Anisotropic multi-phase-field model: Interfaces and junctions

In this paper we bring together and extend two recent developments in phase-field models, namely, a phase-field model of a multiphase system [I. Steinbach et al., Physica D 94, 135 (1996)] and the extension of the Cahn-Hoffman {ξ}-vector theory of anisotropic sharp interfaces to phase-field models [A. A. Wheeler and G. B. McFadden, Eur. J. Appl. Math. 7, 369 (1996); Proc. R. Soc. London, Ser. A 453, 1611 (1997)]. We develop the phase-field model of a multiphase system proposed by Steinbach et al. to include both surface energy and interfacial kinetic anisotropy. We show that this model may be compactly expressed in terms of generalized Cahn-Hoffman {ξ} vectors. This generalized Cahn-Hoffman {ξ}-vector formalism is subsequently developed to include the notion of a stress tensor, which is used to succinctly derive the leading-order conditions at both moving interfaces and stationary multijunctions in the sharp interface limit.