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 curvature-dependent 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 (u(i) del u(j) + u(j) del u(i))/u(i) del u(j) + u(j) del ui\, where u(i) and u(j) are the ith and jth components of the vectorial order parameter u is an element of R-N. 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 three-phase 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. We demonstrate how this influences the solutions. In addition, we calculate the evolution of triple and quadruple junctions in three- and four-phase systems. Finally, we show a simulation of grain growth starting from many grains initially.