Nonlinear dynamics for the solidification of binary melt with a nonequilibrium two-phase zone

Abstract

The processes of crystallization of melts and solutions are usually described by the Stefan frontal model. In application of this model, it is necessary to determine the impurity concentration and temperature of the substance in both the melt and the growing crystal. In addition, the phase-transition boundary is also unknown, and the phase-transition temperature (liquidus) in the two-component system depends on the local impurity concentration and is determined from the phase diagram [1]. As is well known, the solidification of binary melts is sufficiently often accompanied by arising additional supercooled zones, i.e., domains in the liquid phase ahead of the crystallization front, with a temperature in these zones being lower than the liquidus temperature. One of the mechanisms governing the appearance of the supercooling in a melt was indicated in [2] and is referred to as the concentration supercooling. Due to bulk nucleation in the supercooled domain, the spontaneous generation of solid-phase elements in the form of dendrites or particles can begin. Thus, in the concentration-supercooling zone, the substance can exist in both solid and liquid states. This domain is called the two-phase zone. Numerous papers are devoted to investigating the processes of solidification with supercooled domains (see, e.g., [1, 3‐5]). However, in many studies, the twophase zone is considered in the quasi-equilibrium approximation [6]. This implies that the supercooling is completely destroyed due to the release of the crystallization latent heat by growing solid-phase elements. Thus, the substance temperature in the zone is equalized attaining the liquidus temperature. This fact essentially simplifies solving the problem, however, the solution presents no information on the internal structure and the topology of the zone. In addition, in the general case, the supercooling disappears not entirely. Therefore, in constructing a theory of the solidification in the presence of the two-phase domain supercooling, it is necessary to include into the theoretical scheme kinetic factors responsible for the generation of new-phase elements in the zone. In this paper, we offer a method for the theoretical study of the directional solidification process in a binary melt with a two-phase zone in which the supercooling is noticeable. We consider a zone as a suspension of interacting crystalline particles arising and growing in a supercooled melt. The nucleation rate for solid particles is determined by the Frenkel’‐Zel’dovich formula [7‐9]