Morphological stability analysis of partial wetting

We develop a macroscopic static theory of the morphological stability of partial wetting. The system we studied consist of a smooth horizontal solid surface and some non-volatile liquid on it. A necessary condition for the stable equilibrium of such systems is known as the Young condition on the contact angle made at the contact line where the free surface of liquid meets the solid surface. But this condition is local and is not sufficient for the stability. We present a formulation for studying the stability of systems which satisfy the Young condition. Then we apply this to several morphologies of wetting. We find that there are at least two fundamental morphologies that we call a hole and a ridge, which are thermodynamically unstable against certain infinitesimal deformations of the contact lines. The hole type instability has also been found recently D. J. Srolovitz and S. A. Safran, J. Appl. Phyys., 60 (1986), 1]. We also derived a reduced expression for the wetting energy as a functional of the contact line positions under the assumption of almost flat free surface of the liquid. This serves us to understand the characteristic length scale which appears in the ridge type instability. Besides these instabilities there is another category of morphological instability in which the system becomes unstable against an infinitesimal deformation of the free surface of liquid. We show this by an illustrating example in which the instability is described as the so-called tangent bifureation in nonlinear systems.