Fick's first law was generalized so as to describe the coupled diffusion reactions of Mn2+ and Mn3+ ion oxidation in Mn-Zn ferrites where a single spinel phase existed. This involved the successive consideration of thermodynamic and kinetic factors. The modelling of the kinetics was complicated by the fact that oxidation of the 2 ions was markedly interdependent. It was noted that diffusion implied the simultaneous migration of vacancies and cations, so the concept of the so-called locked vacancy was introduced. That is, if a cation underwent diffusion according to another mechanism (due to interactions), the vacancy whose diffusion depended upon the existence of that cation was unable to diffuse. The resultant diffusion equations were solved by using the finite element method.

P.Perriat, B.Gillot: Solid State Ionics, 1993, 67, 35-43