Tarnishing and Scaling Processes - Environmental Degradation of Metals - page 250

to prevent or reduce the oxidation process. The most straightforward approach
is first to establish the level of stress required to prevent the chemical process
of oxidation. This is simply obtained by equating the energy obtained from
stress through the volume change

∆Ω A , with the free energy released ∆ G o ,on

oxidation:
σ∆Ω A ∆

G o (5.114)
to suppress further reaction. Estimates of the stresses needed to satisfy the above
inequality often exceed typical material flow or fracture stresses by over two
orders of magnitude. This suggests that under most experimental or engineering
conditions, the direct effect of stress on the chemical process of oxidation is
negligible.
In actuality, more subtle effects may become important through their action
on the rate of diffusional supply of reacting material. The flux of vacancies ( J v )
through stressed oxide is given by:

D V
ξ

σ H ∆Ω
kT

J V C I exp C II (5.115)

where D v signifies the diffusion coefficient of vacancies.
Considering the same oxygen-deficient oxide in contact with metal (e.g.,
ZrO 2 /Zr), from Eq. 5.115, it is clearly revealed that the flux of anion vacan-
cies, and hence the flux of oxygen ions, becomes zero when the local stress at
the metal-oxide interface is obtained through fulfillment of the following con-
dition:

σ H ∆Ω
kT

C II C I exp 0 (5.116)

Therefore,

kT
∆Ω

ln C II
C I

σ H (5.117)

where σ H is a compressive stress.

Equation 5.117 describes the minimum condition for suppressing oxidation,
at least for the particular example considered. If this equation is rewritten in terms
of PBR and the volume change (

∆Ω ) identified as simply that due to this ratio,

utilizing Eq. 5.113 one obtains:

kT C II

C I

σ H

1) ln

(5.118)

Ω M ( φ

which is the minimum condition to suppress oxidation.
For ZrO 2 film where the principal defects seem to be anion vacancies, its