Shift of the transition temperature and the change in the free energy upon directional solidification of solutions at a given rate

Abstract

Conditions of equilibrium at the interface between the solid and liquid phases in the course of directional solidification of a two-component melt and the rate of the decrease in the free energy of such a system in the case where diffusion in the solid phase can be neglected have been investigated. Using the previously suggested new boundary conditions for the diffusion equation, analytical expressions have been obtained for the shifts of the values of the concentrations in the liquid and solid phases at their interface, as well as for the solidification temperature as compared to the initial equilibrium values corresponding to a zero growth rate. It is shown that the magnitudes of these shifts are directly proportional to the solidification rate and are inversely proportional to the coefficients that describe the transport of atoms of various sorts through the interface. Using the above-mentioned boundary conditions, we have found an expression for the rate of the decrease in the free energy of the system under consideration and have shown that it can be divided into two parts, one of which is determined by the processes at the interface between the phases and the other is controlled by diffusion processes that occur inside the volume of the phases. The results are compared with those obtained in terms of a different approach, which is based on the determination of the balance between the changes in the free energy in the solid and liquid phases far from the solidification front. It is shown that both approaches yield similar results, so that in some cases they can be used as mutually complementary.