Environmental Engineering Reference
In-Depth Information
formation of new oxide at this interface involves a volume increase, work of the
order
is needed to be supplied by the chemical reaction to insert fresh
oxide at this location. The resultant effect is to make it less easy for the reaction
to proceed. The stress dependence of vacancy concentration at this interface is
then given by:
σ
H
∆Ω
σ
H
∆Ω
kT
(
C
I
)
σ
C
I
exp
(5.108)
where
C
I
anion vacancy concentration at the metal-oxide interface. Consider-
ing the flux of vacancies through the stressed oxide, the following stress-depen-
dent oxidation rate expression can be obtained:
d
d
t
D
diss
O
ξ
σ
H
∆Ω
kT
exp
(5.109)
where
D
dis
O
is the diffusion coefficient of oxygen within ZrO
2
equilibrated at its
dissociation oxygen pressure. This expression reduces to the conventional para-
bolic form at zero stress but predicts a lower rate of reaction when
σ
H
∆Ω
is
negative and a higher rate when this stress term is positive.
Figure 5.25 depicts the main volume changes that need to be considered for
the growth of an anion-deficient oxide. Interchange of an oxygen gas atom with
an oxide vacancy produces a volume change as given by:
∆Ω
B
Ω
O
Ω
V
(5.110)
where
Ω
V
are, respectively, the ionic and vacancy volumes in the oxide.
It is this change that also needs to be considered when discussing the influence
of stress on the bias term for diffusion through the oxide layer. A second impor-
tant volume change arises at the oxide-metal interface when oxygen and metal
ions react to form a molecule (MO) of fresh oxide.
∆Ω
A
Ω
MO
Ω
M
Ω
O
and
(5.111)
where
Ω
MO
volume of oxide
volume of metal ion
However, since a vacancy is also expected to be involved with the reaction at
this interface, the total volume change will be given by:
∆Ω
∆Ω
A
∆Ω
B
Ω
M
(5.112)
The values of
∆Ω
B
are not well established, whereas those of
∆Ω
A
are much
better known through the concept of Pilling-Bedworth ratio (
φ
Ω
MO
/
Ω
M
). Uti-
lizing this, one can obtain:
∆Ω
Ω
M
(
φ
1)
(5.113)
A
