Effect of gravity-induced shape change on the diffusion-limited evaporation of thin sessile and pendant droplets

A comprehensive study of the effect of gravity-induced shape change on the diffusion-limited evaporation of thin sessile and pendant droplets on a horizontal substrate is performed. Specifically, theoretical predictions for the evolution, and hence the lifetime, of sessile and pendant droplets evaporating in four modes of evaporation, namely, the constant contact radius (CR), the constant contact angle (CA), the stick-slide (SS), and the stick-jump (SJ) modes, are obtained. In particular, it is shown that gravity-induced shape change can cause quantitative differences in the evolution of sessile and pendant droplets compared to that of a droplet in the absence of (or in the neglect of) the effect of gravity (a "zero-gravity droplet"). For example, whereas sessile and pendant droplets evaporating in the CR mode evolve in qualitatively the same manner as a zero-gravity droplet, the evolution of droplets evaporating in the CA mode is more complicated. Specifically, while a zero-gravity droplet evaporating in the CA mode evolves according to the well-known d^{2} and 2/3 laws, an initially large sessile droplet evolves according to qualitatively different d and 1/2 laws, and an initially large pendant droplet evolves with the contact radius and the volume (but not, of course, the contact angle) behaving as if the droplet was evaporating in the CR mode. It is also found, perhaps somewhat unexpectedly, that the maximum height of a sessile droplet evaporating in the CA mode is a nonmonotonic function of time when the initial volume of the droplet is sufficiently large. Furthermore, it is found that for all four modes of evaporation a sessile droplet always evaporates faster, and hence has a shorter lifetime, than a zero-gravity droplet with the same initial volume, which in turn always evaporates faster, and hence has a shorter lifetime, than a pendant droplet with the same initial volume. It is also shown that for all four modes of evaporation the lifetime of a droplet is a monotonically increasing function of the initial volume of the droplet, that the lifetime of a droplet evaporating in the CA mode is always longer than that of the same droplet evaporating in the CR mode, and that the lifetimes of droplets evaporating in the SS and SJ modes both always lie between the lifetimes of the same droplet evaporating in the extreme modes.