Environmental Engineering Reference
In-Depth Information
will result in the following expression of equilibrium constant [
K
e
(
T
)] at any
temperature (
T
):
[V
Cu
]
2
⋅
p
2
K
e
(
T
)
(5.75)
P
1/2
O
2
Furthermore, the validity of the following charge neutrality condition at the outer
interface of the oxide layer suggests
[V
Cu
]
p
(5.76)
So Eq. 5.75 simplifies to
P
1/2
O
2
[V
Cu
]
4
K
e
(
T
)
⋅
or
[V
Cu
]
P
1/8
O
2
(5.77)
Since virtual electronic current equilibrium prevails in the oxide layer, migration
of Cu
ions through vacant cation sites will become rate limiting for subsequent
thickening of Cu
2
O scale on copper substrate in oxygen atmosphere.
In the expression of Wagner's equation (5.64) for rate constant, since
t
1
t
Cu
t
3
t
h
•
1 and
t
2
t
O
2
0 (almost stationary oxygen sublattice),
|
P
1/8
O
2
(theoretical). However, experimentally obtained parabolic rate constants exhibit
the following oxygen pressure dependence relation:
Z
O
2
|
2; therefore, one may write
σ
t
Cu
σ
Cu
σ
0
Cu
(
P
O
2
1 atm)
⋅
P
1/7
O
2
k
r
(Wagner and Gr ¨ newald [18] at 1273 K).
Subsequent studies at 1123 K by Ananth et al. [21-23] have confirmed the valid-
ity of this relation. In either investigation, with plots of
k
r
vs.
P
1/7
O
2
when extrapo-
lated to
k
r
0, the corresponding values of
P
O
2
agree quite well with the equilib-
rium
P
O
2
at the Cu-Cu
2
O interface calculated from thermodyamic data, lending
support to the defect model as proposed by Wagner and coworkers [18] for
Cu-Cu
2
O-O
2
(g) system.
Slight deviation from theoretically expected pressure dependence relation
(
k
r
O
2
) has been attributed to the presence of some neutral vacancies (V
Cu
)
along with singly charged vacancies whose concentration predominates over the
concentration of neutral vacancies. However, observation of
k
r
P
1/8
P
1/4
O
2
by Mrowec
[24,25] has been attributed to a predominance of neutral vacancies in the Cu
2
O
lattice that is not in accord with reports by Wagner et al. [18], Ananth et al. [21-
23], and Bose et al. [26,27]. Reasons for the discrepancy have been elaborated
in recent publications [21-23,26,27].
Similar oxygen pressure dependence of rate constants can also be assumed as
