A mathematical model for the evaporation of a thin sessile liquid droplet: Comparison between experiment and theory

Abstract

A mathematical model for the quasi-steady diffusion-limited evaporation of a thin axisymmetric sessile droplet of liquid with a pinned contact line is formulated and solved. The model generalises the theoretical model proposed by Deegan et al. [Phys. Rev. E, 62 (2000) 756-765] to include the effect of evaporative cooling on the saturation concentration of vapour at the free surface of the droplet, and the dependence of the coefficient of diffusion of vapour in the atmosphere on the atmospheric pressure. The predictions of the model are in good qualitative, and in some cases also quantitative, agreement with recent experimental results. In particular, they capture the experimentally observed dependence of the total evaporation rate on the thermal conductivities of the liquid and the substrate, and on the atmospheric pressure.

Evaporation of sessile drops under combined diffusion and natural convection

“…Comparing the measured evaporation rates reported in references [11] and [23] to the values computed for diffusion-controlled evaporation supports the hypothesis that an additional transport mechanism, which we suggest is natural convection, is acting to increase the evaporation rate. Using the same data to compare with the results of the combined transport model indicates that the model over predicts the evaporation rate at the small drop sizes and suggests a need for further refinement of the model.…”

“…9 was used to compute the evaporation rates of methanol and acetone drops to compare with the evaporation rates of pinned drops given in references [11] and [23]. For drops ranging in size from 1 to 1.75 mm in radius, the results of the proposed model agree fairly well with the methanol data, with the model results being about 11% higher than the measured values.…”

“…Modeling the dynamics of a sessile drop as the drop first expands to wet the surface and then recedes due to the loss of volume from evaporation has been accomplished recently with very good qualitative agreement between the model and measurements of drop contact line radius and contact angle [1,2,3,4,5,6,7]. Good success has also been achieved in modeling the dynamics of a sessile drop which is pinned to a flat horizontal substrate [8,9,10,11]. In the case of a pinned drop, the radius of the contact line remains constant until sufficient volume is lost due to evaporation to cause the contact line to pull from its original position as the contact area of the drop reduces.…”

“…Using a similar electrostatic analogy, expressions have been derived for the evaporation rate of a sessile drop for two cases, one in which the contact angle remains constant and the contact area reduces, and the second case in which the contact area remains constant, i.e. a pinned drop, and the contact angle reduces [8,11,13,14]. For both of these cases, the rate of evaporation was found to be proportional to the radius of the contact line, in agreement with Stefan's result.…”

“…Other approaches that have been taken to model the evaporation of sessile drops are to use a smoothing function or a separate evaporation model in the region of the contact line [2,3,4,5,6,7,8,11,14], or to use an empirically derived constant of proportionality to relate the total evaporation rate to the transient radius of the drop contact line [1,3,15]. Researchers also have worked to improve the expression for the evaporation rate of sessile drops by accounting for effects such as evaporative cooling [11,15] and parabolic surfaces [11].…”