Crystal Defects Chapter 6

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Crystal DefectsChapter 6

Reality

An ideal crystal can be described in terms a three-dimensionally periodic arrangement of points called lattice and an atom or group of atoms associated with each lattice point called motif:

IDEAL Crystal

Crystal = Lattice + Motif (basis)

Deviations from this ideality.

These deviations are known as crystal defects.

Real Crystal

Is a lattice finite or infinite?Is a crystal finite or infinite?

Free surface: a 2D defect

Vacancy: A point defect

Defects Dimensionality Examples

Point 0 Vacancy

Line 1 Dislocation

Surface 2 Free surface,

Grain boundary

Stacking Fault

Point DefectsVacancy

There may be some vacant sites in a crystal

Surprising Fact

There must be a certain fraction of vacant sites in a crystal in equilibrium.

A Guess

Point Defects: vacancy

What is the equilibrium concentration of vacancies?

Equilibrium?

A crystal with vacancies has a lower free energy G than a perfect crystal

Equilibrium means Minimum Gibbs free energy G at constant T and P

1. Enthalpy H

2. Entropy S

G = H – T S

Gibbs Free Energy G

=k ln W

T Absolute temperature

E internal energyP pressureV volume

k Boltzmann constantW number of microstates11

Vacancy increases H of the crystal due to energy required to break bonds

D H = n D Hf 12

Vacancy increases S of the crystal due to configurational entropy

Number of atoms: N

Increase in entropy S due to vacancies:WkS lnD

Number of vacacies: n

Total number of sites: N+nThe number of microstates:

nN CW!!)!(

!!)!(ln

Configurational entropy due to vacancy

]!ln!ln)![ln( NnnNk 14

Stirlings Approximation

NNNN ln!lnN ln N! N ln N N

10 15.10 13.03

100 363.74 360.51

100!=933262154439441526816992388562667004907159682643816214685\ 9296389521759999322991560894146397615651828625369792082\ 7223758251185210916864000000000000000000000000

WkS lnD ]!ln!ln)![ln( NnnNk

NNNN ln!ln

]lnln)ln()[( NNnnnNnNkS D

fHnH DD

DG = DH TDS

G of a perfect crystal

Change in G of a crystal due to vacancy

DHfHnH DD

TDS]lnln)ln()[( NNnnnNnNkS D

Fig. 6.4 17

fHnH DD

]lnln)ln()[( NNnnnNnNTkHnG f DD

eqnnnG

With neq<<N

Nn feq exp

Equilibrium concentration of vacancy]lnln)ln()[( NNnnnNnNkS D

Nn feq exp

Al: DHf= 0.70 ev/vacancyNi: DHf=1.74 ev/vacancy

n/N 0 K 300 K 900 K

Al 0 1.45x1012 1.12x104

Ni 0 5.59x1030 1.78x10-10

Contribution of vacancy to thermal expansionIncrease in vacancy concentration increases the

volume of a crystal

A vacancy adds a volume equal to the volume associated with an atom to the

volume of the crystal

Contribution of vacancy to thermal expansion

Thus vacancy makes a small contribution to the thermal expansion of a crystal

Thermal expansion =lattice parameter expansion

+Increase in volume due to vacancy

Contribution of vacancy to thermal expansion

NVvNV DDD

V=volume of crystalv= volume associated with one atomN=no. of sites

(atoms+vacancy)

Total expansio

Lattice parameter increase

vacancy

Experimental determination of n/N

Linear thermal

expansion coefficient

Lattice parameter as a function of temperature

Problem 6.2

vacancy Interstitialimpurity

Substitutionalimpurity

Point Defects

Frenkel defect

Schottky defect

Defects in ionic solids

Cation vacancy+

cation interstitial

Cation vacancy+

anion vacancy 25

Line DefectsDislocations

Missing half plane A Defect

An extra half plane…

…or a missing half plane28

What kind of defect is this?

A line defect?

Or a planar defect? 29

Extra half plane No extra plane!

Missing plane No missing plane!!!

An extra half plane…

…or a missing half plane

Edge Dislocation

If a pl

tly in

a crys

The whole of abruptly ending

plane is not a defect

Only the edge of the plane

can be considered as a defect

This is a line defect called an EDGE DISLOCATION33

Callister FIGURE 4.3

The atom positions around an edge dislocation; extra half-plane of atoms shown in perspective. (Adapted from A. G. Guy, Essentials of Materials Science, McGraw-Hill Book Company, New York, 1976, p. 153.)

1 2 3 4 5 6 7 8 9

slip no slip

boundary = edge dislocation

Slip planebBurgers vector

Slip planeslip no slip

Dislocation: slip/no slip boundaryb: Burgers vectormagnitude and direction of the slipt: unit vector tangent to the dislocation line

Dislocation Line:A dislocation line is the boundary between slip and no slip regions of a crystalBurgers vector:The magnitude and the direction of the slip is represented by a vector b called the Burgers vector,

Line vectorA unit vector t tangent to the dislocation line is called a tangent vector or the line vector. 39

1 2 3 4 5 6 7 8 9

slip no slipSlip planeb

Burgers vector

Line vector

Two ways to describe an EDGE DISLOCATION1. Bottom edge of an extra half plane2. Boundary between slip and no-slip regions of a slip plane

What is the relationship

between the

directions of b and t?b t

In general, there can be any angle between the Burgers vector b (magnitude and the direction of slip) and the line vector t (unit vector tangent to the dislocation line)

b t Edge dislocation

b t Screw dislocation

b t , b t Mixed dislocation

Screw Dislo

cation Line

b || t

Screw Dislocation

Slip plane

slipped

unslipped

If b || t

Then parallel planes to the dislocation line lose their distinct identity and

become one continuous spiral ramp

Hence the name SCREW DISLOCATION

Edge Dislocation

Screw Dislocation

Positive Negative

Extra half plane above the slip plane

Extra half plane below the slip plane

Left-handed spiral ramp

Right-handed spiral ramp

b parallel to t b antiparallel to t

Burgers vector

Johannes Martinus BURGERS

Burgers vector Burger’s vector 45

1 82 3 4 5 6 7 9 10 11

23456789

18 234567910

A closed Burgers

Circuit in an ideal crystal

1 82 3 4 5 6 7 9 10 11

1234567

1234568 791011

RHFS convention

Map the same Burgers circuit on a

real crystal

The Burgers circuit fails to close !!

A circuit which is closed in a perfect crystal fails to close in an imperfect crystal if its surface is pierced through a dislocation

Such a circuit is called a Burgers circuit

The closure failure of the Burgers circuit is an indication of a presence of a dislocation piercing through the surface of the circuit

and the Finish to Start vector is the Burgers vector of the dislocation line. 48

Those who can, do. Those who can’t,

teach.G.B Shaw, Man and

SupermanHappy Teacher’s Day

b is a lattice translation

If b is not a complete lattice translation then a surface defect will be created along with the line defect.

Surface defect

N+1 planes

N planes

Compression

Above the slip plane

TensionBelow the slip plane

Elastic strain field associated with an

edge dislocation

Line energy of a dislocation

Elastic energy per unit length of a dislocation line

Shear modulus of the crystalb Length of the Burgers vector

Unit: J m1

Energy of a dislocation line is proportional to b2.

b is a lattice translation

Thus dislocations with short b are preferred.

b is the shortest lattice translation

FCC 11021

DC 11021

NaCl 11021

SC 100

BCC 11121

CsCl 100

A dislocation line cannot end abruptly inside a crystal

Slip plane

slip no slip

Dislocation: slip/no slip boundary

Slip plane

Extra half plane ABCD

Bottom edge AB of the extra half plane is the edge dislocation line

What will happen if we remove the part PBCQ of the extra half plane??56

It can end on a free surface

Dislocation Line AB Dislocation Line APQ

Grain 1 Grain 2

Grain Boundary

Dislocation can end on a grain boundary

A dislocation loop

tNo slip

The line vector t is always

tangent to the dislocation line

The Burgers vector b is

constant along a dislocation

line59

Cylindrical slip plane (surface)

Prismatic dislocation loopCan a loop be

entirely edge?

Example 6.2

Dislocation node

b1 + b2 + b3 = 0

It can end on

Free surfaces

Grain boundaries

On other dislocations at a point called a nodeOn itself forming a loop

Slip planeThe plane containing both b and t is called the slip plane of a dislocation

line.An edge or a mixed dislocation has

a unique slip planeA screw dislocation does not have

a unique slip plane.

Any plane passing through a screw dislocation is a possible slip plane

Dislocation Motion

Glide (for edge, screw or mixed)Cross-slip (for screw only)

Climb (or edge only)

Dislocation Motion: Glide

Glide is a motion of a dislocation in its own slip plane.

All kinds of dislocations, edge, screw and mixed can glide.

Glide of an Edge

Dislocation

Glide of an Edge

Dislocation

crss is

critical

resolved

stress on

the slip

plane in

direction of

Glide of an Edge

Dislocation

crss is

critical

resolved

stress on

the slip

plane in

direction of

Glide of an Edge

Dislocation

crss is

critical

resolved

stress on

the slip

plane in

direction of

Glide of an Edge

Dislocation

crss is

critical

resolved

stress on

the slip

plane in

direction of

Glide of an Edge

Dislocation

Surface step, not a dislocation

A surface step of

magnitudeb is

created if a

dislocation sweeps over the

entire slip plane

slip no slipDislocation

motion

Shear stress is in a direction perpendicular to the GLIDE motion

of screw dislocation

Glide Motion and the Shear StressFor both edge and screw dislocations

the glide motion is perpendicular to the dislocation line

The shear stress causing the motion is in the direction of motion for edge but

perpendicular to it for screw dislocation

However, for edge and screw dislocations the shear stress is in the

direction of b as this is the direction in which atoms move 73

Cross-slip of a screw dislocation

Change in slip plane of a screw

dislocation is called cross-slip

Slip plane 1

Slip pla

Climb of an edge dislocation

The motion of an edge dislocation from its slip plane to an adjacent

parallel slip plane is called CLIMB

Obstacle

climbglide

Slip plane 1

Slip plane 2

Atomistic mechanism of climb

Climb of an edge dislocation

Climb up

Climb downHalf plane shrinks Half plane stretches

Atoms move away from the edge to nearby

vacancies

Atoms move toward the edge

from nearby lattice sites

Vacancyconcentration

goes down

Vacancyconcentration

goes up 77

From Callister

Dislocations in a real crystal can form complex networks

http://www.tf.uni-kiel.de/matwis/amat/def_en/index.html

A nice diagram showing a variety of crystal defects

Surface Defects

External Internal

Free surface Grain boundaryStacking fault

Twin boundary

Interphase boundary

Same phase

Different phases

External surface: Free surface

If bond are broken over an area A then two free surfaces of a total area

2A is created

Area A

Broken bonds

External surface: Free surface

If bond are broken over an area A then two free

surfaces of a total area 2A is created

Area A

Broken bonds

nA=no. of surface atoms per unit areanB=no. of broken bonds per surface atom=bond energy per atom

BA nn21

Surface energy per unit area 83

What is the shape of a naturally grown salt crystal?

Surface energy is anisotropic

Surface energy depends on the orientation, i.e., the Miller indices

of the free surafce

nA, nB are different for different surfaces

Example 6.5 & Problem 6.16 85

Grain 1 Grain 2

Grain Boundary

Internal surface: grain boundary

A grain boundary is a boundary between two regions of identical crystal structure

but different orientation 86

Photomicrograph an iron chromium alloy. 100X.

Callister, Fig. 4.12

Optical Microscopy, Experiment 5

Grain Boundary: low and high angle

One grain orientation can be obtained by rotation of another grain across the grain boundary about an axis through

an angle If the angle of rotation is high, it is called a high angle grain boundary

If the angle of rotation is low it is called a low angle grain boundary

Grain Boundary: tilt and twist

One grain orientation can be obtained by rotation of another grain about an

axis through an angle

If the axis of rotation lies in the boundary plane it is called a tilt boundary

If the angle of rotation is perpendicular to the boundary plane it is called a twist

boundary 89

Edge dislocation model of a small

angle tilt boundary

Grain 1

Grain 2

Tilt boundary

Eqn. 6.7

Or approximatel

Stacking fault

CBACBACBA

ACBABACBA

Stacking fault

FCC FCC

Twin Plane

CBACBACBACBA

CABCABCBACBA

Twin plane

Edge Dislocation

432 atoms

55 x 38 x 15 cm3

Screw Dislocation525 atoms

45 x 20 x 15 cm3

Screw Dislocation(another view)95

A dislocation cannot end abruptly inside a crystal Burgers vector of a dislocation is constant

720 atoms

45 x 39 x 30 cm3Front face: an edge dislocation enters

Back face: the edge dislocation does not come out !! 98

Schematic of the Dislocation Model

Edge dislocation

Screw dislocation

A low-angle Symmetric

Tilt Boundary

477 atoms

55 x 30 x 8 cm3

R. Prasad

Dislocation Models for Classroom Demonstrations

Conference on Perspectives in Physical Metallurgy and Materials Science

Indian Institute of Science, Bangalore

MODELS OF DISLOCATIONS FOR CLASSROOM***

R. Prasad

Journal of Materials Education Vol. 25 (4-6): 113 - 118 (2003)

International Council of Materials Education

Paper is available on Web if you Google “Dislocaton Models”

A Prismatic Dislocation Loop685 atoms

38 x 38 x 12 cm3

Slip plane

Prismatic Dislocation loop

A Prismatic Dislocation LoopTop View

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Crystal Defects Chapter 6

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