Sequences of phase formation in multiphase stressed plates
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INTRODUCTION
MATERIAL systems, such as thermal barrier coatings, metal and ceramic matrix composites, and microelectrochemical system devices, incorporate multiple layers of dissimilar materials. The performance of such systems depends crucially on the stability of these layers, as interdiffusion and the resultant interface motion and phase formation can drastically alter the system’s properties. For example, new oxide phases may nucleate and grow at pre-existing interfaces, and phases initially present may vanish over time. In general, a complete description of interface motion and phase nucleation in a given system involves an understanding of heat and mass diffusion, nonequilibrium thermodynamics of multicomponent systems, reaction kinetics, nucleation kinetics, and time-dependent elastic and plastic stress fields.[1,2] Quantitative descriptions of such complicated systems require an enormous amount of accurate thermodynamic and kinetic information. In order to better generate such information, it is necessary to develop tractable models that capture and identify the main features of the problem. To this end, there has been a great deal of interest in using Cahn–Hilliard-type equations to simulate interface motion in diffusing systems. For layered systems, much of this work is based on the papers of Larche´ and Cahn.[3] They examine a thin plate composed of a binary two-phase alloy with a lattice parameter that depends linearly on composition. Subsequent numerical simulations by Cahn and Kobayashi[4] showed that an initially uniform composition profile decomposed into alternating layers of the two phases, which further evolved (coarsened) as the two phases at the edges of the plate thickened toward the center, consuming the interior phases. The elastic energy of the plate decreased during this coarsening process, as the plate bent in response to the lattice parameter differences of the phases. Subsequently, we[5] WILLIAM C. JOHNSON, Professor, is with the Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 229044745. Contact e-mail: [email protected] PERRY H. LEO, Professor, is with the Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455. Manuscript submitted January 25, 2002. METALLURGICAL AND MATERIALS TRANSACTIONS A
developed an analytic expression for the coarsening rate that confirmed that bending accelerated the coarsening process and predicted that the phases coarsened at a rate linearly proportional to time. We[6] explored how a compliant substrate can influence the microstructural evolution in the plate through its effect on the elastic and free energy of the entire plate (film plus substrate). Cahn–Hilliard-type equations have also been used to study microstructural evolution in a thin plate of a binary system capable of sustaining three isostructural phases.[7] The lattice parameter depended on composition, both through a linear term (Vegard’s law) and a nonlinear quadratic term. It was shown that the resulting co
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