# Materials Science/Structure of Matter
## Structure of Matter
### Atomic Structure and Bonding

Fundamentally, two types of bonding exist- bonds between atoms and bonds between ions. Bonds between atoms of nonmetals are covalent, meaning that they share a pair of electrons in the space between them. These two atoms are bound together and cannot be separated by simple physical means. If these two atoms have similar electronegativity, neither atom has more pull on the electron pair than the other. This type of covalent bond is called Non Polar. Examples of non polar covalent compounds are methane, carbon dioxide and graphite. In graphite, all atoms are identical and so no atom has stronger pull than any of the others. In methane, the carbon-hydrogen bonds are very slightly polar, and the polarities are cancelled because the bonds all point to the same locus. Further there exists a weaker type of bond called hydrogen bonds important in complex molecules such as proteins. These form weak bonds that give complex molecules like Chlorophyll its specific shape and properties. The kinds of bonds and the structure of the molecules affect the microscopic properties of substances.

### Bonding Forces and Energies

Attractive Coulombic Force between charges.
$Z_{1}$ and $Z_{2}$ are the valences of the ions
| | Attractive Forces | Repulsive Forces | Energy |
| Formula | $F_{A}$= ${\frac {1}{4\pi \epsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}$ = $k_{0}{\frac {e^{2}Z_{1}Z_{2}}{r^{2}}}$ | $F_{R}{=}{\frac {-b^{n}}{r^{n+1}}}$ | $E_{net}$ = $\int F_{A}dr+\int F_{R}dr$ = ${\frac {-A}{r}}+{\frac {B}{r^{n}}}$ |
| F/E plot vs r | F vs r -> modulus (stiffness | See left | Thermal expansion coefficiant melt temperature binding energy minimum is equilibrium distance |
Where A= $k_{0}e^{2}Z_{1}Z_{2}$ and B is found using an empirical plot

### Bonding

-Percent Ionic Character - Tells how much of the bond between element A and B is ionic and covalent, based on electronegativity X
$\%IC{=}$ $(1-e^{-(.25)(X_{A}-X_{B})^{2}})*100\%$
| Directional Bonds | Secondary Covalent |
| Non-Directional bonds | Metallic Ionic |
In order of increasing intermolecular force strength:
fluctuating induced dipole < polar molecule induced dipole < hydrogen bonding (permanent dipole moment)

## Structures of Metals and of Ceramics

Common symbols :
| Symbol | Definition | Units (SI) |
| $\rho$ | density | $kg/m^{3}$ |
| n | number of atoms/unit cell | 1 |
| n' | num of formula units/ unit cell | 1 |
| M | atomic weight | g/mol |
| $A_{C}$ | atomic weight of cations in formula unit | g/mol |
| $A_{A}$ | atomic weight of anions in formula unit | g/mol |
| $V_{C}$ | volume of cell | m^3 |
| $N_{A}$ | Avogadro's Constant | atom/mol |

### Density of Metals and Ceramics

| | Metals | Ceramics |
| Density | $\rho {=}{\frac {nM}{V_{C}N_{A}}}$ | $\rho {=}{\frac {n'(\sum A_{C}+\sum A_{A})}{V_{C}N_{A}}}$ |
| How dense | More Dense | Less dense |
| Why dense | metallic bond -> close packing large atomic mass | covalent bonds lighter mass |

### Ceramic Crystal Structure

* In ceramics with ionic charachter, the magnitude of the electrical charge on each ion and the relative sizes of the ions partly determines the structure
* The charges of ions shows the ratio- the crystal must be neutral
* The number of ion neighbors of opposite charge is maximized
* Number of large anions that are able to surround small cation fixed by cation/anion radius ratio
* coordination number increases with ${\frac {r_{cation}}{r_{anion}}}$

#### Table of Major types of Ionic Ceramic Crystal Structures

| Ceramic Structure | geometry | Anion Packing | Coordination number, anions | Coordination number, cations | Structure Stoichiometry |
| Sodium chloride | linear | FCC | 6 | 6 | AX |
| Zincblende | tetrahedral | FCC | 4 | 4 | AX |
| Cesium chloride | tri-planar | Simple Cubic | 8 | 8 | AX |
| Fluorite | Octahedral | Simple Cubic | 4 | 8 | $AX_{2}$ |

#### Atomic Packing Factor

APF= ${\frac {Volumeofatomsinaunitcell}{TotalUnitCellvolume}}=$ $N{\dot {\frac {{\frac {4}{3}}\pi R^{2}}{a^{3}}}}{=}{\frac {V_{s}}{V_{C}}}$

### Table of Metal Crystal Structures

| | Body-Centered Cubic | Face-Centered Cubic | Hexagonal Close Packed | Simple Cubic |
| Coordination Number | 8 | 12 | 12 | 6 |
| Unit cell-radius relationships | $4r=a_{0}{\sqrt {3}}$ | $4r{=}a_{0}{\sqrt {2}}$ | $2r=a_{0}$ $c/a_{0}=1.63$ | $a_{0}=2r$ |
| Volume | $a^{3}$ | $a^{3}$ | $a^{2}c\cos {(30)}$ | $a^{3}$ |
| Stacking Sequence | N/A | A-B-C | A-B | |
| Atoms/ unit cell | 2 | 4 | 6 | 1 |
| Atomic Packing Factor | .68 | .74 | .74 | |
| Close-packed planes | [0001] | [111] | none | |
| Close-packed directions | $<{\bar {1}}120>$ | <110> | <111> | |
| Ceramic Structure | | NaCl, Zincblende | | (Simple cubic)CsCl, Fluorite |

### Miller Indices for Points, Vectors, and Planes

For points and Vectors:
Simpler form:
* find $\Delta x,\Delta y,\Delta z$ scale them to the nearest integer
For Planes:
* If the plane in question passes through the origin, create new origin
* Note the incercepts of the plane in terms of x,y,z
(a) if intersection is entire axis the value is $\infty$
(b) If plane parallels an axis the value is $\infty$
* Take reciprocals of found intercepts
* Reduce to smallest integers
* Enclose in parentheses without commas
[info] [list of directions] [comparison table]

### Linear and Planar densities
## Polymer Structures
### Degree of Polymerization

DP=average number of repeat units per chain
$DP={MolarWeightofpolymer({\bar {M_{n}}}) \over Molarweightofsubstituent({\bar {m}})}$
${\bar {M_{n}}}{=}\sum x_{i}M_{i}<br/>{\bar {M_{w}}}{=}\sum w_{i}M_{i}<br/>{\bar {m}}$ = average molecular weight of repeat unit

### Molecular Structure and Tacity

| Polymer property | Meaning |
| Linear | repeat units joined end-to-end in one chain |
| Branched | Has side-branch chains connected to main chains |
| Crosslinked | Adjacent linear chains connected at various positions by covalent bonds |
| Network | 3D network of multifunctional monomers |
| Isotactic | All substituents on same side of monomolecular backbone |
| Syndiotactic | alternating positions along chain (down/up) |
| Atactic | substituents placed randomly on chain |

### Thermal Behavior

| | Thermoplastics | Thermosets | Elastomers |
| Example | Polyethylene | Polyurethanes | Natural Rubber |
| Response to heating | Heat induced malliability | decompose when heateed |
| Re-shaping | easy to reshape | brittle |
| Crosslinking | minimal; long chains | extensive; covalent bonds |
| Structure | Linear + branched | network w/ cross-links | Thermoplastics or lightly cross-linked thermosets; made of spring-like molecules |
| Cooling response | weak forces reform to new shape | Fast cool -> greater volume Slow cool -> smaller volume; more rigid + dense |

### Copolymers

* Homopolymers are the term for pure polymers
* Copolymers have differing repeat units
Ex: PVC-C-PE is a copolymer (C stands for copolymer)
Types of Copolymers :
| Name | Definition | Example |
| Random | No pattern | AABABBABBBABAA |
| Alternating | Directly alternating units | ABABABABA |
| Block | One block identical, block other | AAAABBBBAAAA |
| Graft | Main homopolymer chain with grafted homopolymer side branches | |

### Crystallinity

* Crystalline regions of polymers charachterized by chain folded structures
* Higher indices of refraction than amorphous
* Electrical insulators
* Mechanically light
* Chemically inert
* Solid at STP
* Low density Polymers --> high optical transparency
* High density Polymers --> opaque
* Higher molecular weight--> less crystalization, longer chains more difficult to align to array
* Heat treating increases % crystallinity
* n= number of repeat units per cell
$n{=}{\rho V_{C}N_{A} \over A}$
% Crystallinity = ${\rho _{c}(\rho _{s}-\rho _{a}) \over \rho _{s}(\rho _{c}-\rho _{a})}\times 100\%$
$\rho _{c}=densityperfectlycrystallinepolymer<br/>\rho _{a}=densitytotallyamorphouspolymer<br/>\rho _{s}=densityofsamplepolymer$

## Crystal Structure

Some of the properties of crystalline solids depends on the crystal structure of the material, the manner in which atoms, ions or molecules are spatially arranged.

## Defects

Defects are the small gaps that develop between crystal layers where the continuation of the layer is interrupted by a boundary of a different crystal layer. Because the crystals do not perfectly align, small gaps are created in between the crystals where they meet. These gaps are called defects. Defects of materials are subject to intense study. However there are some methods to determine the source of defects and, if occurred, the size, shape and position of defects in the materials. There are: destructive testing methods and Non destructive testing methods (NDT).
* material permanently deformed
* can be mixed, many are
* metals many slip planes
| Symbol | Term | Meaning | Notes | Subject |
| $c_{1}$ | weight based impurity | |
| $c_{1}^{'}$ | Atom based impurity |
| | edge dislocation | half plane added b is perpendicular to dislocation line |
| | Screw Disclocation | perfect cut and torsional stress, dislocated | plane of distortion one atom spacing | dislocation line |
| b | bergers vector | amount of distortion, magnitude and direction -> close loop |
| | dislocation line | |
| | slip plane | plane on which dislocation move | plane with highest Planar atomic density, |
| Direction slip | metals most closely packed |
| | imperfection |
| grain boundaries | 4 degrees demarcation |
| crystallites | (grains) |
| | large angle grain boundaries |
| N | num grains/in^2 | $2^{n-1}$ |
| n | ASTM grain size number | |
| etching | to preferentially attack grain area |
| grain size | effect yield strength | inverse sqrt relationship | evident by etching |
| small angle grain boundaries | produced by array of dislocation | not that important |
* covalent, ionic ceramic motion hard
* high force to break bonds
* ionic ceramics have repulsion when move
Metals:
* move in close packed planes
* FCC many close packed planes, directions
* HCP only 1 plane, 3 directions
* many planes-> more plastic
* one plane, more brittle
* move by breaking, remaking atomic bonds
* plastic deformation produce dislocation motion

## Diffusion

When one substance moves into another

## Glossary

| Term | Definition | Notes |
| crystalline material | periodic array of atoms over large atomic distances | |
| unit cells | small, repeating entities of crystalline material | Example |
| mer | unit cell of a polymer | |
| Polymorphism | when metal or nonmetal may have more than one crystal structure | see: allotropy |
| Allotropy | polymorphism for elemental solids specifically | see: Polymorphism |