152
6 Stresses of Anodic Oxide Films Grown on Metal Electrode
dn m
dt
= 2aν exp
−
U b
RT
c m exp
az F ¯
E
RT
−
c m + 2a
dc m
dx
exp
−
az F ¯
E
RT
,
(6.3)
where
dc m
dx
is the concentration gradient of the mobile ion due to diffusion in the film.
Since the electric field is usually high enough to neglect the effect of diffusion
effect (
dc m
dx
= 0), Eq. (6.3) is reduced to
dn m
dt
= 2a νc m exp
−
U b
RT
exp
az F ¯
E
RT
− exp
−
az F ¯
E
RT
.
(6.4)
Moreover, if the electric field is high enough to prevent the ion movement against
the field, Eq. (6.4) can be simplified:
dn m
dt
= 2aνc m exp
−
U b
RT
exp
az F ¯
E
RT
.
(6.5)
The transfer rate of the mobile ion in Eq. (6.5) can be converted to the anodic current
density i a (A m
−2 ) for the film growth:
i a = z F
dn m
dt
= 2aνc m z F exp
−
U b
RT
exp
az F ¯
E
RT
.
(6.6)
The simplified form of Eq. (6.6) which represents the relationship between the anodic
current density and electric field for the growth of the anodic oxide film is expressed
by
i a = i ∗ exp
β ¯
E
,
(6.7)
i ∗ = 2aνc m z F exp
−
U b
RT
,
(6.8)
and
β =
az F
RT
.
(6.9)
In the case where the anodic oxide film is homogeneous, the electric field ¯
E in
the film can be estimated by using the potential drop φ f across the film and the film
thickness d f :
¯
E =
φ f
d f
.
(6.10)
6 Stresses of Anodic Oxide Films Grown on Metal Electrode
dn m
dt
= 2aν exp
−
U b
RT
c m exp
az F ¯
E
RT
−
c m + 2a
dc m
dx
exp
−
az F ¯
E
RT
,
(6.3)
where
dc m
dx
is the concentration gradient of the mobile ion due to diffusion in the film.
Since the electric field is usually high enough to neglect the effect of diffusion
effect (
dc m
dx
= 0), Eq. (6.3) is reduced to
dn m
dt
= 2a νc m exp
−
U b
RT
exp
az F ¯
E
RT
− exp
−
az F ¯
E
RT
.
(6.4)
Moreover, if the electric field is high enough to prevent the ion movement against
the field, Eq. (6.4) can be simplified:
dn m
dt
= 2aνc m exp
−
U b
RT
exp
az F ¯
E
RT
.
(6.5)
The transfer rate of the mobile ion in Eq. (6.5) can be converted to the anodic current
density i a (A m
−2 ) for the film growth:
i a = z F
dn m
dt
= 2aνc m z F exp
−
U b
RT
exp
az F ¯
E
RT
.
(6.6)
The simplified form of Eq. (6.6) which represents the relationship between the anodic
current density and electric field for the growth of the anodic oxide film is expressed
by
i a = i ∗ exp
β ¯
E
,
(6.7)
i ∗ = 2aνc m z F exp
−
U b
RT
,
(6.8)
and
β =
az F
RT
.
(6.9)
In the case where the anodic oxide film is homogeneous, the electric field ¯
E in
the film can be estimated by using the potential drop φ f across the film and the film
thickness d f :
¯
E =
φ f
d f
.
(6.10)