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A ξ-vector formulation of anisotropic phase-field models: 3D asymptotics

Published online by Cambridge University Press:  26 September 2008

A. A. Wheeler  and
G. B. McFadden
A. A. Wheeler
Affiliation:
Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton SO17 1BJ, UK
G. B. McFadden
Affiliation:
National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
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Abstract

In this paper we present a new formulation of a large class of phase-field models, which describe solidification of a pure material and allow for both surface energy and interface kinetic anisotropy, in terms of the Hoffman–Cahn ξ-vector. The ξ-vector has previously been used in the context of sharp interface models, where it provides an elegant tool for the representation and analysis of interfaces with anisotropic surface energy. We show that the usual gradient-energy formulations of anisotropic phase-field models are expressed in a natural way in terms of the ξ-vector when appropriately interpreted. We use this new formulation of the phase-field equations to provide a concise derivation of the Gibbs–Thomson–Herring equation in the sharp-interface limit in three dimensions.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 7 , Issue 4 , August 1996 , pp. 367 - 381
Copyright
Copyright © Cambridge University Press 1996

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