Modeling of nucleation time distribution using Laplace transform of activity density function – Stochastic heterogeneous nucleation

• Published in: Journal of crystal growth Vol. 587; p. 126638
• Main Author: Kubota, Noriaki
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.06.2022; Elsevier BV
• Subjects: A1. Heterogeneous nucleation; A1. Nucleation; A1. Nucleation time; A1. Poisson process; A1. Stochastic model; A1. Stochastic process; Data analysis; Density; Distribution functions; Laplace transforms; Nucleation; A1. Poisson process; A1. Nucleation time; A1. Nucleation; A1. Stochastic process; A1. Stochastic model; A1. Heterogeneous nucleation
• ISSN: 0022-0248; 1873-5002
• DOI: 10.1016/j.jcrysgro.2022.126638

•Various model equations of nucleation time distribution Pr(t) are presented.•Heterogeneous nucleation is assumed in the proposd models.•Model equations of Pr(t) have been derived in a unified manner using the Laplace transformation.•Pr(t) is correlated with the Laplace transform F(t) of the activity density function f(t).•Small number of most ”active” sites govern the nucleation time distribution. This paper deals with heterogeneous nucleation in small samples induced by active sites at isothermal conditions. This type of heterogenous nucleation occurs more easily or earlier, as the active sites increase either in number or activity. The nucleation time, which is defined as the time needed for the detection of a first nucleation, changes significantly sample to sample, and distributes widely. The nucleation time distribution Pr(t) is modelled by introducing the number distribution function of active sites among samples g(p) and the activity density function f(k). Various model equations of Pr(t) are derived in a unified mathematical manner, using the Laplace transform of f(k) functions. This mathematical method enables us to comprehensively understand the theoretical aspect of nucleation time distribution. Once the parameters of a model equation Pr(t) are determined by fitting a theoretical Pr(t) to the experimental Pr(t) data, the functions f(k) and g(p) can be calculated. Data analysis indicates that the small number of “most active” sites governs the nucleation time distribution when active sites distribute randomly among samples. In such cases, the other “less active” sites, even if those exist in great numbers, have no effect on nucleation time.