Lifetime of evaporating two-dimensional sessile droplets.

Abstract

Prevailing diffusion-limited analyses of evaporating sessile droplets are facilitated by a quasisteady model for the evolution of vapor concentration in space and time. When attempting to employ that model in two dimensions, however, one encounters an impasse: the logarithmic growth of concentration at large distances, associated with the Green's function of Laplace's equation, is incompatible with the need to approach an equilibrium concentration at infinity. Observing that the quasisteady description breaks down at large distances, the diffusion problem is resolved using matched asymptotic expansions. Thus the vapor domain is conceptually decomposed into two asymptotic regions: one at the scale of the drop, where vapor transport is indeed quasisteady, and one at a remote scale, where the drop appears as a point singularity and transport is genuinely unsteady. The requirement of asymptotic matching between the respective regions furnishes a self-consistent description of the time-evolving evaporation process. Its solution provides the droplet lifetime as a universal function of a single physical parameter. Our scheme avoids the use of a remote artificial boundary, which introduces a nonremovable dependence upon a nonphysical parameter.