On the theory of the unsteady-state growth of spherical crystals in metastable liquids.

Motivated by a large number of applications, we consider the process of non-stationary growth of spherical crystals in a supercooled binary melt. The moving-boundary problem describing the unsteady-state distributions of temperature and impurity concentration around the growing crystal as well as the dynamics of its radius and growth rate is solved by means of the methods of small-parameter expansion and Laplace-Carson integral transform. We show that the growth rate of crystals contains the main contribution (which is proportional to the supercooling degree Δ) and the first correction (which is proportional to Δ t, where t is time). The second correction is also found. The non-stationary temperature and concentration fields are determined as power functions of Δ and t. We demonstrate that the first corrections to the dynamics of crystal radius R( t) and its growth rate V ( t) play an important role. It is shown that R( t) and V ( t) can change more than twice in comparison with the previously known steady-state solution with the course of time. Such a behaviour will significantly modify the dynamics of a polydisperse ensemble of crystals evolving in a metastable liquid. This article is part of the theme issue 'Heterogeneous materials: metastable and non-ergodic internal structures'.