Free energy of cluster formation and a new scaling relation for the nucleation rate

TL;DR

This paper uses large-scale molecular dynamics simulations to accurately determine cluster formation free energies, deriving a new scaling relation for nucleation rates that links free energy, surface energy, and supersaturation.

Contribution

It introduces a novel analytical formula and a scaling relation for nucleation rates based on extensive simulation data and free energy measurements.

Findings

Critical cluster sizes agree with nucleation theorem estimates.

The new scaling relation collapses nucleation rate data across conditions.

Surface energy ratio remains constant below the triple point temperature.

Abstract

Recent very large molecular dynamics simulations of homogeneous nucleation with $(1-8) \cdot 10^9$ Lennard-Jones atoms [Diemand et al. J. Chem. Phys. {\bf 139}, 074309 (2013)] allow us to accurately determine the formation free energy of clusters over a wide range of cluster sizes. This is now possible because such large simulations allow for very precise measurements of the cluster size distribution in the steady state nucleation regime. The peaks of the free energy curves give critical cluster sizes, which agree well with independent estimates based on the nucleation theorem. Using these results, we derive an analytical formula and a new scaling relation for nucleation rates: $\ln J' / \eta$ is scaled by $\ln S / \eta$, where the supersaturation ratio is $S$, $\eta$ is the dimensionless surface energy, and $J'$ is a dimensionless nucleation rate. This relation can be derived using the free energy of cluster formation at equilibrium which corresponds to the surface energy required to form the vapor-liquid interface. At low temperatures (below the triple point), we find that the surface energy divided by that of the classical nucleation theory does not depend on temperature, which leads to the scaling relation and implies a constant, positive Tolman length equal to half of the mean inter-particle separation in the liquid phase.