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Summary

Valence (of an atom): no. of electrons involved in bonding Electronegativity: tendency of atom to gain electron Metallic bonding: electrostatic attraction between (delocalized) valence electrons positive high modulus Covalent bonding: sharing of valence electrons between atoms Ionic bonding: donatio...

Description

Valence (of an atom): no. of electrons involved in bonding Electronegativity: tendency of atom to gain electron · Metallic bonding: electrostatic attraction between (delocalized) valence electrons & positive nuclei; high young’s modulus · Covalent bonding: sharing of valence electrons between atoms · Ionic bonding: donation of valence electrons, electrostatic attraction binds ions together · van der Waals bonding: secondary bond developed between atoms & molecules due to induced or permanent dipoles London forces: between 2 dipoles in atoms / molecules Keeson forces: between permanently polarized (charges on opposing ends → giving net dipole moment) molecules, due to polarization of molecules / groups of atoms Debye forces: induced dipole interacts with molecule with permanent dipole moment · mixed bonding: 2 or more types of bonds (above) · Interatomic spacing: equilibrium spacing between centers of 2 atoms (= min. interatomic energy of atoms / ions) · Binding energy: energy to separate 2 atoms from equilibrium spacing to infinite distance apart ( ↑ bond energy = ↑ melting point = strength) · Modulus of elasticity: slope of stress – strain curve in elastic region · Yield strength: level of stress, above which will result in permanent deformation · Coefficient of thermal expansion (CTE): amount by which material dimensions change when temperature changes

Einteratomic=

−A B + n m r r

( +¿ ): repulsive

( −¿ ): attractive

F=

d [E ] dr interatomic

Stiffness=

↑

dF d 2 = [E ] dr dr interatomic

steep curve + deep trough = low CTE

Material Classification Monoatomic

Amorphous

liquid crystal

No order

short range order

short range order + long range (for small vol. only)

e.g. argon, neon

e.g. plastics, glasses

e.g. LCD polymers

Metallic glasses: quenching metals on orders of

6

crystalline (single / poly) atoms/ions form regular, repeated 3D pattern poly – : many smaller crystals (grains) w/ varying orientations in space, short range e.g. Si, GaS (single crystalline) e.g. metals, alloys, most ceramics (polycrystalline)

−1

10 K ∙ s

Lattice: a periodic array of points in 3D space Unit cell: smallest group of atoms (like a molecule) that is repeated to form the entire crystal Basis: group of atoms that make up the unit cell, but does not necessarily describe the arrangement of the atoms Crystal systems: unique arrangement of unit cells to fill in a 3D space – cubic / tetragonal / orthorhombic / trigonal / hexagonal / monoclinic / triclinic Bravais lattice: 14 distinct arrangements of lattice points Lattice parameters: describe size and shape of unit cell, including the dimensions of each side and angle between sides Specific number of lattice points define each of the unit cells. Lattice points may be shared by more than 1 unit cell. Lattice per unit cell = no. of lattice points owned by 1 unit cell, in light of the fact that lattice points are being shared by other unit cells. A lattice point at a corner of 1 unit cell is shared by 8 cells, and

1 8

of each corner belongs to 1 particular cell. At each face, the face center atom is shared between 2 cells.

a0 =2 r

Closed packed directions: directions in unit cell along which atoms are in continuous contact (simple cubic:

a0 =2 r √ 2

a0 =

, BCC:

4r √3

, FCC:

)

Coordination number: no. of atoms touching particular atom / no. of nearest neighbors for the particular atom (indicates how tightly / efficiently packed atoms are) Close – packed structure: packing fraction is highest with identical atoms of a fixed size

c =1.633 a

Hexagonal close – packed (HCP) structure: special form of hexagonal structure, where

but will deviate slightly in real metals

Miller index

Directions

Planes

· metals deform easier in directions where atoms are in closest contact

· Surface energy of different faces depends on particular crystallographic planes

· magnetic properties depend on crystallographic directions · Set up coordinate system, where each length of the unit cell is and determine coordinates of points that lie along the direction

1

,

· For each direction, calculate vector

( x y

z )= ( x2

y 2 z 2) − ( x 1

· Deforms along planes that are most tightly packed

y1 z1 )

· Identify points where plane intercepts x , y , z axes in terms of lattice parameters (max. length ¿ 1 ). * if plane passes through origin, then the origin must be shifted · Take reciprocal of each axis intercept, express in the form:

(

· Express column vector above in lowest terms / integers. · Enclose numbers in [square bracket] like this: any negative number with a

¯¿ ´¿

[x y z ]

. Represent

[ x y z ]=k [ x y z ]

*

[ x y z ] ≠−[ x y z ]

*

*

a1 →120 ° →a 2 → 120° → a3 → 120 °→ a1 To find directions,

into

( h' k ' l' )

( ∆ x ∆ y ∆ z )T T

0

0

0

0

0

( 1x y1 1z ) ≠ k( x1 y1 1z ) 0

( ∆ x ∆ y ∆ z )T

number with a

( 1x y1 1z )=−( x1 y1 z1 ) 0

1. use the 3 – axis coordinate system, and find the direction vector

−¿ ¿

· Represent any negative

SPECIAL CASE: hexagonal unit cell

2. Convert the direction vector

)

· Clear fractions but do not reduce to lowest terms / integers

· Groups of equivalent directions  directions of a form, represented with < > bracket

*

1 1 1 x0 y 0 z 0

0

0

0

0

0

¯¿ ´¿

3. Convert into 4 component direction vector

( h k il)

T

where

h=

2 h'3−k '

;

k=

2 k '3−h'

;

i=

−h3' +k '

;

l=l

'

To find planes, 1. use the 4 axis coordinate system and obtain the intersection in each of the axes 2. Miller index is given by:

(a1 a1 a1 1c ) 1

2

3

Plane density: no. of atoms per unit area whose centers lie on the plane, i.e. no. of atoms per face ÷ area of face Packing fraction: area of atoms ÷ area of plane ¿ volume of atoms ÷ unit cell volume ¿ no. of atoms in unit cell ( 3

×

4 πn r 4πr = vol . of unit cell a 03

n

)

3

(assumes atoms are spherical & touch immediate neighbors)

|V 1−V 2|

∆ ( vol. ) =

V1

Planes of a form  groups of equivalent planes that have their particular indices due to orientation of coordinates, represented by the { } bracket In cubic systems, a direction that has the same indices as a plane is ⟂ that plane Close – packed planes & directions [pg. 48]

( 0002 ) planes of the structure  ‘basal’ planes ´ stacking sequence → atoms on plane B fit into spaces created by 2 ‘ A ’ planes · ABABAB…. → AB ´ · ABCABCABC… → ABC stacking sequence → ? Closed – packed planes in hexagonal unit cells are the

( 0001 )

and

Interplanar spacing: distance between adjacent planes of atoms with the same Miller indices (cubic:

d hkl=

a0

√ h +k 2 +l2 2

where

a0

= lattice

parameter) Interstitial Sites There exists gaps between ‘usual’ atoms where smaller atoms may fit Interstitial site: location between ‘normal’ atoms / ions in crystal where another (different) atom / ion is placed, and size of interstitial location is smaller than atom / ion to be introduced · Interstitial atom / ion radius > interstitial site radius: may enter, pushing the surrounding atoms slightly apart · Interstitial atom / ion radius < interstitial site radius not allowed inside, due to ‘rattling’ within the site · Interstitial atom / ion radius >> interstitial site radius: atom / ion will look for site with larger coordination number · Cubic site: an interstitial position with coordination number =8, i.e. atom / ion touches 8 other atoms / ions · Octahedral site: an interstitial position with coordination number = 6, i.e. atom / ion touches 6 other atoms / ions · Tetrahedral site: an interstitial position with coordination number = 4, i.e. atom / ion touches 4 other atoms / ions

Many ionic crystals can be viewed as being generated by close – packing of larger anions into interstitial sites of close – packed anions

→

cations can then be viewed as smaller ions that fit

Crystal Structure Of Ionic Materials ·Many ceramic materials contain significant ionic bonding ·Factors to be considered – ionic radii, electrical neutrality, connection between anion polyhedral, visualization of crystal structures using computers ·Connection between anion polyhedral: corner sharing, edge sharing, face sharing ·Coordination polyhedra (formed by close packing of anions) will share corners, because electrostatic repulsion between cations is reduced considerable in corner sharing polyhedral → more stable crystal structure

Common ceramic structures:

−¿ ¿

- Fluorite: FCC with anions

located at all 8 tetrahedral positions, with compound form

- Perovskite: crystal structure of mineral with compound form

4 +¿ B¿

AB O 3

A X2

, e.g.

, oxygen anions occupy face centers &

Zr O 2

2+¿ A¿

and

ThO 2

ions occupy corners &

occupy octahedral site at the cube’s center

- Corundrum: oxygen anions arranged into hexagon, aluminum cations occupy some of the available octahedral positions Diffraction Techniques For Crystal Structure Analysis · 2 methods: x–ray diffraction , electron diffraction · In a diffractometer, a moving x–ray detector records the angles at which the beam is diffracted, giving a characteristic diffraction pattern · Bragg’s Law:

sin θ=

λ d hkl

Imperfection · Basic types: point , line (dislocation) , surface Vacancy: atom / ion missing from its regular crystallographic site Interstitial defect: point defect due to atom inserted into crystal at an unoccupied point (compression / distortion @ surrounding crystal region) Arrhenius type behavior:

−Q v RT

nvacancy =ninitial ∙ e

Substitutional defect definition: point defect due to atom being removed from lattice point and replaced with a different atom Schottky effect Frenkel effect · Equal number of cations and anions missing from regular lattice sites · cation leaves normal lattice site and occupies interstitial site · creates vacancy in regular lattice structure · found in: ionic, high coordination number & anion size ≅ cation · found in: ionic, low coordination number & anion size >> cation size size · density remains constant

Special case: iron ion of charge ( A ) replacing ion of different charge ( B ) When divalent charge ( 2+¿ ) replaces monovalent charge ( +1 ), another monovalent charge must be removed, which creates a vacancy

1. charge balance must be maintained so crystal has 0 net charge

2. mass is conserved

3. no. of crystallographic sites is conserved

Dislocation definition: a line imperfection, introduced during solidification / deformation & explains deformation & strengthening in metallic materials Mixed dislocation Edge dislocation a mix of both types

Screw dislocation

dislocation due to addition of ‘half a plane’ of atoms between any 2 planes of atoms

lattice is sheared, so part of the lattice is offset by a bit relative to the other

‘the linear defect that forms the extra half a plane’  represented by ‘ᴛ’

Burgers vector Right hand rule: thumb pointing into atom Form a 4 sided loop, where each loop has same number of unit cell lengths ( Identify the side that cannot be closed with ( Burgers vector (

b

n

n

)

) unit cell lengths.

) is the vector from the starting atom to the ending atom, i.e. the vector that completes the ‘circuit’ that is drawn around the

dislocation LINE: SCREW:

b

⟂

b

the dislocation position (dislocation vector into / out of page)

is parallel to line of dislocation

Slip: movement of dislocation, causing metallic material to deform Slip direction: direction in which dislocation moves Slip plane: plane formed by ‘sweep’ Etch: tiny hole created at areas where dislocations meet surface Slip line: visible line produced at surface of material by several thousand dislocations Slip band: collection of many slip lines, often visible Dislocations in metals  mechanism for plastic (permanent) deformation due to slip of numerous dislocations Dislocation density: total length of dislocation line per cubic centimeter Schmid’s Law

F τ plane=σ cos λ cos Ф= 0 cos λ A0

λ : angle between direction Ф : angle between slip plane

σ σ

& slip & normal to

· HCP metals: no cross slip (parallel slip planes) · FCC metals: high ductility ; polycrystalline HCP: brittle ; BCC: middle ground

Surface Defects · boundaries / planes that separate material into regions, each with same structure but different orientation · reduce grain size  number of grains, grain boundary area ↑  dislocation moves short distance before being stopped  strength increased · Hall Petch equation:

σ y =σ 0 +

K √d

 from table of values, write out a column of values of

−1 2

d

and use linear approximation equation

· small angle grain boundary: array of dislocations causing small misorientation of crystals across surface ; caused by edge: tilt boundary // caused by screw: tilt boundaries · twin boundary: plane across which there is a mirror image misorientation

Diffusion · Definition: net flux of ions, electrons, holes & molecules · Types: 1. Self–diffusion: random movement of atoms / vacancies within pure material from 1 lattice position to another 2. Interdiffusion: diffusion of atoms in different directions, diffusion of different atoms in materials

3. Vacancy diffusion: diffusion of atoms when atom leaves regular lattice position to fill vacancy in crystal vacancies 4. Interstitial diffusion: diffusion of small atoms from 1 interstitial position to another, smaller atoms

→

→

counter flow of atoms &

diffuse faster (no vacancy needed)

Activation energy for diffusion · diffusing atom must ‘squeeze’ through surrounding atoms to reach new location, hence requires activation energy · substitutional diffusion requires more energy than interstitial diffusion Rate Of Diffusion · Fick’s 1st Law:

(in atoms

J =−D −3

∙cm

dc dx

D

where

or % atom

−1

∙cm

→

diffusion coefficient

( D=D ∙ e ) −Q RT

and

0

dc dx

→

concentration gradient

)

· Flux: no. of atoms passing through plane of unit area per unit time · Concentration gradient shows how composition of material varies with distance Factors Of Diffusion [Crystalline Materials] · Type of diffusion: volume (moving through crystal) , grain boundary (diffuse along boundaries & interface) , surface (moving along material surface) · Time elapsed: long time to produce uniform structure · Nature of bonding & crystal structure: interstitial diffusion > vacancy / substitutional diffusion, cations have ↑ D than anions, smaller cations = faster · Dependence on concentration of diffusing substance and composition of matrix * main mechanisms for atomic movement: interstitial & vacancy diffusion → substitutional atoms move by vacancy mechanism * activation energy for self – diffusion increases with melting point of metals Use Of Diffusion In Materials Processing · Sintering: high temperature treatment to join small particles e.g. manufacture of ceramic materials – lattice diffusion from bulk of particles into neck region causes densification · Powder Metallurgy: used to produce monolithic metallic parts, similar to sintering

· Grain growth: movement of grain boundaries by diffusion to reduce grain boundary area, causes large grains to grow at the expense of smaller grains

· Diffusion bonding: joining technique where 2 surfaces are pressed at high pressure & temperature

· Melting & Casting: important in solidification of metals & alloys

–

Phase Equilibria Phase Diagram Definition: shows phases (states of matter) and chemical compositions at any combination of temperature & pressure

Phase properties: Same atomic arrangement throughout Same properties throughout Defined interface between plane & adjoining phases

X

: triple point

Gibbs Phase Rule

n ( deg . of freedom )=n 2+C=F+ P C

: no. of components / substances

(variables to be fixed to specify temperature & composition of phase)

P

: no. of coexisting phases

F phase

: no. of independent variables that can be changed to maintain

Special case: either

or

P

is constant

T

→

1+C=F+ P

Solubility Unlimited: no limit to relative composition of substances and cannot be physically separated, e.g. water & alcohol Limited: maximum relative composition between 2 substances, e.g. salt & water

x

–axis : % weight of additive, e.g. C

y –axis : temperature ( T

)

‘line’ represents solubility limit (max. amount of additive that can dissolve into original solution @ given

T

)

Conditions for unlimited solubility (Hume Rothery Rules) in an alloy / ceramic system:

∆

(atomic radii)

≤15 %

same crystal structure

equal valence

approx. equal electronegativity

Solid Solution Strengthening definition: increasing strength of metallic material via formation of solid solution this is achieved by alloying with atoms with large ∆ (atomic radii) and different % weight of alloying substance · Binary phase diagram: phase diagram with 2 (ternary → 3) components · Isomorphous phase diagram: phase diagram where components display unlimited solid solubility · Liquidus temperature: temperature at which 1st solid begins to form during solidification · Solidus temperature: temperature below which all liquid has become solidified

∆=|T liquidus−T solidus|

· Freezing range:

; 2 phases (solid & liquid) coexist in this range

Isomorphous Phase Diagrams region above closed curve (or right of it): solid phase ( region bounded by closed curve: coexistence of solid ( region below closed curve (or left of it): liquid...