Contact angles for evaporating liquids predicted and compared with existing experiments

The stationary meniscus of an evaporating, perfectly wetting system exhibits an apparent contact angle Theta which vanishes with the applied temperature difference DeltaT, and is maintained for DeltaT > 0 by a small-scale flow driven by evaporation. Existing theory predicts Theta and the heat flow q, from the contact region as the solution of a free-boundary problem. Though that theory admits the possibility that Theta and ii, are determined at the same scale, we show that, in practice, a separation of scales gives the theory an inner and outer structure; Theta is determined within an inner region contributing a negligible fraction of the total evaporation, but q, is determined at larger scales by conduction across an outer liquid wedge subtending an angle Theta. at larger scales by conduction across an outer liquid wedge subtending an angle Theta. The existence of a contest angle can thus be assumed for computing the heat flow; the problems for Theta and ii, decouple. We analyse the inner problem to derive a formula for Theta as a function of DeltaT and material properties, the formula agrees closely with numerical solutions of the existing theory. Though microphysics must be included in the model of the inner region to resolve a singularity in the hydrodynamic equations, Theta is insensitive to microphysical detail because the singularity is weak. Our analysis shows that Theta is determined chiefly by the capillary number Ca = mu V-l(l)/sigma based on surface tension sigma, liquid viscosity mu (l) and a velocity scale V-l set by evaporation kinetics. To illustrate this result of our asymptotic analysis, we show that computed angles lie close to the curve Theta = 2.2Ca(1/4) a small scatter of +/- 15% about that curve is the only hint that Theta depends on microphysics. To test our scaling relation, we use film profiles measured by Kim (1994) to determine experimental values of Theta and Cn I these are the first such values to be published for the evaporating meniscus. Agreement between theory and experiment is adequate; the difference is less than +/- 40% for 9 of 15 points, while the scatter within experimental values is +/- 25%.