Coffee-Stain Deposits from Multiple Evaporating Droplets

We build on the recent theoretical work of Wray et al. [J. Fluid Mech. 884 A45 (2020)] concerning the competitive diffusion-limited evaporation of multiple thin, sessile droplets in proximity to each other. We obtain analytical predictions for the density of the contact-line deposits ("coffee-stains") left on the substrate when (initially uniformly dispersed) particles are advected by the evaporation-induced flow. Neighbouring droplets undergoing diffusion-limited evaporation interact via their vapour fields, competing to evaporate, which results in a nonlocal ``shielding'' effect, reducing the rate of evaporation. In particular, our theory confirms that in regions where evaporation is reduced by shielding, the deposition is also reduced. We give explicit results describing the deposits from a pair of identical interacting droplets, and demonstrate the excellent quantitative agreement with experimental findings of Pradhan and Panigrahi [Coll. Surf. A 482 562 (2015)]. We also give corresponding predictions for the case of three droplets arranged in an equilateral triangle.