A slope-dependent disjoining pressure for non-zero contact angles

A thin liquid film experiences additional intermolecular forces when the film thickness h is less than roughly 100 nm. The effect of these intermolecular forces at the continuum level is captured by the disjoining pressure Pi. Since Pi dominates at small film thicknesses, it determines the stability and wettability of thin films. To leading order, Pi = Pi(h) because thin films are generally uniform. This form, however, cannot be applied to films that end at the substrate with non-zero contact angles. A recent ad hoc derivation including the slope h(x) leads to Pi = Pi(h, h(x)), which allows non-zero contact angles, but it permits a contact line to move without slip. This work derives a new disjoining-pressure expression by minimizing the total energy of a drop on a solid substrate. The minimization yields an equilibrium equation that relates Pi to an excess interaction energy E = E(h, h(x)). By considering a fluid wedge on a solid substrate, E(h, h(x)) is found by pairwise summation of van der Waals potentials. This gives in the small-slope limit Pi = (h3)-(B) (alpha(4) - h(x)(4) + 2hh(x)(2)h(xx)), where alpha is the contact angle and B is a material constant. The term containing the curvature h(xx) is new; it prevents a contact line from moving without slip. Equilibrium drop and meniscus profiles are calculated for both positive and negative disjoining pressure. The evolution of a film step is solved by a finite-difference method with the new disjoining pressure included; it is found that h(xx) = 0 at the contact line is sufficient to specify the contact angle.