Nanodroplets on a Planar Solid Surface Temperature, Pressure, and Size Dependence of Their Density and Contact Angles *

The shape and the density of a liquid drop on a planar solid surface, embedded in an inert gas at constant temperature and pressure, were examined on the basis of a microscopic density functional approach that accounts for the entropic (temperature-dependent) and energetic contributions to the free energy of the system. Integro-differential equations describing the profile and the density of a cylindrical (2D) drop were derived by the variational minimization of the Gibbs free energy with respect to both the drop profile and density under the assumption of uniform density. The equations were solved numerically using the constraint of a constant number of molecules N ₁per unit length of the drop. It was shown that for temperatures lower than a certain temperature T _w the free energy against density has generally two minima, representing a stable equilibrium state and a metastable one. One of those minima is located at a density corresponding to the density of a normal liquid, whereas the other one is located at a density comparable to the density of the surrounding inert gas. For this reason, the latter state of the drop cannot be stable. For T > T _w, the minimum corresponding to the liquid state disappears, and no drop can be formed on the surface. The temperature T _w depends on N ₁and the external pressure p and increases when N ₁and p increase. The true wetting angle θ ₀that the drop profile makes with the solid surface depends on the parameters characterizing the microscopic interactions, the density, and the surface densities. If in the thermodynamically stable state |cos θ ₀| > 1, then no drop is formed on the surface. If in that state |cos θ ₀| < 1, then at any pressure the true contact angle decreases when the temperature increases and approaches T _w. However, θ ₀does not reach a zero value for T ≤ T _w but has for T = T _w a discontinuity from a finite to a zero value. The true contact angle also depends on the number of molecules in the drop and on the external pressure. For all considered values of N ₁, p, and microscopic parameters of the intermolecular interactions, the density of the drop decreases with increasing temperature. The rate of decrease is constant for temperatures sufficently far from Tw and increases when T approaches T _w. At a given temperature and pressure, the density of the drop decreases with decreasing N _l. For relatively large drops (N _l ≃ 10¹⁴–10²⁰), the rate of decrease is very small, whereas for small droplets (N _l ≃ 10¹²) it becomes much larger.