Dependence of the surface tension and contact angle on the temperature, as described by the diffuse-interface model

Abstract: Four results associated with the diffuse-interface model (DIM) for contact lines are reported in this paper. First, a boundary condition is derived, which states that the fluid near a solid wall must have a certain density $\rho_{0}$ depending on the solid's properties. Unlike previous derivations, the one presented here is based on the same physics as the DIM itself and does not require additional assumptions. Second, asymptotic estimates are used to check a conjecture lying at the foundation of the DIM, as well as all other models of contact lines: that liquid-vapor interfaces are nearly isothermal. It turns out that, for water, they are not -- although, for a more viscous fluid, they can be. The non-isothermaility occurs locally, near the interface, but can still affect the contact-line dynamics. Third, the DIM coupled with a realistic equation of state for water is used to compute the dependence of the surface tension $\sigma$ on the temperature $T$, which agrees well with the empiric $\sigma(T)$. Fourth, the same framework is used to compute the static contact angle of a water-vapor interface. It is shown that, with increasing temperature, the contact angle becomes either $180^{\circ}$ (perfect hydrophobicity) or $0^{\circ}$ (perfect hydrophilicity), depending on whether $\rho_{0}$ matches the density of saturated vapor or liquid, respectively. Such behavior presumably occurs in all fluids, not just water, and for all sufficiently strong variations of parameters, not just that of the temperature -- as corroborated by existing observations of drops under variable electric field.