Thermodynamically consistent description of the work to form a nucleus of any size

Abstract

A thermodynamically consistent formula for the work W to form homogeneously a one-component gaseous, liquid, or solid nucleus is derived in the scope of the Gibbs theory of capillarity. This formula is applicable to nuclei constituted of whatever number of molecules, because it is valid in the entire range of conditions between the binodal and the spinodal (when such exists) of the nucleating system. Analysis is based on a newly introduced dividing surface called conservative, because its specific surface energy is independent of the nucleus size. The thermodynamically consistent formula for W accounts for the annulment of the nucleation work at the spinodal. For spinodal-unlimited systems it turns into the classical formula for W, thereby justifying the application of the latter to such systems even when the nuclei are of a few molecules only. It is shown that for systems with spinodal, classically, the nucleation rate is lower than that calculated with the aid of the thermodynamically consistent formula for W. The correction is practically only temperature-dependent and vanishes for spinodal-unlimited systems. Expressions are obtained for the excess or deficiency number Δn of molecules in the density fluctuation and for the number nC of molecules constituting the corresponding nucleus defined by the conservative dividing surface. These expressions are valid for any value of Δn and nC and reveal why, as found in experiments and computer simulations, the classical Gibbs–Thomson equation is able to predict the size of nuclei built up of less than a few tens of molecules. The general results are applied to homogeneous one-component nucleation of solids, liquids or gases under isothermal or isobaric conditions.