Heterogeneous nucleation on convex spherical substrate surfaces: A rigorous thermodynamic formulation of Fletcher’s classical model and the new perspectives derived

Abstract

Fletcher's spherical substrate model [J. Chem. Phys. 29, 572 (1958)] is a basic model for understanding the heterogeneous nucleation phenomena in nature. However, a rigorous thermodynamic formulation of the model has been missing due to the significant complexities involved. This has not only left the classical model deficient but also likely obscured its other important features, which would otherwise have helped to better understand and control heterogeneous nucleation on spherical substrates. This work presents a rigorous thermodynamic formulation of Fletcher's model using a novel analytical approach and discusses the new perspectives derived. In particular, it is shown that the use of an intermediate variable, a selected geometrical angle or pseudocontact angle between the embryo and spherical substrate, revealed extraordinary similarities between the first derivatives of the free energy change with respect to embryo radius for nucleation on spherical and flat substrates. Enlightened by the discovery, it was found that there exists a local maximum in the difference between the equivalent contact angles for nucleation on spherical and flat substrates due to the existence of a local maximum in the difference between the shape factors for nucleation on spherical and flat substrate surfaces. This helps to understand the complexity of the heterogeneous nucleation phenomena in a practical system. Also, it was found that the unfavorable size effect occurs primarily when R<5r( *) (R: radius of substrate and r( *): critical embryo radius) and diminishes rapidly with increasing value of R/r( *) beyond R/r( *)=5. This finding provides a baseline for controlling the size effects in heterogeneous nucleation.

Resistance-Spot-Welded AZ31 Magnesium Alloys: Part I. Dependence of Fusion Zone Microstructures on Second-Phase Particles

“…It was recently demonstrated that the larger the nucleant particles, the more effective they are, when the radius of the nucleant particle, R, is smaller than 5r*. [33] However, little further influence of the nucleant particle size on heterogeneous nucleation will be displayed, when R > 5r*. [33] According to Reference 33, the r* for nucleation of Mg could be estimated with the following equation: …”

“…[33] However, little further influence of the nucleant particle size on heterogeneous nucleation will be displayed, when R > 5r*. [33] According to Reference 33, the r* for nucleation of Mg could be estimated with the following equation: …”

Line-tension-induced scenario of heterogeneous nucleation on a spherical substrate and in a spherical cavity

“…Even on a spherical curved surface, the contact angle will be determined from the classical Young equation (Eq. (4)) for flat surfaces 11,18 when the line tension can be neglected (τ = 0). (a) Liquid nucleus on a convex substrate.…”

“…However, the result is general and can be applied to the nucleation of crystal grains or vapor bubbles as well. According to the classical idea of wetting and nucleation theory 8,9,11,18 , the Helmholtz free energy of a nucleus (sessile droplet) is given by ∆F = σ lv A lv + ∆σA sl + τ L,…”

“…Since then, this theory has been applied to study, for example, heterogeneous nucleation in the atmosphere [13][14][15] . Further theoretical analysis has not been conducted until recently, because of the mathematical complexity of the problem [16][17][18][19] . The same problem has also been studied from the standpoint of wetting 20 .…”