Examples for MatSc503
Example 1:
Consider a diffusion couple, A
2
O
3
B
2
O
3
and assume that the diffusivities of
cations, A and B, are much higher than that of oxygen, derive the expression for the
interdiffusion (or chemical diffusion) coefficient in terms of the diffusivities of the
cations, A and B. (You may assume that A
2
O
3
and B
2
O
3
form ideal solid solutions).
Solution:
One may assume that the diffusion coefficient of oxygen is zero.
In this case,
an inert maker placed at the interface of the diffusioncouple will not move at all.
The flux for cations, A and B, are given by
J
A
3
+
=−
C
A
3
+
D
A
3
+
RT
d
µ
A
3
+
dx
+
3
F
d
φ
dx
J
B
3
+
C
B
3
+
D
B
3
+
RT
d
B
3
+
dx
+
3
F
d
dx
where C, D, and μ are the concentration, diffusion coefficient and chemical potential
of respective ions, F is the Faraday constant and
is the electrical potential.
In order to satisfy electrical neutrality condition, the total flux of A and B must be
zero, i.e.,
J
A
3
+
+
J
B
3
+
=
0
From this charge neutrality condition, we can solve for the unknown electrical
potential gradient,
3
F
d
dx
C
A
3
+
D
A
3
+
d
A
3
+
dx
+
C
B
3
+
D
B
3
+
d
B
3
+
dx
C
A
3
+
D
A
3
+
+
C
B
3
+
D
B
3
+
Substituting the electrical potential gradient back into the flux equation for A ions,
J
A
3
+
C
A
3
+
D
A
3
+
C
B
3
+
D
B
3
+
RT C
A
3
+
D
A
3
+
+
C
B
3
+
D
B
3
+
()
d
A
3
+
dx
−
d
B
3
+
dx
Use the GibbsDuhem relation to express
d
B
3
+
in terms of
d
A
3
+
, i.e.,
X
A
3
+
d
A
3
+
+
X
B
3
+
d
B
3
+
=
0
or
d
B
3
+
dx
X
A
3
+
X
B
3
+
d
A
3
+
dx
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View Full DocumentNow the flux equation for A ions looks like
J
A
3
+
=−
C
A
3
+
D
A
3
+
C
B
3
+
D
B
3
+
RT C
A
3
+
D
A
3
+
+
C
B
3
+
D
B
3
+
()
X
B
3
+
d
µ
A
3
+
dx
Express the chemical potential in terms of mole fraction, X,
A
3
+
=
A
3
+
o
+
RT
ln
X
A
3
+
J
A
3
+
C
A
3
+
D
A
3
+
C
B
3
+
D
B
3
+
C
A
3
+
D
A
3
+
+
C
B
3
+
D
B
3
+
X
B
3
+
X
A
3
+
dX
A
3
+
dx
or
J
A
3
+
D
A
3
+
D
B
3
+
X
A
3
+
D
A
3
+
+
X
B
3
+
D
B
3
+
dC
A
3
+
dx
Therefore, the chemical diffusion coefficient is
˜
D
=
D
A
3
+
D
B
3
+
X
A
3
+
D
A
3
+
+
X
B
3
+
D
B
3
+
Example 2:
Outline the experimental procedures that you may take to measure the grain
boundary diffusion coefficient and surface diffusion coeffcient in a given material.
Solution
Measurement of grain boundary diffusion coefficient:
(1) Apply a thin film of tracer to the surface perpendicular to an artificially fabricated
grain boundary or to the surface of a polycrystalline material.
(2) Anneal the sample at a given temperature.
(3) Measure the average concentration,
c
, of tracer atoms in a series of thin slices cut
parallel to the sample surface.
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 Spring '02
 CHEN
 Alloy, chemical diffusion coefficient, BA XA

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