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

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.

[1]  G Grinstein,et al.  Directions in condensed matter physics : memorial volume in honor of Shang-keng Ma , 1986 .

[2]  Charles M. Elliott,et al.  The limit of the anisotropic double-obstacle Allen–Cahn equation , 1996, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[3]  G. Caginalp,et al.  Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations. , 1989, Physical review. A, General physics.

[4]  Fife,et al.  Phase-field methods for interfacial boundaries. , 1986, Physical review. B, Condensed matter.

[5]  R. Aris.  Vectors, Tensors and the Basic Equations of Fluid Mechanics , 1962 .

[6]  L. Modica.  The gradient theory of phase transitions and the minimal interface criterion , 1987 .

[7]  J. Taylor,et al.  Overview No. 98 I—Geometric models of crystal growth , 1992 .

[8]  Nuclear shadowing in a parton recombination model. , 1993, Physical review. C, Nuclear physics.

[9]  Robert V. Kohn,et al.  Local minimisers and singular perturbations , 1989, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[10]  Fife,et al.  Higher-order phase field models and detailed anisotropy. , 1986, Physical review. B, Condensed matter.

[11]  Gunduz Caginalp,et al.  The role of microscopic anisotropy in the macroscopic behavior of a phase boundary , 1986 .

[12]  Jennifer Cahn,et al.  A vector thermodlnamics for anisotropic surfaces—II. Curved and faceted surfaces , 1974 .

[13]  D. W. Hoffman,et al.  A vector thermodynamics for anisotropic surfaces: I. Fundamentals and application to plane surface junctions , 1972 .

[14]  Ryo Kobayashi,et al.  A Numerical Approach to Three-Dimensional Dendritic Solidification , 1994, Exp. Math..

[15]  R. Gomer,et al.  Structure and Properties of Solid Surfaces , 1953 .

[16]  Wheeler,et al.  Phase-field models for anisotropic interfaces. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  J. Keller,et al.  Fast reaction, slow diffusion, and curve shortening , 1989 .

[18]  J. Taylor,et al.  II—mean curvature and weighted mean curvature , 1992 .

[19]  I. Fonseca,et al.  Relaxation of multiple integrals in the space BV(Ω, RP) , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[20]  J. Langer.  Models of Pattern Formation in First-Order Phase Transitions , 1986 .

[21]  John W. Cahn,et al.  Linking anisotropic sharp and diffuse surface motion laws via gradient flows , 1994 .

[22]  Kikuchi,et al.  Transition layer in a lattice-gas model of a solid-melt interface. , 1985, Physical review. B, Condensed matter.

[23]  Herbert Levine,et al.  Pattern selection in fingered growth phenomena , 1988 .

[24]  R. Kobayashi.  Modeling and numerical simulations of dendritic crystal growth , 1993 .

[25]  Peter Sternberg,et al.  Nonconvex variational problems with anisotropic perturbations , 1991 .

[26]  Charles M. Elliott,et al.  The limit of the fully anisotropic double-obstacle Allen-Cahn equation in the nonsmooth case , 1997 .