Short-Circuit Diffusion
in Crystals
3.205 L6 11/5/03 -1-�
"[BCA]
"[BCA] Fig.
Fig. X.Y"
X.Y" are
are from
from
Balluffi,
Balluffi, R.
R. W.,
W., S.
S. M.
M. Allen,
Allen, and
and W.
W. C.
C. Carter.
Carter. Kinetics
Kinetics of
of Materials.
Materials.
New
New York:
York: Wiley-Interscience,
Wiley-Interscience, 2004.
2004.
Courtesy
of
John
Wiley
and
Sons,
Inc.
Used
Courtesy of John Wiley and Sons, Inc. Used with
with permission.
permission.
Today’s topics:�
n
Diffusion spectrum in defective crystals
n
Dislocation core structure and dislocation “short circuits”�
n
Grain boundary structure
n
Grain boundary diffusion mechanisms and phenomena�
n
Some phenomena where short-circuits are important
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Diffusion Paths in Polycrystals�
n
n
n
n
DXL, bulk (lattice) diffusivity
DB, grain boundary diffusivity�
DS, (free) surface diffusivity
DD , dislocation diffusivity
Typical behavior�
in fcc metals�
[BCA] Fig. 9.1*
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Grain Boundary Diffusion in NiO�
n�Diffusion
data from NiO, comparing rates of bulk
diffusion and grain boundary diffusion
[BCA] Fig. 9.3*
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Dislocation “Pipe” Diffusion
Dislocations, especially edge dislocations, can act as shortcircuit diffusion paths.
Dissociated dislocations have cores that are more spread
out, connected by a ribbon of stacking fault:
t̂
A
A
B
tˆ
t̂
A
B
B
A
A
C
B
r
b
Perfect dislocation
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Partial dislocation
Grain Boundary Structure�
Specification of a grain boundary: five degrees of
freedom (at least)
Rotation axis r̂�
Rotation angle q�
Boundary normal n̂�
Example: tilt boundaries�
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Grain Boundary Structure, cont’d
Twist boundaries
Unrelaxed atom positions
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Relaxed atom positions
Mechanism of Grain Boundary Diffusion�
Vacancies and self-interstitials�
Atom jumping events in
S5<001>(310) symmetric tilt
boundary in b.c.c.-Fe calculated by
molecular dynamics using pairpotential model. The ratios of the
scales used in the drawing are
[1 30] : [310] : [001] = 1 :1 : 5 � vacancy trajectory, confined
to grain boundary sites
vacancy/interstitial
pair creation
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Grain Boundaries and Impurity Diffusion�
n� Diffusivity
of Zn along the tilt
axes [110] symmetric tilt boundaries in Al as a function of tilt angle, q. 6
4
n� Orientations
near the center of
the plot, where the boundary
diffusivity is small, correspond
to orientations with a high
density of coincidence sites.
Peaks in the plot are
orientations corresponding to
general boundaries with more
open structure.
3.205 L6 11/5/03 -9-�
2
0
0
60
120
q (degrees)
[BCA] Fig. 9.2
180
Diffusion on Stationary G.B.s�
n
Diffusing species initially coats top surface…�
A regime�
all material�
B regime
boundary region
C regime�
core only�
3.205 L6 11/5/03 -10-�
[BCA] Fig. 9.4
Grain Boundary Diffusion, cont’d�
n�Regime
A: Characteristic diffusion distance in bulk >
grain size s.
[BCA] Fig. 9.4
Fraction of atomic sites in grain boundaries is�h ª 3d s
Effective mean squared displacement is
D�t = D XL (1�
- h)t + D Bht
and for h << 1,
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D = D XL + ( 3d s) D B
Grain Boundary Diffusion, cont’d�
Case C: Characteristic diffusion distance in bulk < atomic spacing <
characteristic diffusion distance in grain boundary.
• �
a diffusing atom visits�
only the grain boundaries�
[BCA] Fig. 9.4
n
Case B: Intermediate regime where
2
XL
2�
l <�
D t<s
•�
a diffusing atom visits�
only a single grain�
[BCA] Fig. 9.4
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B-Type Diffusion, Stationary g.b.s�
[BCA] Fig. 9.7*
n�Solve
two-dimensional diffusion problem for fast boundary
diffusion and relatively slow bulk diffusion, with constant
concentration of diffusant at the surface as illustrated.
È�Ê ˆ1 4 ˘È�
Ê�x1 ˆ˘��
4
XL
c ( x1, y1,t1 ) = expÍ�-Á ˜ y1˙Í1 - erf�
˜˙
Á
pt1 ¯
Ë 2 t1 ¯˚
ÍÎ Ë�
˙˚Î��
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Moving Grain-Boundary Diffusion Source�
[BCA] Fig. 9.8
Steady-state diffusion with respect to a frame at the interface migrating at velocity v :
r
ˆ
d Ê XL dc
-— ⋅ J = - Á-D
- vc˜ = 0
¯
dx Ë
dx
which has the solution
[(
)]
c( x) = c0 exp -v D XL x
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Phenomena Involving Short Circuits…�
Chiang, Birnie, and Kingery Physical Ceramics, 1997
Section 5.3 on “Single-Phase Sintering”
R W Balluffi and A Sutton, Interfaces in Crystalline
Materials, 1995, p. 762–763 explains theory of Coble
creep�
Nieh, Wadsworth, and Sherby, Superplasticity in Metals
and Ceramics, Cambridge University Press, 1997.
Chapter 3 provides an overview of mechanisms of
superplasticity
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