Nanopore formation dynamics during aluminum anodization

Mathematical Modeling of Self‐Organized Porous Anodic Oxide Films

“…(4.1), for which the conduction equation becomes equivalent to Ohm's law = , where is the oxide conductivity. Conservation of charge in the oxide then implies that Laplace's equation for the electric potential , that is, ∇ 2 = 0, governs the electric potential distribution [97,[105][106][107][108][109][110]. While experiments clearly support high-field conduction in anodic oxides, it is interesting that these models nevertheless can predict some key aspects of porous layer formation (see Section 4.5).…”

“…Several aluminum anodizing models have not assumed isopotential interfaces, instead explicitly incorporating kinetics of metal and oxygen transfer reactions [103,[106][107][108][109]. The charge transfer kinetics were represented by typical Butler-Volmer expressions, consistent with the experimental characterization of Våland and Heusler [117].…”

“…As discussed in Section 4.5, the different field dependences of the dissolution and oxidation terms play an important role in pattern formation behavior. Other papers followed the approach of Singh et al [106,107] in calculating the oxide-solution interface velocity based on a Butler-Volmer electrochemical kinetic expression [103,108,109]. Since this expression was closely related to that used for the interface current density (Section 4.4.5), current continuity at the interface produced field-dependent competition effects, similar to those arising from the growth and dissolution terms in Eq.…”

Effect of Electrostriction on the Self-organization of Porous Nanostructures in Anodized Aluminum Oxide

“…with the boundary conditionsû The remaining system of deformation eigenfunctions in the metal substrate are 27) for z ∈ (−∞, 0), and in the oxide layer, at z = 0, (4.32)…”

“…Works such as [25,26,27], however, lack any physical mechanism to damp short-wave disturbances, and in [28], the surface energy effects induced by the deformable interfaces were proposed as a possible mechanism to provide this short-wave cutoff. In [29], the effects of interfacial diffusion at the metal-oxide interface were investigated, and in [30], the effects of a field-dependent oxide conductivity were explored.…”