Simulation of Crystal Growth With Facetted Interfaces
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SIMULATION OF CRYSTAL GROWTH WITH FACETTED INTERFACES ANDREW ROOSEN AND JEAN E. TAYLOR Mathematics Department, Rutgers University, New Brunswick, NJ 08903 ABSTRACT We present a method for simulation of crystal growth using completely facetted interfaces and the definition of curvature appropriate to such interfaces. The representation of these surfaces, as facets with fixed normal directions, variable distances, and semi-fixed adjacencies, is quite different from that in other methods. It is flexible as to possible growth rules, it appears to avoid the problems involved in measuring curvature and in detecting and making topological changes, and it runs relatively fast. We expect it to be a valid method to approximate growth with any isotropic or anisotropic surface energy and kinetics. The use of the method is demonstrated in dendritic crystal growth and in Ostwald ripening. INTRODUCTION. Models of crystal growth usually have a term involving the curvature of the interface, arising from the fact that the mean curvature is -6Area/SVolume, the decrease of area with volume. More generally, when the surface free energy density -f depends on the normal direction n(x) at each point x of the interface, one must consider the variation of surface free energy with volume. For smooth interfaces and smooth surface energy functions, this variation is a local property, and it can be written as a weighted sum of the principal curvatures of the interface (a "weighted mean curvature"). Cahn and Hoffman [1, 2] defined the ý vector as ý(x) = V(Ip1v(p/IpI))fp=n(x), and showed that the weighted mean curvature is the negative of the surface divergence of 4. For a discussion of the various formulations of mean curvature and weighted mean curvature see [3]. For completely facetted interfaces, "weighted mean curvature" cannot be defined in terms of principal curvatures, since they are always zero or infinite. If we wish to measure the variation of surface energy with volume, and if the interface has a surface-free-energyminimizing structure at each point (see [4]), then the types of variations allowed must be restricted to be normal translations of entire facets, with the strips of flat surface near the edges of the facet extended or truncated so as to maintain connectivity. (When we wish to translate just part of a facet, we subdivide the full facet by a transverse edge of "infinitesimal" width, making each subdivision itself a facet.) The rate of decrease of surface energy with volume under the normal translation of a facet can easily be computed (see the next section). This quantity is therefore also called the weighted mean curvature and is denoted by wmc. The weighted mean curvature for facetted interfaces can be used in the same way that mean curvature is used for smooth interfaces. In particular, we present here a method for simulation of dendritic crystal growth and of Ostwald ripening using completely facetted interfaces. As a numerical method, it appears to avoid some of the numerical problems involved in measuring curvature. We
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