\ndistribution function in the bulk as in solution. Thus the only configurational entropy of mixing\ncomes from the increased possibilities for placing the center of mass. The polymers occupy n\nsites in the bulk polymer, and the number of possible locations for the center of mass is thus\nproportional to Nmz. On the other hand, in the mixture the number of locations is proportional to\n"]], ["block_6", ["where the unspecified proportionality factor drops out in the ratio. A similar argument holds for\neach solvent molecule, and therefore\n"]], ["block_7", ["m1 + Nmz. Thus for one polymer molecule (recall Equation 7.2.1)\n"]], ["block_8", ["7.3.2 Flory\u2014Huggins Theory: Entropy of Mixing by a Longer Route\n"]], ["block_9", ["where the total number of lattice sites m is now given by m, +Nm2.\nThere are at least two ways to obtain the Flory\u2014Huggins entropy of mixing; one is rather\ninformal, but simple, the other is more careful in the application of the model, but\nalgebraically tedious. We shall go through both, but you may wish to skip the longer version on the\nfirst pass.\n"]], ["block_10", ["7.3.1\nFlory\u2014Huggins Theory: EntrOpy of Mixing by a Quick Route\n"]], ["block_11", ["chemical respect to the number of\nmoles.) Accordingly, the volume fractions are defined as\n"]], ["block_12", ["Flory\u2014Huggins \n255\n"]], ["block_13", ["ASH-l _k(m1 111$] + m2 ln\u00e9z) \u2014\u2014R(n1 1nd\u201d + n21n\u00a22)\n(7.3.43)\n"]], ["block_14", ["as... =\u2014k(\u00a3;1\u2014lln<;bl N;- lnaz) = \u2014k (a, ma, +% 111(1),)\n(7.3.4b)\n"]], ["block_15", ["as\u201c,l \u2014\u2014klnd>1\n(7.3.3)\n"]], ["block_16", ["ASm; kln(m1 + Nmz) klnNmz \u2014\u2014klnd>2\n(7.3.2)\n"]], ["block_17", ["a \nm\u2018\n"]], ["block_18", ["2\u2014m1+Nm2\n"]], ["block_19", ["1\u2014m1+Nm2\n731\n"]], ["block_20", [{"image_1": "268_1.png", "coords": [101, 508, 306, 538], "fig_type": "molecule"}]]], "page_269": [["block_0", [{"image_0": "269_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["where the last line took a little algebra. Anyway, this now gives us .3m k In 0,0,. To obtain S for\nthe pure polymer and pure solvent, we can set either ml or m; 0 in Equation 7.3.9, respectively:\n"]], ["block_2", ["where the last transformation invokes a dilute solution approximation: iN < m always, and W > N\nfor all but a few values of i. This now allows us to replace the product of powers in the last version\nof Equation 7.3.6 with a ratio of factorials:\n"]], ["block_3", ["where the factor of l/mgl takes care of the fact that the polymers are indistinguishable. Now we\nneed to play a trick to deal with the term inside the product. Consider the following ratio of\nfactorials:\n"]], ["block_4", ["where we neglect the difference between 2 and z\u2014l in one term. The total number of configur-\nations for adding m2 polymers in succession is therefore\n"]], ["block_5", ["where the last step recognizes that m1 2m mgN. Now we are ready for help from Mr. Stirling:\n"]], ["block_6", ["again the total number of sites. Now we add the next polymer segment by segment (and assume f,- is\nconstant throughout the process). The \ufb01rst segment has 771 iN choices, the second has zf,, and the\nthird and all subsequent segments have (2 l)f,\u2014 choices. The number of configurations available to\nthe (i + 1)th chain, 0,, is thus\n"]], ["block_7", ["256\nThermodynamics of Polymer Solutions\n"]], ["block_8", [{"image_1": "269_1.png", "coords": [39, 396, 280, 431], "fig_type": "molecule"}]], ["block_9", ["m_2!( m ) \n1;! (m\u2014iN\u2014N)!:?nj( m ) \n"]], ["block_10", [{"image_2": "269_2.png", "coords": [47, 97, 225, 164], "fig_type": "molecule"}]], ["block_11", ["ln\ufb02l :mllnml \u2014m11nm1=0\n(7310)\n11102 = mzlnmzN m2 1a mgav \u20141)(1\u2014ln(z 1))\n' \u2019\n"]], ["block_12", ["ln\ufb02mt=1nm!\u2014lnm1!\u2014lnm2!+m2(N l)ln(z \nm \n= mlnm m mllnm1+m1\u2014 mzlnmz +m2 +m2(N \u20141)ln(z 1) m2(N ~1)lnm\n"]], ["block_13", ["\u201c(+1 \n"]], ["block_14", ["1\n(Hg\u20141\n1\n3712\u20141\nm iN\nN-1\n0 \u2014\u2014,\n0m = \u2014, H (m \u2014\u2014 7N)(z \u2014\u2014 1)N\"\u2018(\u2014\u2014\u2014\u2014)\n\u201d12.\ni=0\n7712.\n\u00a320\n772\n"]], ["block_15", ["(maiN)!\n__(m\u2014iN)(m\u2014iN-\u20141)---(m\u2014\u2014iN\u2014N)!\n(m\u2014-(i+1)N)! \n(m\u2014 iN\u2014N)!\n=(m\u2014iN)(m\u2014iN\u20141)---(m\u2014\u2014iN\u2014N+ 1)::s(m\u2014\u2014iN)N\n(7.3.7)\n"]], ["block_16", ["1\n2-1 m2(\"\"\u201d\u2019\"2\u201c1\n(m\u2014iN)!\n1\n2\u20141 WW\u201d)\n"]], ["block_17", [".\nN\n= 7<\nm \nH (m \u2014 7N)\n(7.3.6)\n'\ni=0\n"]], ["block_18", ["m iN\nN\u2014l\n% (m WXZ \nDIV\u20141(7)\n(73.5)\n"]], ["block_19", [":(m1+m2)ln(m1+Nm2)H m1 lnml m2 11'l m2(N -\u20141)(1\u2014ln(z 1))\n"]], ["block_20", ["1\nZ _1\nMg(N\u2014I) 012\u20141\n"]], ["block_21", [{"image_3": "269_3.png", "coords": [170, 189, 323, 228], "fig_type": "molecule"}]], ["block_22", ["#777;( m )\n(m\u2014mzN)!\ufb02\u2014m1!m2!(m )\n(7.3.8)\n"]], ["block_23", ["{m!(m\u2014N)!-~(m\u2014(m2\n"]], ["block_24", ["1\nz\u2014l\n\"MN\u2014l)\nm!\nm'\nz\u2014l\nm2(N\u2014l)\n"]], ["block_25", ["\u20141)N)!\n}\n(m \u2014N)!(m 2N)! (m mzN)!\n"]], ["block_26", [{"image_4": "269_4.png", "coords": [264, 456, 436, 491], "fig_type": "molecule"}]], ["block_27", ["(7.3.9)\n"]]], "page_270": [["block_0", [{"image_0": "270_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "270_1.png", "coords": [23, 454, 204, 485], "fig_type": "molecule"}]], ["block_2", ["per site.\nThe main features of Equation 7.3.13a and Equation 7.3.13b are that the entropy terms always\nfavor mixing, the enthalpy opposes mixing when x>0, and the big difference from regular\nsolution theory is the factor of N reducing the polymer contribution to the entropy of mixing.\nThese expressions are very powerful, as they can be used to calculate many thermodynamic\nquantities of interest. For example, in the next two sections we will develop the explicit predictions\nof the model for two experimentally important quantities, the osmotic pressure and the phase\ndiagram, respectively.\nOne further point to bring out now, however, is how problematic the mean-field assumption can\nbe for dilute polymer solutions. This is illustrated schematically in Figure 7.4, where a dilute\nsolution is pictured, along with a trajectory through the solution that happens to pass through\ntwo coils. Also shown is the \u201clocal\u201d monomer concentration along that trajectory. It has two\nregions where the trajectory passes through the coils, and the local concentration of monomers is\nsignificantly higher than the solution average, (:52. However, there are also substantial regions\nbetween coils where the actual monomer concentration is zero. In other words, chain connectivity\n"]], ["block_3", ["The expression for the enthalpy of mixing in Flory\u2014Huggins theory is exactly that for regular\nsolution theory, once the substitutions of (p1 and(152 for x1 and x2 have been made. This can be seen\nbecause the enthalpy was computed on a lattice site basis with only local interactions and the\ncalculation would not be changed by linking the monomers together. (This is not strictly true,\nbecause now for each monomer there are only 2\u20142 neighboring sites that could be occupied by\neither monomer or solvent; two sites are required to be other monomers by covalent attachment.\nHowever, as 2 and Aw do not appear independently in the final expression for A0,\u201c, but are\nsubsumed into the parameter x, and we do not know Aw exactly anyway, we ignore this\ncomplication.) Thus we can write\n"]], ["block_4", ["for the system, and\n"]], ["block_5", ["per site. Combining the expressions for AS\"1 and AH\"1 we arrive at the final result:\n"]], ["block_6", ["for the system, and\n"]], ["block_7", ["Finally, then\n"]], ["block_8", ["7.3.3\nFlory\u2014Huggins Theory: Enthalpy of Mixing\n"]], ["block_9", ["which is exactly the result obtained in the previous section. Next time, you will probably just settle\nfor the simple argument!\n"]], ["block_10", ["Flory\u2014Huggins \n257\n"]], ["block_11", ["AG,1n\nF: \n(7.3.133)\n"]], ["block_12", ["AH,\u201c cplq\ufb01zk\n(7.3.12b)\n"]], ["block_13", ["AH,\u201c \u201d11%k ma\ufb01a/RT\n(7.3.12a)\n"]], ["block_14", ["AG,1n\n(b\nH \n(7.3.13b)\n"]], ["block_15", ["ASm k{(m1 + m2) ln(m1 + mzN) m1 ln m1 m2 ln(m2N)}\n"]], ["block_16", ["N\nN\n: k{m1 Wm)\n+ m, 1,,(ml+_mz) }\nm1\nm2N\n= \u2014\u2014k{m1 lnqbl + m2 lnqbz}\n= \u2014R{n1 ms, + n; 111%}\n(7.3.11)\n"]]], "page_271": [["block_0", [{"image_0": "271_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "271_1.png", "coords": [33, 42, 219, 275], "fig_type": "figure"}]], ["block_2", ["Figure 7.4\nIllustration of the failure of the mean-field assumption in dilute solutions: The local concentra-\ntion along some arbitrary trajectory \ufb02uctuates between higher than average, inside a polymer coil, and lower\nthan average, outside a coil.\n"]], ["block_3", ["guarantees that monomers are clustered in space, whereas the model assumes that the local\nconcentration is uniformly c152 throughout the sample. This turns out to be a major limitation to\nthe quantitative application of Flory\u2014Huggins theory to dilute solutions; however, it also suggests\nthat the theory should get progressively better when the concentration is increased, and the coils\nbegin to interpenetrate. This turns out to be the case.\n"]], ["block_4", ["At this point it is worthwhile to summarize the main assumptions employed in order to arrive at the\nexpression for the free energy of mixing, Equation 7.3.13.\n"]], ["block_5", ["258\nThermodynamics of Polymer Solutions\n"]], ["block_6", ["In this section we consider the osmotic pressure, H, of a dilute, uncharged polymer solution. First,\nwe will develop the virial expansion for I], which is based on general thermodynamic principles\nand therefore completely model-independent. We will learn how measurements of H can be used\nto determine the number\u2014average molecular weight of a polymer, and how the so-called second\nvirial coef\ufb01cient, B, is a diagnostic of the quality of a solvent for a given polymer. Then we will\nreturn to the Flory\u2014Huggins theory, and see what it predicts for H and B. This will lead us to our\nfirst working definition of a theta solvent, a very important concept in polymer solutions. Finally.\n"]], ["block_7", ["7.3.4\nFlory\u2014Huggins Theory: Summary of Assumptions\n"]], ["block_8", ["99:359.\u201c\n"]], ["block_9", ["7.4\nOsmotic Pressure\n"]], ["block_10", ["1.\nThere is no volume change on mixing, and VI 171, V; \n172 are independent of concen-\ntration.\nAS\", is entirely the ideal combinatorial entropy of mixing.\nAH\", is entirely the internal energy of mixing.\nBoth AS\"1 and AH\", are computed assuming entirely random mixing.\nThe interactions are short-ranged (nearest neighbors only), isotropic, and pairwise additive.\nThe local concentration is always given by the bulk average composition (the mean\u2014field\nassumption).\n"]], ["block_11", [{"image_2": "271_2.png", "coords": [71, 45, 203, 183], "fig_type": "figure"}]]], "page_272": [["block_0", [{"image_0": "272_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "272_1.png", "coords": [19, 496, 247, 617], "fig_type": "figure"}]], ["block_2", [{"image_2": "272_2.png", "coords": [24, 425, 256, 662], "fig_type": "figure"}]], ["block_3", [{"image_3": "272_3.png", "coords": [32, 454, 214, 526], "fig_type": "figure"}]], ["block_4", ["Figure 7.5\nSchematic diagram of the osmotic pressure experiment, and the operational definition of the\nosmotic pressure, H.\n"]], ["block_5", ["Rearranging, we obtain\n"]], ["block_6", ["has a dilute polymer solution of known concentration, c (in g/mL), and therefore at the instant the\nsolution is introduced into its compartment, the solvent component has a different chemical\npotential, ul. To reach equilibrium, so that the solvent chemical potential is equal on both sides\nof the membrane, there must be a net \ufb02ow of solvent from the left compartment to the right. This\ncan be easily seen, in that the simplest way to equalize the two chemical potentials would be to\nhave equal polymer concentrations on each side. As the polymer cannot move from right to left,\nsome solvent must move from left to right. However, because the solution in the right is contained,\nthe influx of solvent increases the column of solvent in the tube, i.e., the pressure goes up. This\nincrease in pressure defines the osmotic pressure, H, and ultimately U will oppose further solvent\ntransfer. At equilibrium, then,\n"]], ["block_7", ["From Equation 7.1.12 we recall that the integrand in Equation 7.4.1 is just the partial molar volume\nof the solvent, V1. We can assume this is constant over the relatively small pressure change, H,\n"]], ["block_8", ["and temperature compartments by\na semipermeable membrane; the\nmembrane passes solvent easily, but is impermeable to polymers (e.g., because of their size).\nThere is emerging from the top of each compartment and the height of the fluid in each\ntube reflects the pressure in that compartment. The compartment on the left is full of pure solvent,\nwhich therefore its standard state chemical potential it? (T, p0). The compartment on the right\n"]], ["block_9", ["and thus\n"]], ["block_10", ["the concept of osmotic pressure will turn out later to be central to understanding scattering\nexperiments (see Chapter 8). In short, this is a very important section.\n"]], ["block_11", ["The experiment schematically Figure 7.5. A thermostated chamber at pressure p0\n"]], ["block_12", ["Osmotic \n259\n"]], ["block_13", ["7.4.1\nOsmotic Pressure: General Case\n"]], ["block_14", ["MilTaPO) m (Tape) + UV:\n(7.4.2)\n"]], ["block_15", ["P0+H\n8M1\nmm) m(m + H) MAT-.100) +l\n(7,M\n (7.4.1) \nPo\np\nT\n"]], ["block_16", ["1i1 [1,? 1.1.1 \u2014\u2014RT 1n a1 \u2014RTln71x1 e \u2014RTlnx1\n(7.4.3)\n"]], ["block_17", ["Pure solvent\n#i (T. P0)\nSolution\n"]], ["block_18", ["Semipermeable membrane\n"]], ["block_19", [{"image_4": "272_4.png", "coords": [76, 460, 191, 509], "fig_type": "figure"}]], ["block_20", ["#1 (T, P0 \u2018l' H)\n"]]], "page_273": [["block_0", [{"image_0": "273_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "273_1.png", "coords": [22, 468, 214, 680], "fig_type": "figure"}]], ["block_2", ["The last simplification in Equation 7.4.6 is equivalent to ignoring n2 relative to n1 in the\ndenominator. For a solution with c = 0.01 g/mL, M 100,000 g/mol, and V in mL, n2/V will be\n104, whereas nl/V will be about 10\u20182 (if the solvent density n 1 g/mL and molecular weight\na 100 g/mol), so this is a very reasonable approximation. Substituting Equation 7.4.6 into Equation\n7.4.5 leads to\n"]], ["block_3", ["To replace x2 with \u201cpractical\u201d polymer units, such as c, we recall that c nzM/V where M is the\nmolecular weight and V is the solution volume. Thus\n"]], ["block_4", ["where B is called the second virial coef\ufb01cient: it has units of cm3 mol/gz. The quantity B3, the third\nvirial coefficient, re\ufb02ects ternary or three-body interactions, and is important when (3 is large\nenough and when B is small enough; we will not consider it further here. Equation 7.4.7a is the\ncentral result of this section. The quantity on the left\u2014hand side is measurable; of the quantities on\nthe right, c is determined by solution preparation, and M and B are determined by examining the c\ndependence of H. In Figure 7.6a H/RT is plotted against c for three different values of B: B > 0,\n"]], ["block_5", ["Figure 7.6\nGeneric plots of osmotic pressure versus concentration for dilute polymer solutions: (a) [NET\nversus 6 and (b) H/CRT versus 0, for the indicated values of B. Note the curvature at high c clue to three-body\ninteractions (the 63 term in Equation 7.4.7a).\n"]], ["block_6", ["where we have recalled the definition of the solvent activity, al, from Equation 7.1.13, and\nrecognized that 71, the activity coefficient, will approach\n1\nin sufficiently dilute solution.\nConverting to the polymer concentration, x2 = 1 \u2014- x1, gives\n"]], ["block_7", ["260\nThermodynamics of Polymer Solutions\n"]], ["block_8", ["recalling again that ln(1 \u2014\u2014x) x \u2014x for small x. In order to allow for the effects of finite solute\nconcentration, it is customary to expand the right-hand side of Equation 7.4.4 in powers of x2, a\nvirial expansion:\n"]], ["block_9", ["(a)\n0.9/ml-\n(b)\nc,g/mL\n"]], ["block_10", [{"image_2": "273_2.png", "coords": [35, 463, 354, 625], "fig_type": "figure"}]], ["block_11", [{"image_3": "273_3.png", "coords": [36, 329, 277, 380], "fig_type": "molecule"}]], ["block_12", ["I\u2018l/FIT\nI\u2018l/cFlT\n"]], ["block_13", [{"image_4": "273_4.png", "coords": [42, 462, 178, 610], "fig_type": "figure"}]], ["block_14", ["n2\ncV/M\nN \n2\n:\n__\n7.4.\nn1+n2\n(V/v,)+(cV/M)\nM\n(\n6)\n352\n"]], ["block_15", ["H\n1\nx2\n\u2014:\u2014:'ln1\u2014x\n95:-\n7..44\nRT\nV1\n(\n2)\nV1\n(\n)\n"]], ["block_16", ["%=KZ\u2014+BV\u00a7(\ufb01)2+o--=\ufb01+3c2+33c3+m\n(7.4.7a)\n"]], ["block_17", ["\ufb01:\u2014V\u2018I+BX2+B3X2+\n\"'\n(7.4.5)\n"]], ["block_18", ["H\nx2\n2\n3\n"]], ["block_19", ["I\u2014l\u2014t\u2014II\u2014l\u2014I\u2014II\u2014I\u2014rIUIIuu'nlllluv\u2014I\u2014r\n\"\"\"\"'l\"\"\"\"'I\"\"\"\"'\nI'\n"]], ["block_20", ["I.\n"]], ["block_21", ["l-\n.r\n..\n"]], ["block_22", [{"image_5": "273_5.png", "coords": [169, 461, 393, 639], "fig_type": "figure"}]], ["block_23", [{"image_6": "273_6.png", "coords": [198, 458, 332, 610], "fig_type": "figure"}]]], "page_274": [["block_0", [{"image_0": "274_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["B = O, and B < O. In Figure 7.6b we choose a second format, obtained by dividing Equation 7.4.7a\nthrough \n"]], ["block_2", ["This format often employed in practicereveals clearly that the molecular weight determined\n"]], ["block_3", ["l.\nThe \ufb01rst term in Equation 7.4.7a, c/M, is the number of molecules per unit volume (in\nmol/cm3). Thus, in the very dilute limit, the osmotic pressure is determined only by the\nnumber of solute molecules, whatever they may be. This is characteristic of all colligative\nproperties, including H, freezing point depression, and boiling point elevation. However, as c\nincreases, there will be binary solute\u2014solute interactions. The second term, proportional to c2,\naccounts for these. Now there are three possibilities:\n2.\nIf the polymers are in a good solvent, monomers on different chains are happy enough to be\nsurrounded by solvent; when two coils approach one another, there is steric repulsion (i.e.,\nexcluded volume) and the chains separate. Therefore, the coil\u2014coil interaction is effectively\nrepulsive. This corresponds to B > 0; it means there is an even greater drive for solvent to \ufb02ow\ninto the right compartment and dilute the solution.\n3.\nConversely, if the solvent is poor, such that the polymer is barely able to stay dissolved in\nsolution, monomers find it energetically favorable to be close to other monomers. Thus when\ntwo coils approach one another, there is a tendency to cluster. In this case, B <0 and the\neffectively attractive solute\u2014solute interactions resist the uptake of further solvent.\n4.\nThere is a special intermediate case where B =0. This corresponds to a \u201cnot-very\u2014good\u2018\u201d\nsolvent where the excluded volume and relatively unfavorable solvent\u2014solute interactions\ncancel one another in the net H. This case is given a special name; following Flory, it is\ncalled a theta solvent [3]. We will explore the significance of a theta solvent in more detail\nsubsequently.\n5.\nIt is worth pointing out that there is an analogy with the van der Waals equation of state for 1\nmol of an imperfect gas:\n"]], ["block_4", ["Osmotic \n261\n"]], ["block_5", ["from plotof to the sign of the initial slope.\nSo, the experimental approach is clear enough: Measure H for a series of solutions of known c,\nwith c sufficiently low that higher order terms in the virial expansion are not too important. But,\nwhat does Equation 7.4.7a mean physically?\n"]], ["block_6", ["We noted previously that moles were a troublesome unit for polymers, in part because of the\ninevitable molecular weight distribution. Now we can address the important issue of what average\n"]], ["block_7", ["7.4.1.1\nNumber-Average Molecular Weight\n"]], ["block_8", [{"image_1": "274_1.png", "coords": [40, 510, 194, 541], "fig_type": "molecule"}]], ["block_9", ["H\n1\n__ = _\nB\n. . .\n7.4.7b\ncRT\nM \nC +\n(\n)\n"]], ["block_10", ["1\n\u2014 a RT\nP\n_ + E_L_\n5,2\n_ 2\n+ __ + . . .\n7.4.9\nRT\nV\nV2\nV3\n(\n)\n"]], ["block_11", ["in which B accounts for the excluded volume and a the intermolecular interactions. When this\nequation is expanded in a virial series in the density (UV)\n"]], ["block_12", ["the second virial coefficient is B\u2014(oz/RT). This virial coefficient vanishes at a special\ntemperature, known as the Boyle point, when the excluded volume (B) and interaction (a/\nRT) terms exactly cancel one another. Similarly, the theta temperature for a particular poly-\nmer\u2014solvent system is the temperature at which B vanishes due to a cancelation of effects from\nexcluded volume and net polymer\u2014polymer interactions. It is important to realize, however,\nthat it is not the excluded volume itself that vanishes, but just its effect on the osmotic pressure.\n"]], ["block_13", ["aV2) (V \u2014 ,3) 2 RT\n(7.4.8)\n(P \n"]]], "page_275": [["block_0", [{"image_0": "275_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "275_1.png", "coords": [30, 161, 242, 229], "fig_type": "molecule"}]], ["block_2", ["The three values of Mn are very comparable, as they should be. The values of B are consistent with\nthe notion that cyclohexane is a theta solvent for polystyrene at 345\u00b0C. The experimental value of\nB is essentially zero at this temperature, considering the experimental uncertainty. Note also how\nthe plotting format of Figure 7.7b accentuates the scatter in the data.\n"]], ["block_3", ["For each data set, divide H by CRT, where R 82.1 cm3 atm/K mol and T is in kelvin. The results are\nplotted in Figure 7.7b. Then fit each data set to a straight line by linear regression (a hand calculator\nis sufficient). The results are\n"]], ["block_4", ["0.005\n0.0061\n0.0063\n0.0067\n0.010\n0.0114\n0.0124\n0.0136\n0.015\n0.0173\n0.0187\n0.0203\n0.020\n0.0215\n0.0255\n0.0276\n0.025\n0.0276\n0.0316\n0.0354\n0.030\n0.0332\n0.0381\n0.0421\n0.040\n0.0405\n0.0502\n0.0580\n"]], ["block_5", ["To illustrate the typical magnitude of the osmotic pressure and the extraction of values of Mn and\nB, consider the following data for IT (in atm) for a polystyrene sample in cyclohexane at three\ntemperatures. The data are also plotted in Figure 7.7a.\n"]], ["block_6", ["M will the osmotic pressure experiment measure for a polydisperse sample? The answer is\nappealingly simple and rigorous; as a colligative property, H at infinite dilution depends only\non the number of solute molecules per unit volume, and therefore should determine the number-\naverage molecular weight, Mn. To see that this is indeed so, we can write for very dilute\nsolutions:\n"]], ["block_7", ["c (g/mL)\n20.0\u00b0C\n34.5\u00b0C\n50.0\u00b0C\n"]], ["block_8", ["where we recall the definition of Mn from Chapter 1. Thus a properly conducted osmotic pressure\nmeasurement can determine the absolute value of Mn.\n"]], ["block_9", ["for a distribution of molecular weights, M,-. Next we form the ratio\n"]], ["block_10", ["Solution\n"]], ["block_11", ["262\nThermodynamics of Polymer Solutions\n"]], ["block_12", ["Example 7.2\n"]], ["block_13", [{"image_2": "275_2.png", "coords": [41, 116, 152, 150], "fig_type": "molecule"}]], ["block_14", ["RTC\nc,-\n: __ :\n__\n7.4.\nH\nM\nRT:\nM1\n(\n10)\n"]], ["block_15", ["200\u00b0C:\nSlope B \u20142.0 x 10\u201c4 cm3 mol/g2\n"]], ["block_16", ["345\u00b0C:\nSlope B 1.7 X 10\u20145 cm3 mol/g2\n"]], ["block_17", ["50.0\u00b0C:\n316pe B :1.2 X 10\u20144 cm3 mol/g2\n"]], ["block_18", ["C-\u2014\u20190\nCRT\n-\nE: C;\nE\nZ n;M,-/V\n"]], ["block_19", ["mm\nH\n- _ \n"]], ["block_20", ["1/intercept :Mn 1.97 x 104 g/mol\n"]], ["block_21", ["1/intercept Mn 2.02 x 104 g/mol\n"]], ["block_22", ["1 /intercept Mn 2.00 x 104 g/mol\n"]], ["block_23", ["_ \n__\nl\n\u2014 ZniM, T Mn\n(7.4.11)\n"]], ["block_24", ["____.-\u2014\n"]]], "page_276": [["block_0", [{"image_0": "276_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "276_1.png", "coords": [25, 251, 273, 480], "fig_type": "figure"}]], ["block_2", ["To conclude this section, we return to the Flory\u2014Huggins expression for the free energy of mixing\nfor a polymer solution, Equation 7.3.13, and see what it predicts for H. From Equation 7.1.5 we can\nwrite\n"]], ["block_3", ["Figure 7.7\nOsmotic pressure for polystyrene in cyclohexane plotted as (a) H versus 0 and (b) H/CRT versus c,\nat the indicated temperatures. The straight lines are linear regression fits. The data are provided in Example 7.2.\n"]], ["block_4", ["7.4.2\nOsmotic Pressure: Flory\u2014Huggins Theory\n"]], ["block_5", ["so we need to take the derivative with respect to m of Equation 7.3.13, Le,\n"]], ["block_6", ["Osmotic \n263\n"]], ["block_7", ["(a)\nc,g/mL\n"]], ["block_8", ["\u2014B\u2014 50.0\u00b0C\n4x1o\u20145...........\no\n0.01\n0.02\n0.03\n(b)\nc, g/mL\n"]], ["block_9", ["I\u2018UCRT,\n"]], ["block_10", ["mol/g\n"]], ["block_11", [{"image_2": "276_2.png", "coords": [41, 49, 268, 247], "fig_type": "figure"}]], ["block_12", ["4.5 X 1075 \n+2000\n"]], ["block_13", ["5%:{RT(n11n\u00a21\u2014l\u2014\n712111s + 721(352X)}\n"]], ["block_14", ["5.5><1o-5 \n/\n"]], ["block_15", ["\u2014\n6\n\u201cHi/1 [.Ll M? (\u201c32; AGm)\n(7.412)\n"]], ["block_16", [{"image_3": "276_3.png", "coords": [49, 579, 207, 617], "fig_type": "molecule"}]], ["block_17", ["a\n"]], ["block_18", ["g\n0.04\".\na\nI\n.J\n(U\n:\nEl\n:\n1:\u201c\ng\na\nn\n-\n0.03\n-\nD\n.2\n_\na\nI\n.\nI\n'3'\nI\n(102 '_\na\nI\nI 20.0\u00b0C\n_'\n"]], ["block_19", ["I\nj\nE\n-\n5x10-5>\u00a3\nD\nU\na\nn\na\n\u2014-_\n"]], ["block_20", ["6X10_5\u201cHun-WHHII-Hl-m\n"]], ["block_21", ["E\nP 345\u00b0C\n-\n0.01\n\u2014\nO\n_'\n"]], ["block_22", ["0.05\n:\u2014\nu\n_'\n"]], ["block_23", ["0.07\ufb01r1www.mmlwwlwrr.\n"]], ["block_24", ["0.06 \nPolystyrene in cyclohexane\n_\n..\nE\n-\n"]], ["block_25", ["0\n_ i\n.\ni\nl\nJ\n.\n.\n.\n.\nI\n.\n.\nI\n.\n.\nI\n.\n.\n.\n. '\n"]], ["block_26", ["O\n0.01\n0.02\n0.03\n0.04\n0.05\n"]], ["block_27", [{"image_4": "276_4.png", "coords": [83, 393, 149, 439], "fig_type": "figure"}]], ["block_28", ["+34,5\u00b0C\n"]], ["block_29", ["Polystyrene in cyclohexane\n"]], ["block_30", [{"image_5": "276_5.png", "coords": [181, 171, 234, 221], "fig_type": "figure"}]], ["block_31", ["PI\u201d:\n"]], ["block_32", ["0.04\n0.05\n"]]], "page_277": [["block_0", [{"image_0": "277_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "277_1.png", "coords": [21, 412, 185, 456], "fig_type": "molecule"}]], ["block_2", [{"image_2": "277_2.png", "coords": [25, 63, 253, 140], "fig_type": "figure"}]], ["block_3", [{"image_3": "277_3.png", "coords": [28, 461, 219, 517], "fig_type": "molecule"}]], ["block_4", [{"image_4": "277_4.png", "coords": [32, 308, 202, 366], "fig_type": "molecule"}]], ["block_5", [{"image_5": "277_5.png", "coords": [32, 204, 310, 263], "fig_type": "molecule"}]], ["block_6", [{"image_6": "277_6.png", "coords": [34, 365, 136, 409], "fig_type": "molecule"}]], ["block_7", ["To find the answer, we recall \ngrazml-l\u2014Nn\ufb02\u2014mIL\n"]], ["block_8", ["Proceeding with the differentiation \n"]], ["block_9", ["254\nThermodynamics of Polymer Solutions\n"]], ["block_10", ["This equation has two important features. First, when X: 1/2, then B =0 and we have a theta\nsolvent. Thus x: 1/2 represents a second operational definition of the theta point. For X > 1/2,\nB < 0, and the solvent is poor, whereas for X < 1/2, the solvent is good. Figure 7.8 shows data for\n3(7) for several polystyrenes in cyclohexane and the theta temperature is determined to be 345\u00b0C.\nSecond, 8 is predicted to be independent of molecular weight (note that N mM in Equation 7.4.17\nand thus the M dependence cancels out). It turns out experimentally that this is not quite true; for\nexample B varies approximately as M in good solvents. This incorrect prediction is a direct\nconsequence of the mean\u2014field assumption that we discussed in the previous section.\n"]], ["block_11", ["where we choose to write everything terms polymer concentration. The expansion of\nln(1 (152) \u2014q\u00a72 chi/2 is used to get rid of logarithmkeep the dig/2 term here\nbecause of the virial expansion to second order), over to the side:\n"]], ["block_12", ["Finally, we convert from(152 to c:\n"]], ["block_13", ["similarly,\n"]], ["block_14", ["and Obtain\n"]], ["block_15", ["In this section we examine the phase behavior of a polymer solution, or, more precisely, we\nconsider the temperature-composition plane at fixed pressure and locate the regions where a\n"]], ["block_16", ["7.5\nPhase Behavior of Polymer Solutions\n"]], ["block_17", ["and thus for the Flory\u2014Huggins model,\n"]], ["block_18", [{"image_7": "277_7.png", "coords": [36, 145, 247, 196], "fig_type": "molecule"}]], ["block_19", [{"image_8": "277_8.png", "coords": [40, 63, 220, 143], "fig_type": "molecule"}]], ["block_20", ["H\n1\n(162\n1\n1\n2\n _ z :\u2014 __ _ __\n_ _\n+ . . .\nRT\nv1 N\nv1 (X\n2) $2\n(7'4\u201c)\n"]], ["block_21", ["a\na 1 \na\n\u2014\n2\n\u2014-\n\ufb02=\u2014\u2014\u2014#(\n(\u201951): \u2014i=\n952:\nQM\"\n(7.4.13b)\n8111\n6711\n8111\nN712\nI11\n'\n"]], ["block_22", ["H\nC +\n1\nV \n2 +\n_ 2 _\n_ _\n_..__C\n. . .\nRT\nM\n2\nX\n1W\n(7.4.17)\n"]], ["block_23", ["11171\n{\nm am\n122\n(\u201415%)\n}\n\u2014\u2014\u2014= \n1nd) +\u2014\n+-\u2014\u2014\n\u2014\u2014\n+\n\u2014\nRT\n1\nCb]\n\u201d1\n$2\na\n($2\n\u00a21\u00a22)X\n"]], ["block_24", ["3\u201911\n(111 + Nn2)2\n(\u201dI + \n2 = \n"]], ["block_25", ["_ _ \n2 \nM\n\u2014\nM\n(7.4.16)\n"]], ["block_26", ["1\n_ N2\n1\n\"172\n1\n(2\nM2\n2\nv1 M2\n(\n)\n"]], ["block_27", ["1\n:\u2014\nIn 1\u20141;!)\n+\u00a2 (144)+\n(1)2}\n{\n(\n2)\n2\nN\nX 2\n(7.4.14)\n"]], ["block_28", ["N112\nn1\n(7.4.133)\n"]]], "page_278": [["block_0", [{"image_0": "278_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Figure 7.8\nSecond virial coef\ufb01cient for three polystyrenes in cyclohexane as a function of temperature.\nThe theta temperature based on these data alone lies between 34\u00b0C and 35\u00b0C. (From Yamakawa, H., Abe, F.,\nand Einaga, Y., Macromolecules, 27, 5704, 1994.)\n"]], ["block_2", ["one\u2014phase solution is stable, and where the mixture will undergo liquid-liquid phase separation\ninto two phases. We will do this first for regular solution theory, as a means to illustrate the\nvarious concepts and steps in the procedure. Then we will return to Flory\u2014Huggins theory and\nsee the consequences of having one component substantially larger in molecular weight than\nthe other.\n"]], ["block_3", ["The phase diagram for a regular solution is shown schematically in Figure 7.9. It has the following\nimportant features, which we will see how to calculate:\n"]], ["block_4", ["1.\nA critical point (Tc, x.) such that for T > Tc 3 one-phase solution is formed for all compositions.\n2.\nA coexistence curve, or binodal, which describes the compositions of the two phases xi\u2019 and xi\"\nthat coexist at equilibrium, after liquid\u2014liquid separation at some fixed T< TC. Any solution\nprepared such that (T,x1) lies under the binodal will be out of equilibrium until it has\nundergone phase separation.\n3.\nA stability limit, or spinodal, which divides the two-phase region into a metastable window,\nbetween the binodal and the spinodal, and an unstable region, below the spinodal. The\nsigni\ufb01cance of the terms metastable and unstable will be explained subsequently. Note that\nthe binodal and spinodal curves meet at the critical point.\n"]], ["block_5", ["Qualitatively, of course, we should expect one-phase behavior at high T because AS,1n > 0, and\ntherefore -\u2014TASm contributes an increasingly negative term to A6,\u201c. However, although AGm < 0\nis the criterion for spontaneous mixing, it by no means guarantees a single mixed phase, as we shall\n"]], ["block_6", ["7.5.1\nOverview of the Phase Diagram\n"]], ["block_7", ["Phase Behavior of Polymer Solutions\n265\n"]], ["block_8", ["\u201829\no x 10\u00b0\n'5\nE\nO)\nE\nO\n"]], ["block_9", ["\u201c3 \u20145 x 10'5\n-\n'\n\u2014EB\u2014 M: 9,980\n\u2018\n"]], ["block_10", ["__1.5 \nI\nI\nI\nI\nI\nI\nI\nI\nI\n"]], ["block_11", ["-1 x 10\"4\n\u2014E\u2014 M: 40,000\n\u2018\n"]], ["block_12", [{"image_1": "278_1.png", "coords": [49, 43, 302, 295], "fig_type": "figure"}]], ["block_13", ["Polystyrenes\n'\nin\n5 10\n_'\ncyclohexane\n"]], ["block_14", ["1\nX 10\u20144\nl\nl\nI\n'\nI\nl\n|\n|\n"]], ["block_15", [{"image_2": "278_2.png", "coords": [186, 189, 278, 248], "fig_type": "figure"}]], ["block_16", ["\u2014E\u2014 M: 20,200\n"]]], "page_279": [["block_0", [{"image_0": "279_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "279_1.png", "coords": [23, 38, 242, 255], "fig_type": "figure"}]], ["block_2", ["Figure 7.9\nPhase diagram (temperature versus mole fraction of component 1) for regular solution theory.\nThe binodal (coexistence curve) separates the one-phase region at high temperature from the two-phase region\n"]], ["block_3", ["266\nThermodynamics of Polymer Solutions\n"]], ["block_4", ["at low temperature. The spinodal curve (stability limit) separates the unstable and metastable windows within\nthe two-phase region. The binodal and spinodal curves meet at a critical point.\n"]], ["block_5", ["and recall from its definition (Equation 7.2.9) that )(N l/T. These two functions are plotted in\nFigure 7.10a and Figure 7.10b, respectively. Note that both are symmetric about x1 1/2, and that\nin this format the entropy term is independent of T, whereas the enthalpy term is not (due to x).\nFurthermore, we take X > 0, as expected by the theory. In Figure 7.10c we combine the two terms,\nat two generic temperatures, one \u201chigh\u201d and one \u201clow.\u201d At the higher T, X is so small that AG\u201c.1\nlooks much like the AS\", term; it is always concave up. However, at the lower T, the larger X in the\nenthalpy term produces a \u201cbump,\u201d or local maximum in the free energy. This will turn out to have\nprofound consequences. Note that even at the lower T, AG,n < 0 for all compositions considered in\nthis example.\nPhase separation will occur whenever the system can lower its total free energy by dividing into\ntwo phases. If we prepare a solution with overall composition (x1), and then ask will it prefer to\nseparate into phases with compositions x; and xi\" , we can find the answer simply by drawing a line\nconnecting the corresponding points on the AG,n curve (i.e., AGm(xf) to AGm(xf\u2019)), as shown in\nFigure 7.11a. Because AG\u201c, is an extensive property, this line represents the hypothetical free\nenergy of a combination of two phases, xf and xi\", for any overall composition On) that lies in\nbetween. (Note that the relative proportions of the two phases with compositions x{ and x{\u2019 are\ndetermined once (x1) is selected, by the so-called lever rule.)\nWhat we now realize is that, so long as AG,\u201c is concave up, this straight line will lie above AG,1n\nat (x1) for any choice of xf and x1\u201d , and therefore phase separation would increase the free energy.\n"]], ["block_6", ["see. To begin the analysis, we resolve the two contributions to AGm/RT from regular solution\ntheory (Equation 7.2.1 1):\n"]], ["block_7", ["0\nx1\n1\n"]], ["block_8", ["AS\n_Tm xllnx1+ x2 11l\n(7.5.1)\n"]], ["block_9", ["RT\n"]], ["block_10", ["(To: Xe)\nBinodal\n,\nMetastable\n"]], ["block_11", ["Two-phase\n"]], ["block_12", ["One-phase\n"]], ["block_13", ["Spinodal\n"]], ["block_14", ["Unstable\n"]]], "page_280": [["block_0", [{"image_0": "280_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "280_1.png", "coords": [28, 240, 233, 436], "fig_type": "figure"}]], ["block_2", [{"image_2": "280_2.png", "coords": [30, 42, 227, 436], "fig_type": "figure"}]], ["block_3", ["where the second derivative can be taken with respect to the mole fraction of any component. The\nmeaning of stability is this: In any mixture at a finite temperature, there will be spontaneous, small\nlocal \ufb02uctuations in concentration 8x, such that there are small regions that have x1 bigger than the\naverage, and some regions where it is smaller. Now, by the argument given above, any such\n\ufb02uctuation will actually increase the free energy; the straight line connecting (x1) 8x1 and\n(x1) + 8.151 will fall above AGm(x1). Consequently all these \ufb02uctuations will relax back to (11).\nThe importance of these spontaneous \ufb02uctuations will be taken up again in Chapter 8, where we\nwill show how they are the origin of light scattering.\n"]], ["block_4", ["Thus \u201cconcave up\u201d gives us the criterion for stability of the one\u2014phase solution; the mathematical\nexpression of concave up is\n"]], ["block_5", ["Figure 7.10\nPredictions of regular solution theory for (a) entropy of mixing, plotted as\n\u2014ASm/R; (b)\nenthalpy of mixing, plotted as AHm/RT, for two temperatures; (c) free energy of mixing obtained by\ncombining panels (a) and (b), plotted as AGm/RT.\n"]], ["block_6", ["Phase Behavior of Solutions\n267\n"]], ["block_7", ["J\na:\nF\n2\nE\n'|\nME\n_\nTI,\n25 \u201d\"0 5 _\n j \nE:\n:\n:\n"]], ["block_8", ["AGm/RT\n9;: N\n"]], ["block_9", [{"image_3": "280_3.png", "coords": [38, 29, 232, 250], "fig_type": "figure"}]], ["block_10", ["I\n0.4 :-\n\u2014\n"]], ["block_11", [{"image_4": "280_4.png", "coords": [40, 518, 122, 563], "fig_type": "molecule"}]], ["block_12", [{"image_5": "280_5.png", "coords": [42, 39, 405, 239], "fig_type": "figure"}]], ["block_13", ["\u20140.25 l\n\u2014-\n0 8\n3\n3\n"]], ["block_14", ["_0_75 _\n_\n.\u2014\nHigh T075: 1-5)\n':\n"]], ["block_15", ["(C)\nX1\n"]], ["block_16", ["_-...II..I...II..I...\n0'40\n0.2\n0.4\n0.6\n0.8\n1\n"]], ["block_17", ["(a\nGm)\n>0\n(7.5.2)\n"]], ["block_18", ["(a)\nX1\nX1\n"]], ["block_19", ["_.1\nI\n'\nl\ni\n'\n'\n\u2014l\nI\nl\nI\nl\nl\n1\nl\nl\nI\n'\n'\n'\n-\n"]], ["block_20", ["TIP\n3x?\n"]], ["block_21", ["0\nI\n.\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI r I\nr\u2014r\nr\nJ\n1-2\n_\u2014r\nI\nI\nI\nI\nI\nI\nI\nI\nI\n.\nI\nI\nI\n.\nI\nI\nI\nI\n"]], ["block_22", ["T\n0\n0.2\n0.4\n0.6\n0.8\n1\n(b)\no\n0.2\n0.4\n0.6\n0.8\n1\n"]], ["block_23", ["-\n1\nF\n'\nT\nLow T0: = 2.5)\n\"-\n"]], ["block_24", ["High T(z= 1.5)\n"]], ["block_25", ["i\n0\n-_\n_\n"]], ["block_26", [{"image_6": "280_6.png", "coords": [229, 38, 417, 242], "fig_type": "figure"}]], ["block_27", ["I\nI\nI\nI\nI\nI\nI\n[\nI\nI\nI\nI\nI\nI\nI\nl\nl\nI\nI\n"]]], "page_281": [["block_0", [{"image_0": "281_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "281_1.png", "coords": [26, 44, 226, 321], "fig_type": "figure"}]], ["block_2", ["where we have divided by the total number of moles to get to mole fractions, and where\nAir,- it, pf. Now imagine we draw a straight line that is tangent to AG\", at some composition,\nx\u2019], as shown in Figure 7.11b. This line can be written generically as\n"]], ["block_3", ["Now consider the lower T curve in Figure 7.100, where AGm shows the bump. Here we can see that\nif we prepared a solution with (x1) somewhere near the local maximum of A0,\u201c, we could find an\nx{ and am\u201d such that the straight line between them would fall below the AG\", curve for our (x1),\nand phase separation should occur. In fact, there are many such pairs x1\u2019 and x1\" that would lower\nAGm, so which pair is chosen? We recall the criteria for phase equilibria: T and p must be identical\nin the two phases, and\n"]], ["block_4", ["The chemical potential of component 1 is the same in both phases and the chemical potential of\ncomponent 2 is equal in both phases. (Be careful with this; both relations must be satisfied\nsimultaneously, but it is not an equality between #1 and #2.) It turns out that there will be only\none solution (xl\u2019,x1\u201d) for both of these relations at a particular T, which we can identify by the\ncommon tangent construction. We can write the free energy as the mole\u2014weighted sum of the\nchemical potentials (which are the partial molar free energies, Equation 7.1.5):\n"]], ["block_5", ["Figure 7.11\nGeneric free energy of mixing versus composition curves. (a) If a solution with overall\ncomposition between xf and x1\u201d were to separate into two phases with compositions x{ and xi\u201d, the resulting\nfree energy (the dashed line) would lie above the one-phase case (smooth curve). (b) Tangent construction\nshowing how the chemical potentials of the two components may be obtained for a given composition x{.\n(c) Tangent construction finds the compositions of the two phases xf and xi\" that would coexist at equilibrium,\nfor a system with overall composition between xl\u2019 and x1\u201d . Points a and [9 denote the in\ufb02ection points of AGm,\nwhich separate the metastable (x{ < x1 < (1,!) < x1 < xi\") and unstable regions (a < x1 < 3)).\n"]], ["block_6", ["or\n(7.5.4)\n"]], ["block_7", ["7.5.2\nFinding the Binodal\n"]], ["block_8", ["0\nx1I\nX1\"\n1\n(C)\n"]], ["block_9", ["(a)\ni,\nx,\u00bb\n(b)\nX\u2018\n"]], ["block_10", ["268\nThermodynamics of Polymer Solutions\n"]], ["block_11", ["A(120(1\")\n_\n"]], ["block_12", ["3511209!) \u201d//\n"]], ["block_13", ["AGm\nAGm\n"]], ["block_14", [{"image_2": "281_2.png", "coords": [45, 46, 211, 172], "fig_type": "figure"}]], ["block_15", ["mm) mm\u201d),\n#2060 \u201c20\u201c\u201d)\n(7.5.3)\n"]], ["block_16", ["AG.\u201c xlAP\u2018q + (1 \u2014x1)A#2 A\ufb02z \"I\u2018xlmiui Ape)\n"]], ["block_17", ["AGm mApLI + ngAptz\n"]], ["block_18", ["AG\u201d,\n' \n371109\u201d)\n"]], ["block_19", [{"image_3": "281_3.png", "coords": [52, 169, 218, 296], "fig_type": "figure"}]], ["block_20", [{"image_4": "281_4.png", "coords": [79, 36, 418, 173], "fig_type": "figure"}]], ["block_21", ["A7120(1')\n"]], ["block_22", [{"image_5": "281_5.png", "coords": [227, 44, 397, 164], "fig_type": "figure"}]]], "page_282": [["block_0", [{"image_0": "282_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "282_1.png", "coords": [31, 426, 191, 465], "fig_type": "molecule"}]], ["block_2", ["Returning to Figure 7.11c, we see that there are two in\ufb02ection points, marked a and b, on each side\nof the bump. Between these two compositions, the free energy is concave down, and we say the\nsolution for that (x1,7)-is unstable. What does this mean? For any small local \ufb02uctuation in\nconcentration 8x1, the straight line connecting x1 8x1 and x1 + 8x1 will fall below AGm(x1). These\n\ufb02uctuations will therefore grow in amplitude and spatial extent; the mixture will spontaneously\nphase separate into two phases with compositions x\u2019l and x\u2019l\u2019 . Thus in a region where the AGm curve\nis concave down, the solution is unstable with respect to any \ufb02uctuation in concentration. The\nmechanism by which this phase separation occurs is called Spinoa\u2019al decomposition, and it is quite\ninteresting in its own right. However, in this chapter we are concerned with thermodynamics, not\nkinetics, so we will not pursue this here.\nYou may have noticed that there are two regions on the curve in Figure 7.11c, between x\"1 and a,\nand between b and x\u2019l\u2019 , where the curve is locally concave up, indicating stability, yet we already\nknow that the equilibrium state in these intervals should be liquid\u2014liquid coexistence with\nconcentrations x\u2019l and x\u2019l\u2019. What does this mean? These regions fall between the binodal and\nspinodal, and are termed metastable. They are stable against small, spontaneous \ufb02uctuations, but\nnot globally stable against phase separation. Consequently, a system in the metastable region may\nremain there indefinitely; it requires nucleation of a region of the new phase before separation\n"]], ["block_3", ["The next issue to address is the location of the spinodal, or stability limit. We have already\nindicated the condition for stability, namely Equation 7.5.2. The stability limit, then, is found\nwhere the second derivative of AGm changes sign, which de\ufb01nes an in\ufb02ection point:\n"]], ["block_4", ["In other words, if we tangent to the x1: 0 intercept, we obtain b Au2(x\u20191), and if we\nfollow it to the x1 1 intercept, k + b Anibal).\nThe argument so far applies for any AGm curve. Now if we have a AGm curve with a bump as\nin Figure 7.11c, we can draw one straight line that is tangent to AGm at two particular points,\ncall them x1\u2019 and x1\u201d. From the argument above, the x1:0 intercept gives us both Att2(xf)\nand Auz(x{\u2019), so these two chemical potentials must be equal. By the same reasoning the other\nintercept gives Au\ufb01xf) Auloq\"), and therefore we have shown that x1\u2019 and x1\u201d defined by the\ncommon tangent are indeed the compositions of the two coexisting phases. (Warning: for regular\nsolution theory, where the AGm curve is symmetric, xf and x1\u201d coincide with the local minima in the\nAG,\u201d curve, but this is not generally true.) So, in summary, one can locate the coexistence\nconcentrations by geometrical construction on a plot of AGm versus composition, or one could\ndo it from the analytical expressions for the two chemical potentials. However, the latter is\nalgebraically a little tricky, particularly because of the natural logarithm terms (see, for example,\nEquation 7.4.14).\n"]], ["block_5", ["But this relation holds whatever x\"1 we choose, so we can match the intercepts and slopes to obtain\n"]], ["block_6", ["Where it is the slope and b is the x1 0 intercept. But we chose )1 AGm for x1 xi, so inserting\nEquation 7.5.4 into Equation 7.5.5 we find\n"]], ["block_7", ["7.5.3\nFinding the Spinodal\n"]], ["block_8", ["Phase Behavior of Polymer Solutions\n269\n"]], ["block_9", ["y M \n(7.55)\n"]], ["block_10", ["kx\u2019, + b Ange) + x\u2019l[A,u.1(x'1) Att2(x\u20191)]\n(7.5.6)\n"]], ["block_11", ["5\u2019 A\ufb02zlxl)\nk Au1(x\u20191) Ayala)\n"]], ["block_12", ["(62\u201413(33)\n= 0\non the spinodal\n(7.5.8)\n8x,-\nTa)\n"]], ["block_13", ["(7.5.7)\n"]]], "page_283": [["block_0", [{"image_0": "283_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "283_1.png", "coords": [26, 16, 188, 211], "fig_type": "figure"}]], ["block_2", ["We can understand this by realizing that as T approaches TC from below, one in\ufb02ection point\nmoves to\nthe\nright,\nand\none\nto\nthe\nleft.\nThe\nrate of change of the\nin\ufb02ection\npoint,\n3/3xg(82AGm/3x,2), vanishes when the two meet.\nAlgebraic expressions for the spinodal and the critical point of regular solution theory can be\ndirectly obtained as follows. The chemical potential for component 1 (and of course, by symmetry,\nwe could equally well use component 2) comes from differentiating AGm:\n"]], ["block_3", ["proceeds. Nucleation, and the ensuing process of domain growth, is another interesting kinetic\nprocess that we will not discuss here. However, metastability can be a wonderful thing; diamond is\nmetastable with respect to graphite, the equilibrium phase of carbon at room T and p, but no one\nworries about diamonds transforming to graphite in their lifetime. A mechanical analogy is helpful\nin distinguishing among stable, metastable, and unstable systems, as shown in Figure 7.12. The ball\nin panel (a) may rattle around near the bottom of the bowl, but it will never come out; the system is\nstable. The ball in panel (b) is precariously perched on top of the inverted bowl, and the slightest\nbreeze or vibration will knock it off; the system is unstable. The ball in panel (c) can rattle around\nin the small depression, and may appear to be stable for long periods of time, but with a sufficiently\nlarge impulse it will roll over the banier and downhill to a lower energy state; the system is\nmetastable. Only state (a) is an equilibrium state, but state (c) might not change in our lifetime.\n"]], ["block_4", ["The \ufb01nal feature to locate in the phase diagram is the critical point. We know it lies on the\nspinodal, so it must satisfy Equation 7.5.8. But, we need another condition to make it a single,\nspecial point. The easiest way to visualize this is to return to Figure 7.10c and the plots of AGm at\ndifferent temperatures. Phase separation occurs only when we have the bump in AGm, so the\ncritical point marks the temperature where the bump first appears. This also corresponds to the\ntemperature where the two in\ufb02ection points merge into one and this is determined by\n"]], ["block_5", [{"image_2": "283_2.png", "coords": [34, 559, 215, 594], "fig_type": "molecule"}]], ["block_6", ["7.5.4\nFinding the Critical Point\n"]], ["block_7", ["Stable\nO\n"]], ["block_8", ["270\nThermodynamics of Polymer Solutions\n"]], ["block_9", ["Unstable\n"]], ["block_10", ["Metastable\n"]], ["block_11", ["Figure 7.12\nSchematic illustration of the difference between stable, unstable, and metastable states.\n"]], ["block_12", ["63%\u201c,\n"]], ["block_13", ["3x? \n= O\nat the critical point\n(7-5.9)\n"]], ["block_14", ["\u201911P\n"]], ["block_15", ["O\n"]], ["block_16", ["O\n"]]], "page_284": [["block_0", [{"image_0": "284_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "284_1.png", "coords": [24, 123, 182, 169], "fig_type": "molecule"}]], ["block_2", [{"image_2": "284_2.png", "coords": [29, 248, 199, 277], "fig_type": "molecule"}]], ["block_3", ["Now we can repeat this entire procedure for the Flory\u2014Huggins theory. The main difference will be\nthat the value of N breaks the symmetry of the AS\u201c, expression and will produce an asymmetric\n"]], ["block_4", ["The stability limit can now be obtained by taking the derivative with respect to x1:\n"]], ["block_5", ["Equation 7.5.14a and Equation 7.5.15 constitute two simultaneous equations that can be solved to\nobtain the critical point (see Problem 9): The result is x1 1,, :1/2 (which we could have guessed from\nthe outset, due to symmetry) and Xe :2. This means that unless it costs at least 2kT to exchange one\nmolecule of type 1 with one molecule of type 2, there will be no phase separation. To obtain the\ncritical temperature for a particular system, Tc, we need to know the value of X (i.e., zAw):\n"]], ["block_6", ["Generically, however, we can see that the larger Aw, the larger TC will be, and therefore the larger\nthe two-phase window. If, perhaps due to some specific interactions, Aw happens to be negative,\nthere will be no critical point according to regular solution theory; the system will be completely\nmiscible at all temperatures and in all proportions.\n"]], ["block_7", ["The critical point requires that we differentiate Equation 7.5.13 once more:\n"]], ["block_8", ["Note that this relation can be rewritten in the appealingly symmetric form\n"]], ["block_9", ["Now\n"]], ["block_10", ["where the subscript s denotes the value of X on the spinodal. (You should convince yourself that if\nwe followed the prescription for the stability limit given by Equation 7.5.8, and took the second\nderivative of AGm/kT from Equation 7.2.1 lb with respect to x1 instead of \ufb01rst obtaining [L1, we\nwould get the same relation.) This equation is a quadratic in x1:\n"]], ["block_11", ["7.5.5\nPhase Diagram from Flory\u2014Huggins Theory\n"]], ["block_12", ["so\ni :lnx1+(1\u2014x1)\u2014(1*X1)+X(1*xl)2\nRT\n=lnx1+ )((1-351)2\n(7'5'12)\n"]], ["block_13", ["Phase Behavior Polymer Solutions\n271\n"]], ["block_14", [{"image_3": "284_3.png", "coords": [42, 331, 128, 366], "fig_type": "molecule"}]], ["block_15", ["1\nx? \u2014x1+\u2014=0\n(7.5.l4a)\n2X5\n"]], ["block_16", ["X1\nIQ\n"]], ["block_17", ["6):\nn + n\nH n\nx x\n2\n4:\n1\n2\n21:\n12:12\n(7.5.11)\n3111\n(HI + n2)\n)7]\n722\n"]], ["block_18", ["3\nM1\n1\n3x1 (RT)\nx1\nXS(\n161)\n(7 5 13)\n"]], ["block_19", ["1\na (\u2014\u20142X(1 \u2014x1)) = \u2014\u00a7,+2Xc:0\n(75.15)\n6.17]\n1C]\n1\n"]], ["block_20", [{"image_4": "284_4.png", "coords": [47, 437, 227, 464], "fig_type": "molecule"}]], ["block_21", ["3\n1% :a\u2014nl.{mlnx1+n21n(1\u2014x1)+\u201d1(1_xl)X}\n"]], ["block_22", ["1\n1\n"]], ["block_23", ["__ _ \n_\n_ ___.\n7.5.16\n1%\n2k\n(\n)\nC\n"]], ["block_24", ["121 6x1\n\u201d2\n(9)61\n6x1\nl\u201c(1+)\u201d 3m\n1\u2014x1 3n1+(\nxl)X\nn1 (.3a\n(7'5'10)\n"]]], "page_285": [["block_0", [{"image_0": "285_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "285_1.png", "coords": [17, 199, 217, 269], "fig_type": "figure"}]], ["block_2", [{"image_2": "285_2.png", "coords": [31, 209, 209, 245], "fig_type": "molecule"}]], ["block_3", [{"image_3": "285_3.png", "coords": [34, 500, 300, 550], "fig_type": "molecule"}]], ["block_4", ["phase diagram. We already found the expression for AMI/RT, in Equation 7.4.14, so we can start\nfrom there. We should take derivatives with respect to x1, but in fact we can get away with the\nmuch easier task of differentiating with respect to ($2. This is because 3/811); (8x2/6\u00a22)6/3x2 and\nas we will be setting the expressions equal to zero, 8x2/8gb2 will divide out. Also, 8/6qb2=\n"]], ["block_5", ["Thus the critical polymer concentration depends inversely on x/N. For larger and larger N, qbc will\nbecome lower and lower, approaching 0 as N\u2014roo. Now to complete the analysis, we return to\nEquation 7.5.19 and find Xci\n"]], ["block_6", ["So, as N increases, Xe approaches 1/2. Recall from regular solution theory (which we can obtain\nfrom Flory\u2014Huggins theory by setting N 1) that Xe 2 and gbc 1/2. Thus, as we increase N, the\ncritical concentration moves more and more toward solutions dilute in polymer, and Xe becomes\nsmaller. For a given zAw, therefore, TC increases as N increases, meaning that polymers become\nless likely to form a homogeneous solution at a given T as N increases. This is illustrated in Figure\n7.13 for various values of N.\nRecall an important result from the previous section: At the theta temperature B 0 and X = 1/2.\nNow we have a third definition of the theta temperature; it is the critical temperature for a given\npolymer\u2014solvent system in the limit of infinite molecular weight. A polymer will be completely\nmiscible with a solvent above the theta temperature, but anywhere below the theta temperature, there\n"]], ["block_7", ["The critical point comes from\n"]], ["block_8", ["272\nThermodynamics of Polymer Solutions\n"]], ["block_9", ["or\n"]], ["block_10", ["which is again a quadratic:\n"]], ["block_11", ["We can now substitute this relation into Equation 7.5.17\n"]], ["block_12", [{"image_4": "285_4.png", "coords": [36, 361, 186, 403], "fig_type": "molecule"}]], ["block_13", ["\u2014\u2014 8/8q51, so we can work with 1.1.1. Thus the spinodal curve can be found from\n"]], ["block_14", ["1\n1\n_1(1+x/N)2_1(r+_2_+1)\nN\nx/N\nWm (m2 \n"]], ["block_15", ["63\nA1121\n_ \n_\n_i\n2\nEb; (F)\n_\n6962\n{1171(1\n$2) + 9152 (1\nN) +X\u00a22}\n"]], ["block_16", ["1\nl\nXe \n(7.5.19)\n2 (1 962,92\n"]], ["block_17", ["345%\nRT\n(1 952,192\nX\n(\n)\n"]], ["block_18", ["a \nm1\n1\u00b0\n2(1\u2014,i,,)\nN\u2014l\n(x/N\u20141)(\\/N+1)\nl\n1\n2\ng _\n7.5.22\n1 + m m\n\u2018\n)\n"]], ["block_19", ["_1\n1\n(i520\n+ 1--) \n(7.5.20)\n1 \u20ac523\n(\nN\n(1 <52,c)2\n"]], ["block_20", ["82\n(Am)\n~l\n"]], ["block_21", [{"image_5": "285_5.png", "coords": [52, 312, 217, 347], "fig_type": "molecule"}]], ["block_22", ["1\n2d)\n1\n2\n2c\n"]], ["block_23", [{"image_6": "285_6.png", "coords": [62, 117, 329, 192], "fig_type": "molecule"}]], ["block_24", [{"image_7": "285_7.png", "coords": [98, 152, 311, 185], "fig_type": "molecule"}]], ["block_25", ["\u2014\u2014l\n1\nl\n=.___+1_s_+2\n=\u2014'+\u2014\"\u20182Xs=0\n(7.5.17)\n1\u2014 an\nN\nm\ncm M\n"]], ["block_26", [{"image_8": "285_8.png", "coords": [105, 389, 332, 443], "fig_type": "molecule"}]], ["block_27", ["(7.5.23)\n"]]], "page_286": [["block_0", [{"image_0": "286_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "286_1.png", "coords": [34, 20, 315, 344], "fig_type": "figure"}]], ["block_2", ["is a danger of phase separation. Furthermore, as a polydisperse solution is cooled below the theta\npoint, the higher molecular weight chains will tend to phase separate first, a feature which can be\nused to advantage in fractionation.\nExamples of the phase behavior of polymer solutions are presented in Figure 7.14. The classic\nresults of Shultz and Flory for polystyrene in cyclohexane and polyisobutylene in diisobutyl ketone\nare reproduced in Figure 7.14a and Figure 7.14b, respectively. The data are experimental estimates\nof the coexistence concentrations (binodal), with smooth curves drawn to guide the eye. The\ndashed lines correspond to the predictions of the Flory\u2014Huggins theory. It is clear from these\nfigures that the theory indeed captures the main features of the data, namely a critical concentration\nthat is small and decreases with increasing M, and a critical temperature that increases with\nincreasing M. Neither the shape of the binodal nor the exact concentration dependence of the\ncritical composition is correct, however. The critical temperatures for these two systems are plotted\nas a function of M in Figure 7140, in a format suggested by Equation 7.5.23. The plots are linear,\nand permit reliable determination of the theta temperatures, with values that are in good agreement\nwith those determined by locating B(T) 0 (see Figure 7.8).\nA third system, polystyrene in acetone, is illustrated in Figure 7.15. Here we see phenomena that are\nnot described by the theory at all, namely phase separation upon heating for certain values of M. For\nexample, forM = 10,300 there appear to be two critical temperatures, one just below 0\u00b0C and the other\njust above 140\u00b0C. The solution would be two-phase for temperatures below the former and above the\nlatter and one-phase at intermediate temperatures. For M 19,800, there is no temperature at which a\nsolution with 0.1 < qbz < 0.15 would be one-phase. What are we to make of this? First, there is nothing\nin thermodynamics that forbids this kind of behavior. The Flory\u2014Huggins theory, however, cannot\n"]], ["block_3", ["Figure 7.13\nPhase diagrams for a polymer-solvent system with a theta temperature of 400 K, and N\nvalues of 102, 3 X 102 and 103. Smooth curves are binodals, dashed curves are spinodals. (Reproduced from\nMunk, P. and Aminabhavi, TM. in Introduction to Macromolecular Science, 2nd ed., Wiley, New York, 2002.\nWith permission.)\n"]], ["block_4", ["Phase \n273\n"]], ["block_5", ["T\n(K)\n"]], ["block_6", ["400 \nT: 6\n-\n"]], ["block_7", ["380 \n_\n"]], ["block_8", ["360\n"]], ["block_9", ["340\n"]], ["block_10", ["320\n"]], ["block_11", ["300\n|\nl\nI\n"]], ["block_12", ["r----\u2014---\n"]], ["block_13", ["\u201cq-\n"]], ["block_14", ["-\n"]]], "page_287": [["block_0", [{"image_0": "287_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "287_1.png", "coords": [2, 40, 416, 302], "fig_type": "figure"}]], ["block_2", ["Figure 7.14\nExperimental coexistence curves compared to the Flory\u2014Huggins theory (dashed curves) for\n(a) polystyrene in cyclohexane and (b) polyisobutylene in diisobutylketone. (c) Resulting critical temperatures\nversus degree of polymerization, plotted as suggested by Equation 7.5.23. (From Shultz, AR. and Flory, P.J.,\nJ. Am. Chem. Soc, 74, 4760, 1952. With permission.)\n"]], ["block_3", ["describe it (at least with )(> 0, see the next section) and thus this system illustrates some of the\nqualitative limitations of the theory. The critical point on a phase boundary that separates a two~phase\nregion at low temperature from a one-phase region at high temperature is called an upper critical\nsolution temperature (UCST), whereas a critical point on a phase boundary that separates a two\u2014phase\nregion at high temperature from a one-phase region at low temperature is called a lower critical\nsolution temperature (LCST). Thus for polystyrene in acetone with M:10,300, both a UCST and an\n"]], ["block_4", ["274\nThermodynamics of Polymer Solutions\n"]], ["block_5", ["250,000\na\n\\.\n:9\u201c\nEL 201\n_\ni:\nl-\no\n.\n-.\n_\n.\n\\,\n' -\n89,000\n"]], ["block_6", [{"image_2": "287_2.png", "coords": [40, 49, 255, 281], "fig_type": "figure"}]], ["block_7", ["30\n_\n1,270,000\n\u2014\n"]], ["block_8", ["25 :\n\\\n- .\n\u201c\n"]], ["block_9", ["(a)\n\u20ac92\n"]], ["block_10", ["35\n1\u2014\n1\u2014\nI_\n1\u2014\n"]], ["block_11", ["10.\n\u2018\n"]], ["block_12", ["15\nl:\n\\.\n\u2014\n"]], ["block_13", ["5\n_l\nl\ni\u2014\nI_\n0\n0.1\n0 2\n0.3\n0 4\n0 5\n"]], ["block_14", [".-\n\\\n'\n43,600\n"]], ["block_15", [{"image_3": "287_3.png", "coords": [125, 304, 348, 520], "fig_type": "figure"}]], ["block_16", ["g 3.3 \n_\n"]], ["block_17", ["+50\n3\n8\n3.2 \n_\n"]], ["block_18", ["3-5\nI\nI\n\u2014I\u2019\n\u2014r\n./\n3.4 \n_\n"]], ["block_19", ["O\n3.1 /\n_\n"]], ["block_20", ["30\nl\nl\nl\nl\n0\n0.02\n0.04\n0.06\n0.08\n0.10\n"]], ["block_21", ["(0)\n1__\n1\nW2N\n"]], ["block_22", ["C\n"]], ["block_23", [{"image_4": "287_4.png", "coords": [228, 29, 411, 288], "fig_type": "figure"}]], ["block_24", [{"image_5": "287_5.png", "coords": [235, 156, 439, 286], "fig_type": "figure"}]], ["block_25", [{"image_6": "287_6.png", "coords": [241, 50, 374, 216], "fig_type": "figure"}]]], "page_288": [["block_0", [{"image_0": "288_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "288_1.png", "coords": [25, 102, 281, 274], "fig_type": "figure"}]], ["block_2", ["What's in x?\n275\n"]], ["block_3", [{"image_2": "288_2.png", "coords": [33, 24, 279, 367], "fig_type": "figure"}]], ["block_4", ["LCST are observed; the Flory\u2014Huggins theory is only capable of predicting UCST behavior. (Note a\npossible source of confusion: the lower critical temperature lies above the upper critical temperature.)\n"]], ["block_5", ["As we have seen, the phase behavior of a polymer solution is determined largely by the interaction\nparameter X and by N. In this section we inspect some of the various ways to look at X- This will\ngive us some insight into how intermolecular interactions determine phase behavior. Furthermore,\nit turns out that X is used in the research literature in a range of different specific ways and it is\nimportant to see how that comes about.\n"]], ["block_6", ["We begin by returning to regular solution theory, and the initial definition of X in terms of zAw in\nEquation 7.2.9. If we know something about the intermolecular interactions, we ought to be able to\nsay something more specific about wij rather than just leaving it as a parameter. For London\n(dispersion) interactions, which are caused by spontaneous dipolar \ufb02uctuations on one molecule\ninducing a dipole on another, the interaction energy is\n"]], ["block_7", ["Figure 7.15\nExperimental phase diagrams for polystyrene in acetone. (From Siow, K.S., Delmas, G., and\nPatterson, D., Macromolecules, 5, 29, 1972. With permission.)\n"]], ["block_8", ["7.6.1\nX from Regular Solution Theory\n"]], ["block_9", ["7.6\nWhat's in X?\n"]], ["block_10", ["a\n\u2018:\n.\na,\n70\n:\n:\nw\n1-.\nt\n:'\n19300\n.\n"]], ["block_11", ["F50\nh\n/\u2014-\\.\\\n"]], ["block_12", ["6.0\n(7.6.1)\nWI.\n.\nN\n\u2014-\nJ\nI\u201c\n"]], ["block_13", ["110 \n-\n"]], ["block_14", ["190[\n4,800\n1L\n\ufb01r\n'\no\n140 t\n10300\nL\n"]], ["block_15", ["30\nl.\n..\n"]], ["block_16", ["50 \n"]], ["block_17", ["901 \\x\n\u2014\n"]], ["block_18", ["A?\n10300\n5%\n0 .\n"]], ["block_19", ["Jk\n4\n800\n,%\n"]], ["block_20", ["/\n"]], ["block_21", ["Olga);\n"]], ["block_22", ["0.05\n0.10\n0.15\n0.20\n0.25\n0.30\n"]], ["block_23", ["(02\n"]], ["block_24", ["'\n"]], ["block_25", ["-\n"]], ["block_26", ["-\n"]]], "page_289": [["block_0", [{"image_0": "289_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "289_1.png", "coords": [28, 196, 287, 278], "fig_type": "molecule"}]], ["block_2", [{"image_2": "289_2.png", "coords": [29, 143, 225, 169], "fig_type": "molecule"}]], ["block_3", [{"image_3": "289_3.png", "coords": [30, 196, 290, 282], "fig_type": "figure"}]], ["block_4", [{"image_4": "289_4.png", "coords": [31, 419, 194, 454], "fig_type": "molecule"}]], ["block_5", ["It turns out that the solubility parameter approach to X does not work well as a predictive method\nfor polymer solutions. As a first example, consider polystyrene in cyclohexane, which is a theta\nsolvent at 345\u00b0C (see Figure 7.7 and Figure 7.8). We can calculate X by Equation 7.6.6, using\n"]], ["block_6", ["This is a very simple expression for X that is directly related to fundamental physical quantities (\ufb01vap,\nCED, or 5). Note that1n the Flory\u2014Huggins theory we have assumed no volume change on mixing, so\nV1217] and the lattice size is defined by the solvent size. By examining a table of solubility\nparameters (see examples in Table 7.1), an estimate of X can be quickly obtained. To obtain a\ngood solvent for a polymer, one could begin by seeking solvents with similar solubility parameters. ..\nFor small molecules the heat of vaporization can be measured precisely, but for polymers this is not\nthe case; thus solubility parameter values are obtained indirectly, and can be quite uncertain.\n"]], ["block_7", ["Here we have introduced the solubility parameter, 5, defined as \\/ CED. The CED is usually given in\nunits ofcal/cm3, and thus 5 has the unusual units of (cal/cm?) \u201d2. Now we can rewrite Equation 7.6.3 as\n"]], ["block_8", ["7.6.2\nX from Experiment\n"]], ["block_9", ["This particular \u201cmixing rule\u201d for w,\u2014,- is called the Berthelot rule, and is certainly a plausible starting\npoint. Under this assumption, we have\n"]], ["block_10", ["Now we can see that because Aw can be written as a perfect square, it must be greater than or equal\nto zero, which is why X2 0 in regular solution\ntheory and the Flory\u2014Huggins theory.\nA\nWe can go further with this approach. The molar heat of vaporization for a pure substance, Uvap,\nshould be directly related to w, i.e.,\n"]], ["block_11", ["meaning that zw,,/2 is the interaction energy lost by removing one i molecule from the pure\nsubstance The cohesive energy density (CED)1s defined by dividing U,- m, by the molar volume,\n"]], ["block_12", ["where a,- is the polarizability of molecule i, and n,\u2014 is the distance between molecules i and j.\nBecause wg falls off with distance so rapidly, we are justified in considering only nearest-neighbor\ninteractions (as in regular solution or Flory\u2014Huggins theory). All atoms and molecules experience\nthis attractive (and fundamentally quantum mechanical) interaction, and for many molecules\nwithout strong dipole moments or hydrogen bonds, it is the only interaction that matters. For\nexample, simple alkanes and inert gases form condensed phases primarily because of London\nforces. Now if we have a lattice where we have imposed r,-,- = r},- = n, by design, then\n"]], ["block_13", ["V,\u2014, which1n regular solution theory13 just Avogadro\u2019 3 number times the lattice site volume:\n"]], ["block_14", ["276\nThermodynamics of Polymer Solutions\n"]], ["block_15", [{"image_5": "289_5.png", "coords": [39, 485, 121, 514], "fig_type": "molecule"}]], ["block_16", ["x= Lo] 52?\n(7.6.6)\nRT\n"]], ["block_17", ["CED\u2014UMP = s\u2014ii _=_ 5?\n(7.6.5)\nI\ni\n1'\n"]], ["block_18", ["W12 \u201c2% = \u201c\u20143371- %Z = \u20141/W11W22\n(7.6.2)\n\u201912\nr 11 r22\n"]], ["block_19", ["aiyap \navz\ufb02\n7..4\n2\n(6)\n"]], ["block_20", ["2A\n\u2014\u2014z\nw\n1w\n1\nw\u2014\n\u2014\u2014\u2014~\n\u2014\u2014\n12\n2\n11\nzwzz\n"]], ["block_21", ["= .3 (4m + [(m)2+(m)z])\n(7.6.3)\n"]]], "page_290": [["block_0", [{"image_0": "290_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "290_1.png", "coords": [26, 623, 167, 661], "fig_type": "molecule"}]], ["block_2", [{"image_2": "290_2.png", "coords": [27, 56, 420, 230], "fig_type": "figure"}]], ["block_3", ["What's in x?\n277\n"]], ["block_4", ["Source: EA. Grulke and Immergut, E.H. (Eds), 3rd ed., Wiley, New York, 1989.\n"]], ["block_5", ["V1 m M/p (84 g/mol)/(0.78 g/cm3) 108 cm3/mol and R 1.987 cal/K mol; the result is\nX m 0.14. But, we know that X =0.50 at the theta temperature, so we are off by almost a factor of\n4! To see that this is not an isolated case, consider the following example.\n"]], ["block_6", ["Table 7.1\nSolubility Values for Common Polymers and Solvents (Values for Polymers Are\nRepresentative Only)\n"]], ["block_7", ["What should we conclude from these calculations? First, the solubility parameter approach is\nnot quantitatively reliable for polymer solutions; the predicted value of X can be greater than, less\nthan, or very close to an experimental value. For at least one of the examples above, poly(vinyl\nacetate) in ethanol, the possibility of hydrogen bonding invalidates the basic assumptions of the\ntheory, so we should not be too surprised by this result. For poly(methyl methacrylate) in carbon\ntetrachloride, the predicted X is much too small. Based on comparisons of experiments with many\npolymers and solvents, the following empirical equation is found to be a much more reliable route\nto estimate X when the predicted value is less than about 0.3:\n"]], ["block_8", ["According to the theory, all three X values should be 0.5. In fact, one value is spot on, one is much\ntoo small, and one is much too big.\n"]], ["block_9", ["We will also need molecular weights and densities for the three solvents; they are approximately\n88 g/mol and 1.03 g/cm3 for dioxane; 154 g/mol and 1.59 g/cm3 for carbon tetrachloride; 46 g/mol\nand 0.79 g/cm3 for ethanol:\n"]], ["block_10", ["1 ,4-Polybutadiene\n8.3\nTetrahydrofuran\n9.1\nPolystyrene\n9.1\nChloroform\n9.3\nAtactic polypropylene\n9.2\nCarbon disulfide\n10.0\nPoly(methyl methacrylate)\n9.2\nDioxane\n10.0\nPoly(vinyl acetate)\n9.4\nEthanol\n12.7\nPoly(vinyl chloride)\n9.7\nMethanol\n14.5\nPoly(ethylene oxide)\n9.9\nWater\n23.4\n"]], ["block_11", ["Three other reported theta systems are polyisoprene in dioxane at 34\u00b0C, poly(methyl methacrylate)\nin carbon tetrachloride at 27\u00b0C, and poly(vinyl acetate) in ethanol at 19\u00b0C. Estimate X for each of\nthese cases, using Equation 7.6.6 and Table 7.1.\n"]], ["block_12", ["Polymer\n6 (cal/cm3)\u201d2\nSolvent\n6 (cal/cm3)\u201d2\n"]], ["block_13", ["Poly(tetrafluoroethylene)\n6.2\nn-Hexane\n7.3\nPoly(dimethylsiloxane)\n7.4\nCyclohexane\n8.2\nPolyisobutylene\n7.9\nCarbon tetrachloride\n8.6\nPolyethylene\n7.9\nToluene\n8.9\nPolyisoprene\n8.1\nEthyl acetate\n9.1\n"]], ["block_14", ["Solution\n"]], ["block_15", ["Example 7.3\n"]], ["block_16", ["Av\nx R\u2014T\u2018 (81 82)2\n(7.6.7)\n"]], ["block_17", ["(88/ 1.03)\n_\n(1.987x307)(1\n1)\n05\nPolyisoprene/dioxane: X m\n"]], ["block_18", ["Poly(methyl methacrylate)\u2014carbon tetrachloride: X m\n(8.6 9.2)22 0.06\n"]], ["block_19", ["(46/079)\nPolyvinyl acetate-693\u201c\u201c X mg?)\n"]], ["block_20", [{"image_3": "290_3.png", "coords": [238, 81, 416, 220], "fig_type": "figure"}]], ["block_21", ["(12.7 9.4)2= 1.1\n"]], ["block_22", ["(154/159)\n(1.987 x 300)\n"]]], "page_291": [["block_0", [{"image_0": "291_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Two questions should immediately come to mind. First, have we wasted our time examining the\nFlory\u2014Huggins theory in such detail, given that it fails to describe the thermodynamic properties of\npolymer solutions quantitatively, and, in many cases, even qualitatively? Second, can we do\nanything to rectify the situation? The answer to the \ufb01rst question is simple: no, we have not\nwasted our time. All of the developments that we went through (finding expressions for IT, #4: and\nmapping out the phase diagram, etc.) were model\u2014independent thermodynamics. The model really\nonly entered through the explicit expression for AGm that we used (i.e., Equation 7.3.13), and all\ndifferences between experiment and model predictions are directly attributable to inadequacies of\nEquation 7.3.13. The answer to the second question is not so clear-cut. There are three general\nstrategies employed in the polymer community. One is to try and improve the model, for example\nby identifying further interaction terms, additional sources of entropy, etc. This approach has been\npursued for many years. It has the virtue that it is possible to keep track of both the intended\nmeaning and quantitative effect of any added term in the expression for AGm. It has the drawback\nthat it is not yet generally successful if we try to restrict ourselves to only a small number of new\nterms. A second strategy is known as the \u201cequation of state\u201d approach. In this case, the AGm\nexpression is recast in a general form, with a finite number of parameters. In many cases the\nbehavior of mixtures can be well predicted based on knowledge of the parameters of the pure\ncomponents, so it can be a very practical strategy. One disadvantage is that it is difficult to gain\nmuch physical insight into the underlying molecular processes from the parameter values. The\nthird approach is the one most favored by experimentalists in polymer science, and so we will\nexamine this one a little more carefully. In essence it amounts to using X as a fittingfunction, not a\nnumber, so that X takes on whatever attributes are necessary to describe the data.\nIn general the free energy of mixing can be divided into two parts, an ideal part (superscript id)\nand an excess part (superscript ex):\nA0\u201c, AGE + A0: Mfg rasig + AH: ms:\n(7.6.8)\n"]], ["block_2", ["Furthermore, an ideal solution is defined as one for which\n"]], ["block_3", ["In the poly(methyl methacrylate)\u2014-carbon tetrachloride and polystyrene\u2014cyclohexane cases Equa-\ntion 7.6.7 does a much better job, albeit still not perfect. From Equation 7.6.7 there appears to be a\nnearly constant (and substantial) temperature-independent contribution to X in polymer\u2014solvent\nsystems, which is not anticipated by the regular solution theory approach. The fact that the 0.34\nterm does not have an explicit temperature dependence suggests that it re\ufb02ects an additional\nentropy of mixing contribution, rather than the purely enthalpic X anticipated by the model.\nThis is but one example of the limitations of this theory; another was provided by the\npolystyrene\u2014acetone phase diagrams in Figure 7.15, where the observed LCST behavior cannot\nbe explained by an interaction parameter that follows Equation 7.6.6. Still other problems are\napparent from the solubility parameter values in Table 7.1. For example, poly(ethylene oxide) is\nactually water-soluble, even though the difference between the two solubility parameters is huge.\nThis particular case involves hydrogen bonding and the rather unusual properties of water. In fact,\nthe assumptions of the theory are not consistent with any kind of strong or directional interaction,\nsuch as those involving permanent dipoles or hydrogen bonds. The packing of dipoles can\ninfluence the entropy of mixing through the various orientational degrees of freedom, which are\nnot included in the purely combinatorial entropy of the lattice model. Similarly, the energy of\ninteraction between two dipoles is very sensitive to the relative orientation, and the orientation of\none dipole will be sensitive to the positions and orientations of all neighbors. In short, there are\nmany ways in which the basic theory fails to incorporate important features of real systems,\nespecially when the interactions are strong and directional.\n"]], ["block_4", ["7.6.3\nFurther Approaches to X\n"]], ["block_5", ["278\nThermodynamics of Polymer Solutions\n"]], ["block_6", ["ASE \u2014\u2014k(x1 1a + x2 lnxg),\nM3 = 0\n(7.6.9)\n"]]], "page_292": [["block_0", [{"image_0": "292_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["What's in x?\n279\n"]], ["block_2", ["In other words, the entropy of mixing of an ideal solution is the purely combinatorial entropy of\nmixing, and there is no enthalpy of mixing (i.e., it is an athermal solution). Now we can view\nregular solution theory, as making specific \n"]], ["block_3", ["This Xeff will therefore include all the ingredients that pertain to the system under study, but it is an\nexperimental result not a model prediction. The scientific literature is often very confusing on this\npoint. For example, relatively few authors will actually mention Equation 7.6.11, even though that\nis what they are doing when they fit the data. Furthermore, quite a few will refer to this Xeff as the\n\u201cFlory\u2014Huggins interaction parameter,\u201d which it is not; the Flory\u2014Huggins X is correctly given by\nEquation 7.2.9.\nEmpirically, the Xeff function obtained by fitting data usually follows the form\n"]], ["block_4", ["where Xeff is sometimes resolved into two components, the \u201centhalpic part\u201d Xh and the \u201centropic\npart\u201d Xs- Interestingly, the parameters a and B may each be positive or negative. A negative at\nimplies some kind of specific attractive interaction between the components, such as might occur if\none component was a hydrogen bond donor and the other an acceptor. The sign of B is harder to\ninterpret, but presumably re\ufb02ects details of molecular packing. (Recall that in the lattice model we\nassume that each molecule has the same shape, and fits neatly into the lattice site, so there is no\nentropy associated with how the molecule is oriented within a site. For real molecules anisotropy\nof shape is almost inevitable, but it is not necessarily obvious whether there will be more or less\npacking possibilities per unit volume in the mixture compared to in the pure components, and\nhence the sign of the excess entropy can be hard to predict.) These various possibilities for the\nsigns of a and B give some insight into the various kinds of phase diagram that arise; for example,\nan LCST system would be the natural consequence of having oi < 0 and B > 0.\nOne final point about this experimental Xeff parameter: it very often exhibits a concentration\ndependence, which is equivalent to saying that the excess free energy of mixing is not entirely\nquadratic and symmetric with respect to components 1 and 2 (i.e., not AGE? 0-: (blag). This should\nnot be too surprising, given the great disparity in size and shape between a polymer and a solvent\nmolecule. This feature does have an important practical consequence, however. We can write the\nfree energy of mixing as\n"]], ["block_5", ["and if we actually measure AGm in an experiment, we can extract Xeff directly. However, it is not\neasy to measure AGm; more often we make a measurement that is really sensing a chemical\npotential, such as the osmotic pressure. In this case, if we take Equation 7.4.14 and substitute Xeff\nfor x, we would have\n"]], ["block_6", ["The approach that is adOpted is to de\ufb01ne an effective interaction parameter in of\nthe experimentally accessible free energy of mixing (i.e., the experimentally AGm\n"]], ["block_7", ["less the ideal part from Equation 7.6.9):\n"]], ["block_8", ["quantities, namely\nl\ufb02\u00a7:O,AH\u00a7:xmmU\"\n061m\n"]], ["block_9", [{"image_1": "292_1.png", "coords": [38, 615, 262, 666], "fig_type": "molecule"}]], ["block_10", ["AGE;i\nE \n(7.6.11)\nXeff\n\u00a21\u00a22kT\n"]], ["block_11", ["a\nn\ufb01=f+6=n+xs\nnan)\n"]], ["block_12", [{"image_2": "292_2.png", "coords": [49, 157, 116, 192], "fig_type": "molecule"}]], ["block_13", ["H_V1__\n1\nRT \n{1n(1\u20141\u2014\u2014)+cfqbg}\n(7.6.13b)\nN\n"]], ["block_14", ["A\ufb01i\n\u00a2\nk\u2014T\n= 9\u201851 111q + F2 111952 + Xeff\u00a7b1e1: is proportional to \u2014TS, where S is the entropy lost on stretching. If we\nassume that we begin with a Gaussian chain, we can write\n"]], ["block_7", ["Thus the scaling argument says\n"]], ["block_8", ["and combining Equation 7.7.3 and Equation 7.7.5 we have\n"]], ["block_9", ["where c is the number of segments per unit volume, and v(T) is the strength of the excluded\nvolume interaction between any two segments. (In particular, v B, and the c2 dependence is just a\nre\ufb02ection of the probability of two segments coming into contact. Basically Equation 7.7.2 is the\nsecond term of the virial expansion, Equation 7.4.7a; the first term does not matter because we can\nlet N become very large.) Now 0 N/R3 and when we integrate F05 over the volume of the coil (we\nassume FOS is the same everywhere, i.e., a mean-field approximation within the coil), we gain\nanother factor of R3:\n"]], ["block_10", ["random-walk statistics, and the larger the chain dimensions, the fewer the possible conformations.\nThus entropy resists unlimited coil expansion. The balance between the osmotic drive to swell and\nthe entropic drive to stay coiled up sets the ultimate dimensions. We estimate the \u201cfree energy\u201d per\nunit volume of this chain by a scaling argument, as follows [4]. (A scaling argument means that we\nleave out unimportant numerical factors, and emphasize the dependence of R on the main variable,\nin this case N. Also, we assume R will have the same dependence on N as does Rg.) The osmotic\npart is driven by the segment\u2014segment interactions, which we can write as V(T)(32 per unit volume:\n"]], ["block_11", ["Figure 7.16\nSchematic illustration of the swelling of a coil in a good solvent as a \u201csingle chain osmotic\npressure\u201d experiment.\n"]], ["block_12", ["Excluded Volume and Chains in a Good Solvent\n281\n"]], ["block_13", [{"image_6": "294_6.png", "coords": [36, 526, 221, 591], "fig_type": "molecule"}]], ["block_14", ["F\nN2\nR2\nkt; v(T) E + TV'\n(7.7.6)\n"]], ["block_15", ["Fe]\nR2\n\u2014\u2014 m \u2014_\n7.7.5\nkT\nN\n(\n)\n"]], ["block_16", ["F05\n\u2014 w v T 6\n7.7.2\nkT\n( )\n(\n)\n"]], ["block_17", ["Jeoil\nV(T) Fd(V01ume)\n: V(T) EER3 :\n1411)?\n(713)\n5.515 N\nH\"\n"]], ["block_18", ["17,, \u2014TS \u2014lnP(N,R)\n"]], ["block_19", ["N2\nN2\nN2\n"]], ["block_20", ["Swell\n_\u2014_b.\nC\n"]], ["block_21", ["-\u20143R2\n3\nit2\n"]], ["block_22", [{"image_7": "294_7.png", "coords": [144, 53, 302, 174], "fig_type": "figure"}]]], "page_295": [["block_0", [{"image_0": "295_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "295_1.png", "coords": [34, 348, 234, 399], "fig_type": "molecule"}]], ["block_2", ["The quantities I72 and V, are the molar volumes of the polymer and solvent, reSpectively (which\nare equal to the partial molar quantities in the Flory\u2014Huggins theory). The differences between\nEquation 7.7.9 and Equation 7.7.10 are the term in (13, which arises from an additional entropy\nchange for the chain due to the increased volume accessible to it, and the explicit expression for the\nprefactor. The factor of (1/2 x) in Equation 7.7.10 is proportional to B (recall Equation 7.4.18),\njust as was v(T) in Equation 7.7.9.\nNow we can brie\ufb02y examine the implications of Equation 7.7.10.\n"]], ["block_3", ["R34) NNW, and then raising each side to the fifth power:\n"]], ["block_4", ["To find the desired relation between R and N, we minimize Fto, with respect to R:\n"]], ["block_5", ["The preceding argument is not quite right, but it actually succeeds by a partial cancelation of\nerrors. The osmotic term re\ufb02ects a mean\u2014field argument-\u2014\u2014the probability of segment\u2014segment\ncontact is taken to be uniformly 02 across the coil\u2014which overestimates its importance. On the\nother hand, the elastic part presupposes a Gaussian chain, which is also not quite correct.\nNevertheless, the argument is reasonably simple and it captures the essence of the problem:\n"]], ["block_6", ["1.\nIn a theta solvent, X 1/2, and so a 1; there is no swelling. Thus in a theta solvent the chain\nbehaves as a random walk for all N large enough to be random walks.\n2.\nIn a good solvent, x<1/2, the chain swells (0: >1). In the limit of a very good solvent,\n"]], ["block_7", ["282\nThermodynamics of Polymer Solutions\n"]], ["block_8", ["and so\n"]], ["block_9", ["We can also recast this equation in terms of the expansion factor, a, by dividing each side by\n"]], ["block_10", ["a balance between excluded volume and coil distortion.\nFlory and Krigbaum worked out a much more detailed version of this calculation [5], with\nthe result\n"]], ["block_11", ["where the prefactor CM is given by\n"]], ["block_12", [{"image_2": "295_2.png", "coords": [38, 73, 194, 105], "fig_type": "molecule"}]], ["block_13", [{"image_3": "295_3.png", "coords": [43, 295, 172, 335], "fig_type": "molecule"}]], ["block_14", ["R v(T)1/5N3/5 B(T)1/5N3/5\n(7.7.8)\n"]], ["block_15", ["d\nF\nN2\n2R\nHE our)??? F \n(7.7.7)\n"]], ["block_16", ["a5 v(T)N1/2\n(7.7.9)\n"]], ["block_17", ["C\n=\n2 ,.\n0\n7.7.11\nM (7m) (M2N,.v.)( M i\n(\n)\n"]], ["block_18", ["a5 a3 2CM G @7117:\n(7.7.10)\n"]], ["block_19", [{"image_4": "295_4.png", "coords": [49, 542, 196, 582], "fig_type": "molecule"}]], ["block_20", ["This result, that v=3/5,\nis\nthe classic result for the excluded volume\n(self\u2014avoiding\nwalk) exponent that we cited before. The high N behavior in good and theta solvents is\nillustrated in Figure 7.17, for polystyrene in benzene (a good solvent), cyclohexane (theta),\nand trans-decalin (theta).\n"]], ["block_21", ["0:5 >> a3, and thus a5 N(Equation 7.7.9). Therefore,\n"]], ["block_22", ["N1/2\n0C \n"]], ["block_23", ["Nu oc \n(7.7.12)\n"]], ["block_24", ["R\n5\nN\u201d\n5\n5 =\ng\n1/2\n\u20191\n(Ran)\n0C (NI/2) \u201cN\n"]], ["block_25", ["NV\n"]]], "page_296": [["block_0", [{"image_0": "296_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "296_1.png", "coords": [24, 49, 343, 246], "fig_type": "figure"}]], ["block_2", ["Chapter Summary\n233\n"]], ["block_3", ["and tmns-decalin, both theta solvents. Data obtained by light scattering by several authors. (From Miyake, A.,\nEinaga, Y., and Fujita, H., Macromolecules, 11, 1180, 1978. With permission.)\n"]], ["block_4", ["In this chapter we have covered a great deal of material concerning the thermodynamic prOperties\nof polymer solutions. We have interwoven results that are strictly thermodynamic with those that\ndepend on a particular model, the Flory\u2014Huggins theory. The main points are as follows:\n"]], ["block_5", ["Figure 7.17\nExperimental radii of gyration for polystyrenes in benzene, a good solvent, and cyclohexane\n"]], ["block_6", ["7.8\nChapter Summary\n"]], ["block_7", ["1.\nUsing thermodynamics alone we were able to show how osmotic pressure measurements on\ndilute solutions can be used to determine Mn and the second virial coefficient, B. The latter\ngives direct information about the solvent quality: B > 0 corresponds to a good solvent, and\nB < 0 to a poor solvent.\nThermodynamic arguments were also sufficient to show how the complete phase diagram\n(T, (p) for a binary system could be constructed from an expression for the free energy of\nmixing. The key features of the phase diagram are the critical point, the coexistence curve\n(binodal), and the stability limit (spinodal).\n"]], ["block_8", ["103.\n"]], ["block_9", ["Cyclohexane\n102:\n"]], ["block_10", ["101\n1__I_1_1_|_I_I_LI\n1_\n11IIIIII\n1\n1\n|_II||1\n105\n106\nM\n107\n108\n"]], ["block_11", ["The crossover from theta-like behavior (a z 1) to fully developed excluded volume, a >> 1, is\nvery broad. It depends on both N and X (and therefore T). For example, for a given N the chain\nwill swell progressively as T is increased above T: G), and for a given T> G), a larger N\nchain will swell more than a shorter one. Because of this broad crossover, it is very common\nin experiments to find apparent values of the exponent v falling between 1/2 and 3/5. This\nwill become important particularly in the context of the intrinsic viscosity, as we shall see in\nChapter 9.\nAlthough designed for polymer solutions, Equation 7.7.10 actually hints at a very important\nresult for molten polymers. Suppose the solvent were a chain of the same monomer, but with a\ndifferent degree of polymerization, P. Presumably x~ 0, so the chain should swell. However,\n"]], ["block_12", ["V1 P, so if P exceeds \\/_\nN1n length there should be little or no swelling. In other words,\nchains1n their own melt should be Gaussian. Flory made this very important prediction1n the\n19503, but it was not until the advent of small\u2014angle neutron scattering in the early 19703 that\nthis fundamental result could be confirmed.\n"]], ["block_13", [". 'irans-Decalin\n"]]], "page_297": [["block_0", [{"image_0": "297_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["where TC is the critical temperature for phase separation. For polystyrene with M \u2019\u201c=\u2018-' 3x106,\nShultz and Flory observed TC values of 68\u00b0C and 84\u00b0C, respectively, for cyclohexanone and\ncyclohexanol. Values of Vl for these solvents are about 108 and 106 cm3/m01, and 81 values\nare 9.9 and 11.4 (cal/cm3)1/2, respectively. Use each of these Tc values to form separate\nestimates of 82 for polystyrene and compare the calculated values with each other and with\nthe value of 82 from Table 7.1. Comment on the agreement or lack thereof for the calculated\nand accepted 6\u2019s in terms of the assumptions inherent in this method. If Equation 7.6.7 was\nused instead of Equation 7.6.6, does the agreement among the 82 values improve?\n3.\nThe term (1/2 X) that appears, for example, in the Flory\u2014Huggins expression for B (Equation\n7.4.18) is sometimes replaced by\n;_.=.(._%\n"]], ["block_2", ["W22 contribution continues to be valid. A slight error in counting is made\u2014to allow for\nsimplification of the resulting function\u2014*but this is a tolerable approximation in concentrated\nsolutions. In dilute solutions the approximation introduces more error, but the model is in\ntrouble in such solutions anyhow, so another approximation makes little difference.\n2.\nShow that\n"]], ["block_3", ["3.\nWe developed an expression for the free energy of mixing based on the Flory\u2014Huggins theory.\nThis is a mean\u2014field theory and reduces to the standard regular solution theory when the degree\nof polymerization of the polymer component is set equal to 1. The numerous assumptions of\nthe model were identified.\n4.\nThe Flory\u2014Huggins model was explored in detail, in terms of its predictions for B and for the\nphase diagram. Comparison with experiments reveals that in some systems the Flory\u2014Huggins\ntheory captures the phenomenology in a qualitative manner, but in others it does not. It does\nnot provide a quantitative description for any dilute polymer solution.\n5.\nThe concept of a theta solvent emerges as a central feature of polymer solutions. It has four\nequivalent Operational definitions: (a) the temperature where B 0, (b) the temperature where\nthe interaction parameter X 1/2, (c) the limit of the critical temperature, Tc, as M\u2014+00, and (d) a\nsolvent in which Rg NMW. Physically, a theta solvent is one in which the polymer\u2014solvent\ninteractions are rather unfavorable, so that the chain shrinks to its random~wa1k dimensions.\nThis contraction cancels the effect of the excluded volume interactions, which otherwise swell\nthe chain to self-avoiding conformations: Rg M35.\n6.\nThe Flory\u2014Huggins parameter X can be related to thermodynamic quantities such as the heat of\nvaporization and the cohesive energy density. Only for systems with very weak intermolecular\ninteractions can one h0pe to estimate X reliably based on tabulated thermodynamic quantities, and\neven then there needs to be a substantial, empirical correction term for polymer solutions. This\nmotivates the use of an effective interaction function that can be used to fit experimental data; the\nresulting Xeff has become a commonly used scheme to describe the excess free energy of mixing.\n"]], ["block_4", ["284\nThermodynamics of Polymer Solutions\n"]], ["block_5", ["Problems\n"]], ["block_6", ["1.\nIn the derivation of Equation 7.3.12 it is assumed that each polymer segment is surrounded by\n"]], ["block_7", [{"image_1": "297_1.png", "coords": [45, 637, 140, 672], "fig_type": "molecule"}]], ["block_8", ["To (2/12) 171 (61 62?\n"]], ["block_9", ["2 sites which are occupied at random by either solvent molecules or polymer segments.\nActually, this is true of only (2 2) of the sites in the coordination sphere\u2014(z 1) for chain\nends\u2014since two of the sites are occupied by polymer segments which are covalently bound to\nother polymer segments. Criticize or defend the following proposition concerning this effect:\nThe kinds of physical interactions that we identify as London or spontaneous dipole\u2014dipole\nattractions can also operate between segments which are covalently bonded together, so the\n"]]], "page_298": [["block_0", [{"image_0": "298_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["1\u2018C. Bawn, R. Freeman, and A. Kamaliddin, Trans. Faraday Soc, 46, 862 (1950).\nIWR. Krigbaum and DD. Geymer, J. Am. Chem. Soc, 81, 1859 (1959).\n"]], ["block_2", ["The units of c are mg/mL, and II is in g/cmz. (This unit of g/cm2 is a bit old fashioned; to\nconvert H to dyn/cmz, multiply by the acceleration due to gravity, g 980 cm/sz.)\n5.\nWrite down by inspection the Flory\u2014Huggins theory prediction for the free energy of mixing\nof a ternary solution (polymer A, polymer B, and solvent).\n6.\nThe osmotic pressure of solutions of polystyrene in cyclohexane was measured at several\ndifferent temperatures, and the following results were obtained:I\n"]], ["block_3", ["4.\nThe osmotic pressure of polystyrene fractions in toluene and methyl ethyl ketone was\nmeasured by Bawn et al.\u2019r at 25\u00b0C, and the following results were obtained. Make plots of\n11/0 versus 0, and evaluate Mn for the three fractions in an appropriate way. Do the results\nmake sense? (Hint: If they do not, perhaps a more sophisticated analysis would help.) What\ncan you say about the quality of these solvents?\n"]], ["block_4", ["Problems\n285\n"]], ["block_5", [{"image_1": "298_1.png", "coords": [44, 153, 312, 377], "fig_type": "figure"}]], ["block_6", ["Fraction\n6 (g/cm3)\nl'I/RTcx10\u00b0 (mol/g)\n"]], ["block_7", ["II\n0.0976\n8.0\n0.182\n6.0\n0.259\n8.7\n"]], ["block_8", ["Fraction\n6 (g/cm3)\nII/RTC ><10\u20185 (mol/g)\n"]], ["block_9", ["II\n0.0081\n13.3\n0.0201\n14.2\n0.0964\n14.2\n0.180\n18.7\n0.257\n26.2\n"]], ["block_10", ["III\n1.75\n0.31\n1.41\n0.23\n2.85\n0.53\n2.90\n0.48\n4.35\n0.88\n6.24\n1.11\n6.50\n1.49\n8.57\n1.63\n8.85\n2.36\n"]], ["block_11", ["II\n1.55\n0.16\n3.93\n0.40\n2.56\n0.28\n8.08\n0.95\n2.93\n0.32\n10.13\n1.30\n3.80\n0.47\n5.38\n0.77\n7.80\n1.36\n8.68\n1.60\n"]], ["block_12", ["Fraction\nToluene c\nToluene II\nMEK c\nMEK II\n"]], ["block_13", ["I\n4.27\n0.22\n2.67\n0.04\n6.97\n0.58\n6.12\n0.14\n9.00\n1.00\n8.91\n0.31\n10.96\n1.53\n"]], ["block_14", ["where G) is the theta temperature (K) and d; is a new parameter. This substitution has the effect\nof replacing one parameter (x) with two ((1; and 6)). Why might one do this? How do 11/ and 6)\n\ufb01t into the discussion of effective X parameters in Section 7.6.3?\n"]], ["block_15", ["T 24\u00b0C\n"]], ["block_16", ["T 34\u00b0C\n"]]], "page_299": [["block_0", [{"image_0": "299_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["286\n"]], ["block_2", ["11.\n"]], ["block_3", ["10.\n"]], ["block_4", [{"image_1": "299_1.png", "coords": [39, 29, 240, 245], "fig_type": "figure"}]], ["block_5", [{"image_2": "299_2.png", "coords": [43, 437, 244, 492], "fig_type": "molecule"}]], ["block_6", [{"image_3": "299_3.png", "coords": [49, 370, 174, 411], "fig_type": "molecule"}]], ["block_7", [{"image_4": "299_4.png", "coords": [49, 137, 236, 217], "fig_type": "figure"}]], ["block_8", [{"image_5": "299_5.png", "coords": [52, 41, 232, 112], "fig_type": "figure"}]], ["block_9", ["Plot all of these data on a single graph as H/RTC versus 0, connecting the points so as to\npresent as coherent a display of the results as possible. Evaluate the molecular weights of the\ntwo polystyrene fractions. Criticize or defend the following proposition: These data show that\nthe (9 temperature for this system is about 34\u00b0C. As expected, the range of concentrations\nwhich are adequately described by the first two terms of the virial equation is less for sample\nIII than for sample II. Above this range other contributions to nonideality contribute positive\ndeviations from the two-term osmotic pressure equation. It would be interesting to see how\nthis last effect appears for sample III at 24\u00b0C, but this measurement was probably impossible\nto carry out owing to phase separation.\nBy combining Equation 7.5.23 and Problem 3, the following relationship can be obtained\nbetween the critical temperature T0 for phase separation and the degree of polymerization:\n1-1+; _1_+_1_\nT,_\u00ae\nor; f\n2N\n"]], ["block_10", ["Use the graphical method outlined above to evaluate (9,\n1,11, and X for polyisobutylene in\ndiisobutylketone.\nAssume X for\na polymer\u2014solvent system followed Equation 7.6.12.\nThere\nare four\npossible cases according to whether the parameters a: and B were positive or negative.\nWhat possible phase diagrams could be observed in each case, i.e., UCST only, LCST\nonly, both, neither, etc.\nFill in the steps omitted in the text to derive the relations for the critical mole fraction and the\ncritical X for regular solution theory (see Equation 7.5.15).\nFind the Flory\u2014Huggins expression for AM2(\u00a21), and then find the range of X for which there\ncan be two physically meaningful values of (is, for which Aug is the same. (Hint: Take the\nlarge N limit, but at the correct moment.) What is the signi\ufb01cance of this range of X?\nA polymer\u2014solvent mixture has a critical temperature of 300 K. The polymer is monodisperse\nand has a molar volume equal to three times that of the solvent. If the solvent were replaced\n"]], ["block_11", ["Derive this relationship and explain the graphical method it suggests for evaluating (9 and if}.\nThe critical temperatures for precipitation for the data shown in Figure 7. 14b are the following:\n"]], ["block_12", ["Tc (\u00b0C)\n18.2\n45.9\n56.2\n"]], ["block_13", ["M (g/mol)\n22,700\n285,000\n6,000,000\n"]], ["block_14", ["III\n0.0156\n2.46\n0.0482\n2.24\n0.0911\n3.42\n0.126\n4.96\n0.139\n6.05\n"]], ["block_15", ["Fraction\nc (g/cm3)\nH/RTC ><106 (mol/g)\n"]], ["block_16", ["II\n0.0959\n18.6\n0.178\n28.1\n0.255\n40.0\nIII\n0.0478\n5.50\n0. 125\n1 1.0\n0.138\n13.2\n"]], ["block_17", ["T 44\u00b0C\n"]], ["block_18", ["Thermodynamics of Polymer Solutions\n"]]], "page_300": [["block_0", [{"image_0": "300_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["20.\n"]], ["block_2", ["References\n"]], ["block_3", ["Problems\n237\n"]], ["block_4", ["12.\n"]], ["block_5", ["13.\n"]], ["block_6", ["14.\n"]], ["block_7", ["15.\n"]], ["block_8", ["16.\n"]], ["block_9", ["18.\n19.\n"]], ["block_10", ["17.\n"]], ["block_11", [".U\u2018P\u2018PJE\u2018JZ\u2018\n"]], ["block_12", ["Flory, P.J., J. Chem. Phys., 10, 51 (1942).\nHuggins, M.L., J. Am. Chem. 506., 64, 1712 (1942).\nFlory, P..T., Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY, 1953.\nde Gennes, P.G., Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY, 1979.\nKrigbaum, W.R. and Flory, P.J., J. Am. Chem. 506., 75, 1775 (1953).\n"]], ["block_13", ["by its dimer, what does Flory\u2014Huggins theory predict for the new critical temperature?\n(Hints: It is a good idea to \ufb01gure out which way TC should change before you do the details.\nAlso, remember that although X is dimensionless, it does depend on the chosen site volume.)\nSuppose you have just developed a synthesis method for a new, \ufb02exible polymer, capable of\nproducing high-molecular-weight samples. Provide a step-by-step outline of how you would\ngo about finding a theta solvent for this polymer, and then locating the theta temperature\nprecisely. Note that although there are many ways to proceed, some will be much more labor-\nintensive than others. Try to minimize the amount of effort required.\nThe Flory\u2014Huggins theory may be extended to a binary mixture of different polymers, i.e., an\nA/B blend. The resulting free energy of mixing can be written\n"]], ["block_14", ["qbc) in terms of NA and NB. Show that your results reduce to the solution case when NB \u2014> 1.\nCompare the solution and symmetric blend (i.e., NA=NB) results in the limit of in\ufb01nite\nmolecular weight; what is the crucial difference?\nThe results from Problem 13 imply that the critical temperature for a symmetric polymer\nblend should increase linearly with N. Confirm this result. What would the N dependence be\nfor the critical temperature if in fact X followed Equation 7.6.12?\nProve that for a polymer solution in the athermal limit (AI-1m 0), no phase separation can\noccur in the Flory\u2014Huggins model. (Hint: The sign of AGm is not enough.)\nConsider the Flory\u2014Huggins theory as applied to an AB statistical copolymer dissolved in\nsolution. It is possible to apply the same expression for the free energy of mixing as for a\nhomopolymer, if a new effective X parameter is employed. Develop a simple relation for Xeff\nin terms offA, XA, X3, and XAB, wherefA is the volume fraction of A units in the copolymer,\nand the three X\u2019s refer to polymer A\u2014solvent, polymer B\u2014solvent, and polymer A\u2014polymer B\ninteractions, respectively. Use mean-field approximations in the same spirit as in the original\ntheory. The important feature is that there are unfavorable A\u2014B interactions in the pure\nstatistical copolymer that are diminished in dilute solution. If your result is correct, you can\nuse it to show that, in principle, it should be possible to find a solvent that dissolves the\nstatistical copolymer but neither of the constituent homopolymers, at constant M.\nExtend the derivation for the excluded volume exponent (v: 0.6) given in Section 7.7 to a\nchain in a space of dimensionality d. What is v in two dimensions; why is it different from\n0.6? What does the result for four dimensions tell you?\nRepeat Problem 8 for a polymer\u2014polymer blend at high molecular weight.\nFor a polydisperse sample, what average degree of polymerization would apply in the\ncorresponding Flory\u2014Huggins expression for the spinodal? Prove your answer; one way to\ndo this is to find the spinodal for a binary mixture of two degrees of polymerization.\nRecast Equation 7.7.10 in terms of the second virial coefficient, B(T).\n"]], ["block_15", ["where NA and NB are the respective degrees of polymerization. Find the critical point (Xe and\n"]], ["block_16", ["AGm = \u201d5% lnq +\u00a3\u00a7 In 4313 +X\u00a2A\u00a2B}\n"]]], "page_301": [["block_0", [{"image_0": "301_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Flory, P.J., Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY, 1953.\nGraessley, W.W., Polymeric Liquids and Networks: Structure and Properties, Garland Science, New York,\n2003.\nKoningsfeld, R., Stockmayer, W.H., and Nies, E., Polymer Phase Diagrams, Oxford University Press, New\nYork, 2001.\nKurata, M., Thermodynamics of Polymer Solutions, Harwood, New York, 1982.\nRubinstein, M. and Colby, R.H., Polymer Physics, Oxford University Press, New York, 2003.\nTanford, 0, Physical Chemistry of Macromolecules, Wiley, New York, 1961.\nYamakawa, H., Modern Theory of Polymer Solutions, Harper & Row, New York, 1971.\n"]], ["block_2", ["288\nThermodynamics of Polymer Solutions\n"]], ["block_3", ["Further Readings\n"]]], "page_302": [["block_0", [{"image_0": "302_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["In this chapter we will explore the phenomenon of light scattering from dilute polymer solutions.\nLight scattering is an important experimental technique for polymers for several reasons. First, it\nprovides a direct, absolute measurement of the weight average molecular weight, MW. Second, it\ngives information about polymer\u2014polymer and polymer\u2014solvent interactions, through the second\nvirial coefficient, B (introduced in Chapter 7). Third, under many circumstances light scattering\ncan be used to determine the radius of gyration, Rg (described in Chapter 6) without any prior\nknowledge about the shape of the molecule (e.g., coil, rod, globule). Fourth, the description of\nthe scattering process that we will develop in this chapter may be readily adapted to x-ray and\nneutron scattering, two other techniques in common use in polymer science. In short, the wealth of\ninformation that may be obtained from light scattering more than justi\ufb01es the effort we will need to\nexpend in this chapter to understand how it works. As a \ufb01nal introductory comment, the fact that\nlight scattering can determine MW and B tells us that it is a thermodynamic measurement. The fact\nthat it can measure Rg tells us that it is a structural tool as well.\nA light beam may be described as a traveling electromagnetic wave. For our purposes (i.e.,\npolymers and solvents containing mostly C, H, O, and N atoms), the magnetic component of the\nwave is of no consequence, and so we can represent the wave as\n"]], ["block_2", ["where 50 is the amplitude of the electric field, a) is thefrequency in rad/s, t is time, Fis the position,\nand k is the wavevector. This is illustrated in Figure 8.1. Several comments about Equation 8.1.1\nare appropriate.\n"]], ["block_3", ["1.\nThe amplitude EC, is itself a vector. If we take the wave to be travelling along the x direction,\nEl, lies in the 32\u20142 plane. If E}, lies exclusively along one axis, say 2, then the beam is said to be\nz-polarized or vertically polarized; on the other hand, if 50 is uniformly distributed in the y\u2014z\nplane, then the wave is unpolarized.\n2.\nThe frequency can also be written\n"]], ["block_4", ["where v is the frequency in cycles per second (Hz), 0 is the speed of light in vacuum, and A0 is\nthe wavelength in vacuum.\n3.\nIn the photon picture of light, the energy carried by each photon E 2 hr\u00bb, where h is Planck\u2019s\nconstant (6.63 x 10'34 Is).\n"]], ["block_5", ["8.1\nIntroduction: Light Waves\n"]], ["block_6", ["Ezfocos(wt\u2014\u00a3\u00b0F)\n(8.1.1)\n"]], ["block_7", ["_ \n:2\n_\u2014\n(1)\n77V\nA0\n(8.1.2)\n"]], ["block_8", ["light Scattering by Polymer Solutions\n"]], ["block_9", ["8\n"]], ["block_10", ["289\n"]]], "page_303": [["block_0", [{"image_0": "303_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "303_1.png", "coords": [19, 53, 239, 142], "fig_type": "figure"}]], ["block_2", ["where 5* is the complex conjugate of E, obtained by replacing i ( : \\/\u2014~\u20141) with ~i. The use of the\ncomplex conjugate guarantees that I is a real number, because exp(ix) exp(\u2014\u2014ix) =1. Generally, we\nforget about saying the wave is the real part of the complex form, and just remember that at the end\nof a calculation we convert to intensities by Equation 8.1.5.\n"]], ["block_3", ["Figure 8.1\nSchematic of the electric field component of a z-polarized light wave propagating in the\nx direction.\n"]], ["block_4", ["For visible light )to R: 3500-7000 A, and consequently v c/AO as 1015 Hz (recall 0 3.0x 108 m/s).\nThis means that the oscillations in the electric field amplitude are much too rapid for detectors to\nfollow: photomultiplier tubes, photodiodes, and other photodetectors typically have time constants\non the order of nanoseconds. Instead, the detector actually integrates in time the incident intensity,\nI (energy/area/time), which is proportional to IE EI If12; such detectors are called square-law\ndetectors. We will not worry about the proportionality factors between intensity and electric\nfield squared, because in scattering experiments we will always use the ratio of the scattered\nintensity to the incident intensity, thereby canceling out these prefactors. (Even if we did not do\nthat, the detectors actually generate an electrical current proportional to the incident intensity, and\nit is rare that anyone goes through the trouble of converting this signal into the actual incident\nintensity. Rather, we can calibrate the electrical signal with a reference intensity, as will be\ndescribed in Section 8.7.)\nWe could continue to use the cosine representation of the light wave as in Equation 8.1.3, but it\nturns out the arithmetic is much more convenient if we use complex notation. If you do not recall\nhow complex numbers work, they are reviewed in the Appendix. The advantages stem primarily\nfrom the fact that complex numbers allow us to factor out the temporal and spatial dependences of\nthe wave amplitude, and that squaring the amplitude becomes much easier. Accordingly, the wave\ncan be written as\n"]], ["block_5", ["The essence of the traveling wave is that if we consider a particular instant in time (out \ufb01xed), then\nthe wave oscillates in space with wavelength A; and if we consider a particular point in space (kx\nfixed) the wave oscillates in time with frequency a). It will turn out that it is the spatial dependence,\nI; F or kx, that will play the crucial role in scattering experiments.\n"]], ["block_6", ["4.\nThe phase factor, I: F, is the projection of the wavevector k onto the particular position\nvector of interest. The wavevector points in the direction of propagation, x in this instance, and\n"]], ["block_7", ["where Rel. . .} means the real part of the complex argument. The intensity is proportional to\n"]], ["block_8", ["290\nLight Scattering by Polymer Solutions\n"]], ["block_9", ["_p\n_9\nL50, E\"0{eiwre\u2014iwre\u2014if.Feifof'}| \nI\no Eel\n(3,1,5)\n[Nig*\u00b0\u00a7l:\n"]], ["block_10", ["E Re{l\u00a7\u20180 exp [i(wt i: PM } Re{\u00a7o exp[iwt] exp [\u2014ii: F] }\n(8.1.4)\n"]], ["block_11", ["has amplitude 277M. Thus, our wave could be rewritten as\n"]], ["block_12", ["EOE, t) 2 Ed )2, z) c0s(wt kx) = E; cos (271' [pt 3)\n(8.1.3)\n"]]], "page_304": [["block_0", [{"image_0": "304_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "304_1.png", "coords": [33, 58, 392, 231], "fig_type": "figure"}]], ["block_2", ["Source: From York,1989.\n"]], ["block_3", ["Poly(tetra\ufb02uoroethylene)\n1 .4 1\nMethanol\n1 .326\nPoly(dimethylsiloxane)\n1 .43\nWater\n1 .333\nPoly(ethylene oxide)\n1.46\nEthanol\n1.359\nAtactic polypropylene\n1.47\nEthyl acetate\n1.370\nAmorphous polyethylene\n1.49\nrt-Hexane\n1.372\nPoly(vinyl \n1.49\nTetrahydrofuran\n1.404\nPoly(methyl \n1 .49\nCyclohexane\n1 .424\n"]], ["block_4", ["Table 8.1\nRefractive Indices of Common Polymers and Solvents\n"]], ["block_5", ["1 ,4-Polybutadiene\nl .52\nBromobenzene\n1 .557\nPolystyrene\n1.59\nCarbon disulfide\n1.628\n"]], ["block_6", ["Refractive indices for A, 5393 A, at 20\u00b0C or 25\u00b0C.\n"]], ["block_7", ["Equation 8.1.1 represents a solution to Maxwell\u2019s equations in a homogeneous medium. The\nfrequency of the wave (and also the energy of the equivalent photon, hp) is independent of the\nmedium, but the wavelength is not. The wavelength in the material, A, relative to the wavelength in\nvacuum, A0, is determined by a material property called the refractive index n:\n"]], ["block_8", ["1,4-Polyisoprene\n1 .50\nChloroform\n1 .444\nPolyisobutylene\nl .5 l\nToluene\nl .494\n"]], ["block_9", ["where v is the speed of light in the material. Thus the amplitude of the wavevector, k = Mil, is often\nwritten as 27m/A0. The refractive index is determined, in turn, by the polarizability of the\nconstituent molecules or and their spatial arrangement (density and orientation). Qualitatively,\nthe polarizability re\ufb02ects the ability of the incident electric field to distort the electronic distribu-\ntion within a molecule, and this distortion, in turn, reduces c to v. Consequently, more polarizable\nchemical moieties, such as aromatic rings, generally lead to higher refractive indices than less\npolarizable groups, such as \u2014CH3 or ~CH2~. In the liquid state, all orientations of the individual\nmolecules are equally probable, and we say the liquid is isotropic. (In this case we need not worry\nabout the fact that most molecules are actually anisotropic: the polarizability is different along\ndifferent molecular axes). The magnitude of n ranges from about 1.3 to 1.6 for common polymers\nand aqueous or organic solvents as can be seen from the representative values in Table 8.1. The\nrelevant equation that relates the material property, n, to the molecular property, oz, is a version of\nthe Lorentz\u2014Lorenz equation for an ideal gas (also known as the Clausius\u2014Mosotti equation):\n"]], ["block_10", ["Polymer\nn\nSolvent\nr1\n"]], ["block_11", ["where n is the refractive index of the solution, it, is the refractive index of the pure solvent, and 1/\u20181\u2019\nis the number of solute particles per unit volume.\n"]], ["block_12", ["Basic Concepts of Scattering\n291\n"]], ["block_13", ["Scattering is the reradiation of a traveling wave due to a change in the character of the medium in\nwhich the wave is propagating. The mathematical description of scattering could be largely\ndeveloped without specifying whether we are talking about light waves, sound waves, electrical\nsignals, or waves on a lake, as there is a great deal of commonality to all these phenomena. For\nlight, scattering will be caused by local changes in refractive index or polarizability, due to, for\n"]], ["block_14", ["8.2\nBasic Concepts of Scattering\n"]], ["block_15", [{"image_2": "304_2.png", "coords": [37, 508, 115, 545], "fig_type": "molecule"}]], ["block_16", ["n A\u20140 3\n(8.1.6)\n)1\nv\n"]], ["block_17", [{"image_3": "304_3.png", "coords": [46, 318, 96, 353], "fig_type": "molecule"}]], ["block_18", ["477\n\u2018If\n(8.1.7)\n(If-3\n"]], ["block_19", ["2\n2\ntt\u2014nS\n"]]], "page_305": [["block_0", [{"image_0": "305_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["1 A apart, and we shine A 5000 A light on them (Figure 8.2a). What will the scattering look like?\nEach atom will radiate a scattered wave with the same amplitude and wavelength, but the net\nscattered intensity will be zero. Why? If we select any particular scatterer and particular detection\nangle, we can always find another scatterer that is exactly M2 further away from the detector. The\ntwo waves from these scatterers will cancel each other. We can do this because the array is\nperfectly regular, and because the array is essentially a continuum (1<<5000 A). The only\ndirection where this argument does not apply is forward, i.e., the scattering angle 6:00. Here\nthe phase shift of the scattered waves between two atoms is exactly canceled by the phase shift\nbetween the incident waves arriving at the two atoms. Thus the light beam propagates happily\nstraight through the material. The important lesson is this: there is no scattering from a perfectly\nhomogeneous material.\n"]], ["block_2", ["for the case of molecules in solution. We anticipate that bigger molecules will scatter propor\u2014\ntionally more than smaller ones. On the other hand, suppose we scatter light from objects that are\nconnected to one another, e.g., monomers within one polymer. Now the phases of the waves scattered\nfrom different monomers should be related, because the distance rjk between any two monomersj and\nIt has some preferred value or range of values. This will lead to some interference between\nthese waves and a net loss of scattered intensity. Recall that interference between two waves is\nconstructive only if the difference in distance that the two waves travel to the detector is some integral\nnumber of wavelengths, m; in other words, when krzmh. The interference will be completely\ndestructive if the path difference is an integral number plus one half of A: kr (m + 1/2)/\\. We can\nexpect, therefore, that in order to see significant destructive interference we need the distance\nbetween scatterers to be a significant fraction of A, say a few percent. This will turn out to be\nthe case. For visible light with A0 re 5000 A, only for polymers that are larger than at least 100 A do\nwe have to worry about this kind of interference. You might recall from Example 6.2 that this size\ntypically corresponds to molecular weights in excess of 105 g/mol.\n"]], ["block_3", ["8.2.2\nScattering from a Perfect Crystal\n"]], ["block_4", ["Imagine we have an absolutely perfect, regular array of scatterers (atoms, if you like) placed every\n"]], ["block_5", ["Imagine that we have scattering from randomly placed, noninteracting objects; we should get a\ntotal scattered intensity, [3, proportional to the number of scattering objects:\n"]], ["block_6", ["8.2.1\nScattering from Randomly Placed Objects\n"]], ["block_7", ["example, a dust particle in the air or to a polymer in the solvent. We will only consider\nnonabsorbing media here, so the incident light intensity will either be transmitted or scattered,\nbut not absorbed. In general, the scattered wave propagates in all directions, i.e., it is a spherical\nwave in three dimensions or a circular wave in two. If the incident and scattered frequencies are\nthe same, i.e., no energy is exchanged between the medium and the wave, the scattering process\nis classified as elastic. We will only consider elastic scattering in this chapter. (There is also\nquasielastic scattering, in which very small differences in energy are detected; this is the basis\nof the powerful technique of dynamic light scattering, which will be described brie\ufb02y in\nChapter 9. Raman and Brillouin scattering are examples of inelastic processes, where the\nlight exchanges vibrational or rotational quanta with the molecules in the former, and exchanges\nenergy with traveling density waves or phonons in the latter.) Lastly, scattering is termed\nincoherent if the intensity is independent of the scattering angle, and coherent if it depends\non the scattering angle. The origin of this terminology will become apparent in the following\ndiscussion, but generically coherence implies that there exists some particular phase relationship\namong different waves.\n"]], ["block_8", ["292\nLight Scattering by Polymer Solutions\n"]], ["block_9", ["15 (number of scatterers) >< (scattering power of each object)\n(8.2.1)\n"]]], "page_306": [["block_0", [{"image_0": "306_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["There are two ways by which we could generate scattering from our hypothetical array. One would\nbe to remove a few scatterers at random (Figure 8.2b). Then the \u201cpairing\u2014off\u201d argument fails\nbecause each atom we remove used to cancel a scattered wave from some other atom, but now it\ncannot. Thus we conclude that random \ufb02uctuations in an otherwise homogeneous medium give\nrise to scattering. This scattering is incoherent: because the \ufb02uctuations are random by construc-\ntion, on average there can be no phase relation between them.\nThe second way of obtaining scattering from our hypothetical array would be to make A close to\nthe distance between the scatterers. For our example, if we use x\u2014rays, where )t :1.54 A is a\ntypical value, then the pairing-off argument fails again. There will be particular angles in which\nplanes of atoms are separated by integral multiples of )t and the scattering from different atoms will\nbe in phase, i.e., coherent. This process is called Bragg dz\ufb01\u2018ractz\u2019on, as illustrated in Figure 8.3 and\ndescribed below. The point we want to emphasize now is that there will be coherent scattering\nwhenever there is a spatial correlation between scattering objects on a distance scale comparable to )t.\nBragg diffraction from atomic crystals gives sharp scattering peaks because the spatial coherence\nis very high; the crystal structure is regular over large distances (see Chapter 13). In scattering\nfrom polymers, we will find that the Spatial coherence is lower, but not vanishing; for example,\n"]], ["block_2", ["Figure 8.2\n(a) A light wave propagating through a perfectly regular array of scatterers will not be scattered\nif the distance between neighbors is much less than )t. (b) Random \ufb02uctuations in the perfect array lead to\nincoherent scattering.\n"]], ["block_3", ["8.2.3\nOrigins of Incoherent and Coherent Scattering\n"]], ["block_4", ["(a)\n+++++++++++\n+++++++++++\n+++++++++++\n+++o+++++++\n+++++++++++\n++++++++o++\n+++++++++++\n+++++++++++\n+o+++++++++\n"]], ["block_5", ["m\n+++++++++++\n"]], ["block_6", ["Basic Concepts \n293\n"]], ["block_7", ["+++++++++++\n+++++++++++\n+++++++++++\n+++++++\nt\nm\u2018lll lll'lIl \u201cill\u201c \u201cIll\" \"N\"\n"]], ["block_8", ["III.IHIIMII\nIn...\" ......\n++++++\n+++++++++++\n+++++++++++\n"]], ["block_9", ["l\nl\n|\n"]], ["block_10", ["I\n"]]], "page_307": [["block_0", [{"image_0": "307_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "307_1.png", "coords": [14, 48, 256, 217], "fig_type": "figure"}]], ["block_2", ["294\nLight Scattering by Polymer Solutions\n"]], ["block_3", ["Figure 8.3\nIllustration of Bragg\u2019s law. (a) The incident light wave and the diffracted light wave each makes\nan angle 6/2 with each plane in the crystal. The planes are separated by a distance D, and the direction of the\nscattering vector c? is indicated. (b) The bold line segments correspond to the extra distance the wave travels in\nbeing scattered from the second plane. All four indicated angles are equal to 6/2, and thus each of the bold line\nsegments has length D sin(6/2).\n"]], ["block_4", ["within a coil, the likelihood of two monomers being separated by a certain distance might be given\nby the Gaussian distribution (Equation 6.7.12). This lower degree of spatial coherence will lead to\nscattering that is a smooth function of the scattering angle, rather than sharp peaks, but the\nunderlying phenomenon is the same.\nThe main relationship we will derive in this chapter, known as the Zimm equation (Equation\n8.5.18), re\ufb02ects these two sources of scattering. A polymer solution would be a homogeneous\nmedium, except for two things. There are random \ufb02uctuations in concentration that give rise to\nincoherent scattering, with intensity proportional to the product of concentration and molecular\nweight (Equation 8.4.24b). Then, large polymers have monomers with partially correlated spatial\nseparations that are a significant fraction of A. This correlation leads to interference and coherent\nscattering, which can be used to determine Rg (Equation 8.5.18). The development will proceed in\nthree stages. First we will consider scattering from a single isolated atom or molecule, a result\ngenerally attributed to Lord Rayleigh. In the second stage we will apply this result to a dilute\nsolution of small polymers, to obtain an expression for the incoherent scattering. Finally, we will\nconsider larger molecules and the resulting coherent contribution. To conclude this overview\nsection, we develop Bragg\u2019s equation and define the scattering vector \u00e9\u2019.\n"]], ["block_5", ["Consider a wave incident on a series of equally spaced, parallel planes of scatterers (which could\nbe atoms, but need not be). The planes are separated by a distance D, and the angle between the\nincident wave direction and the scattering planes is 6/2, as shown in Figure 8.3a. What is the\nrelation among D, )t, and 6 for there to be scattering (or diffraction, as it is called in this context) in\nthe direction an angle 6 away from the incident direction? The distance the light travels from the\nsource to the detector is the same for all atoms in a given planej, so there is no problem there. What\nis essential is that the distance traveled by waves scattered from the next further planej + 1 be m\nlarger, where m is an integer. Then the waves scattered from one plane will be in phase with the\nwaves scattered from the next, and there will be constructive interference at the detector. By\nextension, waves from planej + 2 will travel 2m/\u2018t further than from plane j, and therefore will still\nbe in phase, and so will waves from all the planes in the array. This condition is the basis of\nBragg\u2019s law. To state this as an equation, consider the enlarged diagram in Figure 8.3b. By\n"]], ["block_6", ["8.2.4\nBragg's Law and the Scattering Vector\n"]], ["block_7", ["(a)\n(b)\n"]], ["block_8", ["mm\n"]], ["block_9", ["M.\u201c\n9/2\nj+1\n"]], ["block_10", ["j+2\n"]], ["block_11", [{"image_2": "307_2.png", "coords": [245, 44, 419, 197], "fig_type": "figure"}]]], "page_308": [["block_0", [{"image_0": "308_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "308_1.png", "coords": [32, 522, 203, 640], "fig_type": "figure"}]], ["block_2", ["constructing the indicated perpendiculars, it is possible to see that there are two extra segments that\nthe wave must travel when it is scattered off plane j + 1. The length of each of these segments is D\nsin(9/2). Thus Bragg\u2019s law can be written as\n"]], ["block_3", ["we can see that Bragg\u2019s law (Equation 8.2.2) is automatically satisfied. In other words, the\ncriterion for Bragg diffraction is that the scattering vector (determined by the apparatus)\ncoincides with a reciprocal lattice vector (determined by the material).\n3.\nThe magnitude q has dimensions of inverse length. As will become clearer in Section 8.5 and\nSection 8.6, the scattering will be sensitive to structure in the solution on the length scale l/q,\nso the choice of q (which is fixed by choice of )t and 6) ultimately determines what kind of\nstructural information a given scattering experiment can provide.\n"]], ["block_4", ["Figure 8.4\nThe length of the scattering vector is calculated in terms of the scattering angle 6 and the\nmagnitude of the incident and scattered wavevectors, 27r/A.\n"]], ["block_5", ["What is the signi\ufb01cance of (.7? There are three aspects worth bringing out now:\n"]], ["block_6", ["This is illustrated in Figure 8.4, where again 6 is the angle between the scattered and incident\nwaves.\nFor elastic\nscattering,\nboth\nincident\nand\nscattered\nwave\nvectors\nhave\nmagnitude\n27r/)t 27m/)t0, and thus the magnitude of c? can be obtained (see Figure 8.4):\n"]], ["block_7", ["I.\nIt is also known as the momentum transfer vector. Although no energy is exchanged in elastic\nscattering, photons carry momentum and there must be a change in momentum when the\npropagation direction changes. The law of conservation tells us that momentum is transferred\ninto the medium along the direction of \u00a7. In fact, at sufficiently high intensity the resulting\n\u201cradiation pressure\u201d can cause particles to move.\n2.\nThe direction of a\u2019, shown in Figure 8.3a, corresponds to the normal to the planes of scatterers.\nA vector pointing in this direction with amplitude 27r/D is called a reciprocal lattice vector\n(reciprocal because D is in the denominator), and when the magnitudes of these two vectors\ncoincide\n"]], ["block_8", ["A central quantity in any scattering process is the scattering vector, defined by the difference\n"]], ["block_9", ["between the incident and scattered wave vectors:\n"]], ["block_10", ["Basic Concepts of Scattering\n295\n"]], ["block_11", [{"image_2": "308_2.png", "coords": [44, 225, 229, 261], "fig_type": "molecule"}]], ["block_12", ["m)t 2D sin (3)\n(8.2.2)\n"]], ["block_13", ["528\u20148\nman\n"]], ["block_14", ["_,\n2\n,\n6\n4\n,\n6\n|q| E q = 2 (77:) sm (2) 7?.\u2014 srn (2)\n(8.2.4)\n"]], ["block_15", ["271'\n471'\n,\n6\n\u20185 T \n(825)\n"]], ["block_16", ["(2am) sin(9/2)\n"]]], "page_309": [["block_0", [{"image_0": "309_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "309_1.png", "coords": [34, 439, 196, 536], "fig_type": "molecule"}]], ["block_2", ["In the next two sections we will be concerned with incoherent scattering, and will not need to be\nconcerned with the role of q, but then in Section 8.5 and Section 8.6 it will be of central\nimportance.\n"]], ["block_3", ["The derivation of this equation comes from basic electromagnetism, but it involves some rather\nhairy vector calculus, so we omit it; it can be found in many introductory physics texts. However,\nwe can understand the important inverse-dependence on 1. The total energy of the scattered wave\nshould be\nconserved\nFrom Equation 83.2, we see that the intensity will be proportional to\n|E: 0E3 | r 2.The total energy will be proportional to the intensity integrated qverjhesurface\nof a sphere of radius r. As we go further away from the dipole and integrate [ES\n\u00b0 ESI over the\nsurface of the sphere, the area will1ncrease as r2,and thus the total energy will be independent of r.\nNote that (11'/6) has the dimensions of length, since the induced dipole is essentially the product\nof a charge and the distance of charge separation. Therefore we can write the magnitude of the\nacceleration as\n"]], ["block_4", ["The induced dipole moment oscillates at the same frequency w as the field. An oscillating dipole\ninvolves an accelerating charge, and will therefore radiate an oscillating electric field that we will\ncall the scattered wave ES. The magnitude of this wave depends on the direction of observation\n(through an angle ()6 de\ufb01ned below), the speed of light, the distance from the dipole (r), the electron\ncharge (6), and the acceleration of the charge in the dipole (51'):\n"]], ["block_5", ["The ratio of the scattered intensity, 1,, to the incident intensity, 10, is given by\n"]], ["block_6", ["8.3\nScattering by an Isolated Small Molecule\n"]], ["block_7", ["We begin with the fact that an incident light wave Ei will induce a dipole moment 11\u2019 in an atom or\nmolecule, where\n"]], ["block_8", ["296\nLight Scattering by Polymer Solutions\n"]], ["block_9", ["qb l, and there is no scattered wave in the vertical direction 2 (sin qb 0). If we use vertically\npolarized light and detect scattering in the horizontal plane (currently the most commonly\nemployed geometry), then sin qb 1 and life is easy. The scattered intensity, normalized to the\nincident vertically polarized intensity, is given by\n"]], ["block_10", ["where we have used to 277v: Zia-\u20ac010, and as before the dot product is required in taking the\nsquare of the electric field vector.\nIf the incident wave travels along x, and is polarized along 2, then d) is the angle of detection\nrelative to the z-axis (see Figure 8.5a). There is no angular dependence in the x \u2014y plane since sin\n"]], ["block_11", [{"image_2": "309_2.png", "coords": [36, 433, 220, 528], "fig_type": "figure"}]], ["block_12", ["[S___\nEg\u2018ogs\n_62|a|2\nSiab\n[0\nE\"? ii_\nm2\nlEi*l.Ei I\n"]], ["block_13", ["[12 a]; aE\u2019, exp[i(wt 15' F)]\n(8.3.1)\n"]], ["block_14", ["|\u00a7,| =:3}; sins!)\n(8.3.2)\n"]], ["block_15", ["_16ar4a\n_ \u201dT4\nsin (8\n(8.3.4)\n"]], ["block_16", ["_ 6,\n(8:21:04)\n"]], ["block_17", ["dt\n6\n"]], ["block_18", ["d2\n*\n"]], ["block_19", ["|E\u00a7\u201c\u00b0Ei| sin (b\ne2\n#1585)\n"]], ["block_20", ["ad s\n:d\u2014,25i= lEil\n(8.3.3)\n"]]], "page_310": [["block_0", [{"image_0": "310_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "310_1.png", "coords": [14, 26, 281, 212], "fig_type": "figure"}]], ["block_2", [{"image_2": "310_2.png", "coords": [17, 53, 287, 145], "fig_type": "figure"}]], ["block_3", [{"image_3": "310_3.png", "coords": [22, 158, 267, 354], "fig_type": "figure"}]], ["block_4", [{"image_4": "310_4.png", "coords": [24, 234, 278, 334], "fig_type": "figure"}]], ["block_5", [{"image_5": "310_5.png", "coords": [32, 213, 171, 325], "fig_type": "figure"}]], ["block_6", ["Suppose, however, that we are dealing with unpolarized incident light, which is sometimes the\ncase. This we can View as equal parts of vertically and horizontally polarized light. Now we define\nthe scattering angle, 6, between the direction of the transmitted (unscattered) wave (x in this case)\nand the scattered wave to the detector in the x\u2014y plane. For the vertically polarized component of\nthe incident beam, the scattered wave is the same for all 6. For the horizontally polarized part,\nwhich is y-polarized, there will be no scattering along y. Figure 8.5b shows that the sin d) factor\nbecomes sin(7r/2 6) =cos 6 in our nomenclature. Thus the scattered field from the horizontal part\nvaries as cos 6, and the scattered intensity as cos2 6, whereas for the vertical part it is constant. The\nincident intensity was 50% of each, so the net intensity varies as (1 + cos2 6)/2, and we insert this in\nEquation 8.3.4 to obtain\n"]], ["block_7", ["Figure 8.5\nIllustration of the effect of incident beam polarization for a light wave propagating along x, and\nwith scattering being detected in the x\u2014y plane. (a) Vertically (z) polarized beam; q!) is the angle of scattering\nwith respect to z, and sin qb \n1 for all 6 in the x\u2014y plane. (b) Horizontally (y) polarized beam; gb is the angle\nof scattering with respect to y, and therefore sin ch sin(7r/2 6) cos 6 in the x\u2014y plane. The dependence\nof the magnitude of sin qb on 6 is also illustrated for both cases.\n"]], ["block_8", ["(a)\n"]], ["block_9", ["Scattering \n297\n"]], ["block_10", ["(b)\n"]], ["block_11", ["Io,u\n"]], ["block_12", [{"image_6": "310_6.png", "coords": [47, 52, 180, 144], "fig_type": "figure"}]], ["block_13", ["1, _ 8774a2(1 + c032 6)\n"]], ["block_14", ["\u2014\nr2166\n(8.3.6)\n"]], ["block_15", [{"image_7": "310_7.png", "coords": [68, 328, 239, 421], "fig_type": "figure"}]], ["block_16", [{"image_8": "310_8.png", "coords": [82, 340, 215, 414], "fig_type": "molecule"}]], ["block_17", [{"image_9": "310_9.png", "coords": [91, 141, 243, 240], "fig_type": "figure"}]], ["block_18", ["cos2 6\n/ 6\n"]], ["block_19", ["sin2\u00a2=1\\f\nx;\nQ\n\"\n"]], ["block_20", [{"image_10": "310_10.png", "coords": [106, 154, 229, 232], "fig_type": "molecule"}]], ["block_21", ["y\n"]], ["block_22", ["z\nL?\n"]], ["block_23", ["z\ny a\n/\\9\nx\n"]], ["block_24", ["x\n"]], ["block_25", ["y\n"]]], "page_311": [["block_0", [{"image_0": "311_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["In this section we adapt the Rayleigh scattering equation for an isolated object of polarizability a,\nEquation 8.3.5, to the case of a dilute, nonabsorbing polymer solution. Our vertically polarized\nincident light beam will illuminate some region of the sample solution, and the detection optics\n"]], ["block_2", ["Taking 1 Us + (4.1 x 10\u2014 19) J/photon =2.5 ><1018 photons/s. This is a lot of photons, when we bear\nin mind that sensitive photodetectors can count just a few photons per second; but it clearly is not\nenough: 2.5><1018><7.5><10_32 g 2><10\u201413 is still a very small number. Furthermore, the calcula\u2014\ntion gives the total scattered intensity over the surface of a sphere of radius\n1 m, whereas our\ndetector would only collect some small fraction of that. On the other hand, it is worth remembering\nthat if our laser beam encountered a mole of water molecules, the story would be quite different.\nAnd, in particular, sunlight traverses great distances through the atmosphere and encounters many\nmoles of gas phase species, so atmospheric scattering is far from negligible.\n"]], ["block_3", ["after taking care to put all the lengths in centimeters. This is a very small number, which does not\nlook promising. But, even though it is a very small fraction, how much light is going in? Here it is\neasiest to adopt the photon picture. A 1 W laser emits\n1 J/s; how many photons is that? Here we\nrecall the discussion following Equation 8.1.2, and find that 1 photon at this wavelength gives\n"]], ["block_4", ["8.4\nScattering from a Dilute Polymer Solution\n"]], ["block_5", ["where the subscript u denotes unpolarized incident light. This equation is associated with the name\nof Lord Rayleigh, and such scattering from independent polarizable objects is called Rayleigh\nscattering [2]. One interesting feature of Equation 8.3.5 and Equation 8.3.6 is the dependence on\nA54. This strong dependence means that shorter wavelength, higher frequency, or higher energy\nwaves are scattered considerably more. In the atmosphere, molecules and particles scatter blue\nlight down to the earth preferentially over red light, and the sky overhead appears blue. On the\nother hand, when the sun is low in the sky, the blue light is scattered away from the observer and\nthe remaining, transmitted light has a reddish hue.\nWe will use Equation 8.3.5 throughout the rest of this chapter, but it is important to remember\nthat we are assuming vertically polarized incident light, and that the scattering is detected in the\nhorizontal plane.\n"]], ["block_6", ["We\nbegin\nby\napplying\nRayleigh\u2019s\nresult,\nEquation\n8.3.5,\nassuming\npolarized\nlight for\nsimplicity. We need to know the distance to the detector, r, the wavelength of light, A0, and the\npolarizability of water, a. Let us assume we place our detector 1 m from the molecule, and that we\nare using a\n1 W argon laser with A0 2488 nm. The average polarizability of water is given as\n1.65 ><10_24 cm3 [3]. However, it is also instructive to estimate it via Equation 8.1.7, the Lorentz\u2014\nLorenz equation. The refractive index of water is about 1.333 (see Table 8.1), its molecular weight\nis 18 g/mol, and its density is about 1.0 g/cm3, so Equation 8.1.7 gives\n"]], ["block_7", ["which is pretty close to the tabulated value. Using the tabulated value in Equation 8.3.5 we have\n5 g 16 x\n(31:04\nx (1.65 x\n10:4)2 \nlo\n(100)\nx (488 x 10\u20147)\n"]], ["block_8", ["Can we measure the light scattering from a single water molecule in empty space?\n"]], ["block_9", ["Solution\n"]], ["block_10", ["298\nLight Scattering by Polymer Solutions\n"]], ["block_11", ["Example 8.1\n"]], ["block_12", ["hc\n6.63 x 10\u201834 x 3 x 108\n_19\n5\u2014)?0_\n488x10\u20189\n_4.1>< 10\nJ/photon\n"]], ["block_13", ["a X\n18\n.\u2014\n4><3.14\n1.0x6x1023\n"]], ["block_14", ["2 1.86 x 10'24 cm3\n"]]], "page_312": [["block_0", [{"image_0": "312_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "312_1.png", "coords": [28, 428, 248, 475], "fig_type": "molecule"}]], ["block_2", ["The average could be either the time average for one cell, or the ensemble average for the scattering\nvolume, as these two should be equivalent (recall from Chapter 6 that such a system is said to be\nergodic). In the actual measurement, we will record the signal for a finite amount of time, therefore\nperforming a time average, and we will record scattering from all the cells in the scattering volume,\ntherefore performing an ensemble average as well. From Equation 8.3.5, we know that the average\nscattering from any cell will depend on the average of the squared polarizability:\n"]], ["block_3", ["and we simplify this through a very important assumption, that the scattering from \ufb02uctuations in\npressure (6p) and temperature (57) is the same in the neat solvent as in the dilute solution.\nConsequently, when we consider the excess scattered intensity, [ex 130mm\u201c ISOIVCm, only the\nconcentration \ufb02uctuations matter. With these developments Equation 8.3.5 can be transformed into\n"]], ["block_4", ["Therefore the concentration derivative is\nas J}: 92)\n(8.6,\n(9c 1p\u201c 277\n66\nTip\n"]], ["block_5", ["These assumptions are not too restrictive, except that they collectively imply that Rg (and therefore\nthe molecular weight) polymer is not too big. We will deal with the large molecule case in\nthe subsequent sections. Now each cell will have an instantaneous polarizability, which can be\nexpressed the sum of an average value, (0:), plus a \ufb02uctuation, 8a:\n"]], ["block_6", ["There are three terms on the right hand side of Equation 8.4.2. The first will not contribute to the\nnet scattering from the solution, because it is the same for every cell. By the argument given in\nSection 8.2, a completely uniform material does not scatter. The second term is identically zero,\nbecause by definition (50:) :0; the \ufb02uctuations are equally likely to be positive or negative.\nConsequently, we reach the very important conclusion that the scattering is determined entirely by\nthe mean\u2014square \ufb02uctuations in polarizability,\n(80:)2\n.\nNow we need to relate 5a to \ufb02uctuations in t e thermodynamic variables p, T, and c:\n"]], ["block_7", ["will be arranged to collect from some portion of the illuminated region; we call this portion\nthe scattering volume. Now we divide the scattering volume into a large number of imaginary\n\u201ccells\u201d of volumecriteria:\n"]], ["block_8", ["where 1/\u20181\u2019 is the number of cells per unit volume. At this stage we need to work on two terms, (acr/\n8c)\" and ((802). The former is transformed using the Lorentz\u2014Lorenz equation (Equation 8.1.7):\n"]], ["block_9", ["3.\nThe cells are statistically independent, i.e., the \ufb02uctuations in any one cell are uncorrelated\nwith those in any other.\n"]], ["block_10", ["Scattering \n299\n"]], ["block_11", ["1.\n\\l\u2019m\u2019 <.\nEach cell contains many monomers, with a concentration c subject to statistical \ufb02uctuations.\n"]], ["block_15", ["1,,\n16774\n2\n1\n16774\n(Barf\n"]], ["block_16", ["a (a) + \n(8.4.1)\n"]], ["block_17", ["8a\n80:\n6o:\n5\n=\n\u2014\u2014\n\u2014\n5T\n\u2014\n8.4.\na\n(ap)T,c6p\n+\n(8T)p,c\n+\n(6C)T,p66\n(\n3)\n"]], ["block_18", ["or = x1!\n(8.4.5)\n"]], ["block_19", ["[0\nr2/\\,3\n<(\n)\n\u20181;\n72/),3\n(9C\nTip\n(\n)\n\u20181,\n(\n"]], ["block_20", ["(a2) <((a) + 5002)\n: <(a)2> + 2(a)(5a) + <(5a)2>\n(8.4.2)\n"]], ["block_21", [{"image_4": "312_4.png", "coords": [65, 523, 267, 557], "fig_type": "molecule"}]], ["block_22", [":22 113\n"]], ["block_23", ["471'\n"]], ["block_24", ["2\n1\n"]], ["block_25", ["'\n'\n"]]], "page_313": [["block_0", [{"image_0": "313_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "313_1.png", "coords": [30, 107, 168, 165], "fig_type": "molecule"}]], ["block_2", [{"image_2": "313_2.png", "coords": [31, 365, 153, 401], "fig_type": "molecule"}]], ["block_3", ["BZG/c'\ufb01c2 to a virial expansion appropriate for dilute solutions (see Section 7.4). A slight compli\u2014\ncation arises because we have been working with C as the concentration variable, and we need to\nreturn to numbers of moles, Hg, in order to handle the chemical potential. The solution volume, V,\ncan be written in terms of the partial molar volumes (see Equation 7.1.8):\n"]], ["block_4", ["by incorporating Equation 8.4.6 and Equation 8.4.10. The last transformation we need is to relate\n"]], ["block_5", ["which relates the concentration \ufb02uctuations to the associated free\u2014energy penalty. The larger the\ncost in free energy, the smaller the average \ufb02uctuations will be. (Recall from Section 7.5 that for a\nsystem at equilibrium, i.e., one that is stable, 32G/862 must be positive.) However, if the solution\napproaches a stability limit or spinodal, the denominator tends to zero and the \ufb02uctuations, and\ntherefore the scattering, can get very big indeed. The numerator of Equation 8.4.10 just acknow-\nledges the fact that the more thermal energy is available, the larger the \ufb02uctuations will tend to be.\nWe can now recast Equation 8.4.4 as\n"]], ["block_6", ["where P(SC) is the probability of a given \ufb02uctuation Sc. As positive and negative \ufb02uctuations are\nequally probable, P must be symmetric about zero (just like the Gaussian distribution for the end-\nto-end vector in Equation 6.7.1). The size of a \ufb02uctuation is related to the associated \ufb02uctuation in\nfree energy, 5G, through the Boltzmann factor:\n"]], ["block_7", ["The key part of this relation is the so-called refractive index increment, art/ac (where we drop the\nreminder about constant T and p from now on), which can either be measured precisely using a\ndifferential refractometer, or looked up in tables, as will be discussed in Section 8.7.\nNow we return to the concentration \ufb02uctuation term, ((502). We can say that\n"]], ["block_8", ["Note that the first term in Equation 8.4.9 is not symmetric about 5r: = 0, and therefore does not\ncontribute. If we insert the remaining 620/862 term into the exponential and perform the integrals\nin Equation 8.4.7 using the formula cited in Section 6.7, we arrive at the very simple relation\n(see also Problem 3)\n"]], ["block_9", ["and because we need not worry about \ufb02uctuations in volume, dV= 0, and therefore\n"]], ["block_10", ["300\nLight Scattering by Polymer Solutions\n"]], ["block_11", ["and we can expand 5G as a Taylor series (see Appendix):\n"]], ["block_12", [{"image_3": "313_3.png", "coords": [35, 485, 199, 538], "fig_type": "molecule"}]], ["block_13", [{"image_4": "313_4.png", "coords": [44, 277, 227, 310], "fig_type": "molecule"}]], ["block_14", ["2\n2\n2\nE1=W(QE) \n(8.4.11)\n[0\nrZAO\n36\n(azG/acz)\u201c,\n"]], ["block_15", ["\u20145G\nP(6(:) A exp [77,\u2014]\n(8.4.8)\n"]], ["block_16", ["V\ncm, :3a\n(8.4.12b)\nV1\n"]], ["block_17", ["66\n1\n62G\n2\nso (Elma +\n5\u2014! \n(8.4.9)\n"]], ["block_18", ["V \u201dH71 \u2018l' HQVQ\n(8.4.123)\n"]], ["block_19", ["00\nI(5\u20ac)2P(5C) (156\n((56)2 ,0\n(8.4.7)\nIP(56) dSC\n"]], ["block_20", ["2\n_\nH\n((50) > \u2014W\n(8.4.10)\n"]]], "page_314": [["block_0", [{"image_0": "314_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "314_1.png", "coords": [30, 387, 300, 449], "fig_type": "molecule"}]], ["block_2", [{"image_2": "314_2.png", "coords": [32, 283, 168, 326], "fig_type": "molecule"}]], ["block_3", ["where we have used Equation 7.4.7, the virial expansion for H/RT. To conclude,\n"]], ["block_4", ["From Equation 7.4.2 for the osmotic pressure,\n"]], ["block_5", ["wehave\n"]], ["block_6", ["The Gibbs\u2014Duhem relation n1 dul + n; dug :0 at \ufb01xed T and p gives\n%=_\ufb02%\n60\n112\n8C\n"]], ["block_7", ["30\n"]], ["block_8", ["and\n"]], ["block_9", ["Inserting 8.4.13b into 8.4.14 find\n"]], ["block_10", ["At constant T and 7.1.1)\n"]], ["block_11", ["then\n"]], ["block_12", ["Since\n"]], ["block_13", ["Scattering \n"]], ["block_14", [{"image_3": "314_3.png", "coords": [41, 484, 125, 524], "fig_type": "molecule"}]], ["block_15", [{"image_4": "314_4.png", "coords": [44, 231, 154, 270], "fig_type": "molecule"}]], ["block_16", ["32_G__Z 3+2 %__Z\n\u201d1171+\u201d2V'2 %\n862\n\u2014\nM\nV1\n60\n\u2014\nM\nr2217]\n8C\n_:K %\n\u201cCV1\n8c\n"]], ["block_17", ["3,11,]\n_3l'I\n_\n1\n\u2014=\u2014V\u2014=-\u2014VRT\u2014\u2014\n2B\n8c\n13c\n1\n(M+\n6+\n)\n"]], ["block_18", ["n = (M)\nV1\n"]], ["block_19", ["\u00a72_G_ %_Y2% X\n86'2\u2014\n30\nV1\n86\nM\n"]], ["block_20", ["66\nV;\nv\n#2\nM2\u2018TM1\n'\u2014\n66\nV]\nM\n"]], ["block_21", ["V2\nd6 #:1a + 11.2a (M2 71\u2014111)a\n"]], ["block_22", ["dnz _ V\nFIE\u2014M\n"]], ["block_23", ["nzM\n6\u20141\u2014\u2014\nV\n"]], ["block_24", ["kT\n\u2014kT\nckT\n(am/3.22) \n"]], ["block_25", ["c\n1\n\u2018VNa. (1/M+ZBc+-~)\n"]], ["block_26", ["(VN716) (%)\n"]], ["block_27", ["= VRT(l/M + 23.: + . . -)\n"]], ["block_28", ["(8.4.15b)\n"]], ["block_29", ["(8.4.13b)\n"]], ["block_30", ["(8.4.15a)\n"]], ["block_31", ["(8.4.13a)\n"]], ["block_32", ["(8.4.18)\n"]], ["block_33", ["(8.4.19)\n"]], ["block_34", ["(8.4.14)\n"]], ["block_35", ["(8.4.16)\n"]], ["block_36", ["(8.4.17)\n"]], ["block_37", ["301\n"]]], "page_315": [["block_0", [{"image_0": "315_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["This expression is usually rearranged in either of two ways, as follows:\n"]], ["block_2", ["1.\nIn the limit of low concentration Equation 8.4.24b reduces to Ie,~(number of scatter-\ners)><(size of scatterer)~cM, just as advertised for incoherent scattering in Section 8.2.\nNote that Equation 8.4.24a, somewhat perversely, has the experimental signal-the scattered\nintensity\u2014in the denominator, so this result is not so obvious.\n2.\nIn the limit of infinite dilution, light scattering measures M, as does the osmotic pressure.\nThese two experiments are intimately related, because the light scattering is determined by\nconcentration \ufb02uctuations, and their amplitude is related to the associated osmotic cost. In\nother words, one could imagine a semipermeable membrane around each of our fictional cells.\nAny extra polymer in a particular cell will drive up II, whereas a lower-than-average\nconcentration would necessarily cause II to increase in some other cells. Thermal energy\ndrives random \ufb02uctuations, but the osmotic compressibility resists them.\n3.\nThe virial coefficient is obtained from the concentration dependence of the scattering, just as\nin the osmotic pressure experiment. Equation 8.4.24b shows directly that in a good solvent,\nwhen B > 0, the intensity will first increase linearly with c, but the rate of increase will drop\n"]], ["block_3", ["where the second version is obtained by recalling that 1/(1 +x) 1 \u2014x+x2+ -. The former\nversion is more often used when plotting data, because the right hand side should be linear in 6,\nwhereas the latter is more transparent in terms of the physical content. These equations tell us\nseveral important things.\n"]], ["block_4", ["or\n"]], ["block_5", ["to reach the result\n"]], ["block_6", ["where R9 is the so\u2014called Rayleigh ratio. It is the normalized excess scattered intensity per unit\nvolume, with the purely geometrical quantity r factored out; therefore R9 should depend only on\nthe solution and A0, and not the instrument used to measure it. Note that R9 has units of cm\u2018l,\nbecause 1.,x is the scattered intensity per unit volume, whereas I0 is just the incident intensity.\nSimilarly, the purely optical factors can be grouped\n"]], ["block_7", ["and as we have been working with the \ufb02uctuations in a cell of volume \u20181\u2019, we set V \u20181\u2019 in Equation\n8.4.11. The \ufb01nal result is therefore\n"]], ["block_8", ["302\nLight Scattering by Polymer Solutions\n"]], ["block_9", ["where the volume of the fictional cell, \u20181\u2019, has happily disappeared.\nWe now regroup some terms:\n"]], ["block_10", ["KC\n1\n._=._\n23\n8.4.24\nR9\nM \nc +\n(\na)\n"]], ["block_11", ["4\n2\n2\n2\nK E\n77 \u201d (an/ac)\n(3.4.22)\nAONW\n"]], ["block_12", ["R9 KcM{l 230M }\n(8.4.24b)\n"]], ["block_13", ["K\nR9 T\u2014C\u2014\u2014\n(8.4.23)\n__\n23\n. . .\nM \nc +\n"]], ["block_14", ["[ex\n2\n1r E R9\n(8.4.21)\n"]], ["block_15", ["ex\n4\n2\n2\n2\nI\u2014 \nw \" (\u00a7\u201d/3C)\n1\nC\n(8.4.20)\n1.,\nrZAONa,\n(M + 236 + - - .)\n"]]], "page_316": [["block_0", [{"image_0": "316_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "316_1.png", "coords": [29, 395, 174, 429], "fig_type": "molecule"}]], ["block_2", ["_\u2014\nc (g/mL)\n0.0011\n0.0026\n0.0039\n0.0052\n0.0067\n0.0089\n"]], ["block_3", ["1'1.,><105 (cm\u2014l)\n1.87\n3.75\n5.09\n6.06\n7.02\n8.04\n"]], ["block_4", ["Thus light scattering measures the absolute weight average molecular weight, whereas the osmotic\npressure experiment gives the number average (recall Equation 7.4.11). The reason for this\ndifference is that the intensity scattered from an individual polymer is proportional to M, so that\nalthough the \ufb02uctuations are determined by the number of molecules per unit volume, the\nscattering signal is weighted by an additional factor of M.\n"]], ["block_5", ["The following light scattering data were obtained on solutions of polystyrene in toluene at 25\u00b0C.\nA polarized Helium\u2014Neon laser was used (AD: 633 nm); 871/80 was found to be 0.108 mL/g, and\n71: 1.494. Calculate the weight average molecular weight and the second virial coefficient.\n"]], ["block_6", ["when the B term contributes appreciably. On the other hand, in a poor solvent, with B m 0 or\neven negative, the intensity will increase more rapidly with further increase in c. This is\nillustrated schematically in Figure 8.6.\n4.\nWe reiterate that Equation 8.4.24a and Equation 8.4.24b are valid for \u201csmall\u201d polymers only;\nwe will explore this restriction more quantitatively in the following section.\n5.\nAlthough we have stressed the intimate connection between light scattering and osmotic\npressure, there is one important difference. For a dilute but polydisperse sample, we can\n"]], ["block_7", ["Figure 8.6\nThe excess scattered intensity as a function of concentration for different values of the second\nvirial coefficient, B.\n"]], ["block_8", ["and so\n"]], ["block_9", ["Scattering \n303\n"]], ["block_10", ["Example 8.2\n"]], ["block_11", ["q;\n.I'\n.IIII\".\n:\n--.\n|'\nJ\n-\n:\nI'I'J\u2019H\u2019r\n\u2018\n5 :-\n\u201951:?\"\n13> 0\n\u2018:\n"]], ["block_12", ["(t;\n:\noil.\" I J\u201c:\n"]], ["block_13", [":2\n:\nIII-l\n-\n"]], ["block_14", [{"image_2": "316_2.png", "coords": [40, 55, 284, 216], "fig_type": "figure"}]], ["block_15", ["10 :'\nB < 0\ni \n.\nI.\n_f\n\"'\n"]], ["block_16", [{"image_3": "316_3.png", "coords": [43, 445, 211, 489], "fig_type": "molecule"}]], ["block_17", ["15\n_\nI\n_l\u2014I_\n]\nI\nI\nl\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\u2014\nI\n_\nI\n:1\n"]], ["block_18", ["see that\n"]], ["block_19", ["61\u201413(1) R9\n[(26n\n2%:\nMw\n"]], ["block_20", ["\u00b0\n=\n=\n-\n,-\n.42\n1133129\nKcM\n[(20,114\n(8\n5)\n"]], ["block_21", ["715M;\nKC\nKEG; \u201427\n\u20141\u2014\u2014\n(8.4.26)\n1\u2018\n_:_________=\n"]], ["block_22", ["-\nL\n_l\nI\nL\nI\ni.\nI\ni\nI _I_\nI\nI_I_\nI\nI\ni\nI\nI\nI\n-\n00\n2\n4\n8\n10\n"]], ["block_23", ["E\na?\"\n2\n"]], ["block_24", ["C, a.u.\n"]]], "page_317": [["block_0", [{"image_0": "317_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "317_1.png", "coords": [20, 58, 324, 245], "fig_type": "figure"}]], ["block_2", ["We now take into account the finite size of a polymer. Unfortunately, the mathematics is a little\ntedious, but there is no shortcut. On the other hand, the main result will turn out to be relatively\nsimple. We will restrict ourselves to the limit of a very dilute solution, i.e., we only consider one\npolymer at a time. The basic idea was outlined at the beginning of the chapter, namely that if the\ndistance, rjk, between two monomers j and k on a chain is a significant fraction of the radiation\nwavelength, it, then the waves scattered from each monomer will have a phase difference at the\ndetector. This leads to some destructive interference and a net reduction in the scattered intensity.\nThis phase difference will depend on the scattering angle, 9, and therefore the intensity will also\ndepend on 9; thus we are dealing with coherent scattering. Operationally, we can define a form\nfactor for a single polymer, P(9), as\n"]], ["block_3", ["where the Rayleigh scattering is given by Equation 8.4.24. From this relation we can see that\n0\u00a7P(9) _<__ 1. In general, we should worry about interference between waves scattered from\n"]], ["block_4", ["Then we plot KC/Rg versus 0, as shown in Figure 8.7. Linear regression gives an intercept of\n5.46x10\u20186 mol/g and a slope of 7.00><10\"4 mL moi/g2. Thus we obtain\n"]], ["block_5", ["First we need to calculate K by Equation 8.4.22:\n"]], ["block_6", ["8.5\nThe Form Factor and the Zimm Equation\n"]], ["block_7", ["304\nLight Scattering by Polymer Solutions\n"]], ["block_8", ["Figure 8.7\nPlot of data for Example 8.2 according to Equation 8.42a.\n"]], ["block_9", ["Solution\n"]], ["block_10", ["a?\nE\nE\n\u201c:3\n_\n_\n5\u2018\n5 X 10_5 \nlntercept = 5.46 X 10'5\n_-\n"]], ["block_11", ["5 ><10'5 :\u2014\n-:\n9.?\n-\n_\n\u20185\n-\n-\nE\n-\n_\n"]], ["block_12", ["1'5 X 10\u20145\n_\nI\nI\nI\nI\nl\nI\nI\nl\nr\n1\nI\nI\nr\nI\nI\nI\nl\nr\n_l\nJ\n\u2018\n1\n\u2014-\n-I\n"]], ["block_13", ["Actual 16,49)\nP 0 =\n.5.1\n( )\nRayleigh I...(0)\n(8\n)\n"]], ["block_14", ["B 7.00 x 10-4/2 3.5 x 10\u20144 mL moi/g2\n"]], ["block_15", ["M... 1/(5.46 x 10\u20146) 183,000 g/mol\n"]], ["block_16", ["_ ._..\n\u20147\n2\n2\n_\n(6.33 x 10\u20145)4 x 6.02 x 1023 T 1\u201906 x 10\ncm mOI/g\n"]], ["block_17", ["0\n"]], ["block_18", ["0\n0.002\n0.004\n0.006\n0.008\n0.01\n"]], ["block_19", ["E\nSlope = 7.00 x 10-4\nI\n"]], ["block_20", ["E\nE\n"]], ["block_21", ["|-\n"]], ["block_22", ["1\nI\nl\nI\nI\nL\n|\nI\nI\nI\nl\nl\nI\nl\nl\nI\nl\n|\nI\n"]], ["block_23", ["c, g/mL\n"]], ["block_24", ["\u2014\n"]]], "page_318": [["block_0", [{"image_0": "318_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "318_1.png", "coords": [33, 409, 285, 476], "fig_type": "molecule"}]], ["block_2", ["To develop a mathematically precise de\ufb01nition of P09) we consider each monomer to be a\nRayleigh scatterer. electric field scattered by the molecule, Ema, is given by the superposition\nof the fields \n"]], ["block_3", ["where 50,, is the amplitude of the field scattered from each monomer and5}- is the phase of the\nwave from choose to use\nsums overj and k to avoid confusion with i \\/\u2014\u20141.) In Equation 8.5.2 we have assumed that each\nmonomer is identical in scattering power (i.e., a homopolymer). The scattered intensity requires\nthe squaring of the total field:\n"]], ["block_4", ["The Form \n305\n"]], ["block_5", ["different \nbecome the operational of what called the solution structure factor, S(9). However,\n"]], ["block_6", ["because \nthe positions of different polymers and the intermolecular interference terms do not contribute.\n"]], ["block_7", ["8.5.1\nMathematical Expression for the Form Factor\n"]], ["block_8", ["For Rayleigh scattering we assumed that each polymer was very small compared to A, and\ntherefore the phase is the same for each monomer along the chain: 5,- 8k. With this simplification\nwe can write\n"]], ["block_9", ["We have also taken the average (el1(51\u20145))because thatrs the experimentally important quantity.\nEquation 8.5.4 is an alternative definition of P(6), and again we emphasize that it is the single\nchain form factor; the double summation is over the N monomers of one chain. For the general\ncase, we would just extend the double summation over all pairs of monomers in the scattering\nvolume, and then Equation 8.5.4 would define the structure factor.\nThe next step is to develop an expression for 5k 5}, the phase difference between the waves\nscattered from any pair of monomers k and j. Here the scattering vector comes to the rescue,\nbecause this phase difference is very simply expressed as the dot product of \u00a7 with the vector\nseparating monomersj and k:\n"]], ["block_10", ["If we recall the derivation of Bragg\u2019s law in Section 8.2, two particles in a given lattice plane\nautomatically scatter waves in phase with one another. In this case \u00e9\u2019 \u00b0 I}, = 0, because 5 is\nperpendicular to the lattice planes. In other words, the phase difference is determined solely by\nthe component of rj\u2014k that is parallel to the scattering vector.\n"]], ["block_11", [{"image_2": "318_2.png", "coords": [35, 177, 309, 223], "fig_type": "molecule"}]], ["block_12", [{"image_3": "318_3.png", "coords": [41, 319, 168, 362], "fig_type": "molecule"}]], ["block_13", ["5k 5} I? FIk\n(8.55)\n"]], ["block_14", ["N\nN\n_'\nT\n\"'\n\u2014iwt\n-i8-\n_'\niwt\ni5\nIs IEsftot \ns.totl N (53?t\nE\n6\nJ\n\u00b0 (Ems e\nE\ne\n1\u2018\n"]], ["block_15", ["N\nN\n_.\n2\n.\n= IEo.si\nE\nE\n61(8r6\nj=l k=l\n"]], ["block_16", ["M2\nM2\n_.056\u201c\u201d:65\u2019\n(8.5.2)\nj=l\nj=l\n"]], ["block_17", [{"image_4": "318_4.png", "coords": [115, 277, 329, 341], "fig_type": "molecule"}]], ["block_18", ["1\n"]], ["block_19", ["\u2018\nz\n1%\n\n(8.5.4)\n"]], ["block_20", [{"image_5": "318_5.png", "coords": [164, 287, 329, 325], "fig_type": "molecule"}]], ["block_21", ["j=1\n"]], ["block_22", ["(8.5.3)\n"]]], "page_319": [["block_0", [{"image_0": "319_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "319_1.png", "coords": [19, 465, 461, 602], "fig_type": "figure"}]], ["block_2", [{"image_2": "319_2.png", "coords": [26, 624, 236, 675], "fig_type": "molecule"}]], ["block_3", ["Then we can write\n"]], ["block_4", ["where we have looked up the integrals for sin 6 cosk 6. The resulting series in square brackets is, in\nfact, nothing more than\n"]], ["block_5", ["using the transformation to spherical coordinates (if this is unfamiliar, see the Appendix). Then we\neliminate the dot product:\n"]], ["block_6", ["in which q (47ml) sin(6/2) from Equation 8.2.4. If we expand the exponential as a power series\nthis becomes\n"]], ["block_7", ["Now, if we restrict our attention to isotropic samples, such that in spherical coordinates (r,6,(;b) P(FJ}k)\nhas no 6 or d) dependence, then we can get rid of the vectors altogether. First, we recognize that\n"]], ["block_8", ["where P073) is the probability of monomersj and It being separated by a vector i}. For the case of a\nchain in a theta solvent (recall Chapter 6 and Chapter 7), we already know this probability\nfunction: it is a Gaussian function (Equation 6.7.1).\n"]], ["block_9", ["8.5.2\nForm Factor for Isotropic Solutions\n"]], ["block_10", ["This equation represents the standard definition of the form factor. It may appear rather compli-\ncated, because it involves a complex number and two vectors. However, remember that the\ncomplex numbers are just a convenience and they will disappear when we calculate anything\nobservable. Furthermore, the dot product of two vectors is a scalar, so we should be able to get rid\nof the vectors as well. The average in Equation 8.5.6 is\n"]], ["block_11", ["306\nLight Scattering by Polymer Solutions\n"]], ["block_12", [{"image_3": "319_3.png", "coords": [45, 74, 220, 113], "fig_type": "molecule"}]], ["block_13", ["(67302\n(67004\nsin qr k\n2 1\u2014\n+\n+\n_\u20142\u2014\u2014\u2014\n1\n(8.5.11)\n3!\n5!\nqr};c\n"]], ["block_14", ["H\n..\n'\nizqzrf\u2018k cos2 6\nexp[1q\n- rjk] : 1 +1qrjkcos6 +\n2'\n"]], ["block_15", ["exp[iq\u2018 Fir] exp[iqrjk cos 9]\n(8.5.9)\n"]], ["block_16", ["1\nN\n_.\n..\nP(6)=:\n-\n"]], ["block_17", ["dfj-k ax), am dz), :rj, sin 6 d6 dqb drjk\n(8.5.8)\n"]], ["block_18", ["Now we insert this result into Equation 8.5.4:\n"]], ["block_19", ["273'\n\u20196'\n(alpha 8.1) = mam; dry]. Jdd) d6 sm6 + iqrj, cosg \n"]], ["block_20", ["\u2014\u2014]\u2014V-\u20142\u2014 \n(exp[iq Uri)\n(8.5.6)\n"]], ["block_22", ["ll\n"]], ["block_23", ["$62\nq4\ufb01z\n2!3\n 4!5 \nll\n"]], ["block_24", ["0=\u2014-\u2014-\u2014,8\n"]], ["block_25", ["(ah\u2014,8\n"]], ["block_26", ["rah\u2014.58\n"]], ["block_27", ["P(rJ,-k)rj2k eqrJ:(sin 6 + iqrjk cos 6 sin 6 +\n"]], ["block_28", ["P(rjk)2wr-k drjk\n(8.5.10)\n"]], ["block_29", ["0\n"]], ["block_30", ["0\n"]], ["block_31", [{"image_4": "319_4.png", "coords": [198, 558, 342, 597], "fig_type": "molecule"}]], ["block_32", ["-222\n2\n-\n1\nq r- cos 6s1n6+-\u00ab-\n1\"\n2\n)d6\n"]]], "page_320": [["block_0", [{"image_0": "320_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "320_1.png", "coords": [27, 311, 148, 355], "fig_type": "molecule"}]], ["block_2", ["The Form Factor and the Zimm Equation\n307\n"]], ["block_3", ["We now return to the end of the previous section, and insert P(q)=P(9) into the scattering\nequation (Equation 8.4.24a):\n"]], ["block_4", ["Equat10n6.5.8:\n1:11:21\n"]], ["block_5", ["so in the end,\n"]], ["block_6", ["where P(rjk) z 4771'; P\ufb01k).\nThis assumption of isotropy is not at all restrictive for a dilute polymer solution at rest. Note that\nit is not an assumption that the shape of the molecule is spherical, only that on average the\nintrarnolecular bond vectors point equally in all directions. Thus, even a solution of rod\u2014like\nparticles can be isotropic in this sense. We can use Equation 8.5.12 whenever we have information\nabout P(rjk). However, it turns out we can also extract something very useful even if we do not\nknow anything particular about this distribution.\n"]], ["block_7", ["because 1/(1 x) =1 + x + x2 + -.We now use this result to obtain the Zimm equation:\n"]], ["block_8", ["The power series in qr 1,given in Equation 8.5.11, combined with Equation 85.6, gives for P(q):\n"]], ["block_9", ["N\nN\nBut, the average 2 Z is directly related to (82>(2 R2) for any shape, as shown \n"]], ["block_10", ["8.5.4\nZimm Equation\n"]], ["block_11", ["independent of the shape of the particle. Thus if the experiment is designed such that qE < 1, the\nhigher order terms in the expansion Equation 8.5.15 can be neglected, and Rg can be determined\nwithout any prior knowledge of the average conformation. It turns out that for \ufb02exible and semi\u2014\n\ufb02exible polymers this condition is quite often satis\ufb01ed (see Example 8.4 and Problem 6).\n"]], ["block_12", ["in the limit of c \u2014\u2014> 0. Now it is traditional to manipulate 1/P(9) as follows:\n"]], ["block_13", ["8.5.3\nForm Factor as n \u2014> 0\n"]], ["block_14", ["This is the fundamental result for light scattering from dilute polymer solutions [4].\n"]], ["block_15", ["and therefore we have the important result that\n"]], ["block_16", [{"image_2": "320_2.png", "coords": [37, 66, 279, 121], "fig_type": "molecule"}]], ["block_17", [{"image_3": "320_3.png", "coords": [40, 618, 217, 650], "fig_type": "molecule"}]], ["block_18", ["P(q)= A1\u2014,ZZ{1\n<13? >+%...}\n(8.5.13)\n\u20181_\n"]], ["block_19", ["2\nP(q)=1\u2014%R\u00a7+-~\n(8.5.15)\n"]], ["block_20", ["KC\n1\nl\n_ z \u2014 \u2014\u2014\n28\n8.5.16\nR9\nMW 19(0) +\nc +-\n(\n)\n"]], ["block_21", ["1\n1\n1\n2\n:1\n_\n2R2\n8.5.17\n13(9)\nqz\n2\n+ 3 q\ng \n(\n)\n"]], ["block_22", ["Kc\n1\nt]2\n2\n__:__\n\u2014~\u2014~R\n2\n8.5.18\nR9\nMw(1+3\ng+\n)+Bc+\n(\n)\n"]], ["block_23", ["__,\n_,\nsin\nr-\nsin\nr1\n1\nN\nN\n2\nz W ZZ <51)\n(8.5.14)\n)\u2018=1 k=1\n"]], ["block_26", [{"image_5": "320_5.png", "coords": [77, 77, 265, 110], "fig_type": "molecule"}]], ["block_27", ["1\n\u2014\u2014\n313g + - --\n"]], ["block_28", ["1= 1 k:\n6\n"]]], "page_321": [["block_0", [{"image_0": "321_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Equation 8.5.18 provides the basis for a particular method to analyze light-scattering data, the so-\ncalled Zimm plot. We need to perform two extrapolations, to zero scattering angle (6 0\u00b0 or q =0)\nand to zero concentration (23c=0), and the result will be 1 /Mw. Furthermore, the slopes of the\nangle extrapolation and the concentration extrapolation, respectively, will provide values of Rg and\nB, assuming that the range of the independent variable is such that only the first term of the\nrelevant expansion is important. Accordingly, we would plot KC/Rg versus sin2(6/2) + 7c, where -y\nis an arbitrary constant; the sin2(6/2) term is proportional to ([2. The value of -y is chosen to spread\nthe data out. An example is shown in Figure 8.8 for a solution of methylcellulose in water. One\n"]], ["block_2", ["C.\u2014I., and Lodge, T.P., Macromolecules, 32, 7070, 1999. With permission.)\n"]], ["block_3", ["Figure 8.8\nZimm plot for a sample of methylcellulose in water. (Reproduced from Kobayashi, K., Huang,\n"]], ["block_4", ["You should confirm that if you start from Equation 8.5.18, and transform it appropriately you will\nrecover Equation 8.5.19. It is important to point out a subtlety here. You might well ask, if c is high\nenough that polymer\u2014polymer terms contribute to the concentration \ufb02uctuations (i.e., through B),\nwhy don\u2019t we have to worry about interference effects between monomers on nearby chains? The\nanswer is that actually we do, and that is the source of the second P(6) term multiplying B in Equation\n8.5.19. In the simplest approach (which is not simple), the single contact approximation introduced\nby Zimm [4] (Equation 8.5.19) is the result. We should also point out that several texts write the right\nhand side of Equation 8.5.18 incorrectly, as (l/MW + 23c + -) (1 + q2/3R\u00e9 + ).\n"]], ["block_5", ["308\nLight Scattering by Polymer Solutions\n"]], ["block_6", ["8.5.5\nZimm Plot\n"]], ["block_7", ["E\nl\u2018\n.\ngrit-Fir...\u201d-\nI\nJ\n"]], ["block_8", [{"image_1": "321_1.png", "coords": [43, 368, 334, 611], "fig_type": "figure"}]], ["block_9", ["If we recast this relation from Equation 8.4.24b, we obtain\n"]], ["block_10", ["R9 KcMwP(6){1 ZBcMwP(6) . . .}\n2\n2\n(8.5.19)\n_\nq\n2\nq\n2\n\u2014KCMw(1\u2014\u20183\u2014Rg\"'){l_ZBCMw(1_\u00a7'Rgv--)\n...}\n"]], ["block_11", [{"image_2": "321_2.png", "coords": [50, 395, 295, 532], "fig_type": "figure"}]], ["block_12", [{"image_3": "321_3.png", "coords": [52, 72, 315, 127], "fig_type": "molecule"}]], ["block_13", ["10x10\u20145\nl\n1\nI\nI\nl\u2014\nr\nI\u2014I\nT]\nI\nI\nI\n'l\u2014l__'l_\u2018\u2014l'\nFl\nI\nI\nI\nI\u2014\u2018l\nI\nI\nI\nI\n"]], ["block_14", ["_\n,9\n_1\n\u2018e'\nE\nl\n_1\nH\na\n'\nE\nE.\n\u201cg\nE\u201d\n-\n-...__\no)\n-\nE)\nE\no\n3))\n2x106 -\n0\nL0.\n0\n\u2014\n"]], ["block_15", ["G\n.\nI\n'\n\u201dI\n,'\n.\n19:30\"\nj\n4X10F6\n\u2014\ng\n"]], ["block_16", ["r\nwe\" .\n-\n.\nI\n6x10\u20186 \n-\n_ \n.\n,\n\u2014\n"]], ["block_17", ["erg-\u201dff\u201d,\n'\n8X10\u20146 \n.\nO\n.\n.\n...|\n"]], ["block_18", ["ub>0ub>\nub>ub>ub>ub>ub>Iub>ub>ub>ub>ub>\nub>Lub>\nub>ub>nub>ub>\nub>ub>ub>ub>ub>Iub>ub>ub>ub>ub>\nub>Lub>J\nub>a.ub>\nub>tub>\nub>ub>nub>ub>\nub>ub>ub>ub>ub>Iub>ub>ub>ub>ub>\nub>1ub>\nub>ub>ub>ub>ub>Iub>ub>ub>ub>ub>\nub>ub>ub>ub>ub>Iub>ub>ub>ub>ub>\nl J_ub>Lub>\nu\nub>ub>ub>ub>ub>Iub>ub>ub>ub>ub>\nub>ub>ub>ub>ub>Iub>ub>ub>ub>ub>. l\nub>ub>nub>ub>\nub>Lub>\nr\nr\n"]], ["block_19", ["O\n1\n2\n3\n"]], ["block_20", ["_\nf\n.0\nI\nI\ni\n"]], ["block_21", ["'-\n'0',\u201d \n"]], ["block_22", ["-\n\ufb02\n.1\n641500\n"]], ["block_23", [".\nKi\n\u201c'03\n(I?\nn\n'\n(.3\n\u2018\ufb01\u2019\nn\no\no\no\n"]], ["block_24", ["amp-(912) + 30000\n"]]], "page_322": [["block_0", [{"image_0": "322_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "322_1.png", "coords": [25, 438, 341, 577], "fig_type": "table"}]], ["block_2", [{"image_2": "322_2.png", "coords": [25, 409, 381, 611], "fig_type": "figure"}]], ["block_3", ["Solution\n"]], ["block_4", ["The \n309\n"]], ["block_5", ["The scattering data for the methylcellulose sample in Figure 8.8 are given in the following\ntable. The data were taken at 20\u00b0C, with an argon ion laser operating at 488 nm. Under these\nconditions, 72 for water is 1.33, and Biz/ac was determined to be 0.137 mL/g. Calculate MW, Rg, and B.\n"]], ["block_6", ["choice of is l/Ac,Ac is the difference between successive concentrations.\nIn this case all of the angle data for a given concentration \nof the angle concentration. In any event, the choice \ncosmetic. regression on each set of data,\nshifted concentration be fit to a straight line against sin2(6/2), =0\u00b0\nintercept \nangle 6, Kc/Re should be fit to a straight line against 0, and the c=0 intercept recorded (and\nplotted). Then, should be fit to a straight line against 0, and the c 0 intercepts\nto a straight line against sin2(6/2). Both of these lines should meet on the KC/Rg axis, and the\nresulting slopes of these lines are proportional to R: and B, respectively, \nthe proportionality depending on exactly how the data were treated. This process may be under-\nstood in detail by working through Problem 10, or the following example.\n"]], ["block_7", ["The horizontal axis of the Zimm plot requires a choice of the shift factor 7. As the different\nconcentrations are separated by Ac 9:: 0.1 or 0.2 mg/mL, a reasonable choice for 37 would be in the\nrange l/Ac % 5000\u201410,000; in fact, a value of 3000 was selected for Figure 8.8. The next step is to\nperform the first extrapolation to zero angle for each concentration. This may be done directly from\nthe data in the table, using sin2(6/2) as the x axis, by linear regression (\u201cleast squares\u201d). The\nresulting 6=0\u00b0 data are listed below in the second table, and these extrapolations are shown\nin Figure 8.9a, with the 620\u00b0 data highlighted as solid circles. Note that to plot the data in\nFigure 8.9a, each value of sin2(6/2) has to be added to the appropriate value of 7c.\n"]], ["block_8", ["6 (\u00b0).\n311120372)\nKc/R9x10\u00b0\n"]], ["block_9", ["6 (mg/mL)\n0.21\n0.30\n0.50\n0.59\nKc/R9x10\u00b0,6=0\u00b0\n3.42\n3.62\n4.03\n4.13\n"]], ["block_10", ["The second extrapolation is to c=0 for each angle. The resulting linear regression curves and\nintercepts are shown in Figure 8%, and the c = 0 values are listed in the following table.\n"]], ["block_11", ["Example 8.3\n"]], ["block_12", ["30\n0.0670\n3.61\n3.87\n4.21\n4.29\n40\n0.117\n3.86\n4.16\n4.57\n4.71\n50\n0.179\n4.25\n4.43\n4.89\n5.07\n60\n0.250\n4.71\n4.95\n5.38\n5.48\n70\n0.329\n5.14\n5.34\n5.76\n5.84\n80\n0.413\n5.60\n5.78\n6.20\n6.29\n90\n0.500\n6.01\n6.08\n6.59\n6.72\n100\n0.587\n6.37\n6.43\n7.02\n7.14\n110\n0.671\n6.83\n6.89\n7.35\n7.59\n120\n0.750\n6.96\n7.23\n7.66\n7.88\n130\n0.821\n7.49\n7.66\n8.14\n8.25\n140\n0.883\n7.66\n8.00\n8.42\n8.57\n150\n0.933\n7.92\n8.20\n8.65\n8.82\n"]], ["block_13", ["0.21 mg/mL\n0.30 mg/mL\n0.50 mg/mL\n0.59 mg/mL\n"]]], "page_323": [["block_0", [{"image_0": "323_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "323_1.png", "coords": [31, 305, 278, 529], "fig_type": "figure"}]], ["block_2", ["Finally, these two sets of data should be extrapolated to 6 =0 and 6=O\u00b0, respectively. The\nresults are shown in Figure 8.90. From these two straight lines, the desired information can be\nextracted as follows:\n"]], ["block_3", ["Figure 8.9\nConstruction of the Zimm plot of Figure 8.8 as developed in Example 8.3. (a) The extrapolation\nto 9 O for each concentration. (b) The extrapolation to c =0 for each angle.\n(continued)\n"]], ["block_4", ["310\nLight Scattering by Polymer Solutions\n"]], ["block_5", ["I\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\n.I\nI\nI\nI\nI.\nI\nI\nI\nI\nI\nI\nI\nI.\nI.\nI\nI\nI\nC)0\n1\n2\n3\n(a)\nsin2(9/2) + 30000\n"]], ["block_6", ["I\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nL\nI\nI\n.I.\nl\nI\nI\nI\nI\nO0\n1\n2\n3\n(b)\nsin2(9/2) + 30000\n"]], ["block_7", [{"image_2": "323_2.png", "coords": [37, 36, 269, 262], "fig_type": "figure"}]], ["block_8", ["a\n6 \n-\n"]], ["block_9", ["\u00abg\n6 \ufb02\n-\n"]], ["block_10", ["e\n-\n_\n\u00a7\n4 \n-\n"]], ["block_11", ["x\nI\nI\nit?\n-\n.\n\u00a7\n4 \n-\n"]], ["block_12", [{"image_3": "323_3.png", "coords": [44, 334, 270, 455], "fig_type": "figure"}]], ["block_13", ["10\n"]], ["block_14", ["10\n'\n'\n'\n'\nI\n'\n'\n'\n'\nI\n'\n'\n'\n'\nI\n'\n'\n'\n'\nI\n.\n'\n.\n'\nI\n'\n'\n'\n-\n_\nJ\n"]], ["block_15", ["2\n_\n_\n"]], ["block_16", ["8\n..\n_\n"]], ["block_17", ["2\n\u2014\n_\n"]], ["block_18", ["8 \n%\n-\n"]], ["block_19", ["r\nJ\n"]], ["block_20", ["-\n4\n"]], ["block_21", [{"image_4": "323_4.png", "coords": [77, 63, 265, 195], "fig_type": "figure"}]]], "page_324": [["block_0", [{"image_0": "324_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "324_1.png", "coords": [29, 593, 300, 642], "fig_type": "molecule"}]], ["block_2", ["sin2(9/2)\n0.0670\n0.117\n0.179\n0.250\n0.329\n0.413\n0.500\nKc/Rgx 106 6\u2018 =0\n3.28\n3.45\n3.79\n4.31\n4.76\n5.22\n5.54\nsin2(6/2)\n0.587\n0.671\n0.750\n0.821\n0.883\n0.933\nKC/R9X106 I: 0\n5.85\n6.33\n6.48\n7.04\n7.23\n7.47\n"]], ["block_3", ["The angle extrapolation is\n"]], ["block_4", ["therefore\n"]], ["block_5", ["The \n311\n"]], ["block_6", ["Both fits give\na common c=0, 620\u00b0 intercept of 3.03x10\u20186. Thus MW is obtained\nas\n1/(3.03 x10'6)= 3.3 ><105 g/mol.\nThe concentration extrapolation is\n"]], ["block_7", ["Consequently,\n"]], ["block_8", ["and Rg 64 nm.\n"]], ["block_9", ["Figure 8.9 (continued)\n(c) The extrapolations of the 0 0 data to 6 =0, and the 6 =0 data to 0 =0.\n"]], ["block_10", ["0o\n6 \n-\n"]], ["block_11", [{"image_2": "324_2.png", "coords": [38, 528, 300, 587], "fig_type": "molecule"}]], ["block_12", ["x\ni\n_\n9f\n-\n_\n"]], ["block_13", ["(X)\n4//\n"]], ["block_14", [{"image_3": "324_3.png", "coords": [40, 48, 262, 230], "fig_type": "figure"}]], ["block_15", ["I\nI\nI\nI\nl\nI\n_I_\nI\nI\nI\nI\nI\nJ\nI\nI\nI\nI\nO0\n1\n2\n(c)\nsin2(e/2) + 30000\n"]], ["block_16", ["11\u20192 \n3\n=\n0\n2\n8\n3.03 x 10-6 \nX (4 x 3.14 x 1.33)\n4 73 \u201cm\n"]], ["block_17", ["K\n1\n_C 2 _ + 236 = 3.03 x 10-6 + 0.64 x 10-6 (3000c)\n"]], ["block_18", ["B (1/2) x 3000 x 0.64 x 10-6 9.6 x 10-4 mo] cnII3/g2\n"]], ["block_19", ["Kc\n1\n1\n1\n1 166-2712\nR9\nMw( +3q\n3)\nMW< +3\n33\n3111(9/ ) g\n"]], ["block_20", ["1o\n-\n-\n-\n-\nI\n.\n-\n-\n-\nI\n- =\u2014-\n-\nI\n"]], ["block_21", ["8 \ny: 3.03 + 4.82X\n_\n"]], ["block_22", ["y: 3.03 + 0.64x\nj\n24\n-\n"]], ["block_23", ["= 3.03 x 10-6 + 4.82 x 10-6 sin2(6/2)\n"]], ["block_24", ["4.82 x 10-6\n488\n2\n"]], ["block_25", ["_\n"]]], "page_325": [["block_0", [{"image_0": "325_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["We need to calculate Rg for each polymer, and the minimum and maximum q values of the\ninstrument. In Example 6.2, we calculated the unperturbed Rg for polystyrene with M\u2014\n\u2014 105 to be\n8.5 nm. As Rg varies with \\/_\nMwe can estimate R3 for 104 to be \\/_Otirnes smaller, i.e., 2.7 nm, and\nRg for 10\u20185 to3 be \\/_Otimes bigger, or 27 nm. For cyclohexane, n: 1.424 from Table 8.1 (at a\n"]], ["block_2", ["Estimate which of the four regimes would be accessed by light scattering on solutions of\npolystyrene in cyclohexane at the theta temperature (345\u00b0C). Assume M values of 104, 105, and\n106 g/rnol, and that the angular range of the instrument is 25\u00b0\u2014150\u00b0.\n"]], ["block_3", ["qr\u2014I is essentially a \u201cruler\u201d in the sample. The choice of q dictates that the scattering experiment\nwill explore \ufb02uctuations on the length scale cfl. In Bragg diffraction, intensity is found at a\nparticular angle when the relation m}t 20 sin 6/2 is satisfied (Equation 8.2.2), where D is the\nspacing between planes of atoms in the crystal and m is an integer. The Bragg equation just\naccounts for the fact that the phase shift is an integral number of wavelengths when moving from\none lattice plane to the next, and thus the interference is constructive. The Bragg equation for first\norder (m 1) can be rewritten D\u20181 (2/)1) sin(6/2) q/2ar. In other words, scattering is exactly the\nsame process as Bragg diffraction. The only difference is that in scattering, we look for spontan-\neous \ufb02uctuations that just happen to have the right orientation and spacing (2dr/q) to scatter to the\ndetector, whereas in diffraction there are particular lattice planes. Now we can reinterpret the four\nregimes above in terms of the relationship between the spacing of the lines and the size of the\nmolecule. This is illustrated in Figure 8.10, where a set of lines spaced at 277/q is shown with four\ndifferent polymers, illustrating the four size regimes.\n"]], ["block_4", ["1.\nIfn <<1, which means either small molecules or very small scattering angles, then P(q) 2:: 1,\nThus the molecules may be considered as point scatterers, and there is no information on chain\ndimensions. In other words, we may call this the Rayleigh regime, and Equation 8.4.24a and\nEquation 8.4.24b apply.\n2.\nIf n < 1, then only the next term in the power series matters: P(q)~\nW 1 -\u2014 (q2/3)R2. In this\nregime one can obtain the value of Rg without any knowledge about the shape of the\nmolecule. A plot of 1/1ex versus 4]2 should be linear, with slope R2/3 Note that the intensity\nneed not be calibrated to obtain Rg; the units of the intercept cancel out when determining\nthe slope. Alternatively, P(q)~~ exp(\u2014- q2/3R2), and a plot of ln Iex versus i]2 should be linear\nwith slope -\u2014R2/3 This latter format1s termed a Guinier plot, and this regime the Guinier\nregime [5].\n3.\nIf 1 g n _<_ 10, more terms in the power series expansion of P(q) become important, and these\ndepend on the specific shape of the molecule. Accordingly, in this regime the mathematical\nform of P(q) is helpful in distinguishing different molecular shapes. We will identify some\nspecific functional forms below.\n4.\nIf n >> 1, the scattering is dominated by the internal structure of the molecule, and one can\nextract no information about Rg.\n"]], ["block_5", ["A more physical appreciation of these four regimes can be gained from the following viewpoint.\nRecalling the discussion in Section 8.2, the scattering vector has dimensions of inverse length, and\n"]], ["block_6", ["Solution\n"]], ["block_7", ["We continue this chapter with some further discussion of form factors. We return to the expansion\nof Equation 8.5.13, where P(q) is a power series in even powers of gig-k. We can delineate four\ngeneral regimes of behavior, depending on the approximate magnitude of n.\n"]], ["block_8", ["Example 8.4\n"]], ["block_9", ["312\nLight Scattering by Polymer Solutions\n"]], ["block_10", ["8.6\nScattering Regimes and Particular Form Factors\n"]]], "page_326": [["block_0", [{"image_0": "326_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "326_1.png", "coords": [22, 44, 285, 143], "fig_type": "figure"}]], ["block_2", ["R3, to the scattering length scale, I/q.\n"]], ["block_3", ["Therefore for M 104, n ranges from 0.0079x2.7=0.021 to 0.035x2.7=0.095. This falls\nentirely into the Rayleigh regime, and we would not be able to determine R3. For M 105, n\nranges from 0.067 to 0.30. This extends into the Guinier regime, and with precise measurements Rg\ncould be determined. Finally, for M 106, n ranges from 0.21 to 0.95. Although this appears to\nbe in the Guinier regime, in fact the next term in the expansion of P(q) may contribute at larger\nangles, so the lower angle data should be emphasized in the determination of R3.\nThere are two important conclusions to draw from these numbers. First, one can only change\nq by a factor of about 5 in a typical instrument. Second, for \ufb02exible polymers and typical molecular\nweights it is hard to get much beyond the Guinier regime. To access structural details at smaller\nlength scales, it is necessary to increase q substantially. This is done by utilizing x-ray or neutron\nscattering, where the wavelength is only a few angstroms; recall q 1/)\\.\n"]], ["block_4", ["Scattering \n313\n"]], ["block_5", ["slightly different wavelength and temperature, but we are only estimating here). Assuming we have\n"]], ["block_6", ["an argon laser (A0 488 nm), then\n"]], ["block_7", ["The form factors associated with particular distribution functions, P(rjk). have been derived for\na variety of shapes. We will just state the results for three particularly important ones: the Gaussian\ncoil, the rigid rod, and the hard sphere.\n"]], ["block_8", ["Figure 8.10\nIllustration of four regimes of scattering behavior, depending on the ratio of the polymer size,\n"]], ["block_9", ["1.\nFor the Gaussian coil, the form factor is known as the Debye function after it was \ufb01rst\ndeveloped by Debye [6]. It may be written as\n"]], ["block_10", ["231'\n\u20185?\n"]], ["block_11", ["q\n - \n.\nrd\n"]], ["block_12", ["4\n.\n. 24\n2\nX 3 14 X\n1 4\nsin (\u20145) 0.0079 nm\u20181\nqmi\u201c \n488\n2\n4 x 3.14 x 1.424\n.\n150\n_,\nQmax \n488\n3111(7) 0.035 nm\n"]], ["block_13", [{"image_2": "326_2.png", "coords": [48, 151, 288, 254], "fig_type": "figure"}]], ["block_14", ["2\nP(q) = Foe-x\n\u2014 1 +x),\nx a q:\n(8.6.1)\n"]], ["block_15", ["This function applies to chains in a theta solvent, and in the melt. It is important to realize that\nit no longer applies when q\u2018l becomes comparable to the persistence length or statistical\n"]], ["block_16", [{"image_3": "326_3.png", "coords": [52, 153, 162, 257], "fig_type": "molecule"}]], ["block_17", [{"image_4": "326_4.png", "coords": [53, 151, 163, 257], "fig_type": "figure"}]], ["block_18", [".\n/\nmy)\nI\nk)\n(I\n\\\n\\qj\n/\n1q s10\nq>>1\n"]], ["block_19", ["qg \u00ab1\nqt? S1\n"]]], "page_327": [["block_0", [{"image_0": "327_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "327_1.png", "coords": [33, 386, 288, 629], "fig_type": "figure"}]], ["block_2", ["Figure 8.11\nForm factors P(q) for Gaussian coils, hard spheres, and very thin rods (3) as a function\nof (mg)?\n(continued)\n"]], ["block_3", ["The technique of light scattering was first applied to polymer solutions in the 1940s. Early\ninstruments were homemade, and limited by the quality of the available components, such as\nphotodetectors. Two early commercial instruments, the Bryce-Phoenix and the So\ufb01ca, remained in\n"]], ["block_4", ["314\nLight Scattering by Polymer Solutions\n"]], ["block_5", ["segment length; in that regime the chain conformation no longer follows the Gaussian\ndistribution.\n2.\nFor a rigid rod of length L and zero width, the form factor is [7]\n"]], ["block_6", ["where the definite integral is tabulated in most mathematical handbooks.\n3.\nFor a hard sphere of radius R the result (also due to Lord Rayleigh [8]) is\n"]], ["block_7", ["8.7\nExperimental Aspects of Light Scattering\n"]], ["block_8", ["(a)\n(wig)2\n"]], ["block_9", ["1-2lil'llllTllllllllll\n"]], ["block_10", [{"image_2": "327_2.png", "coords": [51, 102, 232, 137], "fig_type": "molecule"}]], ["block_11", ["1/P(q) is plotted versus q\ufb01, i.e., in the form anticipated by the Zirnm plot. In this format all three\nhave the same small q limiting slopes, as they must, but they diverge beyond n a: 1. Note that\nRg :L/m for the rod, and x / 3/5R for the sphere (see Table 6.3), so the different definitions of\nx in Equation 8.6.1 through Equation 8.6.3 need to be handled carefully. In Figure 8.11c, P(q) is\nplotted on a logarithmic axis against n; note the oscillations in P(q) for a sphere. It is clear that\nall three P(q) look similar forn < 1, as expected; this corresponds to regions 1 (\u201cRayleigh\u201d) and\n2 (\u201cGuinier\u201d) above. Beyond that they begin to diverge, corresponding to regimes 3 and 4.\n"]], ["block_12", ["3\n2\n,\n2\nP(q) \n\u20143\n(smx xcosx)\n,\nx E qR\n(8.6.3)\nx\n"]], ["block_13", ["1\nsrnz\nsinx\n2\nqL\nP(q) J\u2014 dz (\u2014>\n,\nx E \n(8.62)\nx\nx\n2\n"]], ["block_14", ["All three P(q) expressions are plotted in Figure 8.11a as a function of (n)2. In Figure 8.11b,\n"]], ["block_15", ["O\nN\n43-\n0)...\n('30\nS\n"]], ["block_16", ["\u2014_\n"]], ["block_17", [".\u2014.\n"]], ["block_18", ["\u2014..\u2014\n"]], ["block_19", ["\u2014-\n"]], ["block_20", ["u\u2014_\n"]], ["block_21", ["\u2014-\n"]], ["block_22", ["\u2014\u2014\n"]], ["block_23", ["I|I_\u2019_LI_IJ_I_|_]_Li_111i11[iI'II\n"]]], "page_328": [["block_0", [{"image_0": "328_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["service in some laboratories for decades. Even with the advent of lasers, high-quality photomulti-\npliers, and sensitive photodiodes, light scattering was largely conducted on custom instrumentation,\nwhich limited its applicability. Two trends in the last two decades have reinvigorated the field,\nhowever. One was the emergence of dynamic light scattering (DLS, see Section 9.5) as an important\ncharacterization technique for polymer, biopolymer, and colloidal solutions. A DLS instrument can\nbe used to measure [(q) as well, and therefore commercial DLS apparatuses now often serve a dual\nrole. The second trend was the emergence of light scattering as an absolute molecular weight\ndetector for size-exclusion chromatography (SEC, see Section 9.7). In this context light scattering\nis finally approaching the status of a \u201croutine\u201d characterization tool for polymer solutions.\n"]], ["block_2", ["Figure 8.11 (continued)\n(b) Inverse form factors as in (a), showing the convergence to a common slope\nof 1/3 in the small n limit. (c) Form factors plotted on a logarithmic scale, showing the oscillations for the\nhard sphere.\n"]], ["block_3", ["(b)\n(CIFI\u2019g)2\n"]], ["block_4", ["(C)\nqFI'g\n"]], ["block_5", ["Experimental of Light Scattering\n315\n"]], ["block_6", ["COII\n\u2019\u2014\n5:\nL.\nI\n._\n"]], ["block_7", ["3\n\u2019\u2014\nI \n_\n4\n-\u2014\u2014\n_\n"]], ["block_8", ["\u20196'-\n_\n_\n"]], ["block_9", ["a: 10 \nE\u2014\n\\\nCo\u201c\n\u2014E\n"]], ["block_10", ["E\n[\u2014\u2014\n/\n"]], ["block_11", [{"image_1": "328_1.png", "coords": [38, 253, 286, 495], "fig_type": "figure"}]], ["block_12", [{"image_2": "328_2.png", "coords": [43, 31, 280, 260], "fig_type": "figure"}]], ["block_13", ["10\nE\nSphere\n\\/ \n_E\n"]], ["block_14", ["10\u20144\n"]], ["block_15", ["10\u20141\n2\u2014\nN \\\n\u20185\nE\n\\\n\u2018 x \nE\nr\n\u2018~~_:\n"]], ["block_16", ["10\n"]], ["block_17", ["100\nl\nI\nl\nI\nI\nI\n|\nIT\n1\u2014 rI\u2014I\n\u2014r T I\u2014I\u2014r\n"]], ["block_18", ["O\n"]], ["block_19", ["6\n_\n"]], ["block_20", ["2 t-\n_\n"]], ["block_21", ["8 l\u2014\n__\n"]], ["block_22", ["_3\n,\u2014\n"]], ["block_23", ["0\n2\n4\n6\n8\n10\n"]], ["block_24", ["E\nI\n"]], ["block_25", ["L\u2014\n_\n"]], ["block_26", ["l-\n_I\n"]], ["block_27", ["I\"\n_\n"]], ["block_28", ["L.\n_.\n"]], ["block_29", ["l_\n_\n"]], ["block_30", ["\u2019\u2014\n"]], ["block_31", ["E\nSphere\n/\n\u2014\n"]], ["block_32", ["0\n2\n4\n6\n8\n10\n"]], ["block_33", ["E\nf \\\nE\n"]], ["block_34", ["E\nE\nI\n\\ f\u2014\n"]], ["block_35", ["I\nI\nl\nI\n|\nI\nI _L _I_ l_l_I\n_I\nL\nl\nI\nI\nI\nI\n"]], ["block_36", ["I\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\nI\n"]], ["block_37", ["I\nI\n1\n\u2014r T 7\u2014I\nl\nI\nI7\nI\nl\n_l\nT\nI_\nI\nl\nl\n"]], ["block_38", ["llllllI\n/ /.\n"]], ["block_39", [".\n_\n"]], ["block_40", ["\u2014\n"]], ["block_41", ["l\nI\n"]]], "page_329": [["block_0", [{"image_0": "329_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "329_1.png", "coords": [19, 474, 339, 630], "fig_type": "figure"}]], ["block_2", ["I\nSource\n:\nin,\n"]], ["block_3", ["where nglass is typically about 1.5 and nah 1.0. This gives a re\ufb02ectivity of about 4% per air\u2014glass\ninterface. The larger the diameter of the sample cell, the further this source of stray light will be\nfrom the scattering volume, and therefore the easier it is to eliminate. Empirically, for a\n1 cm\nsample cell it is very difficult to get to scattering angles below about 25\u00b0 because of this problem.\nNote that there will also be two glass\u2014sample solution interfaces, where for some solutions\n(aqueous ones, for example) there will also be substantial re\ufb02ection.\nThe re\ufb02ection problem is often mitigated, but not completely eliminated, by use of an index\u2014\nmatching \ufb02uid. In this case the sample cell is suspended in a much larger container filled with a\n\ufb02uid of refractive index near that of the glass (silicon oil and toluene are two common examples).\nBy this expedient the main air-glass interfaces can be moved several centimeters from the\nscattering volume. Furthermore, the index-matching bath can provide an excellent heat-transfer\nmedium for controlling the sample temperature. Good temperature stability is important for precise\nmeasurements.\nThe scattered light is collected by some combination of lenses and apertures, and directed onto\nthe active surface of the detector. In routine applications it is usually assumed that the scattered\n"]], ["block_4", ["The basic components of a light scattering instrument are illustrated in Figure 8.12. The typical\nlight source is a laser, although the unique features of a laser source (e.g., temporal and spatial\ncoherence, collimation, high monochromaticity) are not required. The most desirable feature of\nthe source is stability. The source may be vertically polarized or unpolarized, as discussed in\nSection 8.3, but it is important to establish that you have one or the other, and not a partially\npolarized beam. The beam should be collimated before traversing the sample cell. The solution\nis usually contained in a glass cell, and the glass should be carefully selected for clarity and lack\nof imperfections. All other things being equal, the larger the diameter of the sample cell, the\nbetter. The reason is that at each air\u2014glass interface a significant portion of the incident light\nwill be reflected, rather than transmitted. This re\ufb02ected light may find its way to the detector as\nstray light, or back into the scattering volume at a different angle, leading to erroneous\nscattering signals. The re\ufb02ected fraction of the intensity, R, for near\u2014normal incidence is\ngiven roughly by\n"]], ["block_5", ["Figure 8.12\nSchematic diagram of a light scattering photometer.\n"]], ["block_6", ["8.7.1\nInstrumentation\n"]], ["block_7", ["316\nLight Scattering by Polymer Solutions\n"]], ["block_8", ["mM\n(8.7.1)\n(\u201dglass + \u201dair)\n"]], ["block_9", ["Io monitor\n"]], ["block_10", ["I\n\u2018\n"]], ["block_11", ["I\nI, monitor\n"]], ["block_12", ["43/\n/\nI5 detector\n"]]], "page_330": [["block_0", [{"image_0": "330_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["used \nsome situations, such examining rod-like particles, signi\ufb01cant depolarization of the\nscattered polarizer front of detector becomes necessity. Three\ngeneral detection schemes can be envisioned. The simplest is to have a single detector at some\nfixed of \ninformation, detector \nprocess changes in time. The next option is to have a single detector that\ncan rotate which runs through the center of the sample cell. This provides a\ncontinuously variable 9, which is very desirable. In recent years, it has become popular to use a\nmovable Optical collect the light, and keep the actual detector fixed in space. In this\nconfiguration, usuallya photomultiplier, which can be extremely sensitive (capable\nof counting if need be) and which has a wide linear range (output current\nproportional to incident intensity over many orders of magnitude). The third approach is to arrange\nmultiple detectors at fixed angles, which allows for simultaneous detection with the attendant\nincrease in overall signal-to-noise (the \u201cmultiplex advantage\u201d). This technique has only recently\nbecome practical with improvements in photodiode detectors, such that small, inexpensive units\nare sufficiently sensitive and linear. The multiangle light scattering technique is now the basis of a\npopular SEC detector, which can give real-time information about M and Rg for each slice of the\nchromatogram (see Section 9.7).\nThe final instrumental issue is the use of an incident intensity monitor, to follow \ufb02uctuations\nand drift in the source output. Often a small portion of the incident beam is split off, for example\nby inserting a piece of glass in the incident beam at 45\u00b0 to the prOpagation direction, and directed\nto a separate detector. This permits the scattered signal to be normalized to the incident intensity\nin real time, which ultimately leads to a much more reliable R9. For example, when measuring a\ndilute polymer solution the scattered intensity may be less than twice that of the solvent alone.\nThe solvent and solution would be measured at different times, so small but uncontrolled\nvariations in source intensity can severely compromise the reliability of the excess intensity. It\nis also desirable to place a detector to monitor the transmitted beam. This could serve as an\nincident intensity monitor, but by having detectors both prior to and after the sample it is possible\nto detect changes in the total sample scattering, and thereby to assess whether absorption or\nmultiple scattering is a problem.\n"]], ["block_2", ["Referring back to Equation 8.4.22, we see that in order to extract a value of MW, we will need to\nknow the refractive index of the solvent, art/60, 0, A0, and r. All of these are straightforward to\ndetermine, except r. In addition, and most importantly, the Rayleigh ratio involves the ratio of\nthe scattered intensity per unit volume to the incident intensity, and as we noted in the beginning\nof the chapter, we rarely determine a true intensity. Furthermore, we would not know the active\narea of the photodetector, or its quantum efficiency (the fraction of incident photons actually\ndetected).\nThe resolution of these difficulties is actually rather simple. We assume that the detector\nproduces an output signal S (current, voltage, or \u201ccounts\u201d) proportional to the incident intensity:\n"]], ["block_3", ["where the proportionality factor B is independent of the magnitude of S over the relevant range\n(i.e., linear response), and the subscripts denotes \u201cscattering.\u201d As long as we maintain the detector at\na fixed distance r from the scattering volume as the scattering angle is varied, then the detector\n"]], ["block_4", ["8.7.2\nCalibration\n"]], ["block_5", ["light is entirely polarized vertically (for vertically polarized incident light), and a polarizer is not\n"]], ["block_6", ["Experimental Aspects of Light Scattering\n317\n"]], ["block_7", ["SS BSIS\n(8.12)\n"]]], "page_331": [["block_0", [{"image_0": "331_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "331_1.png", "coords": [18, 484, 256, 575], "fig_type": "figure"}]], ["block_2", [{"image_2": "331_2.png", "coords": [18, 465, 282, 649], "fig_type": "figure"}]], ["block_3", ["should collect light over the same range of solid angle fraction at each 6. A similar equation\ndescribes the response of the incident intensity monitor:\n"]], ["block_4", ["We are now in a position to obtain the Rayleigh ratio (units of cm\u201d) as follows:\n"]], ["block_5", ["where there is a single unknown proportionality factor 7. This we obtain by measuring a pure\nsolvent for which the absolute Rayleigh ratio has been measured (by someone who took a lot of\ncare) at the same wavelength and temperature. A second calibration step is required for instruments\nwith multiple detectors, as each detector will have its own value of B (and also a different r and\ncollection solid angle). This is best accomplished by using a solution with relatively high scattering,\nbut one for which the scattering is entirely incoherent. A solution of a moderate molecular weight\npolymer can serve this purpose. Then the signal from each detector can be normalized to a single\nreference detector (usually chosen at 6 90\u00b0) by a multiplicative factor. It should be noted that\nEquation 8.7.2 and Equation 8.7.3 assume that there is no background signal (i.e., a finite 5 when\nthe source is turned off) or stray light (i.e., contributions to S not from sample scattering at angle 6).\nIn fact, all photodetectors have some dark current (output in the absence of input), but this is very\nsmall. (If it is a significant fraction of the scattering signal, then your experiment is in trouble.)\nOther sources of background (e.g., room light) and stray light can be minimized by appropriate\ninstrumental design. However, the simple fact is that the detector is indiscriminate; all photons that\nreach it will be counted as scattering, no matter what their origins, and so great care must be taken to\neliminate background and stray light.\nThere is an additional correction step that must be performed when using a single detector that\nrotates to various 9. Refen\u2018ing to Figure 8.13, the scattering volume is determined by the\nintersection of the incident beam and the collected beam (the latter being set by the geometry of\nthe detection system). Assuming that both are square in cross section, and that the latter is\ncomparable in diameter to the former, we can see that as 6 deviates from 90\u00b0 the volume of\n"]], ["block_6", ["318\nLight Scattering by Polymer Solutions\n"]], ["block_7", ["Figure 8.13\nIllustration of the change in scattering volume with scattering angle, 6, for incident and\nscattered beams with width d.\n"]], ["block_8", ["So 3010\n(8.13)\n"]], ["block_9", ["R9:\n"]], ["block_10", ["Vol d3/sin a\n"]], ["block_11", [": \n__ Siolvent)\n(874)\nSo\n"]], ["block_12", ["\u2014 \n3050\n"]], ["block_13", ["rZUSsolution Igolvent)\nIo\n"]], ["block_14", ["\u20184\n4\n"]], ["block_15", ["d\n"]]], "page_332": [["block_0", [{"image_0": "332_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Usually one is interested in concentrations on the order of 10\u20182 10\u20144 g/mL, and arr/ac is of\norder 0.1, so it is not simply a matter of measuring n directly, because the change in n will only\nappear in the third or fourth place after the decimal. Rather, a di\ufb01ferenrial refractometer should be\nused. A variety of designs are available, either commercially or by relatively simple assembly in\nthe laboratory, but the principle behind the simplest version is illustrated in Figure 8.14. The light\n"]], ["block_2", ["As is evident from the preceding discussion, (Tin/8c\u201c plays a central role in the magnitude of the\nscattered intensity. It may be defined as follows:\n"]], ["block_3", ["a refractive index detector serves as a direct concentration monitor. There are two standard\ndiagnostics for the presence of dust in the sample. The first is to examine the temporal \ufb02uctuations\nin 53. These should be random, and have a root\u2014mean\u2014square amplitude close to x/ (55). If some\ndust is present, the signal is likely to increase suddenly and then decrease suddenly some seconds\nlater, as dust particles drift in and out of the scattering volume. The scattering volume can also be\nexamined visually; the tell-tale bright \ufb02ashes of dust are often easily seen. The second diagnostic is\nto examine 1/109) versus sin2(6/2) (or qz). According to the Zimm equation this should produce a\nstraight line. Dust will increase 1(6) selectively at low 6, resulting in a characteristic downturn in\nthe plot. (Note that this test should only involve the range of 6 deemed to be free of re\ufb02ections.)\n"]], ["block_4", ["where n is the refractive index of the solution, it, the refractive index of the pure solvent, and c is\nthe concentration in g/mL. The initial slope of n(c) versus 6 gives Bit/ac, but is better obtained as\n"]], ["block_5", ["There are two main issues here. First is the choice of solvent, and second is the preparation of\n\u201cdust\u2014free\u201d samples. One may not have the freedom to choose the solvent, but all other things\nbeing equal it is desirable to have |8nl6c| as large as possible, as the intensity is proportional to\n(an/36f. It may also be helpful to choose a solvent with a relatively small R9 of its own, so that the\npolymer contribution to the excess scattering is larger. (You might be wondering \u201cWhy does the\npure solvent scatter at all, given that we emphasized that homogeneous materials do not scatter?\u201d\nThe answer is density \ufb02uctuations, and the scattered intensity is determined by the isothermal\ncompressibility of the solvent, K, in parallel to Equation 8.4.11, the intensity is proportional to\nkTK). Finally, some solvents are easier to make dust-free than others; for example, more polar\nsolvents such as water and THF are often trickier to clean than toluene or cyclohexane. The\npreparation of dust-free samples takes some care and experience. It is essential to remove dust, as\nstray particles that are significantly larger than the polymer molecules will scatter strongly. The\ntwo standard options are filtration and centrifugation, and the fonner is usually preferred. Both are\nless than ideal, in that they may change the concentration of the solution. The use of light scattering\n"]], ["block_6", ["a functioning is to plot 53(6) sin 6 versus 9 for an incoherent scattering solution. This\nproduct should be independent of 6. Ultimately, at high and low 9 the data will deviate from the\nconstant due to re\ufb02ections at the various interfaces discussed above. This measurement will\ntherefore provide a guide as to the reliable range of 6 for the instrument.\n"]], ["block_7", [{"image_1": "332_1.png", "coords": [34, 574, 192, 607], "fig_type": "molecule"}]], ["block_8", ["intersection increases by a factor of l/sin 6. Thus, the signal at any angle should be multiplied by a\nfactor of sin 6 in order to account for this variation in scattering volume. A very good indication of\n"]], ["block_9", ["8.7.4\nRefractive Index Increment\n"]], ["block_10", ["as an SEC detector has a particular advantage here in that the column acts as an excellent filter, and\n"]], ["block_11", ["8.7.3\nSamples and Solutions\n"]], ["block_12", ["Experimental Aspects of Light Scattering\n319\n"]], ["block_13", ["an.\n,\nn n5\n__\n_\nAn,\nE 3313\u201c\nc\n) \u2018l11.%(7)\n(8.7.6)\n"]], ["block_14", ["nzns+\nc+acz+~~\n(8.7.5)\n"]], ["block_15", ["9:\u201d:\n"]], ["block_16", ["BC\n"]]], "page_333": [["block_0", [{"image_0": "333_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "333_1.png", "coords": [33, 23, 280, 234], "fig_type": "figure"}]], ["block_2", ["beam is incident on a divided cell that contains the solvent in one compartment and the solution in\nthe other (or solutions of two different concentrations). The dividing glass surface is angled, so that\nthe beam direction in compartment 2 depends on the refractive index of the \ufb02uid in compartment 2,\nby Snell\u2019s law. The beam is then re\ufb02ected back through both compartments, with a final direction\nthat depends on An. For small An it can be shown that the displacement of the re\ufb02ected beam, d, is\nproportional to An.\nThe value of an/ac depends on T and )t, so it should be measured under the same conditions as the\nlight scattering itself. It is independent of molecular weight for large M, but will develop a significant\nM dependence at low M, so this may need to be taken into account in, for example, SEC detection (see\nSection 9.7). The main source of the M dependence of an/ac lies in the M dependence of the polymer\nrefractive index np, which in tum depends directly on density. The M dependence of the density re\ufb02ects\nprimarily the concentration ofchain ends, and art/ac is therefore usually linear in 1/M,,. It is tempting to\ntry and relate {in/36 to n], n, in some simple way, but this is risky. There is no general, rigorous\nexpression for n(c) that depends only on n3, np, and c, and in fact n need not even vary monotonically\nwith 6. However, it is usually the case that art/ac will increase with the magnitude of up n3, so this\nbecomes a reasonable starting point when choosing a solvent to maximize an/ac. Note that an/ac can\nbe positive, negative, or zero; as it appears squared in the scattered intensity, the sign does not matter. If\none chooses an isorefractive solvent, such that (an/ac z 0, that polymer will be invisible in the\nscattering experiment. This can be used to advantage in examining polymer mixtures and copolymers,\nin order to highlight the behavior of one particular component.\n"]], ["block_3", ["In this chapter we have developed in some detail the equations governing the scattering of light\nfrom dilute polymer solutions. Light scattering is a very powerful experimental tool, providing\nboth thermodynamic and structural information. Recent developments in commercial instrumen-\ntation have accelerated the use of light scattering in routine characterization. The main concepts in\nthe development are the following:\n"]], ["block_4", ["1.\nCompletely uniform materials do not scatter. Scattering in polymer solutions arises from\nrandom \ufb02uctuations in concentration. This scattering is incoherent, which means that the\nintensity is independent of scattering angle.\n2.\nThe incoherent intensity is determined by the mean-square concentration \ufb02uctuations, which\nin turn are set by the ratio of the thermal driving force, H\", to the free energy penalty for a\ngiven \ufb02uctuation, (326/862. It is through this relationship that the scattering experiment\nprovides a measurement of Mw and B.\n"]], ["block_5", ["8.8\nChapter Summary\n"]], ["block_6", ["320\nLight Scattering by Polymer Solutions\n/\n._\n"]], ["block_7", ["Figure 8.14\nIllustration of a refractometer based on a split cell prism.\n"]], ["block_8", ["\u20185\n"]], ["block_9", ["d\n"]]], "page_334": [["block_0", [{"image_0": "334_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["*B.H. Zimm J. Chem. Phys. 16, 1093, 1099 (1943).\nI1.]. Blum and M.F. Morales Arch. Biochem. Biophys. 43, 208 (1953).\n"]], ["block_2", ["3.\nCarry though the integration of Equation 8.4.7 to obtain Equation 8.4.10.\n4.\nAn estimate of RE can be obtained rather simply from the so-called dissymmetry ratio, defined\n"]], ["block_3", ["Draw a plot in polar coordinates of the scattering envelope in the x y plane. How would the\nenvelope of a Rayleigh scatterer compare with this plot? By interpolation, evaluate the\ndissymmetry ratio and Rg. What are some practical and theoretical objections to this procedure\nfor estimating Rg?\n5.\nThe effect of adenosine triphosphate (ATP) on the muscle protein myosin was studied by light\nscattering in an attempt to resolve con\ufb02icting interpretations of viscosity and ultracentrifuge\ndata. The controversy hinged on whether the myosin dissociated or changed molecular shape\nby interaction with ATP. Blum and MoralesI reported the following values of (Kc/R6)C:0\nversus sin2(9/2) for myosin in 0.6 M Kc1 at pH 7.0. Which of the two models for the mode of\n"]], ["block_4", ["1.\nExplain in your own words (i) why the permeation of solvent through a membrane into a\npolymer-rich phase and the amount of light scattered by a polymer solution are related, (ii)\nwhy even though they are closely related, the osmotic pressure and light scattering experi\u2014\nments measure different moments of the molecular weight distribution, and (iii) why the\nosmotic pressure measurement becomes increasingly difficult for degrees of polymerization\n>103 while light scattering becomes harder for degrees of polymerization \u00a3103.\n2.\nEstimate the largest and smallest molecular weight polystyrenes for which one could reason\u2014\nably expect to measure Rg reliably in THF, using an Argon laser at 488 nm, and a usable\nangular range of 30\u00b0 < 6 < 150\u00b0; THF is a good solvent for polystyrene.\n"]], ["block_5", ["3.\nFor polymers that are large enough, a significant phase difference can arise between portions\nof the incident wave that are scattered from different monomers on the same chain. This leads\nto angle-dependent, or coherent, scattering. The underlying process is very similar to Bragg\ndiffraction from crystals, with the key difference being that in polymer solutions there are only\naverage correlations in the positions of the various monomers, rather than a permanent lattice.\n4.\nThe description of coherent scattering is built around the scattering wavevector, (f. The\nmagnitude of this vector depends on wavelength and scattering angle, and has units of inverse\nlength. Depending on the magnitude of the dimensionless product n, the coherent scattering\ncan give information about the internal structure or the overall size of the polymer. It is often\npossible to use light scattering to measure Rg in a completely model-independent way.\n"]], ["block_6", ["Problems\n321\n"]], ["block_7", ["Problems\n"]], ["block_8", ["as Iex(6=450)/Iex(6= 135\u00b0). Explain how this works. Zimm has reported the intensity of\nscattered light ()1 = 364 nm) at various angles of observation for polystyrene in toluene at a\nconcentration of 2><10_4 g/mLT. The following results were obtained (values marked were\nestimated and not measured):\n"]], ["block_9", ["0\n4.29\u201d\u201c\n25.8\n3.49\n36.9\n2.89\n53.0\n2.18\n66.4\n1.74\n90.0\n1.22\n113.6\n0.952\n143.1\n0.763\n180\n0.70*\n"]], ["block_10", ["9 (o)\nIS (a.u.)\n"]]], "page_335": [["block_0", [{"image_0": "335_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["IRE. Steiner and K. Laki, Arch. Biochem. Bt'ophys. 34, 24 (1951).\n1P. Outer, C.I. Carr, and 13.11. Zimm, J. Chem. Phys. 18, 839 (1950).\n"]], ["block_2", ["322\nLight Scattering by Polymer Solutions\n"]], ["block_3", [{"image_1": "335_1.png", "coords": [37, 313, 419, 438], "fig_type": "figure"}]], ["block_4", ["For poly(n-hexyl isocyanate) and poly(methyl methacrylate), estimate the range of molecular\nweights for which the Guinier regime (n < 1) can be accessed. The data in Figure 6.10 and\nTable 6.1 may be useful.\nAggregation of fibrinogen molecules is involved in the clotting of blood. To learn something\nabout the mechanism of this process, Steiner and LakiJr used light scattering to evaluate M and\nthe length of these rod\u2014shaped molecules as a function of time after a change from stable\nconditions. The stable molecule has a molecular weight of 540,000 g mol and a length of\n840 151. The accompanying table shows the average molecular weight and average length at\nseveral times for two different conditions of pH and ionic strength 11. Criticize or defend the\nfollowing proposition: The apparent degree of aggregation x at various times can be obtained in\nterms of either the molecular weight or length. The ratio of the value of x based on M to that\nbased on length equals unity exclusively for end-to-end aggregation and increases from unity as\nthe proportion of edge\u2014to\u2014edge aggregation increases. In the higher pH\u2014lower p. experiment\nthere is considerably less end\u2014to\u2014end aggregation in the early stages of the process than in the\nlower pH\u2014higher p. experiment.\n"]], ["block_5", ["Zimm plots at 546 nm were prepared for a particular polystyrene at two temperatures and in\nthree solvents. The following summarizes the various slopes and intercepts obtainedi:\n"]], ["block_6", ["Slope\nIntercept 9:0\nMethyl ethyl ketone\n0.608\n260\nDichloroethane\n1. 16\n900\nToluene\n1. 14\n1060\n"]], ["block_7", ["Solvent\nIntercept 6:0\nIntercept 9:0\nMethyl ethyl ketone\n0.551\n230\nDichloroethane\n1 .05\n870\nToluene\n1.09\n800\n"]], ["block_8", ["Solvent\n"]], ["block_9", ["ATP interaction with myosin do these data support? Explain your answer by quantitative\ninterpretation of the light\u2014scattering data.\n"]], ["block_10", ["T 67\u00b0C\n"]], ["block_11", ["1 150\n1.63\n1600\n1000\n2.0\n1200\n1670\n2.20\n1900\n2350\n3.30\n2200\n"]], ["block_12", ["sin2(6/2)\n0.15\n0.21\n0.29\n0.37\n0.50\n0.85\nKc/Rgx 107 (Before ATP)\n0.9\n1.1\n1.5\n1.8\n2.2\n2.7\nKc/R9x107 (With ATP)\n1.9\n2.8\n3.7\n4.6\n6.0\n6.8\n"]], ["block_13", ["T:22\u00b0C\n"]], ["block_14", ["1(3)\nMx 10 (g mol\n\u2014 1)\nLength (A)\nr (s)\nMx 10\n\u2014 6 (g mol\n\u2014 1)\nLength (A)\n"]], ["block_15", ["650\n1.10\n1300\n900\n1.10\n1100\n"]], ["block_16", [{"image_2": "335_2.png", "coords": [58, 471, 341, 556], "fig_type": "figure"}]], ["block_17", ["pH 8.40 and p. 0.35 M\npH 6.35 and p. 0.48 M\n"]], ["block_18", [{"image_3": "335_3.png", "coords": [91, 90, 381, 142], "fig_type": "molecule"}]], ["block_19", ["Intercept 6:0\n"]], ["block_20", ["Slope\n"]], ["block_21", ["Slope\nSlope\n"]]], "page_336": [["block_0", [{"image_0": "336_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Problems\n323\n"]], ["block_2", ["10.\n"]], ["block_3", ["11.\n"]], ["block_4", ["12.\n"]], ["block_5", ["13.\n"]], ["block_6", ["14.\n"]], ["block_7", [{"image_1": "336_1.png", "coords": [50, 325, 187, 457], "fig_type": "figure"}]], ["block_8", ["Evaluate MW, Rg, and B from each piece of pertinent data and comment on (a) the agreement\nbetween MW values and (b) the correlation of Rg and B with solvent quality.\nPlot the light scattered intensity (arbitrary units) versus scattering angle that you would\nexpect to see for a very dilute solution of polystyrene with M =4,000,000 in cyclohexane at\n35\u00b0C. Assume the instrument has an angular range of 20\u00b0\u2014150\u00b0. On the same axes show the\nangular dependence of the intensity if the scattering object were a hard sphere with the same Rg.\nFor polystyrene in butanone at 67\u00b0C the following values of Kc/Rgx 10\u00b0 were measured at the\nindicated concentrations and angles. Construct a Zimm plot from the data below using\n7: 100 mL/g for the graphing constant. Evaluate M, B, and R3 from the results. In this\nexperiment A0 546 nm and n 1.359 for butanone.\n"]], ["block_9", ["The slope\u2014intercept ratios have units of cubic centimeters per gram, and the intercepts are\nc/n, where the subscript v indicates vertically polarized light. The following values of n\nand 3n/3c can be used to evaluate K:\n"]], ["block_10", ["Methyl ethyl ketone\n1.378\n0.221\n1.359\n0.230\nDichloroethane\n1 .444\n0.158\n1 .423\n0. 167\nToluene\n1.496\n0. 108\n1.472\n0. 1 18\n"]], ["block_11", ["Draw a sketch of a complete Zimm plot for a high molecular weight polymer in a theta\nsolvent, assuming data were acquired at eight angles and at four concentrations. In a different\ncolor pen, indicate the effect on the data if a small amount of dust were present in the\nsolution. Similarly, in a different color indicate the effect on the data of raising the solution\ntemperature substantially. State any assumptions you make.\nWhen a dilute solution of block copolymers undergoes micellization, i.e., some numbers of\nchains aggregate into a (usually spherical) assembly to shield one block from the solvent it\ndoes not like, the light scattering intensity increases. In fact, if micellization is induced by\nchanging temperature at a fixed concentration, the ratio of the intensity after micellization to\nthat before micellization is a good estimate of the average aggregation number of the\nmicelles. Explain this observation.\nShow that the form factors for the Gaussian coil and the hard sphere do indeed reduce to the\nexpected 1\n\u2014 q\ufb01/3 at low q, using the appropriate series expansions.\nImagine that you perform light scattering measurements on a diblock copolymer, and you\ngenerate a Zimm plot in the standard way. However, one of the blocks is isorefractive with\nthe solvent (an/ac 0), so it does not contribute directly to the scattering signal. Comment\non how the apparent values of MW, Rg, and B (i.e., those obtained by assuming you are\n"]], ["block_12", ["25.8\n\u2014\n1.48\n36.9\n1.84\n1.50\n53.0\n1.93\n1.58\n66.4\n1.98\n1.62\n90.0\n2.10\n1.74\n113.6\n2.23\n1.87\n143.1\n2.34\n1.98\n"]], ["block_13", ["0 (\u00b0)\n1.9x 10'3\n3.8x 10'4\n"]], ["block_14", ["c (g/mL)\n"]], ["block_15", ["72\nan/ac\nn\nan/ac\n"]], ["block_16", ["T = 20\u00b0C\nT 67\u00b0C\n"]]], "page_337": [["block_0", [{"image_0": "337_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["324\n"]], ["block_2", ["15.\n"]], ["block_3", ["16.\n"]], ["block_4", ["17.\n"]], ["block_5", ["18.\n"]], ["block_6", ["19.\n"]], ["block_7", ["looking at a homopolymer) might relate to the values associated with the copolymer as a\nwhole, or those of the two blocks.\nImagine dissolving dilute quantities of two different polymers, A and B, in a solvent and\nmaking light scattering measurements. What is the appropriate equation to describe the\nresults, analogous to the Zimm equation? You may assume that the chains are sufficiently\nsmall that P(q) 1 for both A and B. You will need to consider two concentrations, CA and CB,\ntwo molecular weights MA and MB, two values of one important optical parameter, and more\nthan one second virial coef\ufb01cient. You do not need to do a complicated derivation to get the\nanswer, but it is necessary to consider how the light scattering signal responds to concentra-\ntion \ufb02uctuations, and how many different \u201ckinds\u201d of concentration \ufb02uctuations there are in a\nthree-component mixture.\nSuppose you had a new polymer, in which you knew the monomer structure but not the shape\nof the polymer in solution (e.g., rod, coil, spherical globule, etc.). A good light scattering\nmeasurement would give you MW, Rg, and B. However, from this information alone, you\ncould also infer the shape; explain how, perhaps with numerical estimates of the relevant\nparameters. Note that fitting to the form factor is not the answer; all form factors are the same\nin the small n limit.\nImagine making light scattering measurements on a statistical copolymer of styrene and\nmethyl methacrylate, in a very dilute solution, and at angles such that n << 1. The sample\nhas a mean styrene composition of f (0 c,\u2014, (an/80,,\n(where the subscript i allows for polydispersity in M), and the usual optical constants. In\nreality, the measured intensity may well be larger than this value; the reason lies in the fact\nthat each chain will have a composition that may differ from f. Making reasonable\nassumptions about the M andf dependences of (an/86),, the refractive index increment of\nchain i, develop an expression for the observed intensity, in terms of averages involving 8f,-,\nwhere 5f, f, f, the difference in composition between chain\n1' and the sample average.\nYour result should show the interesting (and correct) prediction that there can be excess\nscattered intensity from a statistical copolymer solution, even when a solvent is chosen\nsuch that (an/8c) 0.\nAn analytical expression for the form factor of a Kratky-Porod worm\u2014like chain (WLC,\nChapter 6.4.2) is not available (although numerical approximations valid in various regimes\nhave been developed). Nevertheless, with a little thought you should be able to make a\nreliable sketch of what P(q) must look like. Take a WLC with L 100 nm and 6p 10 nm.\nUsing a reasonable plotting program, generate a plot of q2P(q) versus q (a so\u2014called \u201cKratky\u201d\nplot) for a Gaussian coil with the same Rg as this WLC. Run the q axis from 0 out to 0.5\naI. Do the same thing for a rigid rid with L 100 nm, and plot it on the same axes. Finally,\nby considering the low and high q asymptotic behavior of the various structures, draw by\nhand a smooth curve representing the WLC.\nThere are often differences in practice between various analysis schemes that are otherwise\nequivalent in principle. For example, use a computer to generate P(q) \u201cdata\u201d for a high\nmolecular weight Gaussian coil from n 0 out to n 2. Add \u201cnoise\u201d to the data with a\nreasonable amplitude using a random number generator. Then, fit the data to a straight line\naccording to the Zimm approach (1/P(q) versus (n)2) and to the Guinier approach (1n\nP(q) versus (n)2. Vary the n range of the data included in the fit at the high n end\n(why is this appropriate?) Which fitting approach is more robust, in terms of returning the\ncorrect Rg value?\n"]], ["block_8", ["Light Scattering by Polymer Solutions\n"]]], "page_338": [["block_0", [{"image_0": "338_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Brown, W. (Ed.), Light Scattering: Principles and Development, Clarendon Press, Oxford, 1996.\nHuglin, M.B. (Ed.), Light Scattering from Polymer Solutions, Academic Press, New York, 1972.\nJohnson, C.S. Jr. and Gabriel, D.A., Laser Light Scattering, Dover Publications, New York, 1994.\nTanford, C., Physical Chemistry of Macromolecules, Wiley, New York, 1961.\nYamakawa, H., Modern Theory of Polymer Solutions, Harper & Row, New York, 1971.\nWyatt, P.J., Analytica Chimica Acta, 272, 1, 1993.\n"]], ["block_2", ["Brandrup, J. and Immergut, E.H. (Eds), Polymer Handbook, 3rd ed., Wiley, New York, 1989.\nStrutt, J.W. (Lord Rayleigh), Phil. Mag. 41, 177, 447 (1871).\nMiller, T.H., CRC Handbook ofChemistry and Physics, 8 1 st ed., pp. 10\u2014160, CRC Press, Cleveland, 2001.\nZimm, B.H., J. Chem. Phys. 16, 1099 (1948).\nGuinier, A., Ann. Phys. 12, 161 (1939).\nDebye, P., J. Appl. Phys. 15, 338 (1944).\nNeugebauer, T., Ann. Physik 42, 509 (1943).\nLord Rayleigh, Proc. Roy. Soc. A84, 25 (1910).\nPOTJQ\u2018P\u2018PP\u2019PT\u2018\n"]], ["block_3", ["Further Readings\n325\n"]], ["block_4", ["References\n"]], ["block_5", ["Further Readings\n"]]], "page_339": [["block_0", [{"image_0": "339_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["This chapter contains one of the more diverse assortments of topics of any chapter in this volume.\nIn this chapter we discuss the viscosity of polymer solutions, the diffusion of polymer molecules,\nthe technique of dynamic light scattering, the phenomenon of hydrodynamic interaction, and the\nseparation and analysis of polymers by size exclusion chromatography (SEC). At first glance these\nseem to be rather unrelated topics, but all are important to molecular weight determination in\nsolution. Furthermore, all share a crucial dependence on the spatial extent of the molecules. In\nChapter 8 we considered in detail how light scattering can provide a direct measurement of the\nradius of gyration. In this chapter the measure of size turns out to be roughly proportional to, but\nnot numerically equal to, the radius of gyration. As the chapter heading suggests, we now consider\nfor the first time in the book the time-dependent properties of polymers and particularly the rate at\nwhich polymer molecules move through a solvent. By emphasizing dilute solutions, the properties\nof individual polymer molecules are highlighted; in Chapter 11 we will consider the dynamic\nproperties of more concentrated solutions and melts.\nA parameter that plays an important role in unifying the concepts of viscosity and diffusion is the\nfriction factor. We will initially define the moiecular friction factor, f, by a thought experiment.\nImagine a polymer molecule dissolved in a solvent. Further imagine that we have a way of pulling\nthis molecule gently, but persistently, in one direction. After an initial start-up period, or transient\nresponse, we would find that if we apply a constant force I? the molecule would move with a constant\nvelocity \u20187 in the direction of the force. The proportionality factor between the applied force and the\nresulting velocity isf = |13|/|v\u2019| The units offare therefore (g cm/s2)/(cm/s) = g/s in the cgs system\n(the SI unit is kg/s), and a typical value for a polymer dissolved in water might fall between 10\u20187 and\n10\u201c5 g/s. The next few sections are concerned with two general questions: how can we gain\nexperimental access to f, and what canf tell us about the polymers themselves? Our starting point\nwill be to consider a single hard sphere. Although this idealization might appear at first to resemble\nthe ideal gas or the ideal solution in thermodynamics\u2014a simple model to use in learning the r0pes,\nbut of limited practical relevancewit is not so. For a rather profound reason, which we shall explore\nin Section 9.6, even \ufb02oppy random coils have friction factors very similar to those of hard spheres of\ncomparable size. However, we will get to that in due course; let us start at the beginning. Before\ndiscussing hard spheres, we need to introduce the viscosity of a \ufb02uid in more concrete terms.\nAs a place to begin, we visualize the \ufb02uid as a set of infinitesimally thin layers moving parallel\nto each other, each with a characteristic velocity. In addition, we stipulate that those \ufb02uid\nlayers that are adjacent to non\ufb02owing surfaces have the same velocity as the rigid surface. This\nis another way of stating that there is no slip at the interface between the stationary and \ufb02owing\nphases, which is a good approximation for the slow \ufb02ows of immediate interest. Now suppose we\nconsider a sample of \ufb02uid that is maintained at constant temperature, sandwiched between two\nrigid parallel plates of area A as shown in Figure 9.1. If a force F in the x\u2014direction is applied to the\ntop plate that plate and the layer of \ufb02uid adjacent to it will accelerate until a steady x\u2014velocity is\nreached (although force and velocity are vectors, we will drop the arrows for the remainder of\nthis chapter). As long as the deforming force continues to be applied, this velocity is unchanged.\n"]], ["block_2", ["9.1\nIntroduction: Friction and Viscosity\n"]], ["block_3", ["Dynamics of Dilute Polymer Solutions\n"]], ["block_4", ["9\n"]], ["block_5", ["327\n"]]], "page_340": [["block_0", [{"image_0": "340_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "340_1.png", "coords": [21, 542, 149, 578], "fig_type": "molecule"}]], ["block_2", [{"image_2": "340_2.png", "coords": [34, 48, 236, 155], "fig_type": "figure"}]], ["block_3", ["This time\u2014independent behavior is called steady \ufb02ow and will be our primary concern. During the\nacceleration that precedes the stationary state, the velocity is a function of time. For our purposes,\nwe shall simply wait until the stationary state is reached and not even question how long it will\ntake. Force per unit area is called stress, and in shear \ufb02ow is given the symbol 0'.\nIn the experiment described in Figure 9.1, the bottom plate remains in place and the nonslip\ncondition stipulated above requires that the layer of \ufb02uid adjacent to the bottom plate also has zero\nvelocity. This situation clearly requires that the velocity of the \ufb02uid varies from layer to layer\nacross the gap between the two rigid plates. To formulate this mathematically, we write that the top\nand bottom of these imaginary \ufb02uid layers are separated by a distance Ay and differ from each\nother in velocity by Av. The ratio Av/Ay has units of reciprocal time and is called either the\nvelocity gradient or the rate of shear. The former name is self-explanatory and the latter may be\nunderstood by considering the actual deformation the sample undergoes under the shearing force.\nDuring a short time interval At, the top layer moves a distance Ax relative to the bottom layer.\nAccordingly, Av may be written as Av :Ax/At and the velocity gradient may be expressed as\nAv/Ay (Ax/At)/Ay (Ax/Ay)/At. The shear displacement Ax divided by the distance over which\nit vanishes to zero, in this case Ay, is called the shear strain, which we represent by 7/. These\nrelationships show that the ratio Av/Ay also describes the rate at which the shear strain develops,\nor, more simply, the rate of shear A'y/At, or 7'1. In summary,\n"]], ["block_4", ["where 7] is called the coef\ufb01cient of viscosity of the \ufb02uid or, more simply, the viscosity. Equation\n9.1.2 implies that the velocity gradient is exactly the same throughout the liquid. As this may not\nbe the case over macroscopic distances, our best assurance of generality is to consider the limiting\ncase, in which Ay and therefore Av approach zero. In the limit of these infinitesimal increments Av\nand Ay become dv and dy, respectively, so Equation 9.1.2 becomes\n"]], ["block_5", ["Figure 9.1\nThe relationship between the applied force F per unit area A and the velocity v used in the\nde\ufb01nition of viscosity.\n"]], ["block_6", ["Now let us invoke our experience with liquids of different viscosities, say, water and molasses, and\nimagine the magnitude of the shearing force that would be required to induce the same velocity\ngradient in separate experiments involving these two liquids. Our experience suggests that more\nforce is required for the more viscous \ufb02uid. Since the area of the solid plates in this liquid sandwich\nis also involved, we can summarize this argument by writing a proportionality relation between the\nshear force per unit area, a, and the velocity gradient\n"]], ["block_7", ["328\nDynamics of Dilute Polymer Solutions\n"]], ["block_8", [{"image_3": "340_3.png", "coords": [38, 428, 124, 472], "fig_type": "molecule"}]], ["block_9", [{"image_4": "340_4.png", "coords": [44, 634, 115, 677], "fig_type": "molecule"}]], ["block_10", [{"image_5": "340_5.png", "coords": [45, 547, 130, 573], "fig_type": "molecule"}]], ["block_11", ["\u00e9i_.__Ax/Ay\nAy_yu\nAt\n(9.1.1)\n"]], ["block_12", ["02nd\u2014yzn'jr\n(9.1.3)\n"]], ["block_13", ["y\nA\n"]], ["block_14", ["F\nAv\n,\n"]], ["block_15", ["dv\n"]], ["block_16", ["l\n\u20191\u2019\nI\n"]], ["block_17", ["I\nA\nxH\u2014+ \n"]], ["block_18", ["*\u2018W\u2019\nX\n"]], ["block_19", ["1\u2014\u00bb!\n"]], ["block_20", ["\u2014v__-}J\nI;\n"]]], "page_341": [["block_0", [{"image_0": "341_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "341_1.png", "coords": [33, 342, 161, 384], "fig_type": "molecule"}]], ["block_2", ["To appreciate this result, we remember that Av as introduced in Figure 9.1 is actually Ax/Ar. Thus\nthe product F(Av/Ay) can be written as F(Ax/At)/Ay, and F(Av/Ay)/A becomes FAx/AAyAt. The\nproduct of a force and the distance through which it Operates equals an energy AE and the product\nof A and Ay equals the volume element of AV, upon which the shearing force described in\n"]], ["block_3", ["Equation 9.1.3 is called Newton\u2019s law of viscosity, and those \ufb02uids that follow it (with\n7?\nindependent of the magnitude of the shear rate) are said to be Newtonian.\nEquation 9.1.3 describes a straight line with zero intercept if 0\u2018 is plotted versus the velocity\ngradient; such a plot is shown in Figure 9.2. Since the coefficient of viscosity is the slope of this\nline, this quantity has a single value for Newtonian liquids. Liquids of low molecular weight\ncompounds solutions are generally Newtonian, but quite a few different variations from\nthis behavior are also observed. We shall not attempt to catalog all of these variations, but shall\nonly mention the other pattern of behavior shown in Figure 9.2. This example of non\u2014Newtonian\nbehavior is described as shear thinning, and is often observed when the material under study is a\npolymer solution or melt. Since Equation 9.1.3 defines the coefficient of viscosity as the slope of a\nplot of 0\u2018 versus velocity gradient, it is clear from Figure 9.2 that shear\u2014thinning substances are not\ncharacterized by a single viscosity. The viscosity at a particular velocity gradient is given by the\nratio a/(dv/dy); inspection of Figure 9.2 reveals that shear-thinning materials appear less viscous at\nhigh rates of shear than at low rates. For the purposes of this chapter and the next, we are only\nconcerned with Newtonian response. For shear-thinning \ufb02uids, Newtonian response can be\nachieved by reducing the shear rate sufficiently to access the linear portion near the origin in\nFigure 9.2. This may be formalized by defining the zero shear viscosity, 7103\n"]], ["block_4", ["In most viscometers it is possible to vary '5! in such a way as to ascertain whether 17 is constant. For\nthe remainder of this book, the symbol 7? refers to the zero shear viscosity, unless explicitly stated\notherwise.\nTo see another interpretation of viscosity, we multiply both sides of Equation 9.1.2 by Av/Ay:\n"]], ["block_5", ["Introduction: and Viscosity\n329\n"]], ["block_6", ["Figure 9.2\nComparison of shear stress versus shear rate for Newtonian and shear\u2014thinning behavior.\n"]], ["block_7", ["PM or\n/\n"]], ["block_8", [{"image_2": "341_2.png", "coords": [40, 442, 278, 647], "fig_type": "figure"}]], ["block_9", ["Av\nFAv\ns\n_=__:\n_\u2014\n-1-\n\u201cA, My\n\"(A33\n(9\n5)\n"]], ["block_10", ["770\nEll/1:161)\n(9.1.4)\n"]], ["block_11", ["Newtonian (high 17)\n"]], ["block_12", [",\nShear thinning\n"]], ["block_13", ["Zero shear limit\n"]], ["block_14", ["dv/dy 7\"\n"]], ["block_15", ["Newtonian (low 17)\n"]]], "page_342": [["block_0", [{"image_0": "342_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["does not tell the whole story, however. Other forces are also operative: external forces of, say,\ngravitational or mechanical origin are responsible for the motion in the \ufb01rst place, and pressure forces\nare associated with the velocity gradient. In general, things are not limited to one direction in space,\nbut the forces, gradients, and velocities have x, y, and 2 components. It is possible to bring these\nconsiderations together in a very general form by adding together all of the forces acting on a volume\nelement of liquid, including viscous forces, and using the net force and the mass of the\nvolume element in Equation 9.2.2. The resulting expression is called the equation ofmotion and is\n"]], ["block_2", ["The deforming forces that induce \ufb02ow in \ufb02uids are not recovered when these forces are removed.\nThese forces impart kinetic energy to the molecules, which is dissipated within the \ufb02uid as heat.\nEquation 9.1.7 implies that at sufficiently high shear rates the amount of heat generated could lead\nto measurable temperature increases; this phenomenon of viscous heating is indeed a concern in\nhigh shear-rate measurements and applications.\nWe conclude this section with a consideration of the units required for r] by Equation 9.1.3. To\ndo this, we rewrite these equations in terms of the units of all quantities except 1). The units of 7\u20191\nmust make the expressions dimensionally correct. Force has units of mass times acceleration, or\ng cm/s2 in cgs units. Since the viscosity gradient has the units 3\u20181 and area is cmz, the dimensional\nstatement of Equation 9.1.3 is\n"]], ["block_3", ["The shearing force F that is part of the definition of viscosity can also be analyzed in terms of\nNewton\u2019s second law and written as\n"]], ["block_4", ["In order to satisfy this equation, 11 must have units g/cm/s, which is defined to be 1 poise (1 P). In\nthe SI system the units are kg/m/s, or pascal seconds (Pa 3). At room temperature, water has a\nviscosity of about 0.01 P or 0.001 Pa 5 and other low molecular weight liquids have comparable\nviscosities. The viscosity of a polymer liquid depends very much on the concentration and\nmolecular weight of the polymer, as we shall see in Section 9.3 and in Chapter 11, and it can be\nmany orders of magnitude larger than 0.01 P.\n"]], ["block_5", ["When the force is divided by the area of a shearing plane, A, to obtain a stress, we would also write\n"]], ["block_6", ["Figure 9.1 operates. Therefore FAx/AAyAt is the same as AE/AVAI. Defining the increment in\nshear energy dissipated per unit volume by the symbol AW, we obtain\n"]], ["block_7", ["If it were complete, Equation 9.2.2 would be a differential equation whose solution would give v, the\nvelocity of the \ufb02owing liquid, as a function of time and position within the sample. Equation 9.2.2\n"]], ["block_8", ["9.2\nStokes\u2019 Law and Einstein's Law\n"]], ["block_9", ["330\nDynamics of Dilute Polymer Solutions\n"]], ["block_10", ["As in the parallel case of going from Equation 9.1.2 to Equation 9.1.3, we take the limit of\ninfinitesimal increments and write\n"]], ["block_11", [{"image_1": "342_1.png", "coords": [39, 150, 117, 189], "fig_type": "molecule"}]], ["block_12", [{"image_2": "342_2.png", "coords": [40, 83, 120, 123], "fig_type": "molecule"}]], ["block_13", [{"image_3": "342_3.png", "coords": [40, 520, 129, 558], "fig_type": "molecule"}]], ["block_14", ["F\nmdv\ndv\n_=__:\n._\n.22\nA\nA dt\nndy\n(9\n)\n"]], ["block_15", ["dv\nF:\n= _\n9.2.1\nma\nmdr\n(\n)\n"]], ["block_16", ["AW\nAv\n2\n37: \n(9.1.6)\n"]], ["block_17", ["dW\ndv\n2\n"]], ["block_18", ["g cm s\u2018z/cm2 (n) 5*1\n"]], ["block_19", ["dt \n(9.1.7)\n"]]], "page_343": [["block_0", [{"image_0": "343_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "343_1.png", "coords": [20, 437, 264, 652], "fig_type": "figure"}]], ["block_2", ["The first of these problems involves relative motion between a rigid sphere and a liquid, as\nanalyzed by Stokes around 1850 [1]. The results apply equally to liquid \ufb02owing past a stationary\nsphere with a steady-state (subscript \u201c0\u201d) velocity v0, or to a sphere moving through\na\nstationary liquid with a velocity \u2014v0; the relative motion is the same in both cases. If the relative\nmotion is in the vertical direction, we may visualize the slices of liquid described above as\nconsisting of a bundle of layers, some of which are shown schematically in Figure 9.3.\nIn the horizontal plane containing the center of the sphere a limiting velocity v0 is reached as the\ndistance r from the center of the sphere becomes large. This would be the observable settling\nvelocity of such a spherical particle, for example. In an infinitesimally thin layer adjacent to the\nsurface, the tangential component of velocity would be that of the solid sphere. This is the no slip\ncondition as it applies to this problem. This means that a velocity gradient exists that is described in\nterms of the distance from the center of the sphere and the component of velocity perpendicular to\nr. The velocity gradient dv/dr should certainly be proportional to v0 and then we need a quantity\nwith units of length to be dimensionally correct. The only relevant length we have is R, the radius\n"]], ["block_3", ["Figure 9.3\nDistortion of flow streamlines around a Spherical particle of radius R. The relative velocity in\nthe plane containing the center of the sphere equals v0 as r \u2014> oo.\n"]], ["block_4", ["the cornerstone of \ufb02uid mechanics. The equation of motion is generally written in terms of vector\noperators and takes on a variety of forms, depending on the system of coordinates and the vector\nidentities that have been employed. If the equation of motion is complicated even in writing, things\nare even worse in the solving. Accordingly, we will not reproduce a full treatment here, but rather\noutline the essential points.\nAs with any differential equation, an important part of solving the equation of motion is\ndefining the boundary conditions. An important boundary condition was introduced above, namely\nthat no slip occurs at the boundary between a moving \ufb02uid and a rigid wall. The \u201cno slip\u201d or\n\u201cstick\u201d condition means that the \ufb02uid layer immediately adjacent to a stationary wall has a\nvelocity of zero, with successive layers away from the wall possessing larger increments of\nvelocity. At a sufficiently large distance from the wall, a net velocity is attained, which is\nunperturbed by the presence of the surface. In Section 9.4 we shall apply this idea to the \ufb02ow of\na liquid through a capillary tube. For now we consider two classic problems involving the effect of\nrigid spheres on the \ufb02ow behavior of a liquid.\n"]], ["block_5", ["9.2.1\nViscous Forces on Rigid Spheres\n"]], ["block_6", ["Stokes\u2019 Law and Einstein's Law\n331\n"]], ["block_7", ["..\nIv.\n"]]], "page_344": [["block_0", [{"image_0": "344_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["The second classic problem arises in describing the viscosity of a suspension of spherical particles.\nThis problem was analyzed by Einstein in 1906, with some corrections appearing in 1911 [2]. As\nwith Stokes\u2019 law, we shall only present qualitative arguments to give plausibility to the final form.\nThe fact that it took Einstein five years to work out the \u201cbugs\u201d in this theory is an indication of the\ncomplexity of the formal analysis. Derivations of both the Stokes and Einstein equations that do\nnot require vector calculus have been presented by Lauffer [3]; the latter derivations are at about\nthe same level of difficulty as most of the mathematics in this book, but are lengthy.\nWe return to Figure 9.1 with the stipulation that the volume of \ufb02uid sandwiched between the\ntwo plates is a unit of volume. This unit is defined by a unit of contact area with the walls and a unit\nof separation between the two walls. Next we consider a shearing force acting on this cube of \ufb02uid\nto induce a unit velocity gradient. According to Equation 9.1.7, the rate of energy dissipation per\nunit volume from viscous forces dW/dt is proportional to the square of the velocity gradient, with\nn, the factor of proportionality. Thus, to maintain a unit gradient, a volume rate of energy\ndissipation equal to n, is required.\nNext we consider replacing the sandwiched fluid with the same liquid, but in which solid\nspheres are suspended at a volume fraction qb. Since we are examining a unit volume of liquid\u2014-\u2014a\nsuspension of spheres in this case\u2014the total volume of the spheres is also qb. We begin by\nconsidering the velocity gradient if the velocity of the t0p surface is to have the same value as\nin the case of the pure liquid. Being rigid objects, the suspended spheres contribute nothing to the\nvelocity gradient. As far as the gradient is concerned, the Spheres might as well be allowed to settle\nto the bottom and then be fused to the lower, stationary wall. The equivalency of the suspended\nspheres and a uniform layer of the same volume are illustrated schematically in Figure 9.4a and\nFigure 9.4b, respectively. Since the unit volume has a unit cross-sectional area, a volume qb fused\nto the base will raise the stationary surface by a distance qS and leave a liquid of thickness 1 d) to\ndevelop the gradient. These dimensions are also shown in Figure 9.4b. If the velocity of the top\nlayer is required to be the same in this case as for the pure solvent, then the gradient in the liquid\nneed only be the fraction 1/(1 (b) of that for the pure liquid. Of course, since (p is less than unity,\nthis \u201cfraction\u201d is greater than unity.\nNow we return to consider the energy that must be dissipated in a unit volume of suspension to\nproduce a unit gradient, as we did above with the pure solvent. The same fraction applied to the\nshearing force will produce the unit gradient and the same fraction also describes the volume rate\n"]], ["block_2", ["This famous equation is Stokes\u2019 law for rigid spheres. We emphasize that the viscosity in Equation\n9.2.3 is that of the medium surrounding the sphere by labeling it with the subscript \u201cs\u201d for solvent,\nand thus\nf :67mg}?\n(9.2.4)\n"]], ["block_3", ["of the sphere, and thus we anticipate that dv/dr is proportional to vo/R. Formal analysis of the\nproblem via the equation of motion verifies this argument from dimensional analysis and provides\nthe necessary proportionality factors as well.\nFrom Equation 9.2.2 we see that the total viscous force associated with this motion equals\n"]], ["block_4", ["for Spherical particles of radius R.\n"]], ["block_5", ["7) x (dv/dr) >< (area), where the pertinent area is proportional to the surface of the sphere and\ntherefore varies as R2. This qualitative argument suggests that the viscous force opposing the\nrelative motion of the liquid and the sphere is proportional to [n(vo/R)] (R2). The complete solution\nto this problem reveals that this is correct; both pressure and shear forces arising from the motion\nare proportional to nRvO, and the total force of viscous resistance is given by\n"]], ["block_6", ["9.2.2\nSuspension of Spheres\n"]], ["block_7", ["332\nDynamics of Dilute Polymer Solutions\n"]], ["block_8", ["Fvis 67rnRv0 =fv0\n(9.2.3)\n"]]], "page_345": [["block_0", [{"image_0": "345_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "345_1.png", "coords": [17, 186, 242, 288], "fig_type": "figure"}]], ["block_2", [{"image_2": "345_2.png", "coords": [26, 54, 220, 305], "fig_type": "figure"}]], ["block_3", [{"image_3": "345_3.png", "coords": [32, 49, 263, 166], "fig_type": "figure"}]], ["block_4", ["d) is small. This being the case, 1/(1\u20149\u20185)2 can be replaced by the leading terms of the series\nexpansion (1+ (,5 + (p2 + -)2 to give\n7,: n,(1+%\u00a2)(1+\u00a2+~)2: n.(1+2.5\u00a2+4\u00a22+~)\n(9-2-7)\n"]], ["block_5", ["This is Einstein\u2019s famous viscosity equation; the following observations are pertinent:\n"]], ["block_6", ["Again, since qS < 1, 17 > 773.\nThis is only one of the contributions to the total volume rate of energy dissipation; a second\nterm that arises from explicit consideration of the individual spheres must also be included. This\nsecond effect can be shown to equal 1.5 (\ufb01ns/(1 q5)2; therefore the full theory gives a value for 7),\nthe viscosity of the suspension:\n"]], ["block_7", ["1.\n1) is the viscosity of the suspension as a whole; 17, is the viscosity of the solvent; and (pi) is the\nvolume fraction occupied by the spheres.\n2.\nThe validity of the derivation is limited to small values of qt\u00bb, 30 Equation 9.2.7 is generally\ntruncated after the first two terms on the right\u2014hand side.\n3.\nThe viscosity does not depend on the radius of the spheres, only on their total volume fraction.\n"]], ["block_8", ["One additional assumption that underlies the derivation of the second term in Equation 9.2.6 is that\n"]], ["block_9", ["Figure 9.4\n(a) Schematic representation of a unit cube containing a suspension of spherical particles at\nvolume fraction d). (b) The volume equivalent to the spheres in (a) is fused to the base, leaving 1 d) as the\nthickness of the liquid layer.\n"]], ["block_10", ["of energy dissipation compared to the situation described above for pure solvent. Since the latter\nwas 77, we write for the suspension, in the case of dv/dy 1,\n"]], ["block_11", ["(a)\n43), Volume occupied by spheres\n"]], ["block_12", ["Stokes' Law and Einstein's Law\n333\n"]], ["block_13", [{"image_4": "345_4.png", "coords": [40, 474, 208, 511], "fig_type": "molecule"}]], ["block_14", ["E;5523E5335g?5533555335553E5255555E;Egigi55555555;E553555;225555355325552355335355-135555 :=-'\nequivalent\nto spheres = q)\n"]], ["block_15", [",..A...............-----..1...;_|_n_n_.........\n"]], ["block_16", ["dW\n1\n\u2014 Z\nZ \n9.2.5\n"]], ["block_17", ["1.5\n1\n.5\nits\naims _\n+0\n(i5\n(92.6)\nn=1_\u00a2+(1_\u00a2)z\u2014m(l_\u00a2)2\n"]], ["block_18", ["<33\n"]], ["block_19", ["Q\nUnit area\n"]], ["block_20", ["I\nl\n"]], ["block_21", ["..\ni Thickness of\n\u2018\n\u2014i \n"]], ["block_22", ["0\nthickness\n"]], ["block_23", ["Thickness of\nImmdz1\u2014a\n"]], ["block_24", [".\n__\n"]], ["block_25", ["Unit\n"]]], "page_346": [["block_0", [{"image_0": "346_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["where we have introduced the intrinsic viscosity, [1)], in place of a\u2019. Note that by dimensional\nanalysis we can see that [17] must have units of inverse concentration, typically mL/g. Conse-\nquently it is not actually a viscosity; rather, it is a coefficient that quantifies the rate at which the\nsolution viscosity increases per g/mL of added solute, when 6 is small. As 6 increases further, the\n"]], ["block_2", ["The intrinsic viscosity [1)], a quantity that will be defined formally below, is a measure of the\nability of added polymer to increase the viscosity of the solution over that of the solvent. It turns\nout that [n] is directly related to the size of the polymer in solution, and before SEC (to be\ndiscussed in Section 9.8) it was the most commonly employed method for determining molecular\nweight. It is still useful in this regard, but it will also provide us with a basis for understanding\nmuch about the behavior of polymers in solution. We begin by proposing that the dilute solution\nviscosity can be written as a power series in concentration, by analogy with the osmotic pressure\n(Equation 7.4.7) and the Einstein relation (Equation 9.2.7):\n"]], ["block_3", ["where a and b are unspeci\ufb01ed coefficients and 773 is the solvent viscosity. We can rearrange\nEquation 9.3.1 by factoring out 17,:\n"]], ["block_4", ["9.3.1\nGeneral Considerations\n"]], ["block_5", ["1.\nThe liquid medium is assumed to be continuous. This makes the results suspect when applied\nto spheres, which are so small that the molecular nature of the solvent cannot be ignored.\n2.\nBoth relationships have been repeatedly verified for a variety of systems and for spheres with a\nwide range of diameters. Despite item (1), both Equation 9.2.4 and Equation 9.2.7 have often\nbeen applied to individual molecules, for which they often work surprisingly well.\n3.\nThe spherical geometry assumed in the Stokes and Einstein derivations gives the highly\nsymmetrical boundary conditions favored by theoreticians. For ellipsoids of revolution having\naxial ratio a/b, friction factors have been derived by Perrin [4] and the coefficient of the first-\norder term in Equation 9.2.7 has been derived by Simha [5]. In both cases, the calculated\nquantities increase as the axial ratio increases above unity. For spheres, of course, a/b :1.\n4.\nIn the derivation of both Equation 9.2.4 and Equation 9.2.7, the disturbance of the \ufb02ow\nstreamlines is assumed to be produced by a single particle. This is the origin of the limitation\nto dilute solutions in the Einstein theory, where the net effect of an array of spheres is treated\nas the sum of the individual nonoverlapping disturbances. When more than one sphere is\ninvolved, the same limitation also applies to Stokes\u2019 law. In both cases, contributions from the\nwalls of the container are also assumed to be absent.\n"]], ["block_6", ["4.\nBy describing the concentration dependence of an observable pr0perty as a power series,\nEquation 9.2.7 plays a comparable role for viscosity as Equation 7.4.7 does for osmotic pressure.\n5.\nThe volume fraction emerges from the Einstein derivation as the natural concentration unit to\ndescribe viscosity. This parallels the way volume fraction arises as a natural thermodynamic\nconcentration unit in the Flory\u2014Huggins theory, as seen in Section 7.3.\n"]], ["block_7", ["We shall make further use of the Stokes equation later in this chapter; for the present, viscosity is\nour primary concern, and the Einstein equation is our point of departure.\n"]], ["block_8", ["9.3\nIntrinsic Viscosity\n"]], ["block_9", ["334\nDynamics of Dilute Polymer Solutions\n"]], ["block_10", ["Both the Stokes and Einstein equations have certain features in common, which arise from the\nhydrodynamic origins they share:\n"]], ["block_11", ["n=ns+ac+b62m\n(9.3.1)\n"]], ["block_12", ["n 173(1 +a'c+b'62~-)\n= 773(1 + Cln] + kh \u20ac2[\ufb0212\u00b0\u00b0)\n(9,3,2)\n"]]], "page_347": [["block_0", [{"image_0": "347_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "347_1.png", "coords": [21, 528, 396, 662], "fig_type": "figure"}]], ["block_2", [{"image_2": "347_2.png", "coords": [32, 359, 180, 390], "fig_type": "molecule"}]], ["block_3", ["Table 9.1\nSummary of Names and De\ufb01nitions of the Various Functions of 17, ns, and c\nin Which Solution Viscosities Are Frequently Discussed\n"]], ["block_4", ["next term, proportional begins contribute. Justthe osmotic \n"]], ["block_5", ["In the literature one encounters a variety of different ways of presenting the solution and\nthese are collected Table 9.1 for reference purposes.\nHaving defined [7}], now we proceed to see what it can tell us about the polymer solute. Equation\n9.3.2, which was just pr0posed on general principles, is analogous to Einstein\u2019s equation (Equation\n9.2.7), which using hydrodynamics. To compare these equations directly, \nconvert the concentration c into a volume fraction, qb. This can be done as follows:\n"]], ["block_6", ["where we have left the solute volume rather vaguely defined. For a rigid sphere, as envisioned by\nEinstein, this volume is simply Avogadro\u2019s number x (417/3)R3. For other shapes, however, or for\n\ufb02exible coils, it is not so clear what volume is appropriate. The molar volume is a thermodynamic\nquantity, but the viscosity measurement is hydrodynamic, not thermodynamic. Accordingly, we\nfinesse this issue by defining a hydrodynamic volume for the molecule, Vh:\n"]], ["block_7", ["which, by comparison with Equation 9.3.2 relates the intrinsic viscosity to the hydrodynamic volume:\n"]], ["block_8", ["Now we make a bold step. We propose that the hydrodynamic volume is proportional to the radius\nof gyration cubed:\n"]], ["block_9", ["the term in re\ufb02ectsinteractions molecules; kind of\nvirial coefficient for viscosity and is known as the Huggins coefficient. We can rearrange Equation\n9.3.2 to obtain a formal definition of [77]:\n"]], ["block_10", ["and we insert Equation 9.3.5 into Equation 9.2.7:\n"]], ["block_11", ["Symbol\nDe\ufb01nition\nCommon name\nIUPAC name\n"]], ["block_12", ["\u201dn1.\n1\nRelative viscosity\nViscosity ratio\n7\u20193\n"]], ["block_13", ["Intrinsic \n335\n"]], ["block_14", ["mp\nI'\u2014 1\nSpecific viscosity\n\u2014\n"]], ["block_15", ["nred\n(1 (\u2014n\u2014 1)\nReduced viscosity\nViscosity number\n773\n"]], ["block_16", ["mm,\n% ln (1)\nInherent viscosity\nLogarithmic viscosity number\n"]], ["block_17", [".\nn\u2014m\n..\n.\n.\n...\n.\n.\n[7}]\n11m)\n\u2014En\u2014\nIntrmsrc v13cosrty\nLimiting Viscosrty number\nc\u2014r\n5\n"]], ["block_18", [{"image_3": "347_3.png", "coords": [46, 463, 96, 498], "fig_type": "molecule"}]], ["block_19", ["a :\ufb01Navl/h\n(9.3.5)\n"]], ["block_20", ["Solute volume\nc\n=\n= \u2014 V\n1\n. .4\n<1\"\nSolution volume\nM(\no ume of a mole of solute)\n(9 3\n)\n"]], ["block_21", ["V1, \u2014R3\n(9.3.8)\n"]], ["block_22", ["5 c\n77: 775(1+\u00a7\u2014M-Navvh+\u201d')\n(9.3.6)\n"]], ["block_23", ["[771211111 (77\u20141173)\n(9.3.3)\nCTO\nC773\n"]], ["block_24", ["[77] 51:3\" g\n(9.37)\n"]], ["block_25", [{"image_4": "347_4.png", "coords": [56, 528, 243, 668], "fig_type": "figure"}]], ["block_26", ["773\n"]]], "page_348": [["block_0", [{"image_0": "348_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "348_1.png", "coords": [24, 194, 160, 234], "fig_type": "molecule"}]], ["block_2", ["Poly(ethylene oxide)\nToluene\n35\n14.5\n0.70\nPoly(ethyleneterephthalate)\nm-Cresol\n25\n0.77\n0.95\n1,4-Polybutadiene\nCyclohexane\n20\n36\n0.70\n1,4\u2014Polyisoprene\nDioxane (0)\n34\n135\n0.50\nPoly(hexamethy1ene adipamide)\nm\u2014Cresol\n25\n240\n0.61\nPoly(dimethy1siloxane)\nToluene\n25\n21.5\n0.65\nPolyethylene\np\u2014Xylene\n75\n135\n0.63\nPolyprOpylene\nCyclohexane\n30\n20.9\n0.76\nPolyisobutylene\nBenzene\n40\n43\n0.60\nPoly(methyl methacrylate)\nChloroform\n25\n4.8\n0.80\nPoly(vinyl chloride)\nChlorobenzene\n30\n71.2\n0.59\nPoly(viny1 acetate)\nEthanol (0)\n56.9\n90\n0.50\nPoly(viny1 alcohol)\nWater\n25\n20\n0.76\nPolystyrene\nToluene\n25\n17\n0.69\nPolystyrene\nCyclohexane (0)\n35\n80\n0.50\nPolyacrylonitrile\ndimethyl formamide\n50\n30\n0.752\nPoly(8-caprolactam)\nm-Cresol\n25\n320\n0.62\n"]], ["block_3", ["Source: From Kurata, M. and Tsunashima, Y., in Polymer Handbook, 3rd ed., Brandrup, J. and Immergut, E.H. (Eds),\nWiley, New York, 1989.\n"]], ["block_4", ["This gives what we were looking for: a direct relation between the intrinsic viscosity and the\nmolecular weight. Using the values of V cited above, we can see that [77] should be independent of\nM for a rigid sphere, increase as Mor MA\"5 for a \ufb02exible chain in a theta solvent or good solvent,\nrespectively, and increase as M2 for a rod. These various possibilities are encompassed by the\ngeneral relation\n"]], ["block_5", ["Table 9.2\nValues for the Mark\u2014Houwink Parameters for a Selection of Polymer\u2014Solvent Systems\nat the Temperatures Noted\n"]], ["block_6", ["a The units for k itself depend on the value of a; the indicated value gives [1]] in mL/g.\n"]], ["block_7", ["We know from Chapter 6, and Equation 6.6.1 in particular, that RgrvM\u201d, where the exponent u\ntakes on various characteristic values (1/3 for a solid sphere, 1/2 for a \ufb02exible coil in a theta solvent\nor in the melt, 3/5 for a \ufb02exible chain in a good solvent, and\n1 for a rigid rod). If we insert this\nrelation into Equation 9.3.8, and the result into Equation 9.3.7, we have\n"]], ["block_8", ["This relationship with a 1 was first proposed by Staudinger [6], but in this more general form it is\nknown as the Mark\u2014Houwiuk equation [7]. The constants k and a are called the Mark\u2014Houwink\nparameters for a system. The numerical values of k and a depend on both the nature of the polymer\nand the nature of the solvent, as well as the temperature; extensive tabulations are available [8,9]\nand Table 9.2 gives a few examples. (Note, however, that the values can vary for a given system\namong different investigators, and that attention must be paid to details such as microstructure,\n"]], ["block_9", ["Polymer\nSolvent\nT (\u00b0C)\nks x 103 (mL/g)El\na\n"]], ["block_10", ["In other words, the volume that matters in the viscosity experiment is not the volume actually\noccupied by the polymer segments (which would be the degree of polymerization times the volume\nof the monomer), but the volume pervaded by the entire molecule. For a random coil this means\nthat we assume that the molecule behaves hydrodynamically like a rigid sphere of radius Rg. This\nmight seem rather far fetched at first glance, but in fact, it is basically correct.\n"]], ["block_11", ["336\nDynamics of Dilute Polymer Solutions\n"]], ["block_12", ["9.3.2\nMark-Houwink Equation\n"]], ["block_13", ["R3\nM31;\n[77] M3\"\u20141\n(939)\n"]], ["block_14", ["[77] kM\u201c\n(9.3.10)\n"]], ["block_15", [{"image_2": "348_2.png", "coords": [96, 419, 453, 638], "fig_type": "figure"}]]], "page_349": [["block_0", [{"image_0": "349_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["molecular weight range, and polydispersity of the samples.) Since viscometer drainage times\n(see Section 9.4) are typically on the order of a few hundred seconds, intrinsic viscosity experiments\nprovide a rapid method for evaluating the molecular weight of a polymer. A possible drawback to\nthe method is that the Mark\u2014Houwink coefficients must be established for the particular system\nunder consideration by calibration with samples of known molecular weight, but given the\nextensive tabulations this is often not a significant limitation.\nThe values of the exponent a in Table 9.2 range between 0.5 and about 1. Although in practice\na m 0.5 in all theta systems, in good solvents the values vary quite a bit. Part of this is a\nconsequence of the slow crossover to good solvent limiting behavior anticipated by the Flory\u2014\nKrigbaum theory (Equation 7.7.10). In other words, even though a solvent might be \u201cbetter-\nthan-theta\u201d for a given polymer, the experimental temperature might not be sufficiently far above\nT 0, or the molecular weight range might not extend to sufficiently high values, to obtain :2 3/5\nand thus a 0.8. In some cases a > 0.8, which can be attributed to a semi\ufb02exible structure (recall\nthat the simple argument above predicts a =2 for a rod). There is another, more complicated\nreason why the exponents can take on a range of values, the phenomenon of hydrodynamic\ninteractions, and this will be explored in Section 9.7.\nEven fractionated polymer samples are generally polydisperse, which means that the molecular\nweight determined from intrinsic viscosity experiments is an average value. The average obtained\nis called the viscosity average molecular weight, MV, which can be derived as follows:\n"]], ["block_2", ["Table 9.3 lists the intrinsic viscosity for a number of polystyrene samples of different molecular\nweights. The M values are weight averages based on light scattering experiments. The values of [n]\nwere measured in cyclohexane at the theta temperature of 35\u00b0C. In the following example we\nconsider the evaluation of Mark\u2014Houwink coefficients from these data.\n"]], ["block_3", ["where n,- is the number of molecules with molecular weight M,- and we assume all species\n1'\nhave the same k and a.\n4.\nCombining Equation 9.3.11 and Equation 9.3.13 we obtain\n"]], ["block_4", ["where the index 2' refers to different molecular weights.\n3.\nWe can now obtain [77] as\n_.\n77\u2014773\n__.\nZQI\u2019UL\u2018\n\u201c'1 \n3936(T)\n\u2014 is(Tl\n_ _ k \nH\nZniMi/V\n_\nEra-M,-\n"]], ["block_5", ["2.\nThe dilute solution viscosity for the polydisperse system can be expressed as\n"]], ["block_6", ["Intrinsic Viscosity\n337\n"]], ["block_7", ["1.\nThe experimental intrinsic viscosity is proportional to some average value, My, raised to the\npower a, according to Equation 9.3.10,\n"]], ["block_8", [{"image_1": "349_1.png", "coords": [38, 415, 218, 485], "fig_type": "molecule"}]], ["block_9", [{"image_2": "349_2.png", "coords": [50, 522, 145, 566], "fig_type": "molecule"}]], ["block_10", ["l/a\nEJliMI-Fa\nMVE\n\u2014\u2019-\u2014\u2018\u2014\n9.3.14\n( $5\"e\n(\n)\n"]], ["block_11", ["For flexible polymers in general, Mn < M,r (MW, and Mv 2MW if a 1. On the basis of this\nlast observation, it can be argued that the Mark\u2014Houwink coefficients should be evaluated\nusing weight average rather than number average molecular weights as calibration standards.\nWe saw in Chapter 8 how MW values can be obtained from light scattering experiments.\n"]], ["block_12", ["n n.(1+ Ze.[n1.\u2014+-~)\n(9.3.12)\n"]], ["block_13", ["[1;] kf\n(9.3.11)\n"]], ["block_14", ["(9.3.13)\n"]]], "page_350": [["block_0", [{"image_0": "350_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "350_1.png", "coords": [0, 59, 343, 321], "fig_type": "figure"}]], ["block_2", ["that corresponds to k=0.088 and (1 =0.497. The exponent agrees with expectation for a theta\nsolvent; the uncertainty in a is at least i0.003. Below M 10,000 the relationship between [17] and\nM becomes more complicated. A full explanation of this dependence is not yet available, but\namong the contributing factors are non-Gaussian conformations for short chains, chain stiffness,\nchain end effects, and modification of the solvent dynamics in the vicinity of the chain.\n"]], ["block_3", ["Table 9.3\nIntrinsic Viscosity as a Function of Molecular Weight for Samples\nof Polystyrene\n"]], ["block_4", ["266\n1.49\n19800\n11.9\n370\n1.92\n44000\n18.0\n474\n2.47\n50500\n19.2\n578\n2.74\n55000\n20.0\n680\n2.93\n76000\n24.5\n904\n3.26\n96200\n26.0\n1,480\n3.70\n1.25 x 105\n29.0\n1,780\n4.08\n1.60 x 105\n34.0\n2,270\n4.50\n1.80 x 105\n35.4\n3,480\n5.43\n2.47 x 105\n42.0\n5,380\n6.59\n3.94 x 105\n54.5\n10,100\n9.00\n4.06 x 105\n55.0\n20,500\n12.3\n5.07 x 105\n60.0\n40,000\n17.2\n6.22 x 105\n66.0\n97,300\n27.3\n8.62 x 105\n78.0\n1.91 x 105\n38.0\n1.05 x 10\u20185\n86.0\n3.59 x105\n51.2\n1.56 x106\n106\n7.32 x 105\n73.4\n1.32 X 106\n98.1\n"]], ["block_5", ["Evaluate the Mark\u2014Houwink coefficients for polystyrene in cyclohexane at 35\u00b0C from the data in\nTable 9.3. How well do the two data sets agree? What is the appropriate set of Mark\u2014Houwink\nparameters for high molecular weight PS in this solvent? Does the exponent agree with expect-\nation? At what molecular weight does this relation break down?\n"]], ["block_6", ["Visually it is clear that the two data sets agree extremely well over the common molecular weight\nrange. Furthermore, for M greater than about 10,000 the data fall on a straight line. A combined\nleast squares fit of the two sets of data in this range gives\n"]], ["block_7", ["The two data sets are plotted in Figure 9.5 in a log\u2014log format. This particular choice is helpful\nbecause adherence to a power law relationship, such as the Mark\u2014Houwink relation, will result in\nthe data falling on a straight line with the exponent as the slope. From Equation 9.3.10\n"]], ["block_8", ["Source: From (a) Yamakawa, H., Abe, F., and Einaga, Y., Macromolecules, 26, 1891, 1993;\n(b) Berry, G., J. Chem. Phys, 46, 1338, 1967.\n"]], ["block_9", ["338\nDynamics of Dilute Polymer Solutions\n"]], ["block_10", ["(a) M...\n[77] (mL/g)\n(b) M...\n[n] (mL/g)\n"]], ["block_11", ["For \ufb02exible coils the value of the Mark\u2014Houwink exponent tells us something about the solvent\nquality, independent of the polymer or solvent. Is there anything similarly general to be extracted\nfrom the proportionality factor k? The answer is yes. In a theta solvent, particularly, it provides\n"]], ["block_12", ["Solution\n"]], ["block_13", ["Example 9.1\n"]], ["block_14", ["log [77] \u2014l.06 + 0.49710gM\n"]], ["block_15", ["log [77] logk + alogM\n"]]], "page_351": [["block_0", [{"image_0": "351_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "351_1.png", "coords": [22, 430, 209, 471], "fig_type": "molecule"}]], ["block_2", [{"image_2": "351_2.png", "coords": [28, 38, 408, 246], "fig_type": "figure"}]], ["block_3", ["Here we have inserted CDC, in place of the numerical prefactor of Equation 9.3.17. More detailed\ntheories give (130 as 2.8 x 1023 as a universal value for all \ufb02exible chains in a theta solvent, which is\n"]], ["block_4", ["Here we have used the subscript 6 to remind us that we are dealing with a theta solvent. Equation\n9.3.17 makes explicit the part of k that depends on the particular polymer in question. For\npolystyrene, M0: 104 g/mol and b: 6.7 x 10\"8 cm (see Table 6.1), and inserting these values\ninto Equation 9.3.17 would predict that k9 $0.12 (for [n] in g/mL). From Example 9.1 above, in\nexperiments k3 0.088, which is only about 30% different from the result utilizing the naive \u201ca\ncoil is pretty much a hard sphere with R =Rg\u201d argument.\nEquation 9.3.17 can be recast in another form, whereby the polymer-specific part is presented as\n(h2)O/M (refer to Table 6.1):\n"]], ["block_5", ["or\n"]], ["block_6", ["We can replace N by M/Mo, where MO is the monomer molecular weight, to obtain\n"]], ["block_7", ["information about the coil dimensions, as the following argument shows. For a theta solvent, we\nknow that R: s/6 (recall Equation 6.5.3). Combining this with Equation 9.3.7 and Equation\n9.3.8 gives\n"]], ["block_8", ["Figure 9.5\nPlot of log [7)] versus logM for the data in Table 9.3. An analysis of the Mark\u2014Houwink\nparameters from these data is presented in Example 9.1.\n"]], ["block_9", [{"image_3": "351_3.png", "coords": [36, 379, 155, 419], "fig_type": "molecule"}]], ["block_10", ["Intrinsic Viscosity\n339\n"]], ["block_11", ["E\n9\u2019\n..\n1\n,\nE\n10\n:L\n13\u2019\n1\n8\u2019\ni\nEVE\na PE,\n3\nYamakawa et al.\n\u2018\n\ufb01x\n-\n"]], ["block_12", ["I\nI\nI IIIII\nI\nI\nI\nIIII\nI\nI\nI\nI\nI\nI IIIII\n102 E\nI-\n-1\nI-\n\u2018\n-\n-\n_\n-\n_\n_\n_\nin] \nmg\ufb01\ns\nas\n-\n"]], ["block_13", [{"image_4": "351_4.png", "coords": [42, 36, 253, 223], "fig_type": "figure"}]], ["block_14", [{"image_5": "351_5.png", "coords": [45, 569, 243, 634], "fig_type": "molecule"}]], ["block_15", ["5N...\nN \n: \u2014-\u2014-V N __ __ _ \n_\n. .15\n[7\u2019]\n2M\n\u201c\n2M\n3\n8\n6\nM63/2\n(9 \n)\n"]], ["block_16", ["b3\n(\u201912)\n3/2\nk0~4.3><10\nMan\u20144.3x\n10\n< M )\n"]], ["block_17", ["100\n1/:\n1|\u201c:a\nI\nn lItIIII\nI IllnIlI\n.\nI .1...\u201c\n102\n103\n104\n105\n106\n"]], ["block_18", ["207m,\nb3\n9 g\u2014_\u2014=4.3 x 1023 \n(9.3.17)\n65/2\n1143/2\nMg/Z\n"]], ["block_19", [".._.\n\u201cif/2\n(9.3.18)\n\u00a2,( M\n"]], ["block_20", ["2077A?av\nb3\n1 2\n"]], ["block_21", ["3/\n"]], ["block_22", ["21, \n- -EEI - -Berry\n"]], ["block_23", ["MW, g/mol\n"]]], "page_352": [["block_0", [{"image_0": "352_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["Figure 9.6\nIntrinsic viscosity versus generation number for polyether dendrimers. The dashed curved\nrepresents the simple calculation described in Example 9.2, scaled by a factor of 5. (Data from Mourey,\nT.H., Turner, S.R., Rubinstein, M., Fr\u00e9chet, J.M.J., Hawker, C.J., and Wooley, K.L., Macromolecules, 25,\n2401, 1992.)\n"]], ["block_2", ["The crucial relation we will invoke is Equation 9.3.9, that is,\n"]], ["block_3", ["and so the key step is to see how M and R grow with generation number n. The two\u2014dimensional\npictures of a dendrimer given in Figure 1.2 and Figure 4.7 suggest a roughly spherical structure,\nwith each generation adding an approximately equal increment to R. Furthermore, as the later\ngenerations have larger numbers of units, one might suspect that they become densely packed. If\nthis were the case, then R would grow as M3, and we would recover the Einstein result for hard\nspheres:\n[17] would be independent of M and therefore\n11. In fact, as the data in Figure 9.6\ndemonstrate, this is not true. Remarkably, [1;] goes through a maximum with increasing :1, near\nn r: 3; how does this arise?\nThe origin of this unusual behavior is rather easily understood. It turns out that, although R does\nincrease roughly linearly with :1, meaning that each generation adds a roughly constant increment\nto the total R, the mass added in each generation grows much more rapidly. Let us consider a\n"]], ["block_4", ["Using the arguments given above, predict the dependence of [n] on generation number for\ndendrimer molecules. Recall from Chapter\n1 and Chapter 4 that dendrimers have a tree\u2014like\nstructure, with each generation adding a new layer of material in a completely regular way.\n"]], ["block_5", ["close to the polystyrene result in Example 9.1. The difference in the numerical value of the\nprefactor arises because in our simple model we assumed that a coil behaves as a hard sphere\nwith R:Rg, whereas in fact it behaves as a hard sphere with R o: Rg and the proportionality\nconstant is a little less than 1. Thus measurements of intrinsic viscosity in a theta solvent gives\naccess to the coil dimensions (i.e., COO, b, or Ep) via Equation 9.3.18.\n"]], ["block_6", ["340\nDynamics of Dilute Polymer Solutions\n"]], ["block_7", ["Solution\n"]], ["block_8", ["Example 9.2\n"]], ["block_9", ["EH4?\n!\n\\\nj\nE\ni\n/\n~01\n_\n/\n"]], ["block_10", ["on\n\"\n\\\n\u2018l\n"]], ["block_11", [{"image_1": "352_1.png", "coords": [39, 388, 260, 623], "fig_type": "figure"}]], ["block_12", ["R3\n[n] M\n"]], ["block_13", ["%\n31\u00b0\n/\n_\n:\nf\ni\n-/\n2__\u2019\n.j\n"]], ["block_14", ["6;\n/-H\\\nL.\n"]], ["block_15", ["-\n/\n5\n1\n5;\n/.\n\\\nJ\n"]], ["block_16", ["7_[\nr\n[I\nr\nFlt\u2014TIT!\n[Ill\n[\nI-\n"]], ["block_17", ["1tl_LlJ_l|_ll_I_l_[llll\nl\n1\nIII\nI\nl\u2018\n"]], ["block_18", ["I-\n/\n\\\n-'\n'\n0\n"]], ["block_19", ["l\u2014\n/\n\\.\n_\n"]], ["block_20", ["[\nl\nI\nl\nI\nI\nI\nO\n1\n2\n3\n4\n5\nIn\n"]], ["block_21", ["0')\n"]]], "page_353": [["block_0", [{"image_0": "353_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["R\n10\n20\n30\n40\n50\n60\nM\n100\n400\n1000\n2200\n4600\n9400\nR3/M\n10\n20\n27\n29\n27\n23\n"]], ["block_2", ["We defined the equation of motion as a general expression of Newton\u2019s second law applied to a\nvolume element of \ufb02uid subject to forces arising from pressure, viscosity, and external sources.\nAlthough we shall not attempt to use this result in its most general sense, it is informative to\nconsider the equation of motion as it applies to a specific problem: the \ufb02ow of liquid through a\ncapillary. This consideration not only provides a better appreciation of the equation of motion, but\nalso serves as the basis for an important technique for measuring solution viscosity. We shall\nexamine the derivation first and then discuss its application to experiment.\nFigure 9.7a shows a portion of a cylindrical capillary of radius R and length L. We measure the\ngeneral distance from the center axis of the liquid in the capillary in terms of the variable r and\nconsider specifically the cylindrical shell of thickness dr designated by the broken line in Figure\n9.7a. In general, gravitational, pressure, and viscous forces act on such a volume element, with the\nviscous forces depending on the velocity gradient in the liquid. Our first task then is to examine\nhow the velocity of \ufb02ow in a cylindrical shell such as this varies with the radius of the shell.\nThe net viscous force acting on this volume element is given by the difference between the\nfrictional forces acting on the outer and inner surfaces of the shell:\n"]], ["block_3", ["From this simple calculation, we see that R3/M goes through a maximum for n 3, in excellent\nagreement with the experiments. The origin of the maximum lies in the fact that M grows\ngeometrically with 22 rather than as a power law. Consequently, M in the denominator eventually\nincreases more rapidly than R3 in the numerator. In other words, the dendrimer density is\nincreasing steadily, as opposed to a hard sphere, for which the density is constant with R. This\ndensity increase cannot persist indefinitely, and in fact dendrimers can typically only be grown out\nto n 6\u20148 before there is no more room on the surface to complete another generation.\n"]], ["block_4", ["In this section we consider two standard techniques for measuring viscosity. The first concerns the\nuse of capillary viscometers for low\u2014viscosity \ufb02uids, such as the dilute polymer solutions of\nrelevance to [n]. The second describes the Couette or concentric cylinder geometry, which is\nmore elaborate but is capable of covering a much wider range of n and \u201c5/.\n"]], ["block_5", ["n\n0\n1\n2\n3\n4\n5\n"]], ["block_6", ["n. : 2, M:400 + 6(100) 1000, and R =20 + 10 30. The multiplier 6 in the calculation of M\narises because the previous generation added 3 units, each of which has two functional groups left\nto react further. The results through it 5 computed in this way are listed in the table below:\n"]], ["block_7", ["simple example. Assume trifunctional reactive unit with M 100 g/mol and assume that each\ngeneration adds 10 A to the radius. We can compute a table of M, R, and R3/M as a function of n.\nFor n 0, M 100, and 10 by assumption. (Note that by convention, the initial dendrimer\n\u201ccore\u201d is labeled generation 0.) For n 1, M: 100 + 3(100) 2400, and R 10 + 10 20. For\n"]], ["block_8", ["9.4.1\nPoiseuille Equation and Capillary Viscometers\n"]], ["block_9", ["9.4\nMeasurement of Viscosity\n"]], ["block_10", ["Measurement of Viscosity\n341\n"]], ["block_11", [{"image_1": "353_1.png", "coords": [35, 149, 254, 220], "fig_type": "molecule"}]], ["block_12", [{"image_2": "353_2.png", "coords": [39, 614, 262, 678], "fig_type": "molecule"}]], ["block_13", ["Fvis,net (Fvis)0uter (FViS)inner\ndv\ndv\n=\n_\n\u2014 2\nL\n-\u2014\n9.4.1\n2770\u2018 + dr) L27 (ml-m;-\n77'!\u201d n (m),\n(\n)\n"]]], "page_354": [["block_0", [{"image_0": "354_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "354_1.png", "coords": [20, 86, 321, 227], "fig_type": "figure"}]], ["block_2", [{"image_2": "354_2.png", "coords": [30, 577, 182, 619], "fig_type": "molecule"}]], ["block_3", [{"image_3": "354_3.png", "coords": [32, 79, 170, 215], "fig_type": "figure"}]], ["block_4", ["Figure 9.7\n(a) Portion of a cylinder of radius R and length L showing (by broken lines) section of thickness\ndr. (b) Profile of \ufb02ow velocity in the cylinder. (From Hiemenz, P.C., Principles of Colloid and Surface\nChemistry, Marcel Dekker, New York, 1977.)\n"]], ["block_5", ["342\nDynamics of Dilute Polymer Solutions\n"]], ["block_6", ["Under stationary-state conditions of \ufb02ow, that is, when no further acceleration occurs, this force\nis balanced by gravitational and pressure forces. For simplicity, we assume that the capillary\nis oriented vertically so that gravity operates downward and, for generality, we assume that\nan additional mechanical pressure Ap is imposed between the two ends of the capillary.\nUnder these conditions, the net gravitational and mechanical forces acting on the volume\nelement equal\n"]], ["block_7", ["Integration in the radial direction (along r) converts Equation 9.4.5 to\n"]], ["block_8", ["where 217Lr dr is the volume of the element, 2m dr is its cross-sectional area, p is the density of\nthe \ufb02uid, and g is the acceleration due to gravity. Under the stationary-state conditions we seek to\ndescribe, Equation 9.4.4 and Equation 9.4.3 are equal, and the following relationship applies to the\nvolume element:\n"]], ["block_9", ["where the length times the circumference of the surface describes the apprOpriate area in Equation\n9.1.2. The relationship between the velocity gradient at the two locations is given by\n"]], ["block_10", ["provided that dr is small. Combining Equation 9.4.1 and Equation 9.4.2 and retaining only those\nterms that are first order in dr give\n"]], ["block_11", ["(a)\n(b)\n"]], ["block_12", [{"image_4": "354_4.png", "coords": [36, 635, 156, 675], "fig_type": "molecule"}]], ["block_13", [{"image_5": "354_5.png", "coords": [40, 327, 185, 365], "fig_type": "molecule"}]], ["block_14", ["Fgrammcch, net (2e dr)pg + (2771' dr)Ap\n(9.4.4)\n"]], ["block_15", ["d2v\ndv\nd\ndv\nF(as, net 27711L[r(d7)dr + (5) Cir] :\n217171;; \n(9.4.3)\n"]], ["block_16", ["dv\nl\nAp\n2\n\u2014 :\n\u2014\n_\u2014\n04.6\nnrdr\n2033+ L}\n(9\n)\n"]], ["block_17", ["4%\n"]], ["block_18", ["(1\ndv\nAp\n_\n_\n=\n_\n.4.5\nndr \n(pg + L )r dr\n(9\n)\n"]], ["block_19", ["dv\ndv\ndzv\n(5?)...\u2014 (aids-Jr\n\u20189'\u201c)\n"]], ["block_20", [{"image_6": "354_6.png", "coords": [68, 395, 296, 427], "fig_type": "molecule"}]], ["block_21", ["7\"\n"]], ["block_22", ["/ 11:7:2:3\"\n4\nME/\n\\i:z\u00e9\u2019i/\n"]], ["block_23", ["\\l\n.l\n4\u2014H\u2014b\u20144\u2014r-b\u2014\n\\ \\l I l\n"]], ["block_24", ["33/\nI\nL/\n"]], ["block_25", ["dr\n"]], ["block_26", [{"image_7": "354_7.png", "coords": [205, 72, 322, 232], "fig_type": "figure"}]]], "page_355": [["block_0", [{"image_0": "355_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "355_1.png", "coords": [26, 174, 158, 212], "fig_type": "molecule"}]], ["block_2", [{"image_2": "355_2.png", "coords": [28, 292, 217, 341], "fig_type": "molecule"}]], ["block_3", [{"image_3": "355_3.png", "coords": [33, 284, 221, 365], "fig_type": "figure"}]], ["block_4", ["This result is called the Poiseuille equation, after the researcher who discovered in 1844 this\nfourth\u2014power dependence of flow rate on radius [10]; the unit of viscosity, poise, is also named\nafter him. The following example illustrates the use of the Poiseuille equation in the area where it\nwas first applied.\n"]], ["block_5", ["Poiseuille was a physician\u2014physiologist interested in the \ufb02ow of blood through blood vessels in the\nbody. Estimate the viscosity of blood from the fact that blood passes through the aorta of a healthy\nadult at rest at a rate of about 84 cm3 s\u2014l, with a pressure drop of about 0.98 mmHg m\u2018l. Use 9 mm\nas the radius of the aorta for a typical human.\n"]], ["block_6", ["The units must be expressed in a common system, with the pressure gradient requiring the most\nmodification:\n"]], ["block_7", ["The pumping action of the heart rather than gravity is responsible for blood \ufb02ow; hence the term\nn can be set equal to zero in Equation 9.4.9 and the result solved for n:\n"]], ["block_8", ["where the at r is used to eliminate the integration constant. the\nvelocity gradient to the radial in the \ufb02uid: itand\n"]], ["block_9", ["This result describes a parabolic velocity profile, as sketched in Figure 9.7b.\nEquation 9.4.8 describes the velocity with which a cylindrical shell of liquid moves through a\ncapillary under stationary\u2014state conditions. This velocity times the cross-sectional area of the shell\ngives the incremental volume of liquid dV, which is delivered from the capillary in an interval of\ntime At. The total volume delivered in this interval, AV, is obtained by integrating this product over\nall values of r:\n"]], ["block_10", ["Because of the nonslip condition at the wall, v 0 when r :R, and the constant of integration can\nbe evaluated to give\nVzma)\n4\"?\n(9.4.8)\n"]], ["block_11", ["has a maximum \nEquation 9.4.6 can be integrated again to give v as a function of r:\n"]], ["block_12", ["Measurement of Viscosity\n343\n"]], ["block_13", ["Example 9.3\n"]], ["block_14", ["Solution\n"]], ["block_15", [{"image_4": "355_4.png", "coords": [38, 559, 116, 599], "fig_type": "molecule"}]], ["block_16", ["R\nV\nL\n1%! \n277(9342410/\n)\nJ (R2 _ ,2), d,\n"]], ["block_17", ["A_p _ 0.98 mmHg X 133.3Nm\u20142\nL \nm\n1 mmHg\n= 131 kg m\u20142 s\u20142\n"]], ["block_18", ["J dv wJ r dr\n(9.4.7)\n"]], ["block_19", ["AprrR4\n7\u2019 SLAV/At\n"]], ["block_20", [",__,, \n(949)\n8171.\n"]], ["block_21", ["4\n"]], ["block_22", ["0\n"]]], "page_356": [["block_0", [{"image_0": "356_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "356_1.png", "coords": [30, 262, 126, 310], "fig_type": "molecule"}]], ["block_2", ["Figure 9.8\nA typical capillary viscometer. (From Hiemenz, P.C., Principles of Colloid and Surface\nChemisty, Marcel Dekker, New York, 1977.)\n"]], ["block_3", ["where A represents a cluster of factors that are constant for a particular apparatus. The constant A\nneed not be evaluated in terms of the geometry of the apparatus, but can be eliminated from\n"]], ["block_4", ["Therefore\n"]], ["block_5", ["The Poiseuille equation provides a method for measuring 1) by observing the time required for a\nliquid to flow through a capillary. The apparatus shown in Figure 9.8 is an example of one of many\ndifferent instruments designed to use this relationship. In such an experiment, the time required for\nthe meniscus to drop the distance between the lines etched at opposite ends of the top bulb is\nmeasured. This corresponds to the drainage of a \ufb01xed volume of liquid through a capillary of\nconstant R and L. The weight of the liquid is the driving force for the flow in this case, so the Ap\nterm in Equation 9.4.9 is zero and the observed \ufb02ow time equals\n"]], ["block_6", ["344\nDynamics of Dilute Polymer Solutions\n"]], ["block_7", ["At 370C, the viscosity of pure water is about 0.69 x 10\u20143 kg m\u20141 s\u2018l; the difference between this\nfigure and the viscosity of blood is due to the dissolved solutes in the serum and the suspended red\ncells in the blood. The latter are roughly oblate ellipsoids in shape.\n"]], ["block_8", ["01'\n"]], ["block_9", [{"image_2": "356_2.png", "coords": [38, 361, 97, 648], "fig_type": "figure"}]], ["block_10", ["D\nf?\n"]], ["block_11", ["V\nA: : (:31);\n(9.4.10)\n"]], ["block_12", ["r) =Ap At\n(9.4.11)\n"]], ["block_13", ["C\n"]], ["block_14", ["_ \n_\n8(84 x10-6 m3s-1)\n: 4,0 x 10*3 kg111_15_I\n(or 0.04 P)\n"]], ["block_15", ["\\/\n"]], ["block_16", [",M\n"]]], "page_357": [["block_0", [{"image_0": "357_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", ["where B AV/Sqr. As above, both A and B can be treated as instrument constants and evaluated by\nmeasuring two liquids, which are known with respect to \u201cn and p and then solving a pair of simultaneous\nequations for A and B. A better strategy might be to choose a capillary sufficiently narrow so that\nA: is long enough to eliminate the second term on the right\u2014hand side of Equation 9.4.13.\nOne limitation of this method is the fact that the velocity gradient is not constant in this type of\ninstrument, but varies with r, as noted in connection with Equation 9.4.6. This would be a concern\nif the viscosity were shear\u2014rate dependent over the relevant range. For dilute solutions, and the\nslow \ufb02ows appropriate to determine the intrinsic viscosity, this usually does not matter.\n"]], ["block_2", ["The second standard geometry for solution viscometry is based on concentric cylinders. The illustration\nthat enabled us to define the coefficient of viscosity also suggests a modification that would be\nexperimentally useful. Suppose the two rigid parallel plates in Figure 9.1 and the intervening layers\nof \ufb02uid were wrapped around the z\u2014axis to form two concentric cylinders, with the \ufb02uid under\nconsideration in the gap between them. The required velocity gradient is then established by causing\none of these cylinders to rotate while the other remains stationary. The velocity is now in the direction\ndescribed by the angle 6 in Figure 9.9a, and its gradient is in the radial direction r. Thus the velocity\ngradient in this arrangement may be written dvg/dr.\nSome of the reasons for our interest in this type of viscometer are the following:\n"]], ["block_3", ["Equation 9.4.11 by measuring both a known (subscript 2) and an unknown (subscript 1) liquid in\nthe same \n"]], ["block_4", ["Note that interval depends on both the density and the viscosity of the \ufb02uid, and the ratio\nn/p is sometimes referred to as the kinematic viscosity.\nIn more precise work an additional term, which corrects for effects arising at the ends of the tube, is\nadded to Equation 9.4. 1 1. This correction\u2014which is often negligible\u2014can be incorporated by writing\n"]], ["block_5", ["transmits a force, which can be measured in terms of the torque on a torsion wire to the suspended\nbob. This arrangement is sketched in Figure 9.9b. In this representation, the outer cylinder has \nradius R and the inner cylinder has a radiusfR, wheref is some fraction. The closer this fraction is\nto unity, the narrower will be the gap between the cylinders and the more closely the apparatus will\napproximate the parallel plate model in terms of which 77 is defined.\nA formal mathematical analysis of the \ufb02ow in the concentric cylinder viscometer yields the\nfollowing relationship between the experimental variables and the viscosity:\nf\ntorque force >< radius :47mLR2w 1\u2014f2\n(9.4.14)\n"]], ["block_6", ["1.\nThe basic design is a direct extension of the discussion of viscosity in Section 9.1 and Section 9.2.\n2.\nThe range of applicability is very wide, extending at least from 7; m 0.01\u2014\u20141010 P.\n3.\nThe design permits different velocity gradients to be considered, so that non-Newtonian\nbehavior (e.g., shear thinning) can be investigated, if desired.\n4.\nThe number of technically important viscosity-measuring devices may be thought of as\nvariants of this basic apparatus.\n"]], ["block_7", ["As a practical matter, the outer cylinder is part of a cup that holds the \ufb02uid, while the inner cylinder\nis a coaxial bob suspended within the outer cup. Suppose the cup is centered on a turntable that\nrotates with an angular velocity to, measured in radians per second. The viscous \ufb02uid now\n"]], ["block_8", ["9.4.2\nConcentric Cylinder Viscometers\n"]], ["block_9", ["Measurement \n345\n"]], ["block_10", ["_ \n\u201d'71 \nB; Kt:\n(9-4-12)\n"]], ["block_11", ["p\n:A\nAt\u2014B\u2014\n1)\n9\nAr\n(9.4.13)\n"]], ["block_12", [{"image_1": "357_1.png", "coords": [49, 83, 114, 117], "fig_type": "molecule"}]]], "page_358": [["block_0", [{"image_0": "358_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "358_1.png", "coords": [13, 66, 411, 237], "fig_type": "figure"}]], ["block_2", [{"image_2": "358_2.png", "coords": [28, 78, 170, 218], "fig_type": "figure"}]], ["block_3", ["asf\u2014> 1.\n4.\nThe area of contact A between the cylinders and the \ufb02uid is\n27TRL; therefore 477LR/\n2R(1\u2014f)=A/(1*f)R-\n5.\nThe product (1 f)R is the width of the gap, 5.\n"]], ["block_4", ["in the limit asf \u2014> 1. This is identical to Equation 9.1.3 and is the result we anticipated in rolling\nFigure 9.1 into a cylinder. Equation 9.4.14 is more general than Equation 9.4.16, since its\napplicability is not limited to vanishingly small gaps.\n"]], ["block_5", ["We now turn our attention to the phenomenon of diffusion. This turns out to be directly related to\nthe viscosity through the friction factor, f. Most of us have a sense of diffusion as a randomization\nprocess: place a drop of food coloring into a glass of water, and before long the entire glass will be\nuniformly colored even without stirring. The time it takes for the color to spread depends on two\n"]], ["block_6", ["This equation can be cast into a more recognizable form by assuming thatfis very close to unity. In\nthat case we have the following:\n"]], ["block_7", ["Since v9 is the difference in velocity between the inner and outer cylinders and 5 is the difference\nin the radial location of the two rigid surfaces, Equation 9.4.15 becomes\n"]], ["block_8", ["346\nDynamics of Dilute Polymer Solutions\n"]], ["block_9", ["1.\nThe radius R applies to the entire \ufb02uid sample. Since torque equals the product of force and R,\ncanceling out one power of R leaves the shearn force acting on the \ufb02uid on the left-hand side\nof Equation 9.4.14.\n2.\nThe remaining factor R times to on the right-hand side of Equation 9.4.14 can be replaced by\nthe linear velocity v9.\n3.\nThe factor 1 \u2014f2 can be replaced by 2(1 \u2014f), since 1 \u2014f2=(1 + f)(1 \u2014f) and (1 \u2014l\u2014 f)\u2014>2\n"]], ["block_10", ["Figure 9.9\nDefinition of variables for concentric cylinder viscometers; (a) the rotating cylinder and (b) the\ncoaxial cylinders.\n"]], ["block_11", ["9.5\nDiffusion Coefficient and Friction Factor\n"]], ["block_12", ["d\nUzngs\ncam)\n"]], ["block_13", ["_ =\n=\n_\n.4.15\nA\n0\nnf5\n(9\n)\n"]], ["block_14", ["Introducing these substitutions in Equation 9.4.14 gives\n"]], ["block_15", ["7Q\n_/\",.,,\u2014\\H\nJ\n"]], ["block_16", [{"image_3": "358_3.png", "coords": [138, 43, 355, 249], "fig_type": "figure"}]], ["block_17", [{"image_4": "358_4.png", "coords": [190, 82, 343, 216], "fig_type": "figure"}]]], "page_359": [["block_0", [{"image_0": "359_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "359_1.png", "coords": [29, 324, 148, 363], "fig_type": "molecule"}]], ["block_2", ["Returning to our glass of water, imagine a single dye molecule somewhere in the glass and further\nimagine that we could actually follow its motion directly in time and space. (This is not too far\nfetched, in fact, with recent developments in single-molecule spectroscopy and microscopy).\nWhat would we see? The molecule is constantly buffeted by solvent molecules, and consequently\nexecutes Brownian motion: it moves tiny distances between collisions, with the instantaneous\ndirection of motion fluctuating randomly. If we watch for a relatively long time interval, how far is\nthe dye molecule likely to move? We already have developed all the mathematics needed to\nanswer this question, when we considered random walks in Chapter 6. Namely, if the molecule\nexecutes a total of N random steps of average length b, the mean square displacement (r2) will be\ns. We can formalize this as follows:\n"]], ["block_3", [{"image_2": "359_2.png", "coords": [35, 507, 234, 541], "fig_type": "molecule"}]], ["block_4", ["Equation 6.7.12 can be recovered by substituting 6D,! =n2 into Equation 9.5.2. Also, recall that\nthe prefactor of 41a emerges because we are interested in the distance from the origin, independ-\nent of direction. Unlike the case of polymer chains, which are self\u2014avoiding walks, a Brownian\nparticle really does execute a random walk.\nThe tracer diffusion coef\ufb01cient is a property of an individual molecule or particle undergoing\nBrownian motion. Its value, however, will generally depend on the size of the molecule and on the\nmedium in which it is diffusing. Einstein showed that there is a beautifully simple relationship\nbetween the friction factor and DI:\n"]], ["block_5", ["things: how rapidly the dye molecules move, and the size of the glass. The former is quantified by a\ndz\ufb01\u2018usz\u2019on coe\ufb01\u2018i\u2019cienz, D, which depends directly on the molecular friction factor. Diffusion is an\nimportant process to understand and control in many polymer applications (e.g., how long will\nsoda in fizz? How long does it take a drug in a skin patch to pass through the\nskin?), and the diffusion coefficient itself is also a useful means of molecular characterization. In\nthis section we will emphasize the diffusion of polymers in dilute solution, but the development\nhas a much broader range of applicability. We begin the discussion by distinguishing two related,\nbut conceptually distinct, diffusion coefficients, which we shall call the tracer diffusion coefficient,\nD,, and the mutual diffusion coefficient Dm. Failure to distinguish clearly between these two\nquantities can be a major source of confusion.\n"]], ["block_6", ["The units of the diffusion coefficient are (length)2/time, or cmz/s in the cgs system. We take the\nlimit of long times just to remove all memory of past collisions and directions. We may consider\nthe factor of 6 in front of Dt to be a historical convention. The average denoted by\n(- -) is\nnecessary because by definition we cannot make any predictions about a single random walk.\nRather, if we watch a particular particle for many time intervals or watch many independent but\notherwise identical particles over a given time interval, we can generate the average mean square\ndisplacement. In fact, based on our work with random walks in Chapter 6, we even know the\ndistribution function that should describe the results of many equivalent experiments: it should be a\nGaussian. Just as we did in Chapter 6, we can ask the following question: if we define the location\nof a particle at time t: O as the origin, what is the probability of finding it a distance r away after\ntime t? This probability is known by a special name, the \u201cvan Hove space\u2014time self-correlation\nfunction,\u201d but it is mathematically equivalent to Equation 6.7.12:\n"]], ["block_7", ["9.5.1\nTracer Diffusion and Hydrodynamic Radius\n"]], ["block_8", ["Diffusion Coefficient and Friction Factor\n347\n"]], ["block_9", ["kT\nDI:\u2014\nf\n(9.5.3)\n"]], ["block_10", ["2\n1\n3/2\n1'2\nP\nz =4\n\u2014 \u2014\n.\n.\n(r\u2019 )\nw r (477 DJ)\nexP<\n4Dtt)\n(9 5 2)\n"]], ["block_11", ["2\n2\nlim (_r_) g E 6D,\n(9.5.1)\nI\u2014roo\nZ\nl\u2018\n"]]], "page_360": [["block_0", [{"image_0": "360_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "360_1.png", "coords": [34, 141, 110, 179], "fig_type": "molecule"}]], ["block_2", ["Now we turn to the mutual diffusion coefficient, Dm, which describes how a collection of\nBrownian particles will distribute themselves in space. In the context of the food coloring analogy,\nDt tells us how rapidly any individual dye molecule explores space, but Dm describes how quickly\nthe entire droplet of food coloring disperses itself. Experience tells us that after a reasonable time\ninterval, the glass of water will be uniformly colored. The underlying reason is that mutual\ndiffusion acts to eliminate any gradients in concentration. Although the individual dye molecules\nare happily diffusing about, largely oblivious of one another, collectively they tend to spread\nthemselves out evenly.\nFick first recognized that mass diffusion was analogous to thermal diffusion and he proposed an\nadaptation of Fourier\u2019s law of heat conduction for the transport of material [1 1]. Speci\ufb01cally, the\n\ufb02ux J (in units of mass per unit area per unit time) across a plane is assumed to be proportional to\nthe gradient in concentration (mass/volume) along the direction perpendicular to the plane. We\nwill restrict ourselves to one-dimensional diffusion along the x-direction, and therefore\n"]], ["block_3", ["In other words, Rh for any polymer is the radius of a hard sphere that would have the same friction\nfactor or diffusivity. In Section 9.3 we saw that \ufb02exible molecules behaved hydrodynamically\nrather like hard spheres with radius proportional to Rg. Does this hold for diffusion as well? Yes,\nindeed. Consequently Rh is directly proportional to Rg, with the proportionality factor depending\non the particular polymer shape. Thus, Rh depends on molecular weight with the same power law\nexponent as does Rg, and Dt exhibits the inverse of that dependence. For \ufb02exible chains, therefore,\nDv\u2018m~ in a theta solvent and Dv\u2018y5 in a very good solvent. Examples are shown in\nFigure 9.10 for polystyrene in cyclohexane at the theta temperature, and in toluene, a good solvent.\nThere are several experimental techniques by which diffusion may be measured, and examples will\nbe given following the next section.\n"]], ["block_4", ["In this expression, called Fick\u2019s first law, the proportionality constant is Dm and it follows that Dm\nalso has units of length2/time. The minus sign in Equation 9.5.6 recognizes the fact that the\ndirection of the \ufb02ow is that of decreasing concentration.\nWe now consider a volume element and the \ufb02ux of solute in and out of that element. Figure 9.1 1\nschematically represents three regions of an apparatus containing a concentration gradient. The end\n"]], ["block_5", ["This simple relation provides a direct connection between the tracer diffusion coefficient, the\nparticle size, and the viscosity of the solvent. We can go an important step further, however. If our\nparticle is not a hard sphere, we can use Equation 9.5.4 to define an equivalent radius in terms of a\nmeasured diffusivity, called the hydrodynamic radius, Rh:\n"]], ["block_6", ["From the definition of the friction factor, we can see that the thermal energy, H\", is playing the role\nof a generalized \u201cforce,\u201d and Dt is the resulting \u201cvelocity.\u201d As T increases, solvent molecules\nmove more rapidly and are more effective at jostling the tracer molecule, so Dt should increase. At\nthe same time, if the solvent viscosity increases, or if the particle increases in size, thenf should\nincrease and Dt will be smaller. If we now consider our tracer molecule to be a hard sphere and the\nsolvent to be a continuum, then we can incorporate Stokes\u2019 law forf (Equation 9.2.4) to arrive at\nthe Stokes\u2014Einstein relation\n"]], ["block_7", ["343\nDynamics of Dilute Polymer Solutions\n"]], ["block_8", ["9.5.2\nMutual Diffusion and Fick's Laws\n"]], ["block_9", [{"image_2": "360_2.png", "coords": [37, 229, 115, 264], "fig_type": "molecule"}]], ["block_10", ["dc\nJ \nDlm dx\n(9.5.6)\n"]], ["block_11", ["D \nkT\nt_6*n\"nSR\n(9.5.4)\n"]], ["block_12", ["6771?, Dt\n"]], ["block_13", ["kT\n"]]], "page_361": [["block_0", [{"image_0": "361_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "361_1.png", "coords": [26, 47, 265, 447], "fig_type": "figure"}]], ["block_2", [{"image_2": "361_2.png", "coords": [32, 40, 289, 262], "fig_type": "figure"}]], ["block_3", [{"image_3": "361_3.png", "coords": [32, 273, 285, 445], "fig_type": "figure"}]], ["block_4", ["Figure 9.10\nMolecular weight dependence of DI for polystyrene in (a) a theta solvent, cyclohexane at 35\u00b0C,\nand (b) a good solvent, toluene. The curve in (a) has a slope of \u2014~0.50, and the curve in (b) approaches a slope\nof \u20140.60 at large M. (Reproduced from Schaefer, D.W. and Han, C.C., Dynamic Light Scattering, R. Pecora\n(Ed), Plenum, New York, 1985. With permission.)\n"]], ["block_5", ["compartments contain the solute at two different concentrations (:1 and (:2, with C2 > CI. The center\nregion is a volume of cross\u2014sectional area A and thickness (13:, along which the gradient exists. The\narrows in Figure 9.11 represent the flux of solute from the more concentrated solutions to the less\nconcentrated one. The incremental change in the total amount of solute dQ in the center volume\nelement per time increment (it can be developed in two different ways. In terms of the \ufb02uxes,\n"]], ["block_6", ["Diffusion \n349\n"]], ["block_7", ["whereas in terms of a concentration change dc in the element of volume A dx,\n"]], ["block_8", ["~99.\n(\\I\nE\n0.\n"]], ["block_9", ["QC\nh-\n$3\n1.0\n"]], ["block_10", ["CUE\n"]], ["block_11", ["c5\u201d\nrx\n-\n._\n$2\n1.0\u2014\n__\n'_\n-\n"]], ["block_12", ["10.0I\u2014\n'\u2014\n33\n_\n_\n"]], ["block_13", ["0.\n_\n_\n"]], ["block_14", [{"image_4": "361_4.png", "coords": [41, 601, 132, 634], "fig_type": "molecule"}]], ["block_15", ["100-0\nlill|\n1\nIII]\nI IIIrI\nI\u2014II]\nI\n1H\n"]], ["block_16", ["100-0\nI\u2014IITI\nI\nI\u2014rII\nI\nil||\nI\nI\u2018ll\n"]], ["block_17", ["dQ\nE; \n-\u2014 10mm\n(9.5.7)\n"]], ["block_18", ["dQ = dc(A dx)\n(9.5.8)\n"]], ["block_19", ["10.0\n"]], ["block_20", ["0.1\nI\nIIII\n|\nIIII\nI\nI_LJ_L_L_LJ_I\n"]], ["block_21", ["L\nl\n0_1\n_I_J_1LLJ_llii\nI_IIII\nI_LJ_IJI\nI\n"]], ["block_22", ["1\n10\n100\n1000\n"]], ["block_23", ["1\n10\n100\n1000\n10,000\n"]], ["block_24", ["J\nPSintqene\n:\n"]], ["block_25", ["\u2014\n.\u2014\n"]], ["block_26", ["l\u2014\n.\u2014\n"]], ["block_27", ["PS in cyclohexane\n\u201dJ\n"]], ["block_28", ["10\u20143MW\n"]], ["block_29", ["10_3Mw\n"]]], "page_362": [["block_0", [{"image_0": "362_0.png", "coords": [0, 0, 463, 716], "fig_type": 0}]], ["block_1", [{"image_1": "362_1.png", "coords": [25, 32, 267, 188], "fig_type": "figure"}]], ["block_2", [{"image_2": "362_2.png", "coords": [32, 519, 130, 553], "fig_type": "molecule"}]], ["block_3", [{"image_3": "362_3.png", "coords": [33, 628, 299, 673], "fig_type": "molecule"}]], ["block_4", ["This differential equation can be solved (sometimes easily, sometimes not) when one speci\ufb01es the\nappropriate initial or boundary conditions (see Example 9.4 and Example 9.5).\nFick\u2019s second law is very useful, but it leaves us with two unresolved issues. First, how is Dm\nrelated to Dt and f? Second, what is the role of thermodynamics in this process? These questions\ncan be answered together. In general, a single phase will be at equilibrium when the temperature,\npressure, and chemical potential of each species are everywhere the same. In other words, diffusion\nactually acts to remove gradients in p. rather than gradients in c. The thermodynamic \u201cdriving\nforce\u201d can be written (recall that the gradient of a potential is a force) as\n"]], ["block_5", ["The areas cancel and the term in square brackets can be recognized as\n"]], ["block_6", ["We divide Equation 9.5.12 by Avogadro\u2019s number to convert the partial molar Gibbs free energy,\nu, to a molecular quantity and the minus sign enters as in Fick\u2019s first law because the force and the\ngradient are in opposing directions. Recalling the definition of chemical potential from Equation\n7.1.13 we write #2 p3 +RTlnag p33 +RTln 720, where (.12 and y; are the activity and\nactivity coefficient, respectively, of the solute (note that when 6 is in g/mL, 'yz is defined\naccordingly). Substituting into Equation 9.5.12 we obtain\n"]], ["block_7", ["Figure 9.11\nSchematic of diffusion with respect to a volume element of thickness dx located at x =0\nbetween regions with concentration cl and c; > cl.\n"]], ["block_8", ["that leads us directly to Fick\u2019s second law (in one dimension):\n"]], ["block_9", ["350\nDynamics of Dilute Polymer Solutions\n"]], ["block_10", ["By combining these two relations, and substituting Fick\u2019s first law for J, we obtain\n"]], ["block_11", ["l</</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub>\nl</</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub>\nI</</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub>\nl</</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub>\nl</</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub>\nl</</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub>\nl</</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub>\n.</</sub>sub>\nl</</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub>\nI</</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub>\nI</</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub>\nI</</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub></</sub>sub><