On Setting of Heat-and-Mass Transfer Problems under Directed   Crystallization
  So far the problem of interface behavior upon phase transition has not yet acquired a satisfactory mathematical formulation due to a variety of the physical phenomena involved. Analytical solutions exist only for elementary problems describing the free interface behavior in directed crystallization conditions, for instance, for those implying a clearly shaped isothermal interface. Numerical calculations of the interface behavior also present significant difficulties since the instability of moving interface does not enable calculations by means of known algorithms. The general solution of this problem does not seem possible now, so quantitative analysis of phase transition is made after significant simplifications of the problem commonly reduced to the so-called quasi-equilibrium problem setting. It was used to study the reasons for the interface instability during phase transition. However, the solutions of the quasi-equilibrium problem are, as a rule, inherently qualitative. In this paper we present a detailed discussion of the boundary conditions of the directed crystallization problem, a formulation of the model considering temperature fields of external sources, the mechanism of attachment of particles to the growing solid surface, the influence of interphase component absorption on the phase distribution ratio of the components as well as the calculation of the period of the morphological interface instability which is made with due regard of all the aforementioned conditions. 
Alex Guskov
cond-mat.stat-mech