Frenkelʹs method and the dynamic wetting of heterogeneous planar surfaces

The relationship between the edge velocity, νE, and the dynamic contact angle, θ, for the spreading of a small spherical cap type droplet on chemically and geometrically heterogeneous surfaces is examined using Frenkelʹs method. In this method, the change in surface free energy is equated to the viscous dissipation caused by Poiseuille flow inside the spherical cap. To describe dynamic wetting of a surface that is heterogeneous due to small variations in the local surface geometry of the solid, we introduce a simple Wenzel type correction for the ratio of the actual to geometric surface areas, r. The rate of change of surface free energy is then (2πr0)γLV(cos θ−rI)νE where r0 is the drop base radius, I=(γSV−γSL)/γLV and the γijʹs are the interfacial tensions. For partial wetting, I=cos θe where θe is the equilibrium contact angle and when the viscous dissipation vanishes, Wenzelʹs relationship linking the equilibrium contact angle on a rough surface to that on a smooth surface is obtained. Using dimensional arguments, we suggest that for a surface with weak geometric heterogeneity, the viscous dissipation is of the form kηr0νE2/tan(θ/2) where η is the viscosity and k is a numerical factor. Balancing the rate of change of the surface free energy with the viscous dissipation gives the edge speed proportional to tan(θ/2)(rcos θe−cos θ), which for small angles and smooth surfaces reduces to the Tanner–de Gennes Law νE∝θ(θ2−θe2). The influence of incomplete penetration of the fluid into the surface structure is also examined. An analogous relationship based on a smooth, but chemically heterogeneous surface is derived. This is shown to give Cassieʹs equation for the equilibrium contact angle. For complete wetting, Frenkelʹs method predicts Tannerʹs law θ∼t−3/10 when the surface is smooth and a modified Tannerʹs law tending towards θ∼t−3/4 when the surface has a weak geometric heterogeneity.