An analytical solution to the nonlinear evolutionary equations for nucleation and growth of particles

In this paper, we consider a mathematical model for the growing crystals in supersaturated or supercooled systems. An integro-differential equation of the Fokker-Planck type is analytically solved using the saddle point method for the Laplace integral. The solution describes an intermediate stage of the phase transition process with allowance for the power law of the growth rate of spherical particles and the nucleation mechanisms known as the Meirs and Weber-Volmer-Frenkel-Zeldovich kinetics. A dynamical dependencies for the supersaturation/supercooling and the distribution function of crystals by their size are obtained. The novelty of the theory is the use of a power law for the growth rate of crystals, which leads to new analytical solutions of the problem.