Some Problems in Simulation of the Thermodynamic Properties of Droplets

Abstract

In this paper, the Gibbs dividing surface method is used to derive a formula to determine curvature-dependent surface tension in a system with two phases. The well-known Tolman formula is a special case of this formula. The problem of a sessile droplet is considered. The Bashforth–Adams equation analogue (in view of curvature-dependent surface tension) is obtained, and the numerical solution of the equation is carried out. It is shown that if the droplet size is not very large relative to the thickness of the surface layer (micro- or nanodroplets), the dependence of the surface tension on the curvature is very important. In addition, the case is considered where the diameters of cylindrical nanodroplets are shorter than the Tolman length.

APPENDIX

The question of the Tolman length, which determines the dimensional effect of the surface tension and the theory application area, is worthy of separate consideration. According to the thermodynamic definition, the Tolman length is numerically equal to the distance between the equimolecular surface and the surface of tension [2]:

$$\delta = {{z}_{e}} - {{z}_{s}},$$

((A.1))

where z_e and z_s prescribe the factors of the equimolecular surface and the surface of tension on one common semiaxis. The equimolecular surface corresponds to the condition Γ = 0. The dividing surface for which there is valid the Laplace equation is the surface of tension. As a rule, the surface of tension is assumed to be a true dividing surface.

The equimolecular surface and the surface of tension are always situated inside the interphase transition layer, therefore, this layer thickness can be assumed as the highest value of the Tolman length δ [2, p. 44]. For not too little drops the Tolman length can be assumed a constant value related to a plane surface. By convention, the Tolman length for the plane dividing surface is as follows:

$$\delta = \frac{\Gamma }{{\Delta n}},\,\,\,\,\Delta n = {{n}_{1}} - {{n}_{2}},$$

((A.2))

where n_1,2 are the volume densities of the equilibrium coexisting phases. There is the conflicting information on the numerical values and even the sign of the Tolman length. It follows from (A.2) that the sign of δ depends on the sign of the Gibbs adsorption on the surface of tension. It is possible to determine the sign of the Tolman length using a well-known formula for the profile of the density distribution on the plane interphase region [3]:

$$n(z) = \frac{{{{n}_{1}} + {{n}_{2}}}}{2} - \frac{{\Delta n}}{2}{\text{tanh}}\left( {\frac{z}{{{{z}_{0}}}}} \right),\,\,\,\,\Delta n > 0,$$

((A.3))

where z is the coordinate; z₀ is the parameter characterizing the slope of the density distribution profile. Formula (A.3) prescribes the density distribution profile symmetric about the point z = 0. More dense and less dense phase are placed on the positive and negative semiaxes. For the Gibbs adsorption we have

$$\Gamma = \int\limits_{{{z}_{i}}}^\infty {[n(z) - {{n}_{1}}]\,dz} + \int\limits_{ - \infty }^{{{z}_{i}}} {[n(z) - {{n}_{2}}]\,dz} ,$$

((A.4))

where z_i prescribes the place of the arbitrary dividing surface. The integration of (A.4) in view of (A.3) yields:

$$\Gamma = {{z}_{i}}\Delta n.$$

((A.5))

For the equimolecular surface and the surface of tension:

$${{z}_{i}} = 0\,\,{\text{and}}\,\,{{z}_{i}} = {{z}_{s}},$$

therefore, from (A.2) and (A.5) for the Tolman length we find:

$$\delta = {{z}_{s}},$$

((A.6))

where z_s is measured from the origin of coordinates, i. e., from the profile center. Formula (A.6) makes possible to deduce that the sign of the Tolman length depends on the fact where the surface of tension is situated. If the surface of tension is near a dense phase (this is, in our opinion, most natural) the Tolman length will be positive. The displacement of the surface of tension into the region of the less dense phase with respect to the equimolecular surface changes the sign of δ for negative one.

Please note that for certain thermodynamic systems the Tolman parameter can be assumed negative, however, here this case is not considered. Thus, according to V.K. Semenchenko [31], for a droplet with the radius r the stability condition has the form ${{({{\partial \sigma } \mathord{\left/ {\vphantom {{\partial \sigma } {\partial r}}} \right. \kern-0em} {\partial r}})}_{{T,p}}} > 0$. It follows that the function σ(r) must be ascending, and for the drop there must be fulfilled a condition of δ > 0.