Crystal Defects Chapter 6
107
Crystal Defects Chapter 6 1
description
Crystal Defects Chapter 6. IDEAL vs. Reality. IDEAL Crystal. 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:. Crystal = Lattice + Motif (basis). - PowerPoint PPT Presentation
Transcript of Crystal Defects Chapter 6
Crystal DefectsChapter 6
1
2
IDEAL
vs.
Reality
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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.
4
Real Crystal
Is a lattice finite or infinite?Is a crystal finite or infinite?
Free surface: a 2D defect
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Vacancy: A point defect
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Defects Dimensionality Examples
Point 0 Vacancy
Line 1 Dislocation
Surface 2 Free surface,
Grain boundary
Stacking Fault
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Point DefectsVacancy
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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
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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
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1. Enthalpy H
2. Entropy S
G = H – T S
Gibbs Free Energy G
=E+PV
=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
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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:
n
nN CW!!)!(
NnnN
!!)!(ln
NnnNk
Configurational entropy due to vacancy
]!ln!ln)
Stirlings Approximation
NNNN ln!lnN ln N! N ln N N
1 0 1
10 15.10 13.03
100 363.74 360.51
100!=933262154439441526816992388562667004907159682643816214685\ 9296389521759999322991560894146397615651828625369792082\ 7223758251185210916864000000000000000000000000
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WkS lnD ]!ln!ln)![ln( NnnNk
NNNN ln!ln
]lnln)ln()[( NNnnnNnNkS D
fHnH DD
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DG = DH TDS
neq
G of a perfect crystal
Change in G of a crystal due to vacancy
n
DG
DHfHnH DD
TDS]lnln)ln()[( NNnnnNnNkS D
Fig. 6.4 17
fHnH DD
]lnln)ln()[( NNnnnNnNTkHnG f DD
0D
eqnnnG
With neq<<N
D
kTH
Nn feq exp
Equilibrium concentration of vacancy]lnln)ln()[( NNnnnNnNkS D
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D
kTH
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
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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
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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
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Contribution of vacancy to thermal expansion
NvV
NVvNV DDD
NN
vv
VV D
D
D
V=volume of crystalv= volume associated with one atomN=no. of sites
(atoms+vacancy)
Total expansio
n
Lattice parameter increase
vacancy
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NN
vv
VV D
D
D
Nn
aa
LL
D
D 33
D
D
aa
LL
Nn 3
Experimental determination of n/N
Linear thermal
expansion coefficient
Lattice parameter as a function of temperature
XRD
Problem 6.2
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vacancy Interstitialimpurity
Substitutionalimpurity
Point Defects
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Frenkel defect
Schottky defect
Defects in ionic solids
Cation vacancy+
cation interstitial
Cation vacancy+
anion vacancy 25
Line DefectsDislocations
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Missing half plane A Defect
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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!
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Missing plane No missing plane!!!
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An extra half plane…
…or a missing half plane
Edge Dislocation
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If a pl
ane e
nds
abrup
tly in
side
a crys
tal w
e
have
a de
fect.
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.)
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1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
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1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
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1 2 3 4 5 6 7 8 9
1 2 3 4 5 6 7 8 9
slip no slip
boundary = edge dislocation
Slip planebBurgers vector
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Slip planeslip no slip
disl
ocat
ion
b
t
Dislocation: slip/no slip boundaryb: Burgers vectormagnitude and direction of the slipt: unit vector tangent to the dislocation line
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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
1 2 3 4 5 6 7 8 9
slip no slipSlip planeb
Burgers vector
t
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
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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
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Screw Dislo
cation Line
b
t
b || t
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3
Screw Dislocation
Slip plane
slipped
unslipped
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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
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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
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Burgers vector
Johannes Martinus BURGERS
Burgers vector Burger’s vector 45
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23456789
18 234567910
11
12
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A closed Burgers
Circuit in an ideal crystal
SF
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15
16
14
15
16
46
12
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1 82 3 4 5 6 7 9 10 11
12
13
14
15
1234567
9
1234568 791011
12
13
14
15
8
16
Sb 1
6
RHFS convention
F
Map the same Burgers circuit on a
real crystal
The Burgers circuit fails to close !!
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A circuit which is closed in a perfect crystal fails to close in an imperfect crystal if its surface is pierced through a dislocation
line
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
b
If b is not a complete lattice translation then a surface defect will be created along with the line defect.
Surface defect
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N+1 planes
N planes
Compression
Above the slip plane
TensionBelow the slip plane
Elastic strain field associated with an
edge dislocation
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Line energy of a dislocation
2
21 bE
Elastic energy per unit length of a dislocation line
Shear modulus of the crystalb Length of the Burgers vector
Unit: J m1
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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
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b is the shortest lattice translation
FCC 11021
DC 11021
NaCl 11021
SC 100
BCC 11121
CsCl 100
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A dislocation line cannot end abruptly inside a crystal
Slip plane
slip no slip
slip no slip
disl
ocat
ion
b
Dislocation: slip/no slip boundary
Slip plane
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A
B
A dislocation line cannot end abruptly inside a crystal
C
D
Q
P
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
A
P
Q
A dislocation line cannot end abruptly inside a crystal
It can end on a free surface
A
B
C
D
Q
P
Dislocation Line AB Dislocation Line APQ
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Grain 1 Grain 2
Grain Boundary
Dislocation can end on a grain boundary
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A dislocation loop
b
b
b
bt
t
t
tNo slip
slip
The line vector t is always
tangent to the dislocation line
The Burgers vector b is
constant along a dislocation
line59
b
Cylindrical slip plane (surface)
Prismatic dislocation loopCan a loop be
entirely edge?
b
Example 6.2
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tt t
b2
b3
b1
Node
Dislocation node
b1 + b2 + b3 = 0
b1
b2
b3
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A dislocation line cannot end abruptly inside a crystal
It can end on
Free surfaces
Grain boundaries
On other dislocations at a point called a nodeOn itself forming a loop
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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
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Dislocation Motion
Glide (for edge, screw or mixed)Cross-slip (for screw only)
Climb (or edge only)
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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.
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Glide of an Edge
Dislocation
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Glide of an Edge
Dislocation
crss
crss
crss is
critical
resolved
shear
stress on
the slip
plane in
the
direction of
b.
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Glide of an Edge
Dislocation
crss
crss
crss is
critical
resolved
shear
stress on
the slip
plane in
the
direction of
b.
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Glide of an Edge
Dislocation
crss
crss
crss is
critical
resolved
shear
stress on
the slip
plane in
the
direction of
b.
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Glide of an Edge
Dislocation
crss
crss
crss is
critical
resolved
shear
stress on
the slip
plane in
the
direction of
b.
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Glide of an Edge
Dislocation
crss
crss
Surface step, not a dislocation
A surface step of
magnitudeb is
created if a
dislocation sweeps over the
entire slip plane
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slip no slipDislocation
motion
Shear stress is in a direction perpendicular to the GLIDE motion
of screw dislocation
tb
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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
1 2
3
b
Cross-slip of a screw dislocation
Change in slip plane of a screw
dislocation is called cross-slip
Slip plane 1
Slip pla
ne 2
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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
glide
Slip plane 1
Slip plane 2
1 2
3 4?
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Atomistic mechanism of climb
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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
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http://www.tf.uni-kiel.de/matwis/amat/def_en/index.html
A nice diagram showing a variety of crystal defects
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Surface Defects
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Surface Defects
External Internal
Free surface Grain boundaryStacking fault
Twin boundary
Interphase boundary
Same phase
Different phases
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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
Area A
Broken bonds
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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
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?
Why?
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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
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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
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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
A
BC
2
2h
b
A
BC
2sin
2
h
b
tanhb
Eqn. 6.7
Or approximatel
y
90
Stacking fault
CBACBACBA
ACBABACBA
Stacking fault
FCC FCC
HCP
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Twin Plane
CBACBACBACBA
CABCABCBACBA
Twin plane
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Edge Dislocation
432 atoms
55 x 38 x 15 cm3
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Screw Dislocation525 atoms
45 x 20 x 15 cm3
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Screw Dislocation(another view)95
A dislocation cannot end abruptly inside a crystal Burgers vector of a dislocation is constant
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A B
CD
P QL
720 atoms
45 x 39 x 30 cm3Front face: an edge dislocation enters
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E
FG
H
R S
Back face: the edge dislocation does not come out !! 98
Schematic of the Dislocation Model
Edge dislocation
bb
GF
A B
RS
MN
D
HE
C
PQL
Screw dislocation
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A low-angle Symmetric
Tilt Boundary
477 atoms
55 x 30 x 8 cm3
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R. Prasad
Dislocation Models for Classroom Demonstrations
Conference on Perspectives in Physical Metallurgy and Materials Science
Indian Institute of Science, Bangalore
2001
101
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”
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A Prismatic Dislocation Loop685 atoms
38 x 38 x 12 cm3
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Slip plane
Prismatic Dislocation loop
104
a b
cd
A Prismatic Dislocation LoopTop View
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Crystal Dislocation Models for TeachingThree-dimensional models for dislocation studies in crystal structures …
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