A multiphase-field model: sharp interface asymptotics and numerical simulations of moving phase boundaries and multijunctions

In this paper we bring together, compare and extend two recent developments in a special formulation of an anisotropic multiphase-field model; namely the results of sharp interface asymptotic analysis by Nestler and Wheeler [ B. Nestler, A.A. Wheeler, Phys. Rev. E 57 (3) (1998) 2602.] and numerical s imulations of moving phase boundaries and multijunctions by Garcke et al. [ H. Garcke, B. Nestler, B, Stoth, Physica D 115 (1998) 87; SIAM J, Appl, Mat h., in press]. First, we present the formulation of the multiphase-held mod el, which includes surface energy anisotropy. Then we state, how the leadin g order conditions at both interfaces and junctions can succinctly be deriv ed in the sharp interface limit by introducing a generalized Cahn-Hoffman z eta-vector formalism and by using the motion of a stress tenser. These anal ytical results contain that the force balance at multijunctions is recovere d, which comprises Young's law and, in the anisotropic case, additional she ar forces. Next, we present numerical simulations of evolving phase boundar ies and junctions, concentrating on the case of isotropic phases. We find t hat our numerical solutions of the multiphase-field model compare favorably with the exact solutions of the sharp interface analytical results. We obs erve that the classical angle conditions at trijunctions are obtained numer ically. Finally, we perform simulations of grain growth evolution and numer ically verify the validity of the qualitative features of the von Neumann l aw. (C) 1999 Elsevier science B.V. All rights reserved.