On Modeling Evaporation of Sessile Drops

Evaporating drops and thin films have numerous applications, from nanometric photoresist films in semiconductor applications to small sessile drops used for analysis of DNA microarrays [1]. Evaporating sessile drops are particularly interesting due to nonuniform drop thickness and the presence of contact lines (separating liquid, gas and solid phase). These factors may lead to nonuniform evaporation along the liquid/gas interface, that induces temperature gradients and related Marangoni effects. The Marangoni effects are a critical ingredient in explaining important problems involving deposits, such as the well know coffee-stain problem [2] and its’ numerous applications [1]. Hence, the benefits of thorough understanding of the evaporation process in sessile drops are evident. Although the problem of an evaporating drop on a thermally conductive solid substrate appears to be rather simple, important aspects regarding this very basic process remain unknown. This is mainly due to the fact that the evaporation phenomena involves an interplay of a number of physical effects, including mass and energy transfer between the three phases, diffusion and/or convection of the vapor in the gas phase, coupled with complex physics of the contact line (see, e.g., [3, 4]). Even if one is to ignore the issues related to the contact line region and the thermal effects in the solid phase, the resulting “2-sided” model that only includes processes in the liquid and the gas phases provides a mathematical obstacle of considerable difficulty [5]. Various simplifications of the 2-sided model have surfaced over the last decade. They rely on estimating the importance of relevant physical processes, thereby reducing the complex problem to a single phase consideration (liquid or gas). However, these estimates involve quantities that are not know precisely enough. The full 2-sided model can be first simplified by noting that the viscosity and thermal conductivity of vapor are small compared to those in the liquid. An additional assumption that the gas phase is free of convection, reduces the 2-sided model to a so called “1.5-sided” model, where only the processes in the liquid and the diffusion of vapor into surrounding gas are considered [6]. A commonly used simplification of the 1.5-sided model is achieved by assuming that the time-scale relevant to diffusion of vapor into surrounding gas is much longer than the one relevant to the phase-change [7, 8]. As a result, the processes in the liquid phase can be ignored. This approach also involves the assumption of thermodynamic equilibrium at the evaporating interface, and critically relies on the relevant thickness of the gas phase, which is unknown. As an outcome, the problem is reduced to a simple Laplace’s equation for concentration of vapor in the gas phase. At this point, one may notice an electrostatic analogy of finding an electric