Non-Parabolic Rate Laws

Metal oxidation is a heterogeneous solid state reaction and starts in the same way as other heterogeneous reactions with nucleation and initial growth. This was discussed in Chapter 6. A time-dependent nucleation rate may dominate the overall growth kinetics of thin Films. Even under an optical microscope (i.e., in macroscopic dimensions), preferential sites of growth can still be discerned [J. Benard (1971)). This indicates that lateral transport on the surface (e.g., at sites where screw dislocations emerge) can possibly be more important for the initial reactive growth than transport across thin oxide layers. 

In electrode kinetics, interface reactions have been extensively modeled by electrochemists [K.J. Vetter (1967)]. Adsorption, chemisorption, dissociation, electron transfer, and tunneling may all be rate determining steps. At crystal/crystal interfaces, one expects the kinetic parameters of these steps to depend on the energy levels of the electrons (Fig. 7-4) and the particular conformation of the interface, and thus on the crystal s relative orientation. It follows then that a polycrystalline, that is, a (structurally) inhomogeneous thin film, cannot be characterized by a single rate law. 

An in-situ characterization of the interface structure of the growing oxide film appears to be necessary for an appropriate modeling, but this is most difficult to achieve [B. Pieraggi, R. A. Rapp (1988)]. 

With this electric potential p(t), we can integrate the Poisson equation (A p = -Qe /(E-e0) el = net charge density) to eventually obtain the concentration of electrons at the film surface (A ). It further follows that Ne(A ) varies with the film layer thickness as A -2. If we now assume that the (catalyzed) rate of dissociation of the adsorbed X2 molecules is proportional to the surface concentration of electrons, and that this dissociation process is rate determining, a cubic rate law for the film growth can be expected (A — At 2 At - t in). In fact, during the oxidation of Ni at temperatures between 250 and 400 °C, an approximately cubic rate law has been experimentally observed. We emphasize, however, that the observed cubic oxidation rate does not prove the validity of the proposed reaction mechanism. Different models and assumptions concerning the atomic reaction mechanism may lead to the same or similar dependences of the growth rate on thickness. 

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