A phase-field model for diffusion-induced grain-boundary motion

We model diffusion-induced grain boundary motion (DIGM) with a pair of differential equations: tau partial derivative phi/partial derivative t = - phi + delta(2) del(2) phi - epsilon partial derivative p(phi, u)/partial derivative phi if - 1 < phi < 1, else partial derivative phi/partial derivative t = 0 partial derivative u/partial derivative t = del.[D(phi)del v], where v = u + epsilon partial derivative p(phi, u)/partial derivative u. Here u represents the concentration of solute atoms, phi takes the values +1 and-1 in the two perfect crystal grains and intermediate values in the boundary between them, tau, delta and epsilon are constants characterizing the material, p(phi, u) is an interaction energy density, and the diffusivity D(phi) is large in the grain boundary (-1 < phi < 1) but zero in the grains (phi = +/- 1). The model is thermodynamically consistent, being derivable from a free energy functional. The aim of the work is to understand what interactions p(phi, u) can or cannot account for the observed results. For small epsilon the speed of travelling wave solutions can be calculated approximately using a successive approximations scheme. The results indicate that the simple interaction, p(phi, u) = u(1 - phi(2)), corresponding to differing solubility in the grain boundary and in the bulk crystal, cannot explain all the observed data. An interaction modelling the elastic coherency strain energy is also considered, and its consequences are consistent with the observed features of DIGM in nearly all cases. (C) 1997 Acta Metallurgica Inc.