A generalized theory of nucleation and growth of crystals in a metastable (supercooled or supersaturated) liquid is developed taking into account two principal effects: the diffusion mechanism of the particle-size distribution function in the space of particle radii and the unsteady-state growth rates of individual crystals induced by fluctuations in external temperature or concentration field. A system of the Fokker–Planck and balance integro-differential equations is formulated and analytically solved in a parametric form for arbitrary nucleation kinetics and arbitrary growth rates of individual crystals. The particle-size distribution function and system metastability are found in an explicit form. The Weber–Volmer–Frenkel–Zel'dovich and Meirs kinetic mechanisms, as well as the unsteady-state growth rates of nuclei (Alexandrov & Alexandrova 2019 Phil. Trans. R. Soc. A 377 , 20180209 ( doi:10.1098/rsta.2018.0209 )), are considered as special cases. Some potential biomedical applications of the present theory for crystal growth from supersaturated solutions are discussed. The theory is compared with experimental data on protein and insulin crystallization (growth dynamics of the proteins lysozyme and canavalin as well as of bovine and porcine insulin is considered). The hat-shaped particle-size distribution functions for lysozyme and canavalin crystals as well as for bovine and porcine insulin are found. This article is part of the theme issue ‘Heterogeneous materials: metastable and non-ergodic internal structures’.

A nonlinear problem with two moving boundaries of the phase transition, which describes the process of directional crystallization in the presence of a quasi-equilibrium two-phase layer, is solved analytically for the steady-state process. The exact analytical solution in a two-phase layer is found in a parametric form (the solid phase fraction plays the role of this parameter) with allowance for possible changes in the density of the liquid phase accordingly to a linearized equation of state and arbitrary value of the solid fraction at the boundary between the two-phase and solid layers. Namely, the solute concentration, temperature, solid fraction in the mushy layer, liquid and solid phases, mushy layer thickness and its velocity are found analytically. The theory under consideration is in good agreement with experimental data. The obtained solutions have great potential applications in analysing similar processes with a two-phase layer met in materials science, geophysics, biophysics and medical physics, where the directional crystallization processes with a quasi-equilibrium mushy layer can occur. This article is part of the theme issue ‘Patterns in soft and biological matters’.

A model is presented that describes nonstationary solidification of binary melts or solutions from a cooled boundary maintained at a time-dependent temperature. Heat and mass transfer processes are described on the basis of the principles of a mushy layer, which divides pure solid material and a liquid phase. Nonlinear equations characterizing the dynamics of the phase transition boundaries are deduced. Approximate analytical solutions of the model under consideration are constructed. A method for controlling the external temperature at a cooled wall in order to obtain a required solidification velocity is discussed.