A complete analytical solution of an integro-differential model describing the transient nucleation of solid particles and their subsequent growth at the intermediate stage of phase transitions in metastable systems is constructed. A Fokker–Plank type equation for the density distribution function is solved exactly for arbitrary nucleation kinetics. A non-linear integral equation with memory kernel connecting the density distribution function and the system supercooling/supersaturation is analytically solved on the basis of the saddle-point method for the Laplace integral. The analytical solution obtained shows that the process at the intermediate stage is divided into three phases: initially the high rate nucleation stage occurs, then this process is accompanied by the particle growth reducing the level of metastability, and finally the mechanism of particle coarsening becomes predominant.

https://iopscience.iop.org/article/10.1088/1751-8113/46/45/455101/pdf

Transient nucleation kinetics of crystal growth at the intermediate stage of bulk phase transitions

This manuscript is devoted to the nonlinear dynamics of particulate assemblages in metastable liquids, caused by various dynamical laws of crystal growth and nucleation kinetics. First of all, we compare the quasi-steady-state and unsteady-state growth rates of spherical crystals in supercooled and supersaturated liquids. It is demonstrated that the unsteady-state rates transform to the steady-state ones in a limiting case of fine particles. We show that the real crystals evolve slowly in a more actual case of unsteady-state growth laws. Various growth rates of particles are tested against experimental data in metastable liquids. It is demonstrated that the unsteady-state rates describe the nonlinear behaviour of experimental curves with increasing the growth time or supersaturation. Taking this into account, the crystal-size distribution function and metastability degree are analytically found and compared with experimental data on crystallization in inorganic and organic solutions. It is significant that the distribution function is shifted to smaller sizes of particles if we are dealing with the unsteady-state growth rates. In addition, a complete analytical solution constructed in a parametric form is simplified in the case of small fluctuations in particle growth rates. In this case, a desupercooling/desupersaturation law is derived in an explicit form. Special attention is devoted to the biomedical applications for insulin and protein crystallization. This article is part of the theme issue 'Patterns in soft and biological matters'.

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Dynamics of particulate assemblages in metastable liquids: a test of theory with nucleation and growth kinetics

Flow-induced morphological instability and solidification with the slurry and mushy layers in the presence of convection

An exact analytical solution of an integro-differential model describing the transient nucleation of solid particles (nuclei) and their growth with fluctuating rates at the intermediate stage of bulk phase transitions in metastable systems is constructed. Two important cases of the Weber–Volmer–Frenkel–Zel'dovich and Mier nucleation kinetics are detailed for supercooled melts and supersaturated solutions.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2013.0647

Nucleation and particle growth with fluctuating rates at the intermediate stage of phase transitions in metastable systems

An exact analytical solution of integro-differential equations, which describe the phase transitions in supercooled binary melts, is constructed with allowance for fluctuating rates of crystal growth. A complete solution of the corresponding Fokker–Planck and balance equations is found for arbitrary nucleation kinetics and growth rates of crystals. In limiting cases, the obtained solutions transform to earlier known solutions for a special form of diffusivity in the space of crystal sizes and purely kinetic and “diffusion-kinetic” growth rates of nuclei.

Nucleation and growth of crystals at the intermediate stage of phase transformations in binary melts

The processes of crystallization of melts and solutions are usually described by the Stefan frontal model. In application of this model, it is necessary to determine the impurity concentration and temperature of the substance in both the melt and the growing crystal. In addition, the phase-transition boundary is also unknown, and the phase-transition temperature (liquidus) in the two-component system depends on the local impurity concentration and is determined from the phase diagram [1]. As is well known, the solidification of binary melts is sufficiently often accompanied by arising additional supercooled zones, i.e., domains in the liquid phase ahead of the crystallization front, with a temperature in these zones being lower than the liquidus temperature. One of the mechanisms governing the appearance of the supercooling in a melt was indicated in [2] and is referred to as the concentration supercooling. Due to bulk nucleation in the supercooled domain, the spontaneous generation of solid-phase elements in the form of dendrites or particles can begin. Thus, in the concentration-supercooling zone, the substance can exist in both solid and liquid states. This domain is called the two-phase zone. Numerous papers are devoted to investigating the processes of solidification with supercooled domains (see, e.g., [1, 3‐5]). However, in many studies, the twophase zone is considered in the quasi-equilibrium approximation [6]. This implies that the supercooling is completely destroyed due to the release of the crystallization latent heat by growing solid-phase elements. Thus, the substance temperature in the zone is equalized attaining the liquidus temperature. This fact essentially simplifies solving the problem, however, the solution presents no information on the internal structure and the topology of the zone. In addition, in the general case, the supercooling disappears not entirely. Therefore, in constructing a theory of the solidification in the presence of the two-phase domain supercooling, it is necessary to include into the theoretical scheme kinetic factors responsible for the generation of new-phase elements in the zone. In this paper, we offer a method for the theoretical study of the directional solidification process in a binary melt with a two-phase zone in which the supercooling is noticeable. We consider a zone as a suspension of interacting crystalline particles arising and growing in a supercooled melt. The nucleation rate for solid particles is determined by the Frenkel’‐Zel’dovich formula [7‐9]

D. L. Aseev

Papers

Nonlinear dynamics for the solidification of binary melt with a nonequilibrium two-phase zone