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Theory of Ostwald ripening in open systems: unifying Lifshitz-Slezov-Wagner and Family-Vicsek scaling approaches

Vladimir G. Dubrovskii*

  • *Contact author: v.dubrovskii@spbu.ru
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Phys. Rev. E 113, 034134 – Published 25 March, 2026

DOI: https://doi.org/10.1103/zhc2-ggh3

Abstract

We study the size distributions (SDs) of two- and three-dimensional nuclei (d=2,3) in the Ostwald ripening (OR) stage in open systems. The size-dependent attachment rates and the time-dependent total amount of monomers are taken in the power-law forms, with indices α and β, respectively. The OR occurs when β<βmax, where βmax depends on α and d. The case of β=0 corresponds to the Lifshitz-Slezov-Wagner (LSW) theory in closed systems. The case of β=1 corresponds to open systems under constant material influx, where the SDs often adopt the Family-Vicsek (FV) scaling form. Using a unified scaling approach, we obtain analytical solutions for the critical size, average size, and the SDs depending on the three parameters d, α, and β. We show that the LSW and FV scaling for the OR process in open systems is essentially the same thing. We analyze the SD shapes and particular solutions under different conditions and reveal the effect of SD narrowing at α<1 as β approaches βmax. The scaled SDs are unimodal in most cases at α<1, and monotonically decreasing at α=1. These results should be useful for modeling the OR process in a wide range of practical systems and interesting from a fundamental viewpoint.

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Phys. Rev. E 113, 034134– Published 25 March, 2026
  • Received 27 November 2025
  • Accepted 4 March 2026
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