A multiphase field concept: numerical simulations of moving phase boundaries and multiple junctions

We present numerical simulations which support the formal asymptotic analysis relating a multiorder parameter Allen-Cahn system to a multiphase interface problem with curvaturedependent evolution of the interfaces and angle conditions at triple junctions. Within the gradient energy of the Allen-Cahn system, the normal to an interface between phases i and j is modeled by the irreducible representations (uiruj ¡ujrui)=juiruj ¡ ujruij, where ui and uj are the ith and jth components of the vectorial order parameter u 2 RN. In the vectorial case, the dependence of the limiting surface tensions and mobilities on the bulk potentials of the Allen{Cahn system is not given explicitly but in terms of all the N components of the planar stationary wave solutions. One of the issues of this paper is to find bulk potentials which allow a rather easy access to the resulting surface tensions and mobilities. We compare numerical computations for planar and circular phase boundaries in two- and threephase systems. The difference is that in a three-phase system, the third phase generally will be present in the interfacial region between two other phases. ... mehrWe demonstrate how this in°uences the solutions. In addition, we calculate the evolution of triple and quadruple junctions in three- and fourphase systems. Finally, we show a simulation of grain growth starting from many grains initially.