Volume term of work of critical nucleus formation in terms of chemical potential difference relative to equilibrium one

TL;DR

This paper revisits the volume term in the work of critical nucleus formation, expressing it in terms of directly measurable chemical potential differences related to supersaturation, refining previous theoretical formulations.

Contribution

The paper derives a new expression for the volume term of nucleus formation work using measurable chemical potential differences, improving theoretical understanding.

Findings

Reformulated W_{vol} in terms of chemical potential difference between reservoir and equilibrium.

Clarified the relation between supersaturation and the work of nucleus formation.

Provided a more accessible way to measure nucleus formation work experimentally.

Abstract

The work of formation of a critical nucleus is sometimes written as W=n{\Delta}{\mu}+{\gamma}A. The first term W_{vol}=n{\Delta}{\mu} is called the volume term and the second term {\gamma}A the surface term with {\gamma} being the interfacial tension and A the area of the nucleus. Nishioka and Kusaka [J. Chem. Phys. 96 (1992) 5370] derived W_{vol}=n{\Delta}{\mu} with n=V_{\beta}/v_{\beta} and {\Delta}{\mu}={\mu}_{\beta}(T,p_{\alpha})-{\mu}_{\alpha}(T,p_{\alpha}) by rewriting W_{vol}=-(p_{\beta}-p_{\alpha})V_{\beta} by integrating the isothermal Gibbs-Duhem relation for an incompressible {\beta} phase, where {\alpha} and {\beta} represent the parent and nucleating phases, V_{\beta} is the volume of the nucleus, v_{\beta}, which is constant, the molecular volume of the {\beta} phase, {\mu}, T, and p denote the chemical potential, the temperature, and the pressure, respectively. We note here that {\Delta}{\mu}={\mu}_{\beta}(T,p_{\alpha})-{\mu}_{\alpha}(T,p_{\alpha}) is, in general, not a directly measurable quantity. In this paper, we have rewritten W_{vol}=-(p_{\beta}-p_{\alpha})V_{\beta} in terms of {\mu}_{re}-{\mu}_{eq}, where {\mu}_{re} and {\mu}_{eq} are the chemical potential of the reservoir (equaling that of the real system, common to the {\alpha} and {\beta} phases) and that at equilibrium. Here, the quantity {\mu}_{re}-{\mu}_{eq} is the directly measurable supersaturation. The obtained form is similar to but slightly different from W_{vol}=n{\Delta}{\mu}.