Formation of self-organized nanoscale porous structures in anodic aluminum oxide

A theory of the spontaneous formation of nanoscale porous structures in aluminum oxide films growing during aluminum anodization is presented. The main elements of this theory are the Butler-Volmer relation describing the exponential dependence of the current on the overpotential and the dependence of the activation energies of the oxide-electrolyte interfacial reactions on the Laplace pressure and the elastic stress in the oxide layer. Two cases are considered, distinguished by whether the elastic stress dependence is significant or not. In the case when the effect of elastic stress is negligible, a linear stability analysis predicts a long-wave instability resulting from the field-assisted dissolution reaction; its competition with the stabilizing effect of the Laplace pressure due to the surface energy provides the wavelength selection mechanism. A weakly nonlinear analysis near the instability threshold reveals that the nonlinear dynamics of the interface perturbations is governed by the Kuramoto-Sivashinsky equation. The spatiotemporally chaotic solutions of this equation can explain the formation of spatially irregular pore arrays that are observed in experiments. In the case when the effect of elastic stress in the oxide layer is significant we show that the instability can transform from the long-wave type to the short-wave type. A weakly nonlinear analysis of the short-wave instability shows that it leads to the growth of spatially regular, hexagonally ordered pore arrays, as observed experimentally.