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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 shortwave type. A weakly nonlinear analysis of the shortwave instability shows that it leads to the growth of spatially regular, hexagonally ordered pore arrays, as observed experimentally.

Abstract. -A theory of the spontaneous formation of spatially regular hexagonal arrays of nanopores in aluminum oxide film growing during aluminum anodization is presented. Linear stability analysis shows that, in certain ranges of the applied voltage and electrolyte pH, the oxide film is unstable with respect to perturbations with a well-defined wavelength. The instability is caused by a positive feedback between the oxidation-dissolution rates and variations of electric field caused by perturbations of the metal-oxide and oxide-electrolyte interfaces. The competition between this instability and the stabilizing effects of the Laplace pressure and elastic stress provides the wavelength selection mechanism. The hexagonal ordering of pores results from the resonant quadratic nonlinear interaction of unstable modes.Spatially regular, hexagonally ordered arrays of nanoscale pores in aluminum oxide can be formed by the anodization of aluminum in acidic electrolytes [1][2][3]. Nanoporous alumina has attracted renewed attention lately as a promising material for the fabrication of new magnetic storage devices, catalytic membranes, and as an inexpensive template for the production of nanoscale particles, wires, and photonic crystals [4].Despite many experimental studies and general understanding of the pore growth mechanism to be associated with electric-field-assisted dissolution of aluminum oxide, the mechanism of self-organization of regular pore arrays is not understood. A model for the steady growth of a single pore, based on the field-assisted dissolution, was proposed in [5]. A similar model was considered in [6] and a long-wave linear stability analysis of the interfaces was performed. It can be shown, however, that the model studied in [6] does not provide a physically justified short-wave cutoff and hence a regularization mechanism needs to be identified that would explain the selection of the pore diameter and interpore spacing. One such mechanism is the dependence of the activation energies of the interfacial reactions on the Laplace pressure at the curved interfaces due to surface energy [7]. An additional important factor may be the elastic stress caused by the volume expansion in the course of the Al → Al 2 O 3 reaction [8]. The presence of elastic stress can significantly affect the morphological stability of the surface [9].c EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx

A general continuum model has recently been proposed for the dynamics of ion intercalation in a single crystal of rechargeable-battery electrode materials [1]. When applied to strongly phase-separating, highly anisotropic materials such as LiFePO4, phase-transformation waves are predicted between the lithiated and unlithiated portions of a crystal. In this paper, we extend the analysis of the wave dynamics, and we describe a new mechanism for current capacity fade through the interactions of these waves with defects in the material.

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