# 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.
${\displaystyle Z_{1}}$ and ${\displaystyle Z_{2}}$ are the valences of the ions

Attractive Forces Repulsive Forces Energy
Formula ${\displaystyle F_{A}}$= ${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}}$ = ${\displaystyle k_{0}{\frac {e^{2}Z_{1}Z_{2}}{r^{2}}}}$ ${\displaystyle F_{R}{=}{\frac {-b^{n}}{r^{n+1}}}}$ ${\displaystyle E_{net}}$ = ${\displaystyle \int F_{A}dr+\int F_{R}dr}$ = ${\displaystyle {\frac {-A}{r}}+{\frac {B}{r^{n}}}}$
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= ${\displaystyle k_{0}e^{2}Z_{1}Z_{2}}$ 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

${\displaystyle \%IC{=}}$ ${\displaystyle (1-e^{-(.25)(X_{A}-X_{B})^{2}})*100\%}$

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

### Density of Metals and Ceramics

Metals Ceramics
Density ${\displaystyle \rho {=}{\frac {nM}{V_{C}N_{A}}}}$ ${\displaystyle \rho {=}{\frac {n'(\sum A_{C}+\sum A_{A})}{V_{C}N_{A}}}}$
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 ${\displaystyle {\frac {r_{cation}}{r_{anion}}}}$

#### 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 Octohedral Simple Cubic 4 8 ${\displaystyle AX_{2}}$

#### Atomic Packing Factor

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

### 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 ${\displaystyle 4r=a_{0}{\sqrt {3}}}$ ${\displaystyle 4r{=}a_{0}{\sqrt {2}}}$ ${\displaystyle 2r=a_{0}}$
${\displaystyle c/a_{0}=1.63}$
${\displaystyle a_{0}=2r}$
Volume ${\displaystyle a^{3}}$ ${\displaystyle a^{3}}$ ${\displaystyle a^{2}c\cos {(30)}}$ ${\displaystyle a^{3}}$
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 ${\displaystyle <{\bar {1}}120>}$ <110> <111>
Ceramic Structure NaCl, Zincblende (Simple cubic)CsCl, Flourite

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

For points and Vectors:

Simpler form:

1. find ${\displaystyle \Delta x,\Delta y,\Delta z}$ scale them to the nearest integer

For Planes:

1. If the plane in question passes through the origin, create new origin
2. Note the incercepts of the plane in terms of x,y,z

(a) if intersection is entire axis the value is ${\displaystyle \infty }$

(b) If plane parallels an axis the value is ${\displaystyle \infty }$

1. Take reciprocals of found intercepts
2. Reduce to smallest integers
3. Enclose in parentheses without commas

[info] [list of directions] [comparison table]

## Polymer Structures

### Degree of Polymerization

DP=average number of repeat units per chain

${\displaystyle DP={MolarWeightofpolymer({\bar {M_{n}}}) \over Molarweightofsubstituent({\bar {m}})}}$

${\displaystyle {\bar {M_{n}}}{=}\sum x_{i}M_{i}
{\bar {M_{w}}}{=}\sum w_{i}M_{i}
{\bar {m}}}$
= 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
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
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

${\displaystyle n{=}{\rho V_{C}N_{A} \over A}}$

% Crystallinity = ${\displaystyle {\rho _{c}(\rho _{s}-\rho _{a}) \over \rho _{s}(\rho _{c}-\rho _{a})}\times 100\%}$

${\displaystyle \rho _{c}=densityperfectlycrystallinepolymer
\rho _{a}=densitytotallyamorphouspolymer
\rho _{s}=densityofsamplepolymer}$

## 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
${\displaystyle c_{1}}$ weight based impurity
${\displaystyle c_{1}^{'}}$ Atom based impurity
b is perpendicular to dislocation line
Screw Disclocation perfect cut and torsional stress, dislocated plane of distortion one atom spacing
b
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 ${\displaystyle 2^{n-1}}$
n ASTM grain size number
etching to preferentially attack grain area
grain size effect yeild 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