We study the microstructures produced by solid-state diffusional phase transformations in two dimensions. The microstructure consists of arbitrarily shaped precipitates embedded coherently in an infinite elastic matrix. This is the problem that Perry Leo discussed some weeks ago. We consider two distinct approaches. The first is a sharp interface method (discussed in brief by Leo in his talk), where the the composition and elastic fields are calculated by using a boundary integral method, and the precipitate-matrix interface is tracked explicitly. The second is a diffuse interface (Cahn-Hilliard) model, where microstructure is captured by the evolution of smooth composition and displacement fields in which internal layers mark the precipitate-matrix interfaces. We discuss both formulations, and show that in the limit of a sharp interface the diffuse interface and sharp interface formulations match exactly. We also consider the numerical implementation of the two models, and we present results comparing precipitate shapes and motions for the two approaches. Finally, we show that elastic inhomogeneity (i.e., different elastic constants for the precipitate and matrix) has an important effect on the microstructures seen in the simulations.
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