Stability of a directional solidification front in subdiffusive media

The efficiency of crystal growth in alloys is limited by the morphological instability, which is caused by a positive feedback between the interface deformation and the diffusive flux of solute at the front of the phase transition. Usually this phenomenon is described in the framework of the normal diffusion equation, which stems from the linear relation between time and the mean squared displacement of molecules 〈x2(t)〉∼K1t (K1 is the classical diffusion coefficient) that is characteristic of Brownian motion. However, in some media (e.g., in gels and porous media) the random walk of molecules is hindered by obstacles, which leads to another power law, 〈x2(t)〉∼Kαtα, where 0<α≤1. As a result, the diffusion is anomalous, and it is governed by an integro-differential equation including a fractional derivative in time variable, i.e., a memory. In the present work, we investigate the stability of a directional solidification front in the case of an anomalous diffusion. Linear stability of a moving planar directional solidification front is studied, and a generalization of the Mullins-Sekerka stability criterion is obtained. Also, an asymptotic nonlinear long-wave evolution equation of Sivashinsky's type, which governs the cellular structures at the interface, is derived.