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 at oms, phi takes the values +1 and-1 in the two perfect crystal grains a nd intermediate values in the boundary between them, tau, delta and ep silon are constants characterizing the material, p(phi, u) is an inter action energy density, and the diffusivity D(phi) is large in the grai n boundary (-1 < phi < 1) but zero in the grains (phi = +/- 1). The mo del is thermodynamically consistent, being derivable from a free energ y functional. The aim of the work is to understand what interactions p (phi, u) can or cannot account for the observed results. For small eps ilon the speed of travelling wave solutions can be calculated approxim ately using a successive approximations scheme. The results indicate t hat the simple interaction, p(phi, u) = u(1 - phi(2)), corresponding t o differing solubility in the grain boundary and in the bulk crystal, cannot explain all the observed data. An interaction modelling the ela stic 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.