Curvature dependence of the interfacial tension in binary nucleation

On the interfacial thermodynamics of nanoscale droplets and bubbles

We present a new self-consistent thermodynamic formalism for the interfacial properties of nanoscale embryos whose interiors do not exhibit bulklike behavior and are in complete equilibrium with the surrounding mother phase. In contrast to the standard Gibbsian analysis, whereby a bulk reference pressure based on the same temperature and chemical potentials of the mother phase is introduced, our approach naturally incorporates the normal pressure at the center of the embryo as an appropriate reference pressure. While the interfacial properties of small embryos that follow from the use of these two reference pressures are different, both methods yield by construction the same reversible work of embryo formation as well as consistency between their respective thermodynamic and mechanical routes to the surface tension. Hence, there is no a priori reason to select one method over another. Nevertheless, we argue, and demonstrate via a density-functional theory (with the local density approximation) analysis of embryo formation in the pure component Lennard-Jones fluid, that our new method generates more physically appealing trends. For example, within the new approach the surface tension at all locations of the dividing surface vanishes at the spinodal where the density profile spanning the embryo and mother phase becomes completely uniform (only the surface tension at the Gibbs surface of tension vanishes in the Gibbsian method at this same limit). Also, for bubbles, the location of the surface of tension now diverges at the spinodal, similar to the divergent behavior exhibited by the equimolar dividing surface (in the Gibbsian method, the location of the surface of tension vanishes instead). For droplets, the new method allows for the appearance of negative surface tensions (the Gibbsian method always yields positive tensions) when the normal pressures within the interior of the embryo become less than the bulk pressure of the surrounding vapor phase. Such a prediction, which is allowed by thermodynamics, is consistent with the interpretation that the mother phase's attempted compression of the droplet is counterbalanced by the negative surface tension, or free energy cost to decrease the interfacial area. Furthermore, for these same droplets, the surface of tension can no longer be meaningfully defined (the surface of tension always remains well defined in the Gibbsian method). Within the new method, the dividing surface at which the surface tension equals zero emerges as a new lengthscale, which has various thermodynamic analogs to and similar behavior as the surface of tension.

Evaluation of surface tension and Tolman length as a function of droplet radius from experimental nucleation rate and supersaturation ratio: Metal vapor homogeneous nucleation

Zinc and silver vapor homogeneous nucleations are studied experimentally at the temperature from 600 to 725 and 870 K, respectively, in a laminar flow diffusion chamber with Ar as a carrier gas at atmospheric pressure. The size, shape, and concentration of aerosol particles outcoming the diffusion chamber are analyzed by a transmission electron microscope and an automatic diffusion battery. The wall deposit is studied by a scanning electron microscope (SEM). Using SEM data the nucleation rate for both Zn and Ag is estimated as 10(10) cm(-3) s(-1). The dependence of critical supersaturation on temperature for Zn and Ag measured in this paper as well as Li, Na, Cs, Ag, Mg, and Hg measured elsewhere is analyzed. To this aim the classical nucleation theory is extended by the dependence of surface tension on the nucleus radius. The preexponent in the formula for the vapor nucleation rate is derived using the formula for the work of formation of noncritical embryo [obtained by Nishioka and Kusaka [J. Chem. Phys. 96, 5370 (1992)] and later by Debenedetti and Reiss [J. Chem. Phys. 108, 5498 (1998)]] and Reiss replacement factor. Using this preexponent and the Gibbs formula for the work of formation of critical nucleus the dependence of surface tension on the radius R(S) of the surface of tension is evaluated from the nucleation data for above-mentioned metals. For the alkali metals and Ag the surface tension was determined to be a strong function of R(S). For the bivalent metals (Zn, Hg, and Mg) the surface tension was independent of radius in the experimental range. A new formula for the Tolman length delta as a function of surface tension and radius R(S) is derived by integration of Gibbs-Tolman-Koenig equation assuming that delta is a monotonic function of radius. The formula derived is more correct than the Tolman formula and convenient for the elaboration of experimental data. Using this formula the values of delta are determined as a function of R(S) from the experimental nucleation data. It is determined that all the metals considered are characterized by strong dependence of delta on radius; for the bivalent metals delta changes sign.

Multicomponent nucleation: Thermodynamically consistent description of the nucleation work

A thermodynamically consistent formula is derived for the nucleation work in multicomponent homogeneous nucleation. The derivation relies on the conservative dividing surface which defines the nucleus as having specific surface energy equal to the specific surface energy sigma0 of the interface between the macroscopically large new and old phases at coexistence. Expressions are given for the radius of the nucleus defined by the conservative dividing surface and by the surface of tension. As a side result, the curvature dependence of the surface tension sigmaT of the nucleus defined by the surface of tension is also determined. The analysis is valid for nuclei of any size, i.e., for nucleation in the whole range of conditions between the binodal and the spinodal of the metastable old phase provided the inequality sigmaT < or = sigma0 is satisfied. It is found that under the conditions of validity of the analysis the nucleation rate is higher than the nucleation rate given by the classical nucleation theory. The general results are applied to nucleation of unary liquids or solids in binary gaseous, liquid or solid mixtures.