Formation of porous metal oxides in the anodization process

A theory of the formation of nanoscale porous structures in oxides of metals grown by anodization is developed. It is shown that a growing oxide layer can become unstable which yields the formation of a spatially irregular array of pores. The instability is shown to result from a nonlinear dependence of electrochemical kinetics at the metal-oxide and oxide-electrolyte interfaces on the overpotential which is governed by the Butler-Volmer relation. The conditions for the instability of the oxide layer are found. The dependence of the oxide conductivity on the electric field is taken into account and is shown to have a destabilizing effect. A weakly nonlinear analysis is performed and it shows that the system evolution near the instability threshold is described by the Kuramoto-Sivashinsky equation. Farther from threshold, in the long-wave approximation, a system of strongly nonlinear equations is derived and solved numerically; this system describes the formation of deep irregular pores. In a particular case, a self-similar solution describing the propagation of a pore with a paraboloidal shape is found.