DOI 10.1140/epjp/i2011-11095-7 Regular Article Eur. Phys. J. Plus (2011) 126: 95 T HE EUROPEAN P HYSICAL JOURNAL PLUS On phase-ﬁeld modeling with a highly anisotropic interfacial energy M. Fleck a , L. Mushongera, D. Pilipenko, K. Ankit, and H. Emmerich b Materials and Process Simulation, University of Bayreuth, Germany Received: 12 May 2011 / Revised: 25 May 2011 Published online: 19 October 2011 – c  Societ`a Italiana di Fisica / Springer-Verlag 2011 Abstract. We report on phase-ﬁeld approaches that allow for anisotropies suﬃciently high so that the interface develops sharp corners due to missing crystallographic orientations. This implies the necessity of a regularization that enforces local equilibrium at the corners, and we use the method of Eggelston et al. (Physica D 150, 91 (2001)), generalized to arbitrary crystal symmetries and rotations of the crystalline axes. Two diﬀerent anisotropic phase-ﬁeld formulations are presented and discussed: The classical model that allows the interface to vary with orientation, and another more recent formulation that has a constant interface width. We develop an explicit ﬁnite-diﬀerence scheme that combines a two-step diﬀerentiation with a stagnation grid formulation. The presented numerical implementation is stable and accurate enough to account for odd crystal symmetries and high angle rotations of the initial crystalline orientation. Even in the case of highly anisotropic interfacial energies, both formulations show excellent agreement with the well-known Wulﬀ construction of the equilibrium shape of a particle embedded in a matrix. 1 Introduction The crystalline nature of solids results in anisotropy of many thermophysical parameters. In particular, the interfacial energy between diﬀerent phases is often found to be a function of the crystallographic orientation of the interface. Using the well-known Wulﬀ theorem, the orientational dependency of the interfacial energy γ (θ) can be related to the equilibrium shape of a particle embedded in a matrix phase, and vice versa [1]. For suﬃciently small anisotropies, the equilibrium shape is smooth. However, as the anisotropy increases the equilibrium particle morphology devel- opes straight edges, known as facets, as well as corners, where certain interfacial orientations are excluded from the particle shape. Phase-ﬁeld models provide a versatile and useful numerical approach to study a wide variety of phase transforma- tion processes in the presence of anisotropic interfacial energies ranging from solid to solid transformations [2–4] to solidiﬁcation [5–7]. The fundamental idea of a phase-ﬁeld model is to include an additional variable, or order parameter φ, that denotes the phases in a multiphase system. The order parameter is constant in each bulk phase, e.g., φ =1 in the matrix phase and φ = 0 in the particle phase. Then, the interface between diﬀerent phases is represented by the transition region, where φ changes smoothly from 0 to 1. The most signiﬁcant computational advantage of a phase-ﬁeld model is that explicit tracking of the interface is unnecessary. In other words, in a phase-ﬁeld simulation freely moving interfaces between diﬀerent phases do not appear as geometric boundaries, i.e. places at which boundary conditions have to be applied explicitly. Instead, all the information about the location of the phase boundaries is implicitly contained in the phase-ﬁeld, which obeys a partial diﬀerential equation that is solved within the whole computational domain. How to incorporate weak anisotropy into phase-ﬁeld models, without intending to describe the development of corners and facets, has been already known for quite a long time (see, for example, [6,8–12]). In these formulations, the anisotropy comes into the phase-ﬁeld model, by allowing the gradient energy coeﬃcient to depend on the interfacial orientation. For this kind of anisotropic phase-ﬁeld models, McFadden et al. [9] could show that the usual Gibbs- Thomson equation for the dependence of the chemical potential on both the interfacial curvature and the orientation is recovered in the sharp interface limit. An undesired side eﬀect of the formulation, using an orientationally dependent a e-mail: michael.fleck@uni-bayreuth.de b e-mail: heike.emmerich@uni-bayreuth.de