A complete analytical solution to the integro-differential model for nucleation and evolution of crystals in a metastable system

Abstract

A nonlinear system of time-dependent integro-differential equations that describes the processes of phase transforma- tions in metastable melts and solutions is investigated. Using the saddle-point method for calculating the Laplace-type integral, a complete analytical solution was constructed, which determines the dynamic dependencies of the crystal size distribution function and the supercooling (supersaturation) of the system. The maximum size of the growing crystals, the average number of crystals and their average size as functions of time are found. It is shown that to correctly describe the evolution of a metastable system, it is necessary to take into account both the fundamental contribution of the saddle-point method and the following four contributions of the asymptotic expansion. The theory under consideration is in good agreement with experimental data. © 2019 Author(s).