Environmental Engineering Reference
In-Depth Information
formation of very thin oxide films and their derivation of rate equation is based
on the following steps:
1.
Adsorption of oxygen on the oxide surface (a few monolayers only) followed
by its ionization due to electrons captured from the metal at the metal-oxide
interface by electron tunneling, which is a quantum mechanical phenomenon.
2.
Formation of cation vacancies at the oxide-oxygen interface.
3.
Subsequent migration of these vacancies through the oxide or, in other words,
the migration of metal ions through these vacant cationic sites under the
action of an electrical field produced by step (1).
4.
Destruction or annihilation of the cationic vacancies at the metal-oxide inter-
face.
100
˚
), the acting electrical
Since thickness of the oxide film is of low order (
field strength (
E
) will be as high as 10
8
V
/
m
. Such an extremely high
field strength will facilitate the movement of migrating species by lowering the
energy necessary for migration.
They also made three more simplifying assumptions:
V
/
ξ
1.
In the thin and very thin films, the space charge may be considered uniform
and thus their effects may be neglected.
2.
The electrical potential difference between the metal-oxide and the oxide-
oxygen interfaces is independent of the oxide thickness. Subsequently, this
concept received support from Fromhold's [34,35] detailed theoretical anal-
ysis.
3.
A uniform defect concentration is maintained in the oxide layer.
Considering the above assumptions, after an elaborate mathematical treatment,
Cabrera and Mott arrived at an equation of the form:
d
d
t
K
exp
ξ
1
ξ
(5.97)
where
K
N
Ων
exp(
W
/
kT
)
qa
kT
⋅
1
2
ξ
1
V
where
ξ
thickness of film in meters at any time (
t
), in seconds,
d
ξ
/d
t
rate of oxidation or rate of film growth at
ξ
,inms
1
N
number of mobile species available per m
2
Ω
volume of oxide per metal ion consumed, in m
3
ν
frequency factor
10
12
s
1
