Approximate solution for the spreading of a droplet on a smooth solid surface

The same approach used by Boender, Chesters, and van der Zanden in the context of an advancing liquid–gas meniscus in a capillary tube is extended to the case of spontaneous spreading of a droplet on an ideal solid surface. The result is an ordinary differential equation for the droplet profile which can be solved if the meniscus inclination f0is specified at some distance ¿ from the solid. As in the capillary-tube case, good agreement is obtained with experimental data obtained by the authors and by others if f0is set equal to the static contact angle (zero in cases investigated experimentally), taking ¿ of the order of a molecular dimension (1 nm). A comparison of predicted dynamic contact angles in the spreading-drop and capillary-tube cases for given values of the capillary number indicates only a weak dependence of the behavior on the system geometry. De Gennes and co-workers have predicted that during the final stages of spreading the inner length scale ¿ should be determined by the effects of disjoining pressure in the thin film adjacent to the contact line rather than by molecular dimensions. The ¿ value implied by their model is derived, thereby establishing the regime of spreading in which such effects should be dominant. The observed behavior in this regime is found to correspond somewhat better with a ¿ value of the order of a molecular dimension, although the differences are small. Although the explanation probably lies in the nonideality of even the smoothest surfaces, this result suggests that the simplest model, based on a single ¿ value of the order of 1 nm, should provide an excellent predictive tool.