|| Theoretical studies of the solidification processes in melts and solutions are usually based on the thermodiffusion Stefan-type models with a planar front. In the course of investigation this model needs to find the concentration of the impurity and temperature of the matter both in the liquid and the growth crystal. The analyses are complicated by the need to apply boundary conditions at solid/liquid interface which are evolving with time and whose position must be found as part of the calculation. The temperature of the phase transition (liquidus) is variable and unknown too. It is dependent on the local impurity concentration and determined from the phase diagram. Some Stefan-type problems have been solved completely in simple geometries in classical studies (for example, solidification from a plane wall, inward and outward crystallization processes of cylinders and spheres). In these early studies the domain of crystallization is divided into two regions: melt (liquid phase) and crystal (solid phase) separated by the moving boundary of phase transition (solidification front). However, the solidification of binary melts is rather frequently accompanied by the appearance of supercooled regions, i.e., regions in the liquid phase, the temperature of which is lower than the equilibrium temperature of phase transition, which depends on the local impurity concentration. One of the supercooling mechanisms termed ‘‘constitutional” was revealed for the first time by Ivantsov. This mechanism occurs in the following manner. When the front moves into the liquid and replaces the solute impurity ahead itself, the solute concentration increases. As a result, the phase transition temperature decreases at the front (accordingly to the phase diagram) and is an increasing function with increasing distance from the phase interface. When the liquid temperature goes below its freezing point a constitutionally supercooled region arises ahead of the front. Elements of the new phase may start spontaneous generation in supercooled zone in the form of dendrites or particles by means of bulk nucleation. In other words, this region (termed ‘‘two-phase” or ‘‘mushy” layer) is consisted of the mixed solid and liquid phases. Solidification problems of binary alloys in the presence of supercooled regions have been studied by a number of investigators. However, with a few exceptions, these works are devoted to analytical and numerical studies of a mushy layer treated as quasiequilibrium. The latter implies that the constitutional supercooling is entirely compensated by a latent heat of crystallization, which is eliminated by growing elements of the new phase. As a result, the mushy layer temperature attains the liquidus temperature. This assumption essentially simplifies the problem under consideration. However, solutions obtained in this manner cannot describe the internal and topological structures of a mushy layer. Furthermore, strictly speaking, the constitutional supercooling ahead of the phase transition front does not disappear completely. For this reason, in order to construct the theory of solidification in the presence of a supercooled region, the theoretical model must include some kinetic factors responsible for the formation of elements of the new phase in a mushy layer.
The process of solidification with a supercooled mushy layer is analytically described on the basis of two joint theories of directional and bulk crystallization for the solidification with a constant rate. Such characteristics as the constitutional supercooling, the solid fraction and the radial density distribution function of solid particles in a mushy layer are found. The complex structure of the non-equilibrium mushy layer is recognized.