PX431 Structure and Dynamics of Solids

PART 2:

Defects and Disorder

Diane Holland P160 d.holland@warwick.ac.uk

2. Defects and disorder (10L)

Lectures 1-2: crystal defects – point, line and planar defects;dislocations and mechanical behaviour

Lectures 3-5: point defects and non-stoichiometry; radiation induceddefects; thermodynamics and stability of defects;elimination of defects

Lectures 6-7: influence of defects on diffusion, ionic conductivity,optical and electronic properties

Lectures 8-10:amorphous materials and glasses – formation andstructure; structural theories; short and intermediaterange ordertechniques for structural analysis – diffraction and thepair distribution function; total scattering; local probes(NMR, EXAFS, Mössbauer, IR and Raman)

References

M.T. Dove, Structure and Dynamics, OUP Appendix A ( 6 pages only!)

S. R. Elliott, The physics and chemistry of solids, WileyChapter 3

W. D. Callister, Materials Science and Engineering, Wiley Chapters 4 & 7

Disorder in crystalline materials

• No perfectly ordered materials• Many materials are technologically of value because

they are disordered/imperfect in some way:

silicon devices – controlled levels of deliberate impurity additions (ppb) p-type : B Si B + h

n-type : P Si P + e

steels – additions of 0.1 to 1 at% other metals to improve mechanical properties and corrosion

resistance

stoichiometric compounds

elements present in simple (small) integer ratios

e.g. NaCl, BaTiO3

non-stoichiometric compounds

non-integer

e.g. Fe0.92O, Ca0.98Y0.02F2.02

Intrinsic defects – do not change overall composition

– stoichiometric defects

Extrinsic defects – created when foreign atom(s) introduced or there is valence change

Types of defect:

Crystal imperfectionsOrientational disorder

Point defects

Crystal imperfections

perfect crystal – all atoms on their correct lattice positions

(actual positions affected by extent of thermal vibrations which can be anisotropic)

imperfect crystal

extended defects

- dislocations

- grain boundaries

- stacking faults

- twinning

Orientational disorder groups of atoms which are non-spherically

symmetric- ammonium salts- linear chains

Point defectsvacancies, interstitials, incorrect atoms

- Schottky

- Frenkel- substitution

Extent of disorder

• Crystal imperfections - depends on preparation and mechanical history

• Orientational disorder - depends on temperature

• Point defects - Schottky and Frenkel normally v. low because formation energy

high

- Frenkel high in certain classes of materials e.g. Superionics

- substitution to high degree in some materials - alloys - spinels

CRYSTAL IMPERFECTIONS

- dislocations- grain boundaries - twinning

Dislocations – linear defectsSource:- growth- stress

Evidence:- metals more deformable than

predicted (but can be strengthened by impurities)

- spiral growths on surface of some crystals

- reactions occur at active surface sites

Types: edge, screw, intermediateTransmission electron micrograph of Ti alloy – dark lines are dislocations

(Callister: Materials Science and Engineering)

Dislocations revealed by etching

‘Etch pits’ produced by preferential etching by acid of the points where dislocations intersect the surface

http://en.wikipedia.org/wiki/Dislocation

Edge dislocation

– partial plane of atoms

– lattice distorted where plane ends

Dislocations characterised by the Burgers vector, b-magnitude and direction found by tracing loop around the dislocation- for metals, b points in a close-packed direction and equals the interatomic spacing

(Callister: Materials Science and Engineering)

Dislocation motion

• – dislocation moves under application of a shear stress (easy for bonds to swap between atoms at dislocation since they are already strained)

(Callister: Materials Science and Engineering)

• Motion of dislocations called slip; the plane over which the dislocation moves is called the slip plane

• For an edge dislocation: b is perpendicular to the dislocation lineb is parallel to the direction of motion of the dislocation line under an applied stress.

(Callister: Materials Science and Engineering)

Screw dislocation• partial slip of a crystal

• on one side of dislocation line, crystal has undergone slip; on other side, crystal is normal

• continued application of shear stress causes dislocation to move through crystal

• b is parallel to dislocation line

(opposite to Edge)• b is perpendicular to motion of

this line (opposite to Edge)

• but b is parallel to direction of shear and slip in both cases

Shear stress

(Callister: Materials Science and Engineering)

Quarter dislocation loop

• combined edge and screw dislocation - pure edge on one face; - pure screw on adjacent face;- mixed in-between

• loops expand easily but asymmetrically because edge moves easier than screw

(Callister: Materials Science and Engineering)

Pinning dislocations

• dislocations make metals easier to deform

• to improve strength of metals, need to stop dislocation motion

trap with:- impurity atoms;- other dislocations (work hardening;

- grain boundaries.

atom trap

(Callister: Materials Science and Engineering)

Effects of crystal structure

• Preferred set of slip planes on which dislocations can occur and also preferred slip directions for dislocation movement slip system

• slip plane – plane having most dense atom packing• slip direction – direction, in plane, having highest linear

density

• Energy required to move dislocation by one unit translation E |b| 2

the most abundant dislocations in a material are those with the smallest value of b

b

Shear in close-packed direction by one unit b = d E d2, where d is the diameter of the sphere (atom)

Shear in non-close-packed direction by one unit b = d 2

E 2d2

In metals, direction of motion of dislocation is usually parallel to one of the directions of close packing

b

2d

b

Tensile F on crystal

F

Slip plane

b

Tensile F

Resolved shear in slip plane

Tensile force produces shear force in slip plane

Stress on plane

SA = F/Asp = F(cos )/A

Critical resolved shear stress - Sb - parallel to direction of slip on slip plane

Sb = SAcos = (F/A)cos cos

- angle between slip direction and stress axis

Maximum value of Sb occurs when = = 45o

giving Sb = ½(F/A)

When slip plane is either parallel or perpendicular to F, the resolved shear stress is 0 and slip cannot occur.

b

Slip plane area Asp

F

Sb

Cross-section of crystal area A

b

Slip plane area Asp

F

Sb

Cross-section of crystal area A

Metal Slip plane Slip direction No. of slip systems

Face-centred cubic

Cu, Al, Ni, Ag, Au {111} <1-10> 12

Body-centred cubic

Fe, W, Mo {110} <-111> 12

Fe, W {211} <-111> 12

Fe, K {321} <-111> 24

Hexagonal Close-packed

Cd, Zn, Mg, Ti, Be {0001} <11-20> 3

Ti, Mg, Zr {10-10} <11-20> 3

Ti, Mg {10-11} <11-20> 6

Slip Systems

• FCC metals are generally more malleable and ductile than HCP or BCC

• BCC metals have many slip systems but planes are not close-packed

• HCP metals have few slip systems

(Callister: Materials Science and Engineering)

FACE-CENTRED CUBIC

AD, AF and DF are the 3 <110> slip directions

ADF and the equivalent upper faces of the octahedron are the 4 {111} slip planes

3 4 12 slip systems

Interfacial (planar) defects

• boundaries separating regions of different crystal structure or crystallographic orientation

• e.g. external surfaces (see final section of module)

Grain boundaries

D = b/

b

Internal surfaces of a single crystal where ideal domains (mosaic) meet with some misalignment: high-angle and small(low)-angle.

NB – in polycrystalline materials, grain boundaries are more extensive and may even separate different phases

Small-angle grain boundary equivalent to linear array of edge dislocations

bonding not fully satisfied region of higher energy, more reactive, impurities present.

(Callister: Materials Science and Engineering)

Twinning

change in crystal orientation during growth

mirror

(Callister: Materials Science and Engineering)