Abstract
Evaporating sessile droplets interact with neighboring droplets via their vapor fields, resulting in nonaxisymmetric evaporative fluxes from their surfaces. One of the consequences of this asymmetry is that, unlike the uniform deposits left at the contact lines of isolated droplets, the deposits left at the contact lines of droplets with neighbors are, in general, nonuniform. In this work we develop a theoretical model for the contact-line deposits from multiple evaporating droplets, and find its predictions for a pair of identical droplets to be in excellent agreement with recent experimental results.
Highlights
Π ak2 − r2 is the local evaporative flux from the kth droplet in isolation, and ψk,n is the angle between the x axis and the line joining the centers of the kth and the nth droplets, shown in Fig. 1(a) and given by tan ψk,n
We gave predictions for the deposits from a pair of identical droplets, which showed that the deposit is reduced the most where the droplets are closest together, and demonstrated excellent quantitative agreement with experimental results of Pradhan and Panigrahi [39]
We gave corresponding predictions for a triplet of identical droplets arranged in an equilateral triangle, which showed that the effect of shielding on the deposit is more subtle in this case
Summary
The evaporation of sessile droplets has been the subject of extensive experimental, numerical and analytical investigation in recent years (see, for example, Refs. [1,2,3,4,5] and the references therein), partly motivated by the wide range of everyday and industrial situations, such as protein crystallography [6], surface patterning [7], ink-jet printing, including that of OLED displays [8], and agrochemical spraying of plants [9], in which it occurs. Du and Deegan [29] examined a two-dimensional 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 [31] investigated a variety of nonaxisymmetric droplets both experimentally and numerically, and found that larger deposits occur where the contact line has the largest curvature (e.g., near the tips of a droplet with a triangular contact line) Their theoretical modeling of the density of the deposit was essentially phenomenological. In the present contribution we build on the work of Wray et al [48] in order to analyze the spatially nonuniform densities of the deposits left on the substrate by the diffusion-limited evaporation of multiple thin droplets with pinned circular contact lines in proximity to each other. In the Appendix we describe the corresponding analysis for the less commonly studied case of large droplets, corresponding to the limit of large Bond number, in which even greater analytical progress is possible
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