Defects Chapter 6 1 2 IDEAL vs Reality 3 An ideal crystal can be described in terms a threedimensionally periodic arrangement of points called lattice and an atom or group of atoms associated with each lattice point called motif ID: 458625
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Slide1
Crystal DefectsChapter 6
1Slide2
2IDEALvs.
RealitySlide3
3An 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)Slide4
Deviations from this ideality.
These deviations are known as crystal defects.
4
Real CrystalSlide5
Is a lattice finite or infinite?
Is a crystal finite or infinite?
Free surface: a 2D defect
5Slide6
Vacancy: A point defect
6Slide7
Defects Dimensionality Examples
Point 0 Vacancy
Line 1 Dislocation
Surface 2 Free surface,
Grain boundary
Stacking Fault
7Slide8
Point DefectsVacancy
8Slide9
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
9Slide10
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
10Slide11
1. Enthalpy H
2. Entropy
S
G
=
H
–
T
S
Gibbs Free Energy
G
=
E
+
PV
=k
ln
W
T
Absolute temperature
E
internal energy
P
pressure
V
volume
k
Boltzmann constant
W
number of microstates
11Slide12
Vacancy increases
H
of the crystal due to energy required to break bonds
D
H = n
D
H
f
12Slide13
Vacancy increases
S
of the crystal due to configurational entropy
13Slide14
Number of atoms: N
Increase in entropy
S
due to vacancies:
Number of vacacies:
n
Total number of sites:
N+n
The number of microstates:
Configurational entropy due to vacancy
14Slide15
Stirlings Approximation
N
ln
N!
N
ln
N
-
N
1 0
-
1
10 15.10 13.03
100 363.74 360.51
100!=933262154439441526816992388562667004907159682643816214685\
9296389521759999322991560894146397615651828625369792082
\
7223758251185210916864000000000000000000000000
15Slide16
16Slide17
D
G
=
D
H
-
T
D
S
n
eq
G
of a perfect crystal
Change in
G
of a crystal due to vacancy
n
D
G
D
H
-
T
D
S
Fig. 6.4
17Slide18
With
n
eq
<<N
Equilibrium concentration of vacancy
18Slide19
Al:
D
H
f
= 0.70 ev/vacancy
Ni:
D
H
f
= 1.74
ev/vacancy
n
/
N
0 K
300 K
900 K
Al
0
1.45x10
-
12
1.12x10
-
4
Ni
0
5.59x10
-
30
1.78x10
-10
19Slide20
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
20Slide21
Contribution of vacancy to thermal expansionThus vacancy makes a small contribution to the thermal expansion of a crystal
Thermal expansion =
lattice parameter expansion
+
Increase in volume due to vacancy
21Slide22
Contribution of vacancy to thermal expansion
V=volume of crystal
v= volume associated with one atom
N=no. of sites
(atoms+vacancy)
Total expansion
Lattice parameter increase
vacancy
22Slide23
Experimental determination of n/N
Linear thermal expansion coefficient
Lattice parameter as a function of temperature
XRD
Problem 6.2
23Slide24
vacancy
Interstitial
impurity
Substitutional
impurity
Point Defects
24Slide25
Frenkel defect
Schottky defect
Defects in ionic solids
Cation vacancy
+
cation interstitial
Cation vacancy
+
anion vacancy
25Slide26
Line DefectsDislocations
26Slide27
Missing half plane
A Defect
27Slide28
An extra half plane…
…or a missing half plane
28Slide29
What kind of defect is this?A line defect?
Or a planar defect?
29Slide30
Extra half plane
No extra plane!
30Slide31
Missing plane
No missing plane!!!
31Slide32
An extra half plane…
…or a missing half plane
Edge
Dislocation
32Slide33
If a plane ends abruptly inside a crystal we have a defect.
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 DISLOCATION
33Slide34
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.)
34Slide35
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
35Slide36
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
36Slide37
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 plane
b
Burgers vector
37Slide38
Slip plane
slip
no slip
dislocation
b
t
Dislocation: slip/no slip boundary
b
: Burgers vector
magnitude and direction of the slip
t
: unit vector tangent to the dislocation line
38Slide39
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.
39Slide40
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
slip
no slip
Slip plane
b
Burgers vector
t
Line vector
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
What is the relationship between the directions of
b
and
t
?
b
t
40Slide41
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
41Slide42
Screw Dislocation Line
b
t
b || t
1
2
3
Screw Dislocation
Slip plane
slipped
unslipped
42Slide43
If b || tThen parallel planes
to the dislocation line lose their distinct identity and become one continuous spiral ramp
Hence the name SCREW DISLOCATION
43Slide44
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
44Slide45
Burgers vectorJohannes Martinus BURGERS
Burgers vector
Burger’s vector
45Slide46
1
2
7
6
5
4
3
8
9
1
8
2
3
4
5
6
7
9
10
11
12
13
1
2
3
4
5
6
7
8
9
1
8
2
3
4
5
6
7
9
10
11
12
13
A closed Burgers Circuit in an ideal crystal
S
F
14
15
16
14
15
16
46Slide47
1
2
7
6
5
4
3
8
9
1
8
2
3
4
5
6
7
9
10
11
12
13
14
15
1
2
3
4
5
6
7
9
1
2
3
4
5
6
8
7
9
10
11
12
13
14
15
8
16
S
b
16
RHFS convention
F
Map the same
Burgers circuit on a
real crystal
The Burgers circuit fails to close !!
47Slide48
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.
48Slide49
Those who can, do. Those who can’t, teach.G.B Shaw, Man and Superman
Happy Teacher’s DaySlide50
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
50Slide51
N+1 planes
N planes
Compression
Above the slip plane
Tension
Below the slip plane
Elastic strain field associated with an
edge dislocation
51Slide52
Line energy of a dislocation
Elastic energy per unit length of a dislocation line
Shear modulus of the crystal
b Length of the Burgers vector
Unit: J m
1
52Slide53
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
53Slide54
b is the shortest lattice translation
FCC
DC
NaCl
SC
BCC
CsCl
54Slide55
A dislocation line cannot end abruptly inside a crystal
Slip plane
slip
no slip
slip
no slip
dislocation
b
Dislocation: slip/no slip boundary
Slip plane
55Slide56
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??
56Slide57
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
57Slide58
Grain 1
Grain 2
Grain
Boundary
Dislocation can end on a grain boundary
58Slide59
A dislocation loop
b
b
b
b
t
t
t
t
No slip
slip
The line vector
t
is always tangent to the dislocation line
The Burgers vector
b
is constant along a dislocation line
59Slide60
b
Cylindrical slip plane (surface)
Prismatic dislocation loop
Can a loop be entirely edge?
b
Example 6.2
60Slide61
t
t
t
b
2
b
3
b
1
Node
Dislocation node
b
1
+ b
2
+ b
3
= 0
b
1
b
2
b
3
61Slide62
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
62Slide63
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 plane
A screw dislocation does not have a unique slip plane.
Any plane passing through a screw dislocation is a possible slip plane
63Slide64
Dislocation MotionGlide (for edge, screw or mixed)
Cross-slip (for screw only)
Climb (or edge only)
64Slide65
Dislocation Motion: GlideGlide is a motion of a dislocation in its own slip plane.
All kinds of dislocations, edge, screw and mixed can glide.
65Slide66
Glide of
an Edge
Dislocation
66Slide67
Glide of
an Edge
Dislocation
crss
crss
crss is critical resolved shear stress on the slip plane in the direction of
b
.
67Slide68
Glide of
an Edge
Dislocation
crss
crss
crss is critical resolved shear stress on the slip plane in the direction of b.
68Slide69
Glide of
an Edge
Dislocation
crss
crss
crss is critical resolved shear stress on the slip plane in the direction of b.
69Slide70
Glide of
an Edge
Dislocation
crss
crss
crss is critical resolved shear stress on the slip plane in the direction of b.
70Slide71
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
71Slide72
slip
no slip
Dislocation
motion
Shear stress is in a direction perpendicular to the GLIDE motion of screw dislocation
t
b
72Slide73
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
73Slide74
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 plane 2
74Slide75
Climb of an edge dislocationThe motion of an edge dislocation from its slip plane to an adjacent parallel slip plane is called
CLIMB
Obstacle
climb
glide
glide
Slip plane 1
Slip plane 2
1
2
3
4
?
75Slide76
Atomistic mechanism of climb
76Slide77
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
77Slide78
From Callister
Dislocations in a real crystal can form complex networks
78Slide79
http://www.tf.uni-kiel.de/matwis/amat/def_en/index.html
A nice diagram showing a variety of crystal defects
79Slide80
Surface Defects80Slide81
Surface DefectsExternal
Internal
Free surface
Grain boundary
Stacking fault
Twin boundary
Interphase boundary
Same phase
Different phases
81Slide82
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
82Slide83
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
n
A
=no. of surface atoms per unit area
n
B
=no. of broken bonds per surface atom
=bond energy per atom
Surface energy per unit area
83Slide84
What is the shape of a naturally grown salt crystal?
Why?
84Slide85
Surface energy is anisotropicSurface energy depends on the orientation, i.e., the Miller indices of the free surafce
n
A
, n
B
are different for different surfaces
Example 6.5 & Problem 6.16
85Slide86
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
86Slide87
Photomicrograph an iron chromium alloy. 100X.
Callister, Fig. 4.12
Optical Microscopy, Experiment 5
87Slide88
Grain Boundary: low and high angleOne 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
88Slide89
Grain Boundary: tilt and twistOne 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
89Slide90
Edge dislocation model of a small angle tilt boundary
Grain 1
Grain 2
Tilt boundary
A
B
C
2
h
b
A
B
C
Eqn. 6.7
Or approximately
90Slide91
Stacking faultC
B
A
C
B
A
CB
A
A
C
B
A
B
A
C
B
A
Stacking fault
FCC
FCC
HCP
91Slide92
Twin PlaneC
B
A
C
B
A
CB
A
C
B
A
C
A
B
C
A
B
C
BACBA
Twin plane92Slide93
Edge Dislocation
432 atoms
55 x 38 x 15 cm
3
93Slide94
Screw Dislocation
525 atoms
45 x 20 x 15 cm
3
94Slide95
Screw Dislocation (another view)
95Slide96
A dislocation cannot end abruptly inside a crystal
Burgers vector of a dislocation is constant
96Slide97
A
B
C
D
P
Q
L
720 atoms
45 x 39 x 30 cm
3
Front face: an edge dislocation enters
97Slide98
E
F
G
H
R
S
Back face: the edge dislocation does not come out !!
98Slide99
Schematic of the Dislocation Model
Edge dislocation
b
b
G
F
A
B
R
S
M
N
D
H
E
C
P
Q
L
Screw dislocation
99Slide100
A low-angle Symmetric Tilt Boundary
477 atoms
55 x 30 x 8 cm
3
100Slide101
R. PrasadDislocation Models for Classroom DemonstrationsConference on Perspectives in Physical Metallurgy and Materials Science
Indian Institute of Science, Bangalore
2001
101Slide102
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”
102Slide103
A Prismatic Dislocation Loop
685 atoms
38 x 38 x 12 cm
3
103Slide104
Slip plane
Prismatic Dislocation loop
104Slide105
a
b
c
d
A Prismatic Dislocation Loop
Top View
105Slide106
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