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

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

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Directional Bonds	Secondary Covalent
Non-Directional bonds	Metallic Ionic

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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	Octohedral	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:

1. find $\Delta x,\Delta y,\Delta z$ 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 $\infty$

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

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

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

${\bar {M_{n}}}{=}\sum x_{i}M_{i}<br/>{\bar {M_{w}}}{=}\sum w_{i}M_{i}<br/>{\bar {m}}$ = 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 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