3) Crystal Defects

Give the equation for change in energy caused by vacancies

U = ε_f N; ε_f = vacancy formation energy; N = number of vacanies

Why don’t perfect crystals form

imperfections increase the entropy o

Give the equations for entropy

S = kln(g); g = (Na + Nv)!/(Na!Nv!); Na = number of atoms; Nv = number of vacancies

Give the equation relating energy to vacancy formation

Nv/(Na + Nv) = exp(-ε_f/kT); ε_f = vacancy formation energy

Give the equation for free energy and what happens at eqm

U-TS; at eqm d(U-TS)/dNv = 0

Interstitial atoms

atom not on lattice site; adopts site with largest free volume; pushes neighbours slightly costing energy

How does weak atomic bonding affect vacancy formation

easier for vacancies/interstitials to form; less E required to overcome E loss of bonds

How does density affect interstitials

low density makes it easier for interstitials to form; more space for it to sit

What are substitutional impurities

different atoms replaces native atom on a site; requires similar size and chemistry

Give the equation for the formation of point defects

N_eq = nt exp(-ε_f/kT); N_eq = defects at eqm; nt = lattice sites; ε_f = formation energy

Describe the diffusion of point defects

diffusion low in solids; defects have higher E; easier to escape and diffuse; controlled by bond strength

Interstitial displaces a native atom becoming a substitution is more likely to occur when:

interstitial E of native atom < impurity; subs E of impurity < native atom

Interstitial diffuses into another interstitial site is more like occur when

large free volume in the lattice structure; impurity is smaller than native atom

Give the equation for Ficks law of diffusion

D = A exp(-ε_a/kT); D = diffusion rate (m²/s); ε_a = activation energy; A = prefactor (depends on diffusion distance, lattice, vibrations etc)

How can defects aggregate

vacancy and interstitial annihilate; vacancies aggregate forming large voids; interstitials aggregate forming high density regions

Why might defects aggregate

double vacancy breaks fewer bonds so more energetically favourable; but requires atoms with fewer interactions (not favourable); double interstitials disrupt fewer atoms; hindered by diffusion; aggregation requires work

Describe defects in an ionic structure

substitutions must be of same charge; imbalance of charge is not energetically favourable; interstitial sites are v limited as ions must have opposite charge to neighbours

What is Kroger Vink terminology

M^C_S; M = species (atoms, vacancies, e or h); C = resulting charge (.=+ve, ‘=-ve); S = lattice occupied (atom or i)

Charge neutral pairs

ceramics need to be neutral; formation of a charged defect will require one of opposite charge too

Schottky pairs

anion and cation vacancy

Frenkel pairs

anion/cation vacancy and same interstitial

How is diffusion of defects different in ceramics

dominated by energy of coulomb repulsion

Clustering

multiple defects (of opposite charge) aggregate together inducing large relaxations in surrounding lattice; if enough can form effectively different lattice (and phase separation); likelihood influenced by strength of attraction (size, charge, mobility)

What makes clustering more likely in ceramics

greater difference between defect and lattice; but harder to form defects due to larger activation energy

How can defects affect lattice strain

inbuilt strain from large/small ions can be relieved by defects if of a better size for the site

Give the equation for gibbs free energy

G = U + PV - TS; lower G is more favourable

U

sum of internal energies; find lowest energy arrangement of atoms; maximise low energy interactions; clustering defects to reduce loss of bonds

PV

generally small; crystals dont compress or expand much

TS

disordered system has higher entropy; prefers no clustering

What factors govern mixing in solid solutions

same as defects: similar sized cations have greatest solubility

Vergards law

lattice parameter is weighted mean of components lattice parameter; rarely accurate (better in ordered systems) as atoms not linearly interpolatable

Vergards law equation

a = xA aA + (1-xA)aB; xA = mole fraction of A present; aA/B = lattice parameter of A/B

List the larger defects found in crystals

stacking faults; voids; grain boundaries; twin boundaries (two orientations); dislocations

Stacking faults

disruption in layers; fcc/ccp: missing layer (intrinsic) or extra layer (extrinsic)

What are dislocations

line defects in crystal created by natural shear within crystal; separates slipped and unslipped region

Slip

one crystal plane slides over another

Slip plane

crystal plane on which the slip occurs; its the plane separating the two perfect crystals that slips; normally highest atom density (close packed planes)

Line vector

t; vector along dislocation line; +ve when looking along the line

Burgers vector

shows slip caused by dislocation movement; want as small as possible to minimise energy; must be conserved

How to define a Burgers vector

draw a RH loop around the disloc along atom bonds; draw same loop on a perfect crystal; b vector joins end to start on the perfect lattice

A burgers vector of an edge dislocation is

perpendicular to the dislocation line

A burgers vector of a screw dislocation is

parallel to the dislocation line

Use this image to explain how an edge dislocation forms

slice from edge to line AA; apply shear force perpendicular to the slice in opposite directions at top and bottom; AA is the dislocation line and AABB is the slip plane

Use this image to explain how a screw dislocation forms

slice from edge to line AA; apply shear force parallel to line of dislocation (AA); slip plane is AABB

What are mixed dislocations

when disloc forms a loop or ends on two perpendicular edges; at one edge stress is perpendicular to the dislocation line (edge disloc); at the other edge stress is parallel to dislocation line (screw disloc); in between it is mixed

What happens to dislocations at a boundary

must end at a surface (or loop) to conserve Burgers vector; collecting together can form a low angle boundary

How do dislocations affect lattice stress

can relieve stress from lattice mismatches; shift the atoms to reduce distortion lowering the energy

What is glide

dislocation motion in the slip plane; edge dislocations glide in their plane; screw dislocations can cross slip to another plane

Sessile dislocations:

cannot glide

Glissile dislocations

can glide

What is slip

movement of many dislocations resulting in permanent deformation; concentrated in slip bands

What is a slip system

slip plane+direction; specific to crystal system

Wide dislocation core =

distortion spread over larger area; lower energy to move as less lattice distortion; plastic deformation easier

Wide space slip planes

atoms have more space to move without repulsion from neighbours; lower energy for movement

Describe the slip systems in fcc

close packed on the {111} planes; there are 4 of these planes and 3 slip directions on each one so 12 slip systems; fcc tends to be soft

What are partial dislocations

sometimes favourable to slip in smaller steps; AB stacking: B moves onto C site instead of another B; lower energy to move; causes stacking fault; two partials

Describe the energetics of dislocations

b vectors are conserved; dislocations can combine (or annihilate) or split

Give the equation for Franks rule for elastic energy of a dislocation

E = alpha Gb²; alpha=parameter:0.5-1; G = shear modulus; b=burgers vector

What is the condition for it to be energetically favourable for a dislocation to split

b1² > b2² + b3²; splitting reduces energy

What is the condition for it to be energetically favourable for dislocations to combine

b1² < b2² + b3²; combining reduces energy

What is cross slip

screw disloc moves to a different slip plane due to local stresses; as line and burgers vectors are parallel there are infinite planes containing both (an edge dislocation cannot do this)

What is climb

edge dislocation moves up or down a plane by gaining a vacancy (moves up) or extra atom (moves down)

What is a dislocation lock

a sessile disloc; two dislocs on different glide planes intersect producing a b vector that doesnt allow glide along either plane