Environmental Engineering Reference
In-Depth Information
molecular forces on new oxide molecules, which forms the second molecular
layer. The second molecular layer thus formed, while subjected to influences
other than those associated with bulk thermodynamic properties of the oxide in
question, the crystal structure and the lattice constant in the second molecular
layer, may again be quite different from those found in bulk quantities of the
oxide. Each successive molecular layer can thus be viewed as a new substrate
for formation of yet another molecular layer. So the initial influence of the metal
surface can be propagated over distances within the newly formed oxide layers,
which are large relative to those expected from the usual range of molecular
forces. At each stage of new oxide layer formation, however, the driving force
due to bulk thermodynamic properties always exists in competition with the in-
fluence of the metal substrate. It is thus intuitive that as the oxide film becomes
thicker and thicker, the influence of the original metal substrate becomes weaker
and weaker, so that eventually the crystal structure and lattice constant of the
oxide must approach those characteristic of the bulk oxide.
A simple mathematical model has been advanced by Fromhold [52] to explain
the epitaxial influences of metal-oxide combination. In this model it has been
assumed that there exists a possibility of some relaxation of constrained lattice
toward the bulk lattice constant with each new epitaxial layer being formed. It
is further assumed that the amount of this relaxation with formation of each new
layer is proportional to the existing deviation from the bulk lattice constant, then
one can write the following proportionality relation:
d
d
[
a
(
ξ
)
a
o
]
[
a
(
ξ
)
a
o
]
(5.105)
ξ
where
a
o
the bulk lattice constant, and
a
(
ξ
)
the oxide lattice constant at distance
within the oxide measured in a
direction perpendicular to the metal substrate.
ξ
Further more, on denoting the proportionality constant determining the rate of
such relaxation by
1/
δ
, Fromhold finally arrived at the following expression:
ξ
δ
a
(
ξ
)
a
o
[
a
(o)
a
o
] exp
(5.106)
where
a
(o)
lattice constant of the oxide at the metal-oxide interface. Equation
5.106 can be further simplified to
ξ
δ
a
(
ξ
)
a
o
1
γ
exp
(5.107)
where
γ
[
a
(o)
a
o
]/
a
o
.
