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

Abstract

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 Π. Since Π dominates at small film thicknesses, it determines the stability and wettability of thin films. To leading order, Π = Π(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 Π = Π(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 Π 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 Π = B h 3 (α 4 - h 4 x + 2hh 2 xh xx ), where a 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.