Three Equations of Oxidation

11.4.1 Three Equations of Oxidation

The three main equations that express thickness y of fi lm or scale forming on any

PROTEC TIVE AND NONPROTEC TIVE SC ALES

Figure 11.3. Equations expressing growth of fi lm thickness, y , as a function of time of oxida- tion, t .

The particular equation that applies depends, in part, on the ratio Md / nmD and the relative thickness of the fi lm or scale.

For the linear equation, the rate of oxidation is constant, or dy/dt = k and y = kt

+ const, where k is a constant. Hence, the thickness of scale, y , plotted with time, t , is linear (Fig. 11.3 ). This equation holds whenever the reaction rate is constant at an interface, as, for example, when the environment reaches the metal surface through cracks or pores in the reaction - product scale. Hence, for such metals, the ratio Md / nmD is usually less than unity. In special cases, the linear equation may also hold even though the latter ratio is greater than unity, such as when the con- trolling reaction rate is constant at an inner or outer phase boundary of the reac- tion - product scale; for example, tungsten fi rst oxidizes at 700 – 1000 ° C, in accord

with the parabolic equation, forming an outer porous WO 3 layer and an inner compact oxide scale [14] . When the rate of formation of the outer scale becomes equal to that of the inner scale, the linear equation is obeyed.

For the parabolic equation, the diffusion of ions or migration of electrons through the scale is controlling, and the rate, therefore, is inversely proportional to scale thickness.

dy k ′

or y 2

2 kt ′+ const

222 OXIDATION

Accordingly, if y 2 is plotted with t , a linear relation is obtained (Fig. 11.3 ). This equation holds for protective scales corresponding to Md / nmD

> 1 and is appli- cable to the oxidation of many metals at elevated temperatures, such as copper, nickel, iron, chromium, and cobalt.

For relatively thin protective fi lms, as when metals oxidize initially or when oxidation occurs at low temperatures, it is found that

dy k ′′

( const )

or y = ′′ k ln

dt

This is called the logarithmic equation. Correspondingly, if y is plotted with log ( t + const), or with log t for t

> > const, a linear relation is obtained. First reported by Tammann and K ö ster [15] , the logarithmic equation has been found to express the initial oxidation behavior of many metals, including Cu, Fe, Zn, Ni, Pb, Cd, Sn, Mn, Al, Ti, and Ta. The logarithmic equation can be derived on the condition that the oxidation rate is controlled by transfer of electrons from metal to reac- tion - product fi lm [9] when the latter is electrically charged — that is, when it con- tains a space charge throughout its volume. The preponderance of electric charge of usually negative sign in oxides near the metal surface, similar to the electrical double layer in aqueous electrolytes, has been demonstrated experimentally. Therefore, any factor changing the work function of the metal (energy required to extract an electron), such as grain orientation, lattice transition, or magnetic transition (Curie temperature), changes the oxidation rate, as has been observed

[16] . When the thickness of the fi lm exceeds the maximum thickness of the space - charge layer, diffusion or migration through the fi lm then usually becomes con-

trolling, the parabolic equation applies, and the factors just mentioned, such as grain orientation and Curie temperature, no longer affect the rate. On this basis, metals forming protective fi lms fi rst follow the logarithmic equation and then the parabolic (or linear) equation.

If, for thin - fi lm behavior, migration of ions controls the rate, and the prevail- ing electric fi eld within the fi lm is considered to be set up by gaseous ions adsorbed on the outer surface, the rate of ion migration is an exponential func- tion of the fi eld strength, and the inverse logarithmic equation [10] results:

const − k ln t

This relation has been reported to hold for copper and iron oxidized at low tem- peratures [17] . It is often diffi cult to distinguish between the logarithmic and the inverse logarithmic equations because of the limited range of time over which data can be accumulated for thin - fi lm behavior, with either equation apparently applying equally well. This situation also makes it diffi cult to evaluate other types

of equations that have been proposed, such as the cubic equation, y 3 = kt + const. Data obeying this equation can also be represented, in many cases, by a two - stage logarithmic equation where an initial lower rate is followed by a fi nal higher rate

WAGNER THEORY OF OXIDATION

[16] (Fig. 11.3 ). The higher rate is ascribed to formation of a diffuse space - charge layer overlying an initially constant charge - density layer [9] . The oxidation rate for thin - or thick - fi lm conditions increases with tempera- ture, obeying the Arrhenius equation