Thin Films in Partial Wetting: Internal Selection of Contact-Line Dynamics

When a liquid touches a solid surface, it spreads to minimize the system's energy. The classic thin-film model describes the spreading as an interplay between gravity, capillarity, and viscous forces, but it cannot see an end to this process as it does not account for the nonhydrodynamic liquid-solid interactions. While these interactions are important only close to the contact line, where the liquid, solid, and gas meet, they have macroscopic implications: in the partial-wetting regime, a liquid puddle ultimately stops spreading. We show that by incorporating these intermolecular interactions, the free energy of the system at equilibrium can be cast in a Cahn-Hilliard framework with a height-dependent interfacial tension. Using this free energy, we derive a mesoscopic thin-film model that describes the statics and dynamics of liquid spreading in the partial-wetting regime. The height dependence of the interfacial tension introduces a localized apparent slip in the contact-line region and leads to compactly supported spreading states. In our model, the contact-line dynamics emerge naturally as part of the solution and are therefore nonlocally coupled to the bulk flow. Surprisingly, we find that even in the gravity-dominated regime, the dynamic contact angle follows the Cox-Voinov law.