What you have to learn from this chapter

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Chapter 6. Elements of Grain Boundaries
What you have to learn from this chapter? ?
Dislocation Model for Grain Boundary ? Low
Energy Dislocation Structure ? Stress Field,
Energy, and Surface Tension of a Grain
Boundary ? Special Boundary Coincident site
boundary ? Grain Boundary and Mechanical
Properties
2
Most engineering objects are polycrystalline
materials. Grain boundaries play an important
part in determining the properties of a material
e.g. resistivity, diffusion, fracture mechanism,
etc.
?
? Small-Angle Grain Boundary Consists of an
array of edge dislocation! The angle
between two grains ? the spacing between
dislocations d the Burgers vector b
b
? /2
A real idea case, difficult to find in real case.
Experimental evidence of small angle boundary
see Fig. 6.3, 6.4 (not a simple tilt boundary,
involving twisting).
2d
3
? Degrees of freedom of a grain boundary
See Fig, 6.5. 2 degrees of freedom in tilting
with respect to the two tile axis that could
be defined on the grain boundary plane. 1
degree of freedom in twist. 2 degrees of
freedom in grain boundary itself (asymmetrical).
? ? /2
B
?
D1
?
A
?
D2
? -? /2
? /2
? /2
h
? /2
C
D
How many planes that could be put in AB?
How many planes that could be put in CD?
4
The number difference gt dislocation number (one
for each extra plane!)
? Stress Field of a Grain Boundary The
grain boundaries do not possess long-range
stress fields. To demonstrate this claim,
take the simple small angle boundary as a
case.
y -(nd-y)
a

nd-y
nd
p
y
d
x
Summation of all shear stresses that act on point
p from n -? to ?
5
??? (1)
gt
??? (2)
gt
Differentiating equation (1) with respect to p gt
(3)
gt
6
Differentiating equation (2) with respect to p gt
(4)
gt
Using eq. (3)
For x gtgt d/2?
gt
In the textbook, y 0
gt
gt
Using eq. (4)
gt
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Stress of a single edge dislocation
The boundary stress approaches that of a single
dislocation as x become very small. See Fig. 6.7
(A). CRSS critical resolved shear stress. ?
Fig. 6.7 of the textbook assume d22b gt
boundary stress drop to negligible level at x
about several tens of b.
Boundary stress
Shear stress, ?xy
CRSS
x
Number of b from boundary
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? Grain Boundary Energy Consider the work
required to separate a set of dislocation
array (one positive and the other negative
array) form r0 to ?
r0 the distance where the core energy of
two dislocations should cancel each other!
Attractive force
?
?xyb
wgb the energy per unit length per dislocation
?b the energy per unit area per dislocation
9
gt
When ?0 ltlt 1,
?? b/d
? a factor accounting for the core energy
Shockley-Read equation
?1lt?2lt?3
?max
Large angle equation
1
?3
See Fig.6.8!
?2
?
?
? /?max
?1
Small angle equation
? (degrees)
0
? (degrees)
? (degrees)
30
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? Low-Energy Dislocation Structure
? Consider the case of simple tilt boundary again!
Follow previous calculation

when nd lt x gt ?nxy gt0 when nd gt x gt ?nxy lt0 gt
Only small n gt large positive contributions to
shear stress
n
?
-
0
The overall effect of shears stress field is
small at reasonable x. (already shown in previous
section!!)
? The other two stress components (normal
stresses) gt the stress equation is
anti-symmetric gt ?xx(x,y)-?xx(x,-y)
?yy(x,y)-?yy(x,-y)! When n 0, ? 0xx(x,y)?
0yy(x,y) 0, when n ? 0 gt ? nxx(x,y) -?-
-nxx(x,y) ?nyy(x,y) -?- -nyy(x,y)!
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? Incorporation of a random dislocation into a
tilt boundary Energy per unit length of a
boundary dislocation wgb Energy per unit
length of a random dislocation we.
1
wgb/we 1 no energy difference for a dislocation
to be in a grain boundary or to be a random
one. wgb/we lt1 a dislocation could lower its
energy by being part of a grain boundary.
wgb/we
d
Dislocation spacing
Even with d 500b gt a dislocation could still
lower its energy significantly by being part of
the tilt grain boundary. As d ? wgb/we ? gt Tilt
boundary attracts edge dislocations!
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? The tilt boundary is one of a very large number
of dislocation arrays which have the property
of having a low strain energy. gt
Kuhlmann-Wilsdorf proposed low energy
dislocation structures (LEDS). ? Grain and
subgrain boundary structures normally belong
to LEDS!
Slip plane
Taylor Lattice earliest form of LEDS
Kink band
? When the plastic deformation involves a number
of different slip planes and Burgers vector
gt more complicated LEDS structure and
difficult to perceive! Plastic deformation gt
forms cells with a low internal dislocation
density and boundaries composed of
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dislocation tangles. See Fig. 6.15. Plastic
deformation ? cell size ? of cell ? ? ??
-1/2 ? average cell diameter ? a
constant ? dislocation density
? Dynamic Recovery ? Recovery after
plastic deformation, a considerable
amount of energy is released by the local
rearrangement of the dislocations in the
tangles and further release of energy
occurs when LEDS are formed. gt the
process typically requires thermal
energy gt Annealing is the typical process for
recovery (static) to happen! ?
Recovery could also occur during plastic
deformation (no matter the temperature
of the sample is low or high) gt dynamic
recovery (a very important factor
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in the deformation process) gt affects
the ? - ? curve (lower the effective
rate of work hardening) ? Dynamic recovery
occurs most strongly in metals of high
stacking fault energy and shows limited effect
on metals of low stacking fault energy!
See Fig. 6.16. Why? The major mechanism
involving dynamic recovery is thermally
activated cross-slip. (Why ?)
? Surface Tension of the Grain Boundary ?
The solid grain boundaries process a surface
energy gt surface tension (equivalent to
that of a liquid). In solid, surface
tension is a confusion term! Surface
energy is the energy required to create a unit
surface. Surface tension is the work
required to create a unit surface. It
costs less work to create a low energy
surface!
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? Typical value of surface tension of
external metal surface 1.5 J/m2 of
liquid surface 0. 1 J/m2 of large
angle grain boundary 0. 4 J/m2.
?b
Crystal 1
Force balance requires
c
?a
Crystal 2
a
b
?c
Crystal 3
? the surface energy of grain boundary
Grain boundary
? The kinetics of reaching force balance
configuration is mainly controlled by the
grain boundary motion. The grain boundary
mobility is definitely a function of
temperature. The thermodynamics of the grain
boundary movement is to reduce the total energy
of the system move through highly deformed
region and leave behind strain free region
reducing the boundary
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area (by straighten the boundaries, or grain
growth), etc. ? In pure metal, low angle
boundaries are seldom observed. gt Fig. 6.8 gt
all grain boundaries have the same energy! gt
120o is the balanced angle between three grains
? Boundaries Between Crystals of Different
Phases ? force balance
180o
?fs
?fs
?12
?
Phase 1
?11
Grain 2
?b
?
Grain 1
Phase 2
?
Phase 1
?12
0
Force balance requires
?12 / ?11
0.5
? When ? ? 0, the second phase tends to form a
thin film between crystals! If ?12/?11 lt 0.5
gt the second
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phase penetrate the single phase boundary gt
isolated the crystals of the first phase
(occurs even at very small amount of second
phase) ? E.g. Bi (a brittle metal) in Cu (a
ductile metal) gt ? very small ?12 gt ? 0 gt
when Bi could form a continuous film (lt0.05)
around Cu crystal gt Cu loses its
ductility. ? Another important metallurgical
case second phase impurities remain in the
liquid state until a temp. well below the
freezing point of the major phase! ? A small
amount of harmful impurities could do a lot of
damage to the plastic properties of metals. High
interfacial energy gt the liquid tends to form
discrete globules see Fig. 6.21 gt less
significant effect on the plastic properties
of the metals!
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? Grain Size ? An important parameter in a
lot of properties, but difficult to
define precisely. ? Most generally accepted
method linear intercept method mean
grain intercept average distance
between grain boundaries along a line laid on a
photograph (SEM, TEM). ?
Quantitative metallography has shown
where Sv is the surface area of grain boundaries
per unit volume.
? Effect of Grain Boundaries on Mechanical
Properties ? Polycrystalline metals always
show a strong effect of grain size on
hardness. gt
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?2
?3
Slope kH
?1
Hardness, H
Flow-stress, ?
?3 gt?2 gt ?1
H0
d-1/2
d-1/2
H0 not necessarily the hardness of single crystal
Hall-Petch equation
Rationale Considering the grain boundaries as
a barrier to the dislocation movement gt a
pile up at one grain could generate
sufficiently large stresses to operate sources
in an adjacent grain at a yield stress! There
are also other interpretations based on grain
boundary dislocations.
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Linear arrays of edge dislocations piled-up
against barriers under an applied shear stress
?1 the stress experienced by the leading
dislocation! ?0 a backward force due to
the internal stress produced by the
obstacle! n number of dislocations in a line
?
Barrier
S
?x
?
The leading dislocation move ?x (a small
displacement) gt all other dislocations move
?x gt the work done is n? b?x. The increase in
the interaction energy between the leading
dislocation and ?0 is ?0 b?x. In equilibrium
gt?1 ?0 gt n? b?x ?1 b?x gt ?1 n? .
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gt The stress at the head of the pile-up is
magnified to n times the applied stress. The
pile up exerts a back-stress ?b on the source.
The dislocation could continue generating
dislocation as long as (? - ?b) gt ? (critical
stress for source operation)! gt number of
pile-up dislocations allowed over a region L
(Eshelby et.al)
If ?2 gt critical shear stress ?1 gt S2
operates gt move and generate Dislocations!
?
S1
S2
?
gt
m constant
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? Coincidence Site Boundaries ? A very
important form of grain boundary first
observed in annealed cold rolled fcc metal (Cu)
gt the two stages annealing a lower
temperature anneal followed by a high
temperature anneal (secondary
recrystallization) gt the secondary
recrystallized crystals have specific
orientation relation to the old
crystals. gt could be characterized by
coincidence sites of atoms in two orientation.
gt coincidence site boundaries!
23
38o
22o
The above two cases are rotation of (100)
plane! See Fig. 6.25.
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? Any two lattices with a specific rotation
relation gt form a boundary with certain
coincidence sites gt the more the
coincidence sites gt the more important
the boundary ? higher boundary
mobility gt a rapid rate of grain
growth. ? Density of Coincidence Sites in
previous two figures, one is 1/13 of the
density of the original lattice, the other
is 1/5 of the density of original lattice.
Fig. 6.25 is 1/7! How to quickly count it?
? The reciprocal of the density is commonly used
as a parameter to describe the boundary
?13, ?5, ?7.
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? The Ranganathan Relations ? Ground rules
for rotation to generate a coincidence
sites in a 3-D lattice 4 factors (1)
axis to rotate, hkl (2) rotation
angle, ?, about axis hkl (3) the
coordinates of a coincidence site in the
coincidence site (hkl) net, (x,y)
(4) ? A the reciprocal of the density of
coincidence sites in (hkl) net.
A1/2ap structural periodicity ? The
relation between factors the length ratio
y/xN1/2
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? Twist Boundaries Examples 1, see Fig.6.26.
Simple cubic lattice rotation axis lt100gt gt
N1 take x2 and y1 gt ? 53.1o, ?5
take x3 and y1 gt ? 36.9.1o, ?10 see Fig.
6.27. 10/2 gt 5!
? A could only be odd number divided the A
value by smallest 2n to get odd number. ?
Twist Boundaries Examples 2, see Fig.6.25
and Fig. 6.28. FCC lattice rotation axis
lt111gt gt N3 outline (111) as a tetragonal
unit cell, see Fig. 6.28 take x9 y1 ? ?
21.8o, ?84 84/127 ? ?7 take x5 y1 ?
? 38.2o, ?28 28/47 ? ?7
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? Tilt Boundaries Examples 1, see Fig.6.29
Simple cubic lattice rotation axis lt100gt gt
N1 take surface steps x2 and y1and join
two surface gt the tilt boundary gt ?
53.1o, ?5 51/2a p take x3 and y1 gt
? 36.9o, ?10 10/25, ?5 51/2a p see
Fig. 6.30. ? Cases of overlapping atoms, see Fig.
6.31. gt Relieving by relative translation of
the lattice above and below the boundary, see
Fig. 6.32.
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(111) x9 y1 ? ? 21.8o, ?84 84/127 ? ?7
(X)
31/2B
b
31/2b
b
B
31/2b
?
31/2b
b
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?3(111)1-10 G.B of (SrTiO3)
Z. Zhang, et. al., Science, 302, 846 (2003)