title: Stability of a directional solidification front in subdiffusive media.
abstract: 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.
authors.affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel and Department of Mathematics, The college of Sakhnin-Academic College for Teacher Education, Sakhnin 30810, Israel.
authors.forename: Mohammad Abu
authors.initials: MA
authors.lastname: Hamed
authors.affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel and Minerva Center for Nonlinear Physics of Complex Systems, Technion-Israel Institute of Technology, Haifa 32000, Israel.
authors.forename: Alexander A
authors.initials: AA
authors.lastname: Nepomnyashchy
mesh_terms: D000497:Alloys / Q000737:chemistry*; D001672:Biocompatible Materials / Q000737:chemistry*; D003198:Computer Simulation; D003460:Crystallization / Q000379:methods*; D004058:Diffusion; D008956:Models, Chemical; D017711:Nonlinear Dynamics; D044367:Phase Transition; D012996:Solutions / Q000737:chemistry*; D013696:Temperature
chemical_list: D000497:Alloys; D001672:Biocompatible Materials; D012996:Solutions