Environmental Engineering Reference
In-Depth Information
has been related to the self-diffusion coefficients,
D
1
and
D
2
of metal and non-
metal ions, respectively, for an oxide of
t
e
1. Using Einstein's relation,
D
i
drift velocity of species i per unit force; Wagner finally
obtained the following expression for
k
r
:
B
i
kT
, where
B
i
a
(o)
O
a
(i)
O
|
Z
1
|
k
r
C
equiv
D
1
D
2
dln
a
O
(5.88)
|
Z
2
|
or
a
(i
M
a
(o)
O
D
1
|
Z
2
|
k
r
C
equiv
D
2
dln
a
M
(5.89)
|
Z
1
|
where
C
equiv
(concentration in kg equivalent of compound)
⋅
m
3
D
1
and
D
2
self-diffusion coefficient of cations and anions, respectively, in
the compound layer, expressed in m
2
s
1
Z
1
and
Z
2
valence of metal and nonmetal atoms, respectively,
a
(o)
O
and
a
(i
O
activity of oxidant at the outer oxide-oxygen and the inner
metal-oxide interfaces, respectively
a
(o
M
and
a
(i
M
activity of metal atoms at the outer oxide-oxygen and the inner
metal-oxide interfaces, respectively
Wagner's theoretical relation (5.88) for parabolic rate constant has been success-
fully tested and found to be valid for a large number of metal-oxidant systems.
However, his expression considered only lattice or volume diffusion in the grow-
ing product layer but did not take into account the diffusion through grain bound-
ary and other easy diffusion paths. In many a situation, porosities and microchan-
nels are developed within the scale where this model is not fully satisfied.
Moreover, no specific mention has been made regarding the temperature of appli-
cability of this theoretical relation. Subsequent experimentations have established
that the temperature of validity of this mechanism will vary from one system to
another.
5.5.2 Determination of Self-Diffusion Coefficient from
Parabolic Rate Constant
Fueki and Wagner (31) and subsequently Pettit (32) evaluated self-diffusion coef-
ficient from the knowledge of oxygen pressure dependence of rational rate con-
stant (
k
r
) by following an inverse procedure, i.e., by differentiating Eq. 5.88 as
proposed by Wagner (30), which correlates
k
r
and diffusivity values of ionic
species. One would obtain the following expression by differentiating Eq. 5.88:
