Gas–liquid nucleation at large metastability: unusual features and a new formalism

Abstract

Nucleation at large metastability is still largely an unsolved problem, even though itis a problem of tremendous current interest, with wide-ranging practical value,from atmospheric research to materials science. It is now well accepted that theclassical nucleation theory (CNT) fails to provide a qualitative picture and givesincorrect quantitative values for such quantities as activation-free energy barrier andsupersaturation dependence of nucleation rate, especially at large metastability.In this paper, we present an alternative formalism to treat nucleation at largesupersaturation by introducing an extended set of order parameters in terms of thekth largest liquid-like clusters,where k = 1 is the largestcluster in the system, k = 2 is the second largest cluster and so on. At low supersaturation, the size of the largestliquid-like cluster acts as a suitable order parameter. At large supersaturation, thefree energy barrier for the largest liquid-like cluster disappears. We identify thissupersaturation as the one at the onset of kinetic spinodal. The kinetic spinodal issystem-size-dependent. Beyond kinetic spinodal many clusters grow simultaneouslyand competitively and hence the nucleation and growth become collective. Inorder to describe collective growth, we need to consider the full set of orderparameters. We derive an analytic expression for the free energy of formation of thekth largest cluster. The expression predicts that, at large metastability (beyond kineticspinodal), the barrier of growth for several largest liquid-like clusters disappears, and allthese clusters grow simultaneously. The approach to the critical size occurs by barrierlessdiffusion in the cluster size space. The expression for the rate of barrier crossingpredicts weaker supersaturation dependence than what is predicted by CNT at largemetastability. Such a crossover behavior has indeed been observed in recent experiments(but eluded an explanation till now). In order to understand the large numericaldiscrepancy between simulation predictions and experimental results, we carried outa study of the dependence on the range of intermolecular interactions of boththe surface tension of an equilibrium planar gas–liquid interface and the freeenergy barrier of nucleation. Both are found to depend significantly on the range ofinteraction for the Lennard-Jones potential, both in two and three dimensions.The value of surface tension and also the free energy difference between the gasand the liquid phase increase significantly and converge only when the rangeof interaction is extended beyond 6–7 molecular diameters. We find, with thefull range of interaction potential, that the surface tension shows only a weakdependence on supersaturation, so the reason for the breakdown of CNT (withsimulated values of surface tension and free energy gap) cannot be attributed tothe supersaturation dependence of surface tension. This remains an unsettledissue at present because of the use of the value of surface tension obtained atcoexistence.