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'.
Keywords: crystal growth; metastable liquid; nucleation; phase transformation.