Motion of Interfaces Governed by the Cahn--Hilliard Equation with Highly Disparate Diffusion Mobility

We consider a two-phase system governed by a Cahn--Hilliard type equation with a highly disparate diffusion mobility. It has been observed from recent numerical simulations that the microstructure evolution described by such a system displays a coarsening rate different from that associated with the Cahn--Hilliard equation having either a constant diffusion mobility or a mobility that degenerates in both phases. Using the asymptotic matching method, we derive sharp interface models of the system under consideration to theoretically analyze the interfacial motion with respect to different scales of time $t$. In a very short time regime, the transition layer stabilizes into the well-known hyperbolic tangent single-layer profile. On an intermediate $t=O(1)$ time regime, due to the small mobility in one of the phases, the sharp interface limit is a one-sided Stefan problem, determined by data in the phase with constant nonzero mobility. On a slower $t=O(\varepsilon^{-1}) $ time scale, the leading order dynamics is a one-sided Hele--Shaw problem. When this one-sided Hele--Shaw dynamics is equilibrated, the system evolves in $t=O(\varepsilon^{-2})$ time scale according to the combination of a one-sided modified Mullins--Sekerka problem in the phase with nonzero constant mobility and a nonlinear diffusion process that solves a quasi-stationary porous medium equation in the phase with small mobility. Scaling arguments suggest that there should be a crossover in the coarsening rate from $t^{1/3}$ to $t^{1/4}$.