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 ρ0 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 contactline dynamics. Third, the DIM coupled with a realistic equation of state for water is used to compute the dependence of the surface tension σ on the temperature T , which agrees well with the empiric σ(T ). Fourth, the same framework is used to compute the static contact angle of a watervapor interface. It is shown that, with increasing temperature, the contact angle becomes either 180 • (perfect hydrophobicity) or 0 • (perfect hydrophilicity), depending on whether ρ0 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.

Capillary condensation of saturated vapor in a corner formed by two intersecting walls

“…Indeed, since the integrand in this term involves dρ/dz, it is small everywhere except near the interface. f , on the other hand, is order-one everywhere between the interface and substrate -which is an asymptotically wider region due to estimate (42). Thus, any characteristic of the solution in the latter region can be calculated with the integral term in Eq.…”

“…In addition to the advantage of simplicity, condition (16) has the advantage of applicability under the same assumptions as the DIM itself [42]. Furthermore, since the expected effect of spontaneous condensation depends only on the curvature of the meniscus interface (as argued in the introduction), the model used for the boundary condition is not essential: condensation occurs at the liquid/vapor interface, so the fluid-substrate interaction does not affect it too much.…”

“…If this condition does not hold, the substrate becomes either perfectly hydrophobic (if ρ 0 ≤ ρ v ) or perfectly hydrophilic (if ρ 0 ≥ ρ l ) [42]. In the former case, condensation cannot occur on the substrate (as it repulses the liquid phase), whereas the latter implies immediate condensation regardless of all other parameters.…”

“…As for l i [which also appears in expression (78)], one can use numerous indirect estimates (e.g., Refs. [22,24,42]) suggesting that…”

The dynamics of liquid films, as described by the diffuse-interface model

“…It has been shown, however, that in some, if not most, common fluids including water, interfaces are not isothermal 15,16 , and it is unclear how this affects liquid films. A thin liquid layer can behave differently from the general case -especially, if the substrate is maintained at a fixed temperature, acting as a thermostat for the adjacent fluid.…”