Statistical Thermodynamics of Spherical Droplets

Abstract

The model employed by Hill for investigating the nature of the transition region between two phases in a one‐component system is extended to spherical interfaces. The surface tension and location of Gibbs' dividing surfaces in droplets with a radius of 3 to 25 molecular diameters (the molecules are hard spheres with an attractive potential) are calculated at a reduced temperature of 0.6, providing the numerical basis for investigating the dependence of surface tension on droplet size. It is found that the surface tension in all cases is about 35% of that of a plane interface, but is almost independent of droplet size. The values of the surface tension relative to that of a plane interface are compared with those obtained from nucleation experiments in jets, using the Becker—Doering theory of nucleation; the agreement is poor, which is not surprising, considering the rather complicated substances used for the experiment. From the values of the surface tension at the Gibbs surface of tension, the work of formation, W/kT, of a droplet of the size in question can be calculated. Since the method here amounts to finding the critical nucleus in a supersaturated system, this quantity determines a ``critical nucleus for condensation''; from this can be found the ``critical supersaturation'' to be compared with existing cloud‐chamber measurements on low‐molecular‐weight vapors. Assuming that thermal fluctuations will suffice to initiate condensation if W/kT is less than 75, the critical nucleus for condensation is found to contain about 50 molecules, and the corresponding critical supersaturation is found to be S=2.6, in reasonable agreement with most experiments. Some questions concerning the applicability of thermodynamic concepts to microscopic droplets are briefly discussed.