b - Materials Science

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Crystal
Defects
Chapter 6
1
IDEAL
vs.
Reality
2
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)
3
Real Crystal
Deviations from this ideality.
These deviations are known as crystal defects.
4
Is a lattice finite or infinite?
Is a crystal finite or infinite?
Free surface:
a 2D defect
5
Vacancy: A point defect
6
Defects Dimensionality
Examples
Point
0
Vacancy
Line
1
Dislocation
Surface
2
Free surface,
Grain boundary
Stacking Fault
7
Point
Defects
Vacancy
8
Point Defects: vacancy
A Guess
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.
9
Equilibrium?
Equilibrium means Minimum Gibbs free
energy G at constant T and P
A crystal with vacancies has a lower
free energy G than a perfect crystal
What is the equilibrium
concentration of vacancies?
10
Gibbs Free Energy G
G=H–TS
1. Enthalpy H =E+PV
2. Entropy S
=k ln W
T Absolute temperature
E internal energy
P pressure
V volume
k Boltzmann constant
11
W number of microstates
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
13
Configurational entropy due to vacancy
Number of atoms:
N
Number of vacacies: n
Total number of sites: N+n
The number of microstates:
W
( N  n)!
Cn 
n! N !
N n
Increase in entropy S due to vacancies:
( N  n)!
n! N !
 k[ln(N  n)! ln n! ln N!]
DS  k ln W  k ln
14
Stirlings Approximation
ln N! N ln N  N
N
ln N!
N ln N N
1
0
1
10
15.10
13.03
100
363.74
360.51
100!=933262154439441526816992388562667004907159682643816214685\
9296389521759999322991560894146397615651828625369792082\
15
7223758251185210916864000000000000000000000000
DS  k ln W  k[ln(N  n)! ln n! ln N!]
ln N ! N ln N  N
DS  k[(N  n) ln(N  n)  n ln n  N ln N ]
DH  n DH f
16
Change in G of a crystal due to vacancy
DG
DH
DH  n DH f
G of a
perfect
crystal
DG = DH  TDS
neq
n
TDS
Fig. 6.4
DS  k[(N  n) ln(N  n)  n ln n  N ln N ]
17
Equilibrium concentration of vacancy
DS  k[(N  n) ln(N  n)  n ln n  N ln N ]
DH  n DH f
DG  nDH f  Tk[(N  n) ln(N  n)  n ln n  N ln N ]
DG
n
0
n  neq
 DH f
 exp 
N
 kT
neq



18
With neq<<N
 DH f
 exp 
N
 kT
neq
Al:
Ni:



DHf= 0.70 ev/vacancy
DHf=1.74 ev/vacancy
n/N
0K
300 K
900 K
Al
0
1.45x1012 1.12x104
Ni
0
5.59x1030 1.78x10-10
19
Contribution of vacancy to thermal expansion
Increase 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
20
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
21
Contribution of vacancy to thermal expansion
V  Nv
DV  N Dv  V DN
DV
V
Total
expansion
V=volume of crystal
v= volume associated with
one atom
N=no. of sites
(atoms+vacancy)
Dv
DN


v
N
Lattice
parameter
increase
vacancy
22
Experimental determination of n/N
DV
Dv DN


V
v
N
3DL 3Da
n


L
a
N
n
 DL Da 
3


N
a 
 L
Problem
6.2
Linear
thermal
expansion
coefficient
Lattice
parameter as a
function of
temperature
XRD
23
Point Defects
Interstitial
impurity
vacancy
Substitutional
impurity
24
Defects in ionic solids
Frenkel
defect
Cation vacancy
+
cation interstitial
Schottky
defect
Cation vacancy
+
anion vacancy
25
Line Defects
Dislocations
26
Missing half plane A Defect
27
An extra half plane…
28
…or a missing half plane
What kind of
defect is this?
A line defect?
Or a planar
defect?
29
Extra half plane
No extra plane!
30
Missing plane
No missing plane!!!
31
An extra half plane…
Edge
Dislocation
32
…or a missing half plane
This is a line defect called an
33
EDGE DISLOCATION
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.)
34
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
935
1
1
2
2
3
3
4
4
5
6
7
8
9
5
6
7
8
936
1
2 3
4
5
6
7
Burgers vector
b
8
9
Slip plane
slip
no slip
boundary = edge dislocation
1
2
3
4
5
6
7
8
937
Slip plane
t
b
no slip
dislocation
slip
Dislocation: slip/no
slip boundary
b: Burgers vector
magnitude and
direction of the slip
t: unit vector
tangent to the
dislocation line
38
Dislocation Line:
A dislocation line is the boundary between
slip and no slip regions of a crystal
Burgers vector:
The magnitude and the direction of the
slip is represented by a vector b called
the Burgers vector,
Line vector
A unit vector t tangent to the dislocation
line is called a tangent vector or the line
vector.
39
Two ways to describe an EDGE DISLOCATION
1. Bottom edge of an extra half plane
2. Boundary between slip and no-slip regions of a slip plane
1
2
3
4
Burgers vector
b
5
6
7
Line vector
slip
8
9
Slip plane
no slip
t
What is the
relationship
between the
directions of
b and t?
b t
1
2
3
4
5
6
7
8
9
40
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
41
Screw Dislocation
b || t
Slip plane
t
unslipped
slipped
b
1
2
3
42
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
43
Positive
Edge
Dislocation
Screw
Dislocation
Extra half
plane above
the slip plane
Left-handed
spiral ramp
b parallel to t
Negative
Extra half
plane below
the slip plane
Right-handed
spiral ramp
b antiparallel to t
44
Burgers vector
Johannes Martinus
BURGERS
Burger’s vector
Burgers vector
45
S
F
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
1
9
2
8
3
A closed
Burgers
Circuit in an
ideal crystal
7
6
5
4
4
5
6
7
3
8
2
1
9
16
15 14
13
12
11
10
9
8
7
6
5
4
3
2
1
46
The Burgers circuit fails to close !!
F b
9
1
2
3
4
5
6
7
8
9
10 11
12 13
14
15
16
S
1
2
8
3
 same
Map the
Burgers circuit on a
real crystal
7
6
5
4
4
5
6
7
3
8
2
1
9
16 15
14
13
12
11
10
9
8
7
6
RHFS convention
5
4
3
2
1
47
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 Superman
Happy Teacher’s Day
b is a lattice translation
b
Surface defect
If b is not a complete lattice translation then a surface
defect will be created along with the line defect.
50
Elastic strain field associated with an
edge dislocation
N+1 planes
Compression
Above the slip plane

Tension
Below the slip plane
N planes
51
Line energy of a dislocation
Elastic energy per unit length of a
dislocation line
1
2
E  b
2

Shear modulus of the crystal
b
Length of the Burgers vector
Unit: J m1
52
Energy of a dislocation line
is proportional to b2.
Thus dislocations with
short b are preferred.
b is a lattice translation
b is the shortest lattice
translation
53
b is the shortest lattice
translation
SC
100
BCC
1
111
2
FCC
1
110
2
DC
1
110
2
NaCl
1
110
2
CsCl
100
54
A dislocation line cannot end
abruptly inside a crystal
Slip plane
Slip plane
b
no slip
dislocation
slip
Dislocation:
slip/no slip
boundary
slip
no slip
55
A dislocation line cannot end
abruptly inside a crystal
C
Q
D
B
P
A
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 dislocation line cannot end
abruptly inside a crystal
C
Q
Q
D
B
P
P
A
A
Dislocation Line AB
Dislocation Line APQ
It can end on a free surface
57
Dislocation can end on a grain boundary
Grain
Boundary
Grain 1
Grain 2
58
The line vector
t is always
tangent to the
dislocation line
A dislocation loop
t
b

b t
t b
slip

t
No slip
b
The Burgers
vector b is
constant along a
59
dislocation line
Can a loop be
entirely edge?
Prismatic dislocation loop
Example 6.2
b
b
Cylindrical slip plane (surface)
60
Dislocation node
b2
t
Node
t
t
b1
b3
b2
b1
b1 + b2 + b3 = 0
b3
61
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 node
On itself forming a loop
62
Slip plane
The 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 plane
A screw dislocation does not have a
unique slip plane.
Any plane passing through a screw
dislocation is a possible slip plane
63
Dislocation Motion
Glide (for edge, screw or mixed)
Cross-slip (for screw only)
Climb (or edge only)
64
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.
65
Glide of
an Edge
Dislocation


66
Glide of
an Edge
Dislocation
crss
crss is
crss
critical
resolved
shear
stress on
the slip
plane in
the
direction
of b.
67
Glide of
an Edge
Dislocation
crss
crss is
crss
critical
resolved
shear
stress on
the slip
plane in
the
direction
of b.
68
Glide of
an Edge
Dislocation
crss
crss is
crss
critical
resolved
shear
stress on
the slip
plane in
the
direction
of b.
69
Glide of
an Edge
Dislocation
crss
crss is
crss
critical
resolved
shear
stress on
the slip
plane in
the
direction
of b.
70
Glide of
an Edge
Dislocation
A surface
step of
magnitude
b is
created if
a
dislocation
sweeps
over the
entire slip
plane
crss
Surface
step, not a
dislocation
crss
71
slip

b
Dislocation
motion
no slip
t
Shear stress is in a direction
perpendicular to the GLIDE motion of
screw dislocation
72
Glide Motion and the Shear Stress
For 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
Slip plane 1
b
1
2
3
Change in slip
plane of a screw
dislocation is
called cross-slip
74
Climb of an edge dislocation
The motion of an edge dislocation
from its slip plane to an adjacent
parallel slip plane is called CLIMB
Slip plane 2

1
?
glide

3
glide
climb

4

2
Obstacle
Slip plane 1
75
Atomistic mechanism of climb


76
Climb of an edge dislocation
Climb up
Climb down
Half plane shrinks
Half plane stretches
Atoms move away
from the edge to
nearby vacancies
Atoms move toward
the edge from
nearby lattice sites
Vacancy
concentration
goes down
Vacancy
concentration
goes up 77
Dislocations in a real crystal can form
complex networks
From Callister
78
A nice diagram showing a variety of crystal
defects
http://www.tf.uni-kiel.de/matwis/amat/def_en/index.html
79
Surface
Defects
80
Surface Defects
External
Free surface
Internal
Grain boundary
Stacking fault
Same
phase
Twin boundary
Interphase
boundary
Different
phases
81
External surface: Free surface
Area A
Broken
bonds
If bond are broken over
an area A then two free
surfaces of a total area
2A is created
Area A
82
External surface: Free surface
nA=no. of surface atoms per
unit area
nB=no. of broken bonds per
surface atom
=bond energy per atom
1
  n A nB 
2
Surface energy
per unit area
Area A
Broken
bonds
Area A
If bond are broken over an
area A then two free surfaces
83
of a total area 2A is created
What is the shape of a naturally grown
salt crystal?
Why?
84
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
Internal surface: grain boundary
Grain
Boundary
Grain 1
Grain 2
A grain boundary is a boundary between two
regions of identical crystal structure but
86
different orientation
Optical Microscopy,
Experiment 5
Photomicrograph an iron
chromium alloy. 100X.
Callister, Fig. 4.12
87
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
88
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
Tilt boundary
Edge dislocation
model of a small angle
tilt boundary

Grain 1
C
C
B


A
Grain 2
b

 sin
2h
2
Or
approximately
b B

2
2h
b
 tan 
h
Eqn. 6.7
90
A
Stacking fault
C
B
A
C
B
A
C
B
A
FCC
Stacking
fault
A
C
B
A
B
A
C
B
A
FCC
HCP
91
Twin Plane
C
B
A
C
B
A
C
B
A
C
B
A
Twin
plane
C
A
B
C
A
B
C
B
A
C
B
A
92

Edge Dislocation
432 atoms
55 x 38 x 15 cm3
93
Screw Dislocation
525 atoms
45 x 20 x 15 cm3
94
Screw Dislocation (another view)
95
 A dislocation cannot end
abruptly inside a crystal
 Burgers vector of a
dislocation is constant
96
B
A
L
P
Q
C
D
Front face: an edge
dislocation enters
720 atoms
45 x 39 x 30 cm3
97
G
F
R
S
E
H
Back face: the edge dislocation does not
come out !!
98
F
G
Screw
dislocation
B
A
S
b
N
M
b
L
P
E
D
R
Q
H
Edge dislocation
C
Schematic of the Dislocation
Model
99

A low-angle
Symmetric
Tilt Boundary



477 atoms
55 x 30 x 8 cm3
100
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”
102


A Prismatic Dislocation Loop
685 atoms
38 x 38 x 12 cm3
103
Slip plane
Prismatic
Dislocation
loop
104
d
c
a
b
A Prismatic Dislocation Loop
Top View
105
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107

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