Paralinear growth kinetics of two compound layers

If the ApBq layer grows in the diffusion controlled regime with regard to components A and B, while the ArBs layer grows in the reaction controlled 

Since equation (2.40i) which is similar to equation (2.38) does not contain y, it can be solved separately from equation (2.402). It is clear that under constant temperature and pressure conditions the thickness of the ApBq layer asymptotically tends with passing time to the maximally possible value (see Section 2.5) 

A system of differential equations of this type appears to have been first proposed by J. Loriers in 1949 (see Ref. 13) to describe paralinear growth kinetics of two oxide layers. The term paralinear growth, being a combination of the words parabolic and linear, means that some initial portion of the time dependence of the total thickness or mass of two compound layers is almost parabolic and then there is a gradual transition to linear kinetics. 

Indeed, from the system of equations (2.42), it is easily seen that at low values of t and consequenly x (klBl + k[A2)/x (rg/p)k 0B2 and therefore 

In the oxidation of metals, paralinear growth kinetics of oxide layers are known to be a quite usual phenomenon. Such a dependence is observed much less frequently with metallic systems due to three reasons. Firstly, the duration of investigations of the process of oxidation of metals is far longer than that in examining the solid-state interaction of two metals. Secondly, the minimal measurable thickness (or mass) of compound layers which can be detected using available techniques is in the former case much less than in the latter. Thirdly, since this anomalous dependence has no satisfactory explanation from a diffusional viewpoint, experimentalists investigating metallic systems probably prefer not to accentuate on it. 

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