INST. OF STAND & TECH NIST

PUBLICATIONS e guide

1.1.107 514SS7

: Molecular Mass

j Distribution

3 Measurement by

Mass Spectrometry

True Molecular Mass Distribution

William E. Wallace and

Charles M. Guttman

Special

Publication

960-21

NIST Recommended Practice Guide

Special Publication 960-21

Molecular Mass

Distribution

Measurement by

Mass Spectrometry

William E. Wallace and

Charles M. Guttman

Polymers Division

Materials Science and Engineering Laboratory

National Institute of Standards and Technology

April 2010

U.S. Department of Commerce

Gary Locke, Secretary

National Institute of Standards and Technology

Dr. Patrick D. Gallagher, Director
Certain equipment, instruments or materials are identified in this paper in order

to adequately specify the experimental details. Such identification does not imply recommendation by the National Institute of Standards and Technology

nor does it imply the materials are necessarily the best available for the

purpose.

Official contribution of the National Institute of Standards and Technology; not

subject to copyright in the United States.

National Institute of Standards and Technology

Special Publication 960-21

Natl. Inst. Stand. Technol.

Spec. Publ. 960-21

40 pages (April 2010)

CODEN: NSPUE2

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 2001

For sale by the Superintendent of Documents

U.S. Government Printing Office

Internet: bookstore.gpo.gov Phone:(202) 512-1800 Fax:(202) 512-2250

Mail: Stop SSOP, Washington, DC 20402-0001
Preface: Purpose of the Guide

This NIST Recommended Practice Guide presents a method that uses mass

spectrometry to make a quantitative determination of the molecular mass

distribution of narrow polydispersity synthetic molecular materials. The

main obstacle for converting an oligomer-resolved mass spectrum (MS) into a valid molecular mass distribution (MMD) is the systematic uncertainty in the ion intensity axis of the mass spectrum. This guide demonstrates how to make gravimetric mixtures of analytes having the same general chemistry

but differing in molecular mass distribution to 1) ascertain if mass bias is present for a given set of measurement conditions, 2) create a calibration curve

to properly convert the MS to a valid MMD, and 3) determine systematic

uncertainties for the calibration curve when certain specific conditions are met.

Within the framework of this Recommended Practice Guide two broad classes

of mixtures exist: those where oligomer peaks in the mass spectrum cannot

be attributed to a particular component of the mixture and those where they

can. For example, mixtures of two polymers having the same repeat units

and the same end groups fall into the former category (see Figure 4 ), while

if the components have the same repeat units but different mass end groups

they fall into the latter category (see Figure 5). However, if the components

have the same repeat unit and the same end group but the mass distributions are far enough apart such that they do not overlap, their mixture falls into the

second category: each oligomer can be assigned to a specific component of the

mixture (see Figure 6). The first type is referred to as "indistinguishable" and

"overlapping" and the second type as "distinguishable" or "non-overlapping".

The first type is most often encountered in practice. Two molecular materials

may be made that have different average molecular masses, or a broad

molecular mass material may be separated, often chromatographically. In such

situations when the mixtures are indistinguishable and overlapping, moments

of the molecular mass distribution of the pure components and of gravimetric

mixtures are used to create a calibration curve. It is this case that is covered in Section 4 of this Recommended Practice Guide.

The second type of mixture, distinguishable or non-overlapping, is encountered

less often. It often requires a separate synthesis to create a different chemical

end group, or perhaps chemical modification after synthesis, to systematically

change the mass of one component. However, the advantage of such a

situation is that an absolute molecular mass distribution standard can be

created. Appendix A and references therein describe this situation. NIST

Standard Reference Material 2881, "A Polystyrene Absolute Molecular Mass

iii
Distribution Standard," was created this way. This second method is useful

in creating standards, especially of proprietary materials, that may be used to calibrate other types of instruments, for example, chromatography equipment.

Section 1 introduces the broad classes of molecular materials to which this method applies and the basic concepts of polydispersity. molecular mass

distribution, and mass moments. Section 2 gives a conceptual overview

of mass spectrometry and introduces the concept of mass bias. Section 3 describes basic concepts in metrology and delves into the mathematical

rationale behind the method for indistinguishable, overlapping mixtures.

Section 4 gives step-by-step instructions of how to implement the approach

described in Section 3. Appendix A presents the full theory' used to determine systematic uncertainties

for the calibration curve when an absolute molecular mass distribution standard

is desired and requires distinguishable or non-overlapping mixtures. Appendix

B gives a computational approach to implement the method found in Section

4 that can easily be implemented in software. Appendix C lists documentary

standards in force when this document was prepared for the measurement of

molecular mass distribution by mass spectrometry.

This document is available online at: http://wvvw.nist.gov/public_affairs/practiceguides/practiceguides.htm

More information of the NIST Quantitative Polymer Mass Spectrometry

Project can be found at: http://www.nist.gov/maldi

iv
Table of Contents

Preface: Purpose of the guide iii

1. The Molecular Mass Distribution 1

1.1. Importance to molecular materials 1

1 .2. Definition of mass moments 1

1 .3. Measurement methods 2

2. Mass Spectrometry Overview 3

2.1. Ion creation, separation, and detection 3

2.2. Mass resolution requirements 4

2.3. Mass bias 5

3. The Metrology behind Molecular Mass Distribution

Measurement by Mass Spectrometry 6

3.1 . Random and systematic measurement uncertainties 6

3.2. Systematic uncertainties in mass spectrometry 7

3.3. Linearity in signal response versus analyte concentration 9

3.4. Example for mixtures of monodisperse components 10

3.5. Example for mixtures of polydisperse components 11

3.6. Calculating the correction factor for each oligomer 14

4. A Step by Step Procedure for Quantitative Molecular Mass

Distribution Measurement 16

5. Appendix A — Full mathematical theory for

distinguishable/non-overlapping mixtures 19

6. Appendix B — Computer code to obtain estimate of Q/k from

experimental data 24

7. Appendix C — Documentary standards 29

8. References 30

V
Figure 1 — Steps of a complete measurement: data collection, data analysis,

and uncertainty analysis.

Figure 2— Steps in a mass spectrometry experiment including the three core

functions of the mass spectrometer: ion creation, separation by mass to charge ratio, and ion detection.

Figure 3— Schematic illustration of type A (' random") and type B ("systematic")

uncertainties in molecular mass distribution measurement.

Figure 4— Schematic illustration of an indistinguishable and overlapping

mixture of two components where the peak intensities in the

mixture are simply sums of component oligomer intensities.

Figure 5— Schematic illustration of a distinguishable but overlapping mixture

of two components.

Figure 6— Schematic illustration of a non-overlapping mixture of two

components which is distinguishable by definition.

vi
1 The Molecular Mass Distribution

1 The Molecular Mass Distribution

1.1 Importance to molecular materials

As an unavoidable consequence in their synthesis, man made molecular

materials (polymers, dendrimers, macromers. nanoparticles, nanotubes. etc.) are rarely obtained as pure, single molecular mass entities. Rather they are formed as mixtures of molecules having multiple molecular masses with

regularly repeating mass differences indicative of the molecular building blocks

from which they were created. This common feature of molecular materials is called polydispersitv and is described fully and quantitatively by the material's

molecular mass distribution (MMD). Research chemists compare the measured

MMD to predictions from mathematical models of reaction kinetics in order

to a gain fundamental understanding of the synthesis mechanism. Chemical

engineers use the measured MMD for process control in large scale industrial operations. Materials scientists find that the MMD is important in determining

material properties (mechanical, rheological, thermodynamic, etc.) and use it in materials processing and in consumer product design. For this last reason

the MMD is used as a common specification between buyers and sellers in domestic and international commerce. To this end the National Institute of

Standards and Technology (NIST) has been a provider of molecular mass

standards and advances in molecular mass distribution measurement for over 50 years.

1.2 Definition of mass moments

Various averages, known as molecular moments (MM), where the entire shape

of the distribution is reduced to a single number, serve as useful numerical

simplifications of the MMD. Measuring and computing these summary

statistics has historically comprised the core of the analysis of molecular

materials. The two most common measures of the MMD are the number

average molecular mass, Mn

, and the mass average molecular mass, Mw . [

l ] A^=5>,"i Zwi (i)

PD = MH /Mn (3) where m, is the mass of a discrete oligomer i. n, is the number of molecules

at the given mass mi; and PD defines the polydispersity index. When the

polydispersitv index is equal to one (i.e., in statistical terms the variance

1
The f distribution

of MMD is zero), all of the polymer molecules in a sample are of the same

molecular mass and the polymer is referred to as monodisperse. Generally

the mass moments of polydisperse molecular materials give an incomplete

description of the overall MMD because all the information in a complex

curve is reduced to a single value, or a small set of values. Properties like melt viscosity, tensile strength, and impact strength often depend on the

tails of the MMD rather than its central portion that is defined by Mn and

Mw . Consequently, it is critically important to measure the entire MMD.

Furthermore, it is not uncommon, due to purposeful blending of materials, or due to the details of the synthetic chemistry (for example from two competing

mechanisms going on simultaneously), that the polymer MMD will be

multimodal. If so the central moments, Mn and Mw, will be exceedingly poor

representations of the MMD and must be used as descriptors of a material's

molecular mass distribution only with great caution.

1.3 Measurement methods

Chromatographic separations are typically used to measure the MMD with

size-exclusion chromatography and high-performance liquid chromatography

being the most useful for organic molecular materials. However, each suffers from the fact that it is not absolute, that is, each requires calibrants of known

molecular mass, that are chemically and architecturally identical to the material

being studied for proper mass calibration. On the other hand, methods to determine the absolute molecular mass of polydisperse samples return only one moment of the MMD. Mn is obtained by measurement of colligative properties

like osmotic pressure, or alternatively by end group analysis via spectroscopic

or titration techniques. Mw is obtained typically by light scattering or by

ultracentrafugation. These measurements may yield absolute moments but,

because they return only a single moment, are incomplete descriptors of the

MMD. Recently it has been demonstrated by NIST that mass spectrometry

(MS) can be used to determine the absolute MMD. [2] MS combines the mass

dispersive aspects of chromatography with the absolute aspects of the moment

measuring methods. This practice guide describes how mass spectrometry can

be used to find a material's absolute MMD conceptually illustrated in Figure 1.

2
Mass Spe

3) Uncertainty Analysis

Reduced

Data

mil

Analyte

1) Data Collection

HI Mass Spectrum

i, ili,

2) Data Analysis

Figure 1 — Steps of a complete measurement: data collection, data analysis, and

uncertainty analysis.

2. Mass Spectrometry Overview

2.1 Ion creation, separation, and detection

Mass spectrometry (MS) is an analytical technique for the determination of

the mass of individual molecules. As such it is well suited to the study of the

chemical composition of molecular materials. In principle MS entails the

ionization of chemical compounds in vacuum to generate isolated charged

molecules and subsequently separating them by their mass-to-charge ratio. Thus, the mass spectrometer must perform three basic functions: create gas-phase ions, separate these ions by their mass to charge ratio, and detect

the ions once separated. A mass spectrometry experiment schematically

shown in Figure 2. adds to these three basic functions two additional steps:

sample preparation and data analysis. Samples often need some sort of

chemical or physical manipulation before they can be introduced into the mass spectrometer. An example of sample preparation could be dissolution

of the analyte into a solvent with perhaps the addition of an agent to assist

in gas-phase ionization. Energy input is required to drive the molecules into the vacuum and to ionize them. This energy can come from many sources,

for example, lasers in the case of matrix-assisted laser desorption ionization
(MALDI) or high electric field strengths in the case of electrospray ionization

(ESI). The mass spectrometer itself produces a mass spectrum which in its most simple form is a histogram of the number of ions detected versus ion mass to charge ratio. This mass spectrum is then converted into a molecular mass distribution, typically using a computer code. Some computational steps

for this may include baseline subtraction and peak integration.

Figure 2— Steps in a mass spectrometry experiment including the three core functions of the mass spectrometer: ion creation, separation by mass to charge ratio, and ion detection.

2.2 Mass resolution requirements

Mass spectrometers can be thought of as modular instruments where an ion source is coupled to ion separation optics and from there to an ion detector to create a complete instrument. There are about a dozen fundamental ionization

methods embodied in a bewildering array of ion sources. These ion sources may be coupled to about a half-dozen prominent ion separation methods

that are embodied in a wide array of ion optic configurations. Lastly, while

the types of basic ion detection technologies is few er than a half-dozen their number multiplies the possible configurations for any given mass spectrometer.

However, in the end no matter the configuration of the mass spectrometer

only one factor matters for measurement of the molecular mass distribution

of a molecular material by mass spectrometry: can the instrument resolve

the individual oligomers in the sample? Here to "resolve" means having the

ability to separate the mass spectrometric peaks to baseline of tw o oligomers
Mass Bz

whose mass differs by one unit of their (typically) periodic mass spacing. Thi

ability is required for the quantitation methods described in the next section.

An instrument's mass resolution is defined as the smallest mass difference

Am between two equal magnitude peaks such that the valley between them is a specified fraction of the peak height. [3] Similarly, mass resolving power is defined as the observed mass divided by the difference between two masses

that can be separated: m/Am where the method by which Am was obtained

and the mass at which the measurement was made must also be reported. In the method described in this practice guide oligomer mass resolution is a key

requirement.

2.3 Mass bias

The determination of the MMD from a well-resolved mass spectrum

depends on accounting for the mass bias in the measurement. Mass bias

in the systematic over or under counting of specific parts of the molecular mass distribution by the mass spectrometer. Here "specific parts" can

refer to the high mass or low mass parts of a mass spectrum, or to specific

types of oligomers as defined by, for example, end group or molecular

architecture. Mass bias can occur in any of the three basic functions of the mass spectrometer, as well as in the sample preparation or data analysis. By

systematic it is meant that the bias is an inherent aspect of the measurement

method and not simply due to imperfect counting statistics. In the latter case

taking more data will resolve the problem, in the former case taking more

data is not a solution. For systematic bias the magnitude of the bias must be

found and a correction applied, otherwise the measured MMD is of little use. How to perform this correction is what this Recommended Practice Guide

demonstrates.

To achieve the best results in measuring the absolute molecular mass

distribution with mass spectrometry, it is best to minimize as far as possible

the systematic mass bias. It is generally not possible to eliminate this bias

completely; therefore, control experiments must be undertaken that elucidate

the source of mass bias and determine ways to minimize it. This will help

insure that the systematic bias is not so great as to defy correction, for exampl

in the case were certain oligomers are not detected at all which represents a degree of bias that is impossible to correct. Due to the enormous number

of mass spectrometry approaches, this guide does not describe how to best tune an instrument and analyze any given sample to reduce bias. Previous

work in instrument tuning [4], and data analysis [5] suggests some ways to approach this task. Sample preparation is an often overlooked source of mass
>nt by MS

bias through such effects as mass-dependent solubilities. The analyst should

carefully consider all preparation steps and be cognizant of possible sources of mass bias. In Section 4 methods are given to determine how much mass bias is inherent in your chosen method. And how to correct for it if it is within defined

limits. Reducing mass bias will ultimately give a more accurate measured

molecular mass distribution by keeping the correction factors as small as

possible.

3. The Metrology behind Molecular Mass Distribution

Measurement by Mass Spectrometry

3.1 Random and systematic measurement uncertainties

Fundamental metrological principles identify two types of measurement

uncertainty type A and type B. Type A refers to uncertainty that can be

evaluated by the statistical analysis of a series of observations, whereas ty pe

B refers to uncertainty that cannot be evaluated by the statistical analysis of

a series of observations. [6] Generally, type A is spoken of as statistical or random uncertainty and type B as systematic uncertainty. Their differences

applied to the mass spectrometry of organic molecular materials are shown

in Figure 3. This guide is concerned with the determination of type B. Type

A uncertainty that can be determined (and reduced) by repeat measurements

is not explicitly discussed here; however, interlaboratory comparisons [2,7]

have been used to address the statistical uncertainty in the MMD determination

of organic molecular materials measurement by mass spectrometry. It will

be noted that measurement repeatability is critical. If the operator cannot

repeat the measurement from run to run and from day to day the chances of

measuring the correct molecular mass distribution decrease dramatically.

The measurement method must be repeatable and reliable before it can be

considered for quantitative, much less standards, work.

6
Metrology behind M

r

i ill II li;llllJ hi,,,

nil! illIII

Type A, random uncertainty

1 + 1 Type B, systematic uncertainty

iilllllllllillllll r , i

lllllllll

i

1

s

Figure 3— Schematic illustration of type A (random") and type B (''systematic") uncertainties in molecular mass distribution measurement.

3.2 Systematic uncertainties in mass spectrometry

Mam of the physical quantities measured for which uncertainties are determined, for example, the mass of an object, the w avelength of light, or the viscosity of a solution, are single point measurands: therefore, the instrument

calibration and estimates of uncertainty need only be in a single dimension.

Specifically, this applies to the absolute molecular moment determination

methods discussed in Section 1.3. The molecular mass distribution is a two dimensional quantity of which the mass spectrum is its (imperfect)

representation. Thus, both the mass axis and the signal axis (i.e.. the intensity

of the ion signal at a given mass) have to be calibrated separately, and their associated ty pe B uncertainties considered separately.

Mass axis quantification is the most easily performed of the two and is not

a significant source of uncertainty in determining the MMD from the mass

spectrum. Calibration of most mass spectrometers is usually done w ith biopolymers of know n molecular masses. These biopolymers are selected

because they typically provide a single major peak whose mass is known

accurately: thus, mass axis quantification is quite straightforward. Calibration
Metroloc lent by MS

must be done using at least two or three of these biopolymers that span the mass range of interest. More calibration points would increase calibration accuracy.

Calibration of the mass axis can also be done by combining a single

biopolymer with a molecular material calibrant. If this material is close to, or

identical to, the material under study then, in general, inaccuracies in mass axis

calibration will be minimized. The oligomeric masses, m„ with n repeat units

of mass r and masses of the end group, mend, of the polymer calibrant are given

by:

mi= nr + merul+ madduct (4) where madduct refers to the mass of any charged or neutral atoms or molecules

non-covalently bound to the analyte. This may be, for example, any salts added to the sample preparation to encourage charging of the analyte.

Thus, calibration of the mass axis using a homopolymer calibrant (for example)

reduces to determining n for one of the peaks. This is accomplished through

use of the biopolymer mass as follows. The main peak from the biopolymer is assigned to its mass. The biopolymer peak will either lie between the masses

of two n-mers of the calibrant, or exactly correspond to the mass of an n-mer.

If it is at exactly the same mass as one of the w-mers of the calibrant, use

equation 4 to find the degree of polymerization, n, for the «-mer. If the peak

of the biopolymer lies between the masses of two n-mers of the calibrant, use

equation 4 to find n,, the mass of the n-mer whose mass is less than that of

the repeat unit lower than the mass of biopolymer. Find additional calibration

points by selecting calibrant peaks at intervals between five to ten repeat units

less than and greater than n, and compute masses from equation 4. Generally,

a total of four or five calibration points are selected. For obtaining a MMD of a

typical organic material, mass accuracy of better than a few mass units may not be necessary since polymer MMD are often not critically dependent on such

accurate masses.

Calibration of the signal axis is much more difficult and will occupy most of

the rest of this practice guide. There are many systematic uncertainties that can arise in the signal axis quantitation as discussed in the previous section. It would be an insurmountable task to try to quantify each of these uncertainties

individually. Instead, the systematic bias in the signal axis is determined

heuristically by a gravimetric calibration technique. By mixing together

samples having different molecular mass distributions in carefully prepared

8
Metrology behind MMD Measurem

gravimetric ratios, a mixture's molecular mass distribution can be controlled.

By comparing the gravimetric ratios to the total signal intensity in the mass

spectrum, a calibration curve for the signal axis can be obtained.

3.3 Linearity in signal response versus analyte concentration

To estimate the level of uncertainty in an instrumental method, a theoretical construct is needed to determine how type B uncertainties affect the final measurand. Assume that there is a point in the experimental parameter space

(sample preparation, instrument operation, and data analysis) where the signal

intensity, Si , for an oligomer of mass mi is linearly proportional to ni , the

number of polymer molecules at that oligomer mass. Mathematically this is given by:

St = kni (5)

where for a narrow enough range of m, it is assumed that k is a constant independent of irij and the range of linearity, n, < n , is about the same for all molecules in the sample.

If the measurement is performed in the linear region for all the oligomers of the

sample, the overall signal from the quantity of analyte introduced into the mass

spectrometer is given by:

ZSi mi = n mi (6)

with iij rrij summed over all z. From this it can be derived that: Z^/Z^ =kYj ni mi jkYj ni (7)

The right hand side of the equation is the exact Mn of the polymer (see equation

1) independent of k since k in numerator and denominator cancels out. The

same holds for equations for Mw and all higher moments. This is generally true when the measurements are made in the linear range of analyte versus signal

strength. However, it is well known that the mass spectra of wide polydispersity

analytes give poor representations of the molecular mass distribution due to large systematic uncertainties in the signal axis. That is, if the values of the m, span too great a mass range then the values for k and/or the n saturation limits must change dramatically, otherwise mass spectrometry would be able to obtain

the MMD correctly for very broad distribution analytes.

9
Metre Joe d HMD Measurement by MS

If k is not a constant independent of i then, and if the measurements are made

in a linear concentration range for each oligomer /' (that is n,< no), then:

where k, is now a function of the oligomer i for a fixed experimental method:

sample preparation, instrument operation, and data analysis.

3.4 Example for mixtures of monodisperse components

The simplest example of gravimetric quantitation to test mass spectrometer

response is to create a mixture of two monodisperse compounds: species 1 as a standard and species 2 as the analyte whose concentration is sought. If there is no systematic bias in the measurement then the ratio of S2/S] is directly

proportional to the gravimetric mass ratio G2/G, where G, is defined as the

gravimetric mass of each species.

The signal from such a mixture, call it A, is:

S. = ki ni (8)

SA = k] nl +k2 n2 (9)

The mass moments would be:

(10)

[km^ + k2m2n2

(11)

The gravimetric mass of species i is

(12)

Substituting into equation (11)

(13)

(kfix +k2 G2

10
Metrology behind MMD Measurement by MS

To simplify this let the mass fraction X be

G, (14)

Substituting equation (14) into equation (13) and dividing numerator and

denominator by (G, +G2 ) yields: ^

[kx m,X + k2 m2 (\-Xj) (m, X + 6 m2 (\-X))

wA ~ (kx X + k2 {\-X)) (X + 0(\-X))

where

6 =k2 /k} (16)

In this way the mass bias in the mass spectrum is reduced to a single metric, 6. 9 equals one for an unbiased system. If species 2 is overcounted with respect

to species 1 9 will be greater than one, if species 2 is undercounted 9 will be

less than one. The further 9 is from one the greater the systematic bias in the mass spectrum.

3.5 Example for mixtures of polydisperse components

For most mixtures encountered, any given oligomer peak in the mass spectrum

cannot be assigned exclusively to one or the other component of the mixture.

In fact, a given oligomer peak may have contributions from both components

in the mixture. Typically these overlapping molecular mass distributions are made up of indistinguishable-oligomer components, that is, each component

of the mixture has some (but not all) oligomers that are identical to those in the other component as illustrated in Figure 4. This means that in this case

the mass moments of the mixtures must be calculated and used to create a calibration curve. A full theory for the atypical case of distinguishableoligomer mixtures (shown in Figure 5), or non-overlapping molecular mass

distributions (shown in Figure 6), where each oligomer peak can be assigned

to a specific component is given in Appendix A. In this special case, true type

B uncertainties can be given for each oligomer in the target material and a true absolute molecular mass standard can be created. Refer to Appendix A for full details.

11
©gy behind liilllD Measurement by MS

An Indistinguishable, Overlapping Mixture

mass

Figure 4— Schematic illustration of an indistinguishable and overlapping mixture of two components, where the peak intensities in the mixture are simply sums of component

oligomer intensities.

A Distinguishable, Overlapping Mixture

II, Component A Component B jl

III j|

illl 1 mass

Figure 5— Schematic illustration of a distinguishable but overlapping mixture of two components.

12
Metrology behind MMD M

A Nonoverlapping Mixture

Distinguishable (by definition)

mass

Figure 6— Schematic illustration of a non-overlapping mixture of two components which

is distinguishable by definition.

Equation (15) can be extended to a gravimetric mixture of polydisperse

components by substituting the experimental average molecular mass of

each pure component derived from its mass spectrum. This leads to the mass

moments:

fix + k^l-xj) {X + 8Q-X))

Where qrepresents a given gravimetric mixture. In equation 1 7 kx and k2 replace kx and k2 used in the monodisperse example and are the mass-average means over each component of the mixture which is conceptually similar to the mass-average molecular mass. Likewise, Xis now calculated from the

gravimetric amounts of each component in the mixture. The mass moments of

the pure components are from their mass spectra using equation (18):

M7q =5X^7 £v* (is)

13
easurement by MS

To obtain an estimate of the value of 6, the minimum value of the sum of

squares is found. The sum of squares over all mixtures q is expressed as:

'xp Y (19)

The simplest way to solve this equation is to insert an arbitrary value for 8

(typically 9=1 ) and calculate a value for 55e then increment 8 and recalculate

SSq. This most basic iterative process will yield an optimal value typically in a few steps and can easily be encoded in spreadsheet software. Recall that values

of 9 near one indicate systems with little bias in the mass spectrum.

3.6 Calculating the Correction Factor for Each Oligomer

Once 8 has been calculated and found to be near one, the next step in the

process is to calculate the various kx in order to correct the molecular mass

distribution. If the kx are a smoothly and slowly vary ing function of i (or m,). a Taylor expansion on kx may be made around a mass peak near the center of the

MMD, termed M . The center is used to assure that the function is changing as

little as possible over the entire width of the MMD; however, mathematically

the choice is arbitrary. Thus:

K

= + Q(mi ~ M ) + higher order terms in nr (20)

S. = k ni + Q(mj - M )ni + higher order terms in m, (21)

where k and Q are the first two coefficients in the Taylor expansion. They are also functions of all the experimental conditions: the instrument parameters,

the sample concentrations, and the sample preparation method. (By k it is not meant the k of the zeroth index oligomer but rather the zeroth derivative

of the Taylor expansion). In this way the entire physics of the experiment is folded into these two coefficients. From these assumptions, and dropping the

higher order terms in equation (21) one can derive the following important

relationship:

>l + (Q/kn )(PD M° -Mn )\ Mexp = M° <^ - — ^ ^ — [ (22)

vg

\ + (Q/k )(Ml-M ) where M^p is the mass spectral mass-average molecular mass for the mixture

of analytes given in equation (18). PDW is mass average polydispersity

(M/Mw) and is taken here to be the experimentally measured value (M/^/M^).

14
Metrology behind MMD Measi

Equation (22) is then solved for M°wq for various values of Q/k at a fixed M

chosen as described below for the values of the mixtures described by q = A, B,

C, etc. and for the initial components of the mixtures described as j = 1 and j = 2.

For a gravimetric mixture A, M^v -° is calculated from the values for the

individual components M°w] and M°w2 computed for each Q/k using a simple

weighted average:

Mgrav_0 = Q M + ^^

M (23)

G,+G2 w] G,+G2 wl

where G, is the gravimetric mass of component 1 in the mix, and G2 is similarly defined.

For each Q/k the sum of squares, SS,Q/k ] , is computed as:

(24)

where the sum is taken over all measured mixtures. The Q/k which gives the

minimum value of the SS,QI

. is then taken as the best fit. As with equation

(19), solution of equation (24) required iteration over incremented values of Q/k .

Dropping the higher order terms and rearranging equation (21) yields: —J- = \ + —(mi -Ma ) kn n. kri (25)

Equation (25) shows us how to apply the correction factor Q/k to each

oligomer i to arrive at a more reliable measure of the molecular mass

distribution. If Q/k were equal to zero, then the mass spectrum would show

no mass bias and St = k n,. This would mean that the peak areas are directly

proportional to the oligomer concentrations in the sample. If Q/k is non-zero,

then mass bias is present. If M is taken at the middle of the distribution being

calibrated, then the sign of Q/k along with where the mass ml of an oligomer i is greater than or less than M determines if the correction to the ion intensity is positive or negative.

15
"ill Step by Step P Quantitation

4. Step by Step Procedure for Quantitation

The steps of the method can be summarized as follows:

1 . Obtain at least two samples having different molecular mass distributions

but with otherwise very similar, if not identical, properties.

For example, these could be polymers with different degrees of

polymerization or nanoparticles with different levels of functionalization.

The different samples could be obtained directly by synthesis or by

separation of a single broader molecular mass sample. Two samples

are required at a minimum, but additional samples will allow for more

calibration points. If possible the only difference between the two should

be molecular mass. Any other differences, for example, different functional

groups may contribute to mass bias in an uncontrolled way.

2. Take mass spectra of each sample endeavoring to keep all experimental

conditions constant. As much as possible keep all aspects of the measurement constant. This

includes sample preparation, instrument settings, and data analysis. Also,

measurements should be made contemporaneously to keep constant any

variables that may change over time. These variables could be sample

preparation conditions, for example water absorption into samples or

solvents, or time drift in instrument settings.

3. Use a laboratory balance to make carefully controlled gravimetric

mixtures of two samples in several well-spaced ratios. The balance needs to be calibrated and accurate to about least 0.1% of the

total mass measured. Any gravimetric errors are carried through the entire

analysis. Making stock solutions and then mixing solution volumes can be more accurate than repeated weighing of small amounts of material.

Generally, as a practical matter, final weights must be at least 25 mg.

4. Take mass spectra of each mixture using the same experimental

conditions as used for the pure components.

The instrument settings may not be optimal for the mixtures, but they

must be held constant to satisfy the self consistency of the method. If the

experimental conditions are such that some oligomers of the mixture have

16
Step by Step Procedure for Quantitation

disappeared (as compared to the pure component measurements), then

compromise experimental conditions must be found. If this occurs, then it suggests strong mass bias in the measurement.

5. From the mass spectra calculate the mass-average molecular masses of

the pure components and of the mixtures.

Be careful in the application of "black box" software for this step. Unseen

algorithms for data processing can lead to substantial errors in converting

the mass spectrum to a molecular mass distribution. [7] Smoothing

can introduce mass bias into a spectrum that is not a product of the

measurement itself but of the data analysis method applied. [5]

6. Use equation (19) to iteratively calculate the minimum value of SSe at a given 6. The most direct way to do this is to set up a simple spread sheet. Start with 6=1 and change it systematically by small steps until a minimum in SSe is found. If 9 is between 0.5 and 2 then the possibility exists that the

molecular mass distribution can be corrected. If not, the results should be

treated with caution, and the error is too great to be corrected using only

the linear term in the Taylor expansion. See Appendix B for an example of

computer code to make this calculation.

7. Choose M , a mass near the center of the average molecular masses of the

two components.

The exact choice of M is not critical; however, the correction to the

distribution will be more accurate near M and less accurate the farther any

given oligomer mass is from M . If a certain mass range is more critical, then

choose M at the center of that range.

8. Use equation (24) to iteratively calculate the minimum value of SS(Q/k ) at a given Q/k .

See Appendix B for computational assistance.

9. Use equation (25) and the value for Q/k to correct the ion intensities S, in the mass spectrum to arrive at a new molecular mass distribution.

17
ntitation

Individual oligomer intensities may increase or decrease depending on

whether they were under counted or over counted in the mass spectrum.

At this stage the analyst should have a good feel for the degree of mass bias in the mass spectra. Furthermore, if this bias is not too large it can be corrected

using the methods outlined in this section. If the bias is large, higher order

terms in equations (20) and (21 ) need to invoked; however, methods to determine the values of the higher order coefficients have not been created. This is a fruitful topic for future research.

The procedures outlined in this section do not provide systematic uncertainties

for the corrected values. The corrected mass spectrum is closer to the true molecular mass distribution, but just how close is it? In order to determine

this, the procedures of Appendix A must be invoked. These procedures

require distinguishable or non-overlapping mixtures as well as numerical

instrument optimization [4] to determine the systematic uncertainties inherent

in the instrument. This requires extra effort on the part of the analyst, but a molecular mass distribution with both type A (random) and type B (systematic)

uncertainties is a very useful calibration standard for mass spectrometry and

any other molecular mass measurement technique.

18
Appendix A

5. Appendix A — Full mathematical theory for distinguishable/

non-overlapping mixtures

If the desired outcome of a quantitation effort is the certification of an absolute

molecular mass distribution reference material, including both type A and type

B uncertainties, then calibration of the signal intensity axis must be performed

using mixtures of analytes where each oligomer peak in the mass spectrum

can be identified with a specific component of the gravimetric mixture. These

are termed distinguishable-oligomer mixtures. The non-overlapping aspect

of the mixture can be created in two ways. The first is by having molecular mass distributions that are well separated so that the highest mass oligomers

of one are lower than the lowest mass oligomers of the other. The difficulty

here is the requirement of a linear (not quadratic or higher) correction term. Over wide mass ranges higher order correction terms are more likely to be

required. The second is by using components that are chemically similar with one distinguishing mass feature. For example, in the case of NIST Standard

Reference Material 2881 [2] polystyrene homopolymers with different

alpha end groups (butyl and octyl) but the same omega end group were used

to positively identify which oligomers belong to which component in the

gravimetric mixture. Identification was possible even though the mass ranges

of the components overlapped. The inert nature of the alkyl end groups was

shown not to introduce mass bias into the mass spectrometry experiments.

Starting with equation 8 from Section 3.3, S,=kA, if an assumption is made that

kj is a slowly varying function of i (hence also of m^, then a Taylor expansion

around a mass peak near the center of the MMD, termed M , can be made. The

center of the mass spectrum is used to assure that the function is changing as

little as possible over the entire width of the MMD. Then:

Sf = k n. + Q{mi - M )nt + higher order terms in m- (A. 1

)

Here Q and k are functions of M as well as of all the experimental conditions:

the instrument parameters, the sample concentrations, and the sample

preparation method. (By k it is not meant the k of the zeroth index oligomer

but rather the zeroth derivative of the Taylor expansion). In the experimental

procedures, once the instrument parameters and experimental preparation

methods are optimized, every attempt should be made to keep them constant

to insure experimental reproducibility. Variation in the machine parameters can

affect the variation of Q/k and thus the type B uncertainty.

19
Appendix A

The implications of the model embodied in equation (A. 1 ) will now be

explored and it will be shown how small linear shifts of the calibration constant Q over limited mass ranges effects quantities derivable from mass spectral data.

First the total signal, the total detected mass, and the mass ratios of mixtures

will be considered, and it will be shown how these quantities relate to the true MMD of the analyte.

The total signal, ST, from the polymer is given by:

ST = XSt

, = K + 0( M

{) n - M n (A. 2)

while the total mass of polymer detected, G^xp

, is given by:

G;

xp =

y£mi Sl =k i M°n ^ni +QM°(M°w -Mo )^ni (A.3)

where M°n and MQware defined in equations (A.4) and (A. 5), and are the true number average and mass average molecular masses.

K° =ZW<VZ"W (A.5)

PD°n =MJMn (A.6)

M°z = ^mi ~ni I ^mi 2 ni (A. 7)

PD°w =MJMw

(A.8)

where mx is the mass of a discrete oligomer, n, is the number of molecules at the

given mass mx . The experimental moments from mass spectrum are defined as Mn , Mw, and Mz, while the true values are given as Mfl°, M^and M°. PDn defines

the polydispersity index which is a measure of the breadth of the polymer

distribution. When PDn is equal to one (i.e., in statistical terms the variance

20
of MMD is zero), all of the polymer molecules in a sample are of the same

molecular mass and the polymer is referred to as monodisperse.

Multiplying equations (A.4) and (A. 5) together gives:

M°w M°n = £m~ni I ]T w. (A .9)

i i Then taking the ratio of equations (A.2) and (A. 3), one obtains:

m:p =i>,s,/I>, ( a. io)

with the result that:

{(}HQ'K)(M°n -M )l

< An > where Men xp

is the experimentally-measured M\ .

For use later in this section by the same algebra is obtained:

ATxp =Yj mfSi /Yj m,Si (A. 12)

with the result that:

ro \(l + (Q/ko )(M° Mexp z -Mo ) = M

(\ + (Q/ko )(M:-Mo ) (A.13)

All higher moments may be obtained in a similar way and have a similar form.

Equation (A.13) gives by simple division:

\\ + (Qlko )(M°z -Mo )\ {AA4)

which yields:

M° = Mexp

{ 1 (Q/ko )M°,(PDw -\)

\ + (Q/ko )(M°z -Mo ) (A.15)

Equation (A.l 5) states that the deviation of the mass moment measured by mass spectrum from the true mass moment is a function of the polydispersity

(PD) (arising from that moment) divided by a correction term arising from how

far that moment is from the mass Mo around which the Taylor expansion to obtain k and Q is centered. In equation (A.15) the reader should notice that if

21
Appendix A

M° is close to Mo the term in (Qlk )(M° - Ma

) is small compared to 1 and the

result depends only on the polydispersity of the polymer.

Since the method depends on gravimetrically mixing analytes to obtain

estimates of Q/k , it is necessary to consider the equations relating to these

mixtures. Equation (A. 3), states that the MS measured total mass, Gr

exp

, is proportional to the true mass, G°

:

GT= M^ni (A. 16)

G-p = ko G°T {

1 + (Q I ko ){Ml - M )}

(A. 1 7)

Consider now a mixture of the chemically identical analytes with functional

groups having different masses, or two different molecular mass analytes

having distributions that are well separated, such that each oligomer in the mass spectrum can be assigned to a specific polymer in the mixture. Call them

analyte A and analyte B that will make up the components of the gravimetric

mixtures. Then the measured ratio of the masses of each is: G

exp _ koA GTA G

exp TB koB^TB 1

1 +(&/*„ XJ»C -K)

Notice the expansions are performed for both polymer distributions A and B

around the same M . Note QA ,QB ,koA and koB are all functions of Mo.

22
Appendh

Thus from equation (A. 18):

G°TA \

\ + (Q/ko )(M:,A -Mo ) G°TB \\ + (Q/ko )(M:B -Mo ) (A. 19)

Simple algebra leads us to:

; exp qo

TA _ ^TA

'exp s~<o 'TB ^TB (Q' WML -Kb)

l + (Q/k )(M:B -MJ

(A.20)

What are measured are from mass spectrometry versus gravimetrically

determined. The calculated slope is:

)tice if M°wB is close

to Mo the term in {Ql

o )(M°wB -o

) is small compared to 1 which means the

slope depends only on the difference (M°wA -M°wB ) and, thus, (Q/ka ) maybe

easily calculated. This concept with these equations can be used to obtain

estimates of {Q/ko )m a self consistent approach to the data analysis. Lastly,

remember that the gravimetric calibration of the signal axis using chemically

identical analytes can avoid the issues pertaining to the uncertainties arising

from ablation, ionization, and detection. However, uncertainties in sample

preparation as well as data analysis repeatability and consistency still affect the

gravimetric calibration techniques.

23
Appendi)

6. Appendix B — Computer Code to Obtain Estimate of Q/ko

from Experimental Data

This section gives sample Visual Basic (VBA) code to solve for M°w by

varying Q/k (referred to as Qset in the code) assuming PD = PDexp . Using

equation 22 from Section 4 for the M*xp of a mixture made up of components

j = 1 and 2:

'

\ + (Q/k )(PDwq M: Mexp q -K)} (EU)

wq = Mlwq

where the subscript q can take on values of A, B, C, etc. for various gravimetric

mixtures of components 1 and 2. Assuming PDwq = PDTMP

, then M° is quadratic

and is solved for each j. From equation (23) of Section 4:

AfgrBV-° = — M\ + —^—M\ ( B- 2 ) wq G,+G: w] G,+G2 u2

Equation (24) from Section 4 is then invoked. That is, the sum of squares of the

difference between M^TM-° and M°wq with PD = PDexp is calculated for any

Q/k for all the mixtures:

sv.u = I(M»T-°-<) <B3)

all _q Equation (B.3) is solved iteratively on a simple spread sheet with Q/ko varied in steps over an expected range. M is arbitrarily chosen near the middle of the

component spectrum to be quantified. Once the code calculates a range of

SS(Q/k ) values, the minimum is found by simple inspection. This minimum

gives the slope of the correction line, Q/k , to be applied to each peak in the mass spectrum. If the minimum is too broad the algorithm can be repeated

with a smaller step size around the minimum.

24
Appendix B

Sample VBACode:

' All lines with single quote mark, " ' " , at beginning of line are comments

4 All line with Cells function or with Irow or Icol variables in them are for

1 input or output onto the spreadsheet

4 Input Experimental MMD moments of mixtures as well as Mn, Mw and Mz

* of two species being mixed

4 Input gravimetric mixtures fractions of each of the mixtures

4 Need to have at least 3 mixtures

4 Vary Q/ko (as Qset) to compute Mno (or Mntrue) assuming PDo=PDexp .

4 Using Mno for each species in mixture

4 and fraction of each species computing M_grav computing M grav n

4 moment

4 Try to optimized quadratic difference between Mn grav and Mn true for

4 mixtures of first and last species

4 The next few lines of code take equation B.l and solve the quadratic in MwjO

4 (called M_true(2,ik))for

4 where ik=j

4 equations assume PDo is known and is PDexp

4 equations Rw=M_exp(2,ik)*Qset

4 equations for Mw Uw=l/(2*PD(2,ik)*Qset)

4 equation square Vw=((l-Ro*Qset-Qset*M_exp(2,ik))A2+2*

(l-Ro*Qset)*Qset*M_exp(2,ik)

£ ifVw<0 then stop 003

4 equation M_true(2,ik)= Uw*((l-Ro*Qset-Qset*M_exp(2,ik))+sqr(Vw))

4 equation M_true(2,ik)= Uw*((l-Ro*Qset-Qset*M_exp(2,ik))-sqr(Vw))

4 check on whether + or - is correct by seeing if in limits

4 The next few lines of code take equation B.l and solve the quadratic in MnjO

4 (called M_true(l,ik))for

4 where ik=j

4 equations for Mn Un=l/(2*PD(l,ik)*Qset)

4 equation Vn=(( 1 -Ro*Qset-Qset*M_exp( 1 ,ik))A2+2*( 1 -Ro*Qset)*Qset*M_

4 exp(l,ik)

4 ifVn<0 then stop 003

4 equation M_true(l,ik)= Un*((l-Ro*Qset-Qset*M_exp(l,ik))+sqr(Vn))

4 equation M_true(l,ik)= Un*((l-Ro*Qset-Qset*M_exp(l,ik))-sqr(Vn))

4 check on whether + or - is correct by seeing if in limits equation for polydispersity

4 equation PD_exp(i,ik)=PD_true(i,ik)*(l-Ro*Qset+Qset*M_true(i+2,ik))*(l4 Ro*Qset+Qset*M_true(i,ik))/((l-Ro*Qset+Qset*M_true(i+l,ik))A

2)

25
Appendix B

'Input Data for run other the raw data

LimitUp = 1 147 'data from range of distribution—generally upper limit of

smallest peak seen LimitLow = 720 'data from range of distribution—generally lower limit of

smallest peak seen Mo = 950

Mo = Cells(Irowin - 4, Icolin + 1

)

' The values below are established for our specific data—reader needs to estimate his own limits LowQSet = -0.0043

StepQset = 0.00012

IstepMaxQset = 100

QupperLim = Abs( 1 / (Mo - LimitUp))

QLowLim = Abs(l / (Mo - LimitLow))

QLim = QupperLim

If QLim > QLowLim Then QLim = QLowLim

' fix up bounds on limits and outputs

'output starts here

Cells(Irowin - 5, 1) = "MacroName="

Cells(Irowin - 5, 3) = "Opt_Qset_Moments_Mixtures_Plus()"

Cells(Irowin - 4, l) = "Mo="

Cells(Irowin - 4, 2) = Mo

Cells(Irowin - 4, 3) = "QLim="

Cells(Irowin - 4, 4) = QLim

' Icol set previously in software

IcolO = Icol + 2

For IQavg = 1 To IstepMaxQset

' setQ/ko

Qset = LowQSet + StepQset * IQavg

' this section is shifting active cells for output

irowStart = Irowin + IQavg * (NumSheet + 7)

Cells(irowStart - 3, IcolO) = "Qset="

Cells(irowStart - 3, IcolO + 1) = Qset

icol = IcolO

Irow = irowStart

For ik = 1 To NumSheet

PDw(ik) = M_exp(3, ik) / M_exp(2, ik)

' equations to solve quadratic in Mwo for various values of Qset

Uw = 1 7 (PDw(ik) * Qset)

26
Ww = -( 1 - Mo * Qset - Qset * M_exp(2, ik)) / 2 Vw = Ww A

2 + PDw(ik) * (1 - Mo * Qset) * Qset * M_exp(2, ik)

IfVw<OThen

Cells(Irow, IcolO + 5) ~ " Vw negative-thus skip"

GoTo stop Vn

End If

4 M_true(2, ik) is Mw

° M_true(2, ik) = Uw * (Ww + Sqr(Vw))

4 input Cells(Irow, icol + 1) = M_true(2, ik)

*M_true(2, ik) = Uw * (Ww - Sqr(Vw))

4 check on whether + or - is correct by seeing if in limits

4 output Mw

° spread sheet

4 Cells(irow + 1, icol + 1) = M_true(2, ik)

Irow = Irow + 1 Next ik Cells(irowStart - 2, icol + 1) = "Mw true"

icol = icol + 2

Irow = irowStart

For ik = 1 To NumSheet

4 calculate Mw

°-^rav using equation B2

4 FractionSeriesl(ik) refers to mass fraction of series

4 M_grav(2,ik) is Mw°-rav M_grav(2, ik) = (FractionSeriesl(ik) * M_true(2, 1) + (1 - FractionSeriesl(ik))

* M_true(2, lend))

4 Output Mw -^av to spread sheet

Cells(Irow, icol + 1) = M_grav(2, ik)

Irow = Irow + 1 Next ik Irow = irowStart

SumVar2 =

For ik = 1 To NumSheet

M_Var(2, ik) = (M_true(2, ik) - M_grav(2, ik)) A

2

SumVar2 = SumVar2 + M_Var(2, ik)

4 Output squared differences

Cells(Irow, IcolO + 7) = M_Var(2, ik)

Irow = Irow + 1 Next ik Irow = Irow + 1 output Sum of squares are in equation B4

27
Appendix B

Cells(Irow, IcolO + 7) = SumVar2

Irow = I row + 1

' Label column

Cel ls( irow Start - 2, icol + 1) = "Mwgrav"

stopVn:

Next IQavg

End Sub

28
7. Appendix C — Documentary standards

Documentary standards for quantitative molecular mass distribution

measurement by mass spectrometry in force at the time of publication of this Recommended Practice Guide.

1. ASTM International D7034-05 "Standard Test Method for Molecular Mass

Averages and Molecular Mass Distribution of Atactic Polystyrene by Matrix

Assisted Laser Desorption/Ionization (MALDI) Time of Flight (TOF) Mass

Spectrometry (MS)''; http://www.astm.org/

2. International Organization for Standardization (ISO) DIS 10927

"Determination of the molecular mass and molecular mass distribution of

polymer species by matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDITOF-MS)"; http://www.iso.org/

3. Deutsches Institut furNormung (DIN) 55674 "Synthetic Polymers

- Determination of molecular mass and molecular mass distribution of

polymers by matrix assisted laser desorption/ionization-time-of-flight-mass

spectrometry"; http://www.din.de/

29
8. References

[1] According to International Organization for Standardization standard

31-8 (31. Quantities and units — Part 8: Physical chemistry and

molecular physics), the term "molecular mass" has been replaced by

"relative molecular mass," symbol Mr

. If that nomenclature and notation were followed in this publication, one should write Mr u instead of the

historically conventional Mw for the weight average molecular mass with

similar changes for Mn , and Mz . Mw would be called the "mass average

relative molecular mass.*' The conventional notation, rather than the ISO

notation, has been used in this publication.

[2] CM. Guttman, K.M. Flynn, W.E. Wallace, A.J. Kearsley, Quantitative

Mass Spectrometry and Polydisperse Materials: Creation of an Absolute

Molecular Mass Distribution Polymer Standard, Macromolecules 42

(2009) 1695-1702.

[3] International Union of Pure and Applied Chemistry (IUPAC)

"Standard Definitions of Terms Relating to Mass Spectrometry";

http://www.iupac.org/web/ins/2003-056-2-500

[4] W.E. Wallace, CM. Guttman, K.M. Flynn, A.J. Kearsley, Numerical

optimization of matrix-assisted laser desorption/ionization time-of-flight mass spectrometry: Application to synthetic polymer molecular mass

distribution measurement, Anal. Chim. Acta 604 (2007) 62-68.

[5] W.E. Wallace, A.J. Kearsley, CM. Guttman, An Operator-Independent

Approach to Mass Spectral Peak Identification and Integration, Anal.

Chem. 76 (2004) 2446-2452.

[6] Guide to the Expression of Uncertainty in Measurement, International

Organization for Standardization, Geneve, Switzerland, 1993.

[7] CM. Guttman, S.J. Wetzel, W.R. Blair, B.M. Fanconi, J.E. Girard.

R.J. Goldschmidt, W.E. Wallace, D.L. VanderHart, NIST-Sponsored

Interlaboratory Comparison of Polystyrene Mass Distribution Obtained

by Matrix-Assisted Laser Desorption/ionization Time of Flight Mass

Spectrometry, Anal. Chem. 73 (2001) 1252-1262.

30

4 II" 1 cites.
Monties

i
April 2010