Evaporation of a sessile droplet on a slope

Abstract

We theoretically examine the drying of a stationary liquid droplet on an inclined surface. Both analytical and numerical approaches are considered, while assuming that the evaporation results from the purely diffusive transport of liquid vapor and that the contact line is a pinned circle. For the purposes of the analytical calculations, we suppose that the effect of gravity relative to the surface tension is weak, i.e. the Bond number (Bo) is small. Then, we express the shape of the drop and the vapor concentration field as perturbation expansions in terms of Bo. When the Bond number is zero, the droplet is unperturbed by the effect of gravity and takes the form of a spherical cap, for which the vapor concentration field is already known. Here, the Young-Laplace equation is solved analytically to calculate the first-order correction to the shape of the drop. Knowing the first-order perturbation to the drop geometry and the zeroth-order distribution of vapor concentration, we obtain the leading-order contribution of gravity to the rate of droplet evaporation by utilizing Green’s second identity. The analytical results are supplemented by numerical calculations, where the droplet shape is first determined by minimizing the Helmholtz free energy and then the evaporation rate is computed by solving Laplace’s equation for the vapor concentration field via a finite-volume method. Perhaps counter-intuitively, we find that even when the droplet deforms noticeably under the influence of gravity, the rate of evaporation remains almost unchanged, as if no gravitational effect is present. Furthermore, comparison between analytical and numerical calculations reveals that considering only the leading-order corrections to the shape of the droplet and vapor concentration distribution provides estimates that are valid well beyond their intended limit of very small Bo.

Evaporating droplets on inclined plant leaves and synthetic surfaces: Experiments and mathematical models

“…We note that our YLP approach is similar to that used in a very recent study by Timm et al [52] (which was published after our model was developed), who also consider evaporating droplets on an incline. Like our YLP model, they assume the contact line is circular, solve the Young-Laplace equation to determine the shape of the droplet, and compute the flux of liquid at the air/droplet interface by solving Laplace's equation outside the droplet.…”

Drying Droplets: A Review on its Numerical and Experimental Studies to Remove Coffee Ring Effect

“…On experiments of droplet evaporation on a structured superhydrophobic surface that displays very high contact angle (CA ∼ 160 deg), and negligible CAH (<1 deg) [11]. The droplet evaporation is observed to occur in a constant-contact-angle mode, with contact radius shrinking for almost the entire duration of evaporation [16].…”

“…Drying of droplet is actually a complex, non-equilibrium and difficult-to-control process [8], the control of the deposition morphology of droplet has significant influence. Further studies on this non-uniform redistribution process indicate that inner flows, including capillary flow [9] and Marangoni flow [10], dynamics of the three-phase contact line [11], and particle-particle/particle-interface interaction [12] will influence the final particle distribution.…”

Contact-line deposits from multiple evaporating droplets

“…where the depth-averaged radial and azimuthal velocities ūk and vk are given by (26). Equation (30) may be solved by the method of characteristics:…”

“…The vast majority of the previous work on deposition from evaporating sessile droplets has, for obvious reasons, focused on axisymmetric deposits from axisymmetric droplets. There has, however, been some work on non-axisymmetric deposits from non-axisymmetric droplets (see, for example, [9,10,[22][23][24][25][26]). In particular, Du and Deegan [23] examined a twodimensional droplet on an inclined substrate numerically, and found that, depending on the initial volume of the droplet and the angle of inclination of the substrate, the larger deposit can occur at either the upper or the lower contact line, while Sáenz et al [25] investigated a variety of non-axisymmetric droplets both experimentally and numerically, and found that larger deposits occur where the contact line has the largest curvature (e.g.…”