Profile and contact angle of small sessile drops. A more general approximate solution

Abstract

The second-order differential equation describing the profile of a sessile drop has been derived in polar coordinates using the criterion of minimum free energy and applying the calculus of variations. Application of a form of perturbation theory leads to an approximate solution valid for drops of sufficiently small maximum diameter. In the case of drops of contact angle > 90°, this solution can be exploited directly to obtain the contact angle from a knowledge of drop height, maximum diameter and diameter at the plane of contact with the solid. If this last datum is lacking, the contact angle can still be obtained by a reiterative method or graphically. For contact angles < 90° this last procedure must be used and thus little advantage is gained over a solution previously obtained in cartesian coordinates. Although the solution is less accurate than data obtained from numerical integration, its relative simplicity should prove useful for the objective determination of contact angles.