Geometric motion for a degenerate Allen-Cahn/Cahn-Hilliard system: The partial wetting case

Abstract

Using formal asymptotics we demonstrate that in a low temperature coarsening limit, a degenerate Allen-Cahn/Cahn-Hilliard system yields a geometric problem in which small particles whose shape evolves according to surface diffusion move along a surface where the chemical potential is quasi-static, which itself moves by motion by mean curvature. The degenerate Allen-Cahn/Cahn-Hilliard system was developed in [J.W. Cahn, A. Novick-Cohen, Evolution equations for phase separation and ordering in binary alloys, J. Stat. Phys. 76 (1994) 877–909] to describe simultaneous ordering and phase separation, and within this context the particles which contain a minor disordered phase are embedded along grain boundaries which partition the system into two ordered phase variants. The limiting problem, though, can also be viewed as a diffuse interface approximation for various problems in materials science in which surface diffusion and motion by mean curvature are coupled, see, for example, [J. Kanel, A. Novick-Cohen, A. Vilenkin, A numerical study of grain boundary motion in bicrystals, Acta Mat. 53 (2004) 227–235; A. Novick-Cohen, Order-disorder and phase separation: modeling grain sintering, in: E. Inan, K.Z. Markov, (Eds.), Proceedings of the Ninth International Symposium (CMDS9), Istanbul, Turkey, World Scientific Publishing Co. 1999, pp. 56–68]. The present analysis extends a previous study [A. Novick-Cohen, Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system, Physica D 137 (2000) 1–24] which focused on the complete wetting limit and on motions in the plane; here we treat the partial wetting case and our analysis accommodates motion in three dimensions.