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In the framework of density functional theory ~DFT!, two patching models for the density profile of spherical liquid droplet are developed. The patching is based on analytical expressions of the asymptote of the density profiles. The first model leads to analytic expressions of the Tolman length and the effective rigidity constant, from which the temperature dependence of the Tolman length and effective rigidity constant can be determined. The second model is developed particularly for small spherical droplets, from which the dependence of chemical potential and the surface tension of the droplet on the radius are obtained. The results are compared with numerical DFT calculations. © 1999 American Institute of Physics.@S0021-9606~99!51847-3# I. INTRODUCTION This article continues the theme of our previous work 1 ~hereafter referred as Paper I!, that is, a study of surface tension of liquid droplets and related properties. A motivation of the study is that it is relevant to the gas‐liquid nucleation, one of our recent research interests. It is well-known that in the classical theory of homogeneous nucleation 2 the surface tension of a droplet is taken to be that of planar surface, whereby the dependence of surface tension on the radius of the droplet is not taken into account ~the capillarity approximation!. It turns out that the resulting rates of the nucleation based on the capillarity approximation can be orders of magnitude different from measured ones in experiments. An improvement over the capillarity approximation is to account for the effect of size dependence of the surface tension. Density functional theory ~DFT! of nucleation is such a theory that implicitly incorporates this effect. 3,4 In fact, DFT is particularly useful to calculate the surface tension in the limit of an infinitely large droplet and to obtain the limiting behavior of related surface properties. 5‐7 However, substantial numerical calculations involved in DFT limit its wide applicability. On the other hand, simple phenomenological theory of nucleation such as the classical one not only offers analytical formulas for evaluating the rate of nucleation but is also easy to apply. Therefore, a wellgrounded analytical formula for the surface tension of a droplet will benefit future development of better phenomenological theory of nucleation. In Paper I we obtained analytical expressions for the surface tension of planar liquid‐vapor interface and the Tolman length within the framework of DFT. We developed a theoretical approach—the patching model for the asymptote of density profile, from which a rational analytical expression for the density profile is obtained. This analytical expression can then be placed into the statistical mechanics formulas for the surface tension and Tolman length. The Tolman length is the coefficient in the first-order curvature correction to the surface tension. In this work, we will extend the approach in Paper I and study not only the large droplets but also small droplets with size comparable to critical droplets in gas‐liquid nucleation at high supersaturation. A different analytical expression for the Tolman length will be present. Moreover, we will present an analytical expression for the second-order curvature correction—the so-called effective rigidity constant. The latter also plays an important part in the dependence of surface tension on the radius of a droplet. Statistical mechanics formulas for both the Tolman length and the effective rigidity constant were originally derived by Blokhuis and Bedeaux. 8 With the two curvature corrections, the formula for surface tension provides a reasonable description of the size dependence of surface tension of large droplets. We note that these curvature corrections have been calculated numerically 9,10

A patching model for surface tension of spherical droplet and Tolman length. II