The viscous flow of oxide driven by the electrostriction force during the growth of anodic aluminum oxide is treated by linear stability and scale analysis. A uniform oxide film is unstable to a periodic deformation. The restabilization of pore growth is examined by scale analysis of the steady pore diameter. The pattern spatial period and pore radius predicted by the analysis are compared with experimental observations reported in the literature.Anodizing is an electrochemical process technology that imparts an integral oxide coating to the surface of aluminum and its alloys. The metal is immersed in acid solution and an anodic current is passed through the metal-solution interface. An oxide coating is formed on the metal surface as the metal is consumed by electrochemical oxidation. The driving force in anodizing is enormous, ranging from 10 to more than 100 V. In a phase-growth process driven so far from equilibrium, we expect instability, dynamic restabilization, and pattern formation. On a scale of tens to hundreds of nanometers, the anodic aluminum oxide ͑AAO͒ coating is in fact highly ordered, with an array of high aspect ratio pores centered within hexagonal cells.The emergence of a pore array in AAO has been attributed to locally accelerated dissolution, 1 mechanical stresses originating with mismatches in a specific volume between the metal substrate and the oxide film, 2 and electrokinetic forces. 3 Stability analyses have been formulated based on these driving forces. 2-7 However, the most recent experimental and computational results make it clear that oxide flow is an important element of the process. 8-10 Based on these developments and on an electrochemical and fluid mechanical model of oxide growth described below, we propose that anodizing be viewed as a pattern formation process governed by a flow instability of the oxide film and its dynamic restabilization into cellular growth. 11,12 Pattern formation growth or deposition processes are characterized by high driving force, rapid entropy generation, and the emergence of structures that have no obvious relationship to the microstructure of the component material. 13 They typically progress through stages of compact growth, instability, and dynamic restabilization. The patterns are generated dynamically in the sense that they depend not on equilibrium properties of the material but on dissipative rate processes such as heat transfer, diffusion, or viscous flow. These characteristics are in contrast to self-assembly processes in which highly structured molecules aggregate into equilibrium structures.The hexagonal cell structure of the thicker oxide layers produced in anodizing is the signature of a flow process. The notion that oxide deformation induced by electrostatic forces could account for a breakdown of passive films was introduced by Sato 14 and a tracer study by Skeldon et al. 8 found evidence for flow in the AAO layer. Skeldon et al. considered the possibility that the flow is driven by electrostriction and reasoned that if the oxide laye...
Internal waves in a two-layer fluid are considered. The layers have different values of the buoyancy frequency, assumed to be constant in each layer. The density profile is chosen to be continuous across the interface and the flow is Boussinesq. The solution is an expansion in the wave amplitude, similar to a Stokes expansion for free surface waves. The results show that the nonlinear terms in the interfacial boundary conditions require higher harmonics and result in nonlinear wave steepening at the interface. The first few harmonics are scattered by the interface, whereas the higher harmonics are evanescent in the vertical. The second-order correction to the wave speed is negative, similar to previous results with a rigid upper boundary.
[1] A sequence of nine weather balloons were launched recently over the island of Hawaii during the nights of 12, 13, and 17 December, 2002, providing measurements of ascent rate, horizontal wind speed and direction, temperature, and other quantities. The measurements show short intervals of altitude with a large increase in ascent rate, occurring only near the tropopause, indicating regions of strong upward air velocity at this location. The large ascent rates correlate well to the strength of a jet stream, and with the presence of a local critical level, indicating mountain waves as the primary cause. No corresponding decreases in ascent rate were measured, suggesting strong threedimensional effects.
Weakly nonlinear internal gravity waves are treated in a two-layer fluid with a set of nonlinear Schrodinger equations. The layers have a sharp interface with a jump in buoyancy frequency approximately modeling the tropopause. The waves are periodic in the horizontal but modulated in the vertical and Boussinesq flow is assumed. The equation governing the incident wave packet is directly coupled to the equation for the reflected packet, while the equation governing transmitted waves is only coupled at the interface. Solutions are obtained numerically. The results indicate that the waves create a mean flow that is strong near and underneath the interface, and discontinuous at the interface. Furthermore, the mean flow has an oscillatory component that can contaminate the wave envelope and has a vertical wavelength that decreases as the wave packet interacts with the interface.
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