Curvature Corrections Remove the Inconsistencies of Binary Classical Nucleation Theory

The study of nucleation in fluid mixtures exposes challenges beyond those of pure systems. A striking example is homogeneous condensation in highly surface-active water-alcohol mixtures, where classical nucleation theory yields an unphysical, negative number of water molecules in the critical embryo. This flaw has rendered multicomponent nucleation theory useless for many industrial and scientific applications. Here, we show that this inconsistency is removed by properly incorporating the curvature dependence of the surface tension of the mixture into classical nucleation theory for multicomponent systems. The Gibbs adsorption equation is used to explain the origin of the inconsistency by linking the molecules adsorbed at the interface to the curvature corrections of the surface tension. The Tolman length and rigidity constant are determined for several water-alcohol mixtures and used to show that the corrected theory is free of physical inconsistencies and provides accurate predictions of the nucleation rates. In particular, for the ethanol-water and propanol-water mixtures, the average error in the predicted nucleation rates is reduced from 11–15 orders of magnitude to below 1.5. The curvature-corrected nucleation theory opens the door to reliable predictions of nucleation rates in multicomponent systems, which are crucial for applications ranging from atmospheric science to research on volcanos.

Figure 1

Helfrich coefficients for the water-ethanol mixture at 260 K along the path of constant liquid composition. The experimental value of the surface tension is taken from Ref. [16].

Figure 2

Properties of critical droplets in the water-ethanol mixture at 260 K, corresponding to a nucleation rate $J^{\mathrm{spec}}={10}^{13}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$ for CNT and c-CNT, with experimental data from Ref. [16]. Top: onset activities. Middle: excess ethanol content. Bottom: excess water content.

Figure 3

Onset activities for water-propanol at 260 K, corresponding to a nucleation rate $J^{\mathrm{spec}}={10}^{13}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$ for CNT and c-CNT, with experimental data from Ref. [44].

Figure 4

Effect of Helfrich expansion on surface tension (top) and adsorption (bottom) for droplets of ethanol liquid mole fraction 0.1. The dashed line is the capillary approximation (CNT), and the marked critical droplet corresponds to a gas state yielding $J^{\mathrm{CNT}}={10}^{13}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$, having the indicated number of water particles in the interior ($\mathrm{\Delta }{N}_w^{\mathrm{int}}$) and the surface ($N_w^{\mathrm{sur}}$). The full line is the Helfrich expansion (c-CNT), and the marked droplets correspond to the same gas state.