https://doi.org/10.1140/epjs/s11734-026-02329-4
Regular Article
Population-balance theory of particle growth with fluctuations from a saturated solution
Laboratory of Multi-Scale Mathematical Modeling, Department of Theoretical and Mathematical Physics, Ural Federal University, Lenin Avenue, 51, 620000, Ekaterinburg, Russian Federation
a
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Received:
17
March
2026
Accepted:
17
April
2026
Published online:
29
April
2026
Abstract
In this paper, we analyze the population-balance model for the evolution of numerous particles of different sizes during the intermediate step of volumetric phase transition. We use the approximation of spherical particle shapes and neglect interactions between them, considering the average interparticle distance to be much greater than the characteristic radius of the particles. The population-balance model under consideration includes a second-order Fokker–Planck equation for the distribution function, an integro-differential equation for mass conservation, initial and boundary conditions. This model is analytically solved in the case when the growth rate of particles solely depends on their radius. The particle-radius distribution, solution supersaturation, and the moments of the distribution function are derived analytically for various mass flows entering and leaving the solution during product crystal growth. Numerical calculations are carried out for the growth of insulin crystals in the case of two-step crystallization mechanism.
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© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2026
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
