(715c) On the Surface Thermodynamics of Nanoscale Droplets and Bubbles

The thermodynamic properties of small systems are well known to be very different from their macroscopic counterparts. One area, among many, where such differences need to be understood in greater detail is nucleation and growth. A widely used description of nucleation is classical nucleation theory (CNT), which in its original version assumes the embryos (either droplets or bubbles) that form within the metastable mother phase have the same thermophysical properties as their corresponding bulk phases. This assumption, however, is no longer valid as the metastable phase approaches the spinodal, where the inhomogeneous nature of the embryo cannot be ignored. Various modifications have been introduced into CNT, such as accounting for, though to a limited extent, the dependence on curvature of the surface tension of finite-sized embryos. Further improvements to CNT can be made, but a better understanding of the surface thermodynamics of nanoscale embryos must first be obtained.

In previous work, Hill[1] presented a straightforward analysis of the surface thermodynamics of embryos maintained at the same chemical potential of the surrounding mother phase. For the case of large embryos that contain a bulk fluid at their centers, all the well-known relations of surface thermodynamics are recovered (e.g., Laplace relation). Hill then extended this formalism to the description of nanoscale embryos, for which their centers did not contain some portion of a uniform density. Here, a reference pressure is introduced, which denotes the pressure of the corresponding ?bulk? embryo at the same chemical potential of the mother phase. While Hill's method is self-consistent, in that the resulting definition of the surface tension is designed to recover the free energy of the system, his approach does not yield physically meaningful limits of the thermophysical properties of unltrasmall embryos. In particular, Hill's method does not generate a surface tension that smoothly vanishes at the spinodal (as physical arguments suggest).

We therefore present another approach, though still based on Hill's analysis, for describing the surface thermodynamics of nanoscale embryos. In this alternative method, the value of the transverse or normal pressure at the center of the embryo becomes the key quantity of interest for describing the surface thermodynamic properties of the embryo. With this pressure, the resulting surface tension now smoothly vanishes at the spinodal. We utilize density-functional theory (DFT) to generate the density and transverse pressure profiles of embryos within both the metastable Lennard-Jones fluid and a model water-like fluid. From these profiles, we analyze the behavior of various surface thermodynamic properties, including the surface tension and the radius of the Gibbs surface of tension surface, over a broad range of degrees of metastability. Interestingly, and despite what was implied by previous modifications to CNT, the first- through third-order corrections to the curvature dependence of the surface tension, based on the Tolman length, fail to capture the behavior of the surface tension even at modest degrees of metastability. Finally, we also discuss the extension of this formalism to the case where the embryos are neither in chemical nor mechanical equilibrium with the surrounding metastable phase. Additional terms appear in the standard surface thermodynamic relations, and some interesting limits for macroscopic sizes are considered. We also discuss the implications of this work for the surface thermodynamics of nanoscale sessile drops, where the generalized Young equation for the line tension may need to be modified.

[1] Hill, 1952, J. Phys. Chem. 56, 526-531.