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A \(\xi\)-vector formulation of anisotropic phase-field models: 3D asymptotics. (English) Zbl 0909.35160

The presence of surface energy anisotropy in the solidification of a pure material is a phenomenon of both practical and theoretical importance. Various ad hoc modifications to phase-field models have been proposed to represent surface energy anisotropy. Dendrites are three-dimensional and a sharp interface model requires the representation of their interfaces as two-dimensional surfaces. A phase-field model provides a setting for computation of three-dimensional dendrites [see R. Kobayashi, Exp. Math. 3, 59-81 (1994; Zbl 0811.65126)]. To assess such computations, it is important to know the free boundary problem approached in the sharp interface limit.
This paper provides a new formulation of a class of phase-field models that allow anisotropic surface energy and interface kinetics in terms of the Hoffman-Cahn \(\xi\)-vector [see J. W. Cahn and D. W. Hoffman, Surface Sci. 31, 368-388 (1972), Acta Met. 22, 1205-1214 (1974)]. The authors exploit this representation to show that in the sharp interface limit these phase-field models yield, at leading order, a free boundary problem in three dimensions which correctly accounts for the surface energy and kinetic anisotropy.

MSC:

35R35 Free boundary problems for PDEs
80A22 Stefan problems, phase changes, etc.
74A15 Thermodynamics in solid mechanics

Citations:

Zbl 0811.65126
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References:

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