Multi - Phase - Field Model for the Motion ofMultiple

A general class of anisotropic multi-phase-eld models is proposed to describe the motion of multiple interfaces that may intersect at multiple junctions. The model reduces to classical sharp interface models in the limit of vanishing interfacial thickness. Numerical simulations show that our model is capable to recover generic features of grain growth such as crystalline curvature ow, an anisotropic as well as a crystalline version of Young's law at triple junctions, wetting phenomena and the (in{)stability of quadruple junctions. The phase-eld methodology has recently emerged as a powerful mathematical and computational modelling technique to capture a wide variety of crystal growth and solidiication structures. In this Letter, we present the formulation of a general anisotropic and crystalline multi-phase-eld model which has the exibility to be applied either to simulate multi-phase alloy systems or to model growth phenomena in polycrystalline microstructures. The proposed model uses N order parameters to describe diierent phases in alloy systems or diierent orientational variants in a polycrystal. To design the multi-phase-eld model, an approach by Steinbach et al. 1] has been extended to incorporate eeects of smooth and crystalline time relaxation and surface energy anisotropy 2]. In phase eld models the interface has a positive but small thickness and it can be shown by formal asymptotic expansions that classical physical laws can be recovered in the sharp interface limit of vanishing interfacial thickness. In 2], the relation of the diiuse interface model to a sharp interface model in the asymptotic limit is shown for smooth anisotropies. In particular, an anisotropic form of the Gibbs-Thomson equation at interfaces and an anisotropic equilibrium condition at triple junctions, known as Young's law, with additional shear forces due to surface energy anisotropy is derived. By introducing generalized Cahn-Hooman ~-vectors and a stress tensor formalism for the multi-phase-eld model, an alternative way to obtain the sharp interface equations has been in