Environmental Engineering Reference
In-Depth Information
d
d
t
D
v
e
V
kT
1
ξ
C
v
Ω
(for metal vacancies)
(5.100)
and
d
d
t
D
i
e
V
kT
1
ξ
C
i
Ω
(for interstitial ionic species)
(5.101)
where
C
v
concentration of cation vacancies, in (number)
⋅
m
3
C
i
concentration of interstitial ions in (number)
⋅
m
3
D
v
diffusion coefficient of vacancies in m
2
s
1
Ω
volume of oxide per metal ion consumed in m
3
E
V
/
ξ
field strength in V
⋅
m
1
D
i
diffusion coefficient of ionic species in m
2
s
1
k
Boltzmann constant, in 1.38
10
23
J
⋅
K
1
ξ
instantaneous oxide thickness in meters.
Roy et al. [38] tested and established the validity of this rate law for Cu-O
2
(g)
system in the temperature range of 348-374 K under the action of an externally
impressed direct current during film growth by connecting the substrate metal to
the negative terminal of a direct current source. Similar mechanism is also re-
ported to be operative during oxidation of chromium-doped copper. Subsequent
study by Bose et al. [27] has further confirmed that even at a temperature of 523
K the contribution to the cuprous oxide growth on copper by Wagner's mecha-
nism is hardly 3%.
Changeover in the mechanism of film growth process from Hayfield's loga-
rithmic to Cabrera-Mott's parabolic one has been attributed to the huge supply
of electrons at the oxide-oxygen interface, thereby altering the rate-controlling
step from electron availability at the outer surface to the electrical field-induced
ion migration.
Cabrera and Mott [7] also derived a cubic rate equation for a situation when
the concentration of charged defects at the outer interface of the film, which is
responsible for the electrical field creation, becomes dependent on the thickness
of the layer. Assuming electrons to be available at the outer film-gas interface
from the metal substrate by thermionic emission, the rate of arrival of these elec-
trons will go on decreasing with increasing film thickness and hence the defect
concentration at the outer interface will change accordingly. Taking this into
consideration, the following cubic rate equation has been proposed for the growth
of a metal-deficient oxide layer on a metal:
d
d
t
3
D
v
Ω
V
2
Q
RT
1
ξ
exp
(5.102)
4
π
akT
2
