Materials for engineering, 3rd Edition - (Malestrom)

Metals and alloys 123
The EMF (E) is related to the free energy of formation of the oxide (∆G)
by the equation:
E = –∆G/NF [3.8]
where N is the number of moles and F is the Faraday constant.
If the resistance of the cell to ionic migration is R i and to electronic
migration is R e , then the rate of film thickening will depend on the current
flowing (I), given by Ohm’s Law:
I = E/(R i + R e ) [3.9]
The movement of electrons, cations and anions all contribute to the current
in proportions represented by their respective transport numbers (n), where
n a + n c + n e = 1 [3.10]
If the electrical conductivity of the film material is κ, for a film of area A and
thickness y we can write:
R e = y/κn e A
R i = y/κ (n a + n c ) A
and equation [3.9] may be written:
I = EκA n e (n a + n c )/y [3.11]
If J is the equivalent weight of the film substance, whose density is ρ, then
a current I in time dt produces a volume of film given by:
I dt J/Fρ
which is equal to Ady.
Substituting equation [3.11] for I, we obtain:
dy/dt = EκAn e (n a + n c )J/Fρy [3.12]
Thus
y 2 = kt
where k is a constant and a parabolic law of film thickening is predicted,
with the rate dependent upon the electrical properties of the film substance
and (through equation [3.8]) the free energy of formation of the film substance.
Almost any metal will obey a parabolic law of film thickening over a
limited range of temperature and it depends upon the assumption that the
integrity of the film is perfect. In the first instance this will depend on
whether the volume of the oxidation product is greater or less than the
volume of metal replaced. This ratio, known as the Pilling–Bedworth ratio is
less than unity for metals such as K, Na, Ca, and Ba, whereas metals of
engineering interest have a ratio greater than unity, implying that most oxide
films commonly encountered are in a state of compression. Brittle films may