The evaporation of multiple nearly-circular droplets and applications to the coffee stain effect

The diffusion-limited evaporation of droplets has received significant interest in recent years due to its fundamental significance in a variety of industrial processes such as printing, patterning, DNA microarray technologies and the manufacture of OLEDs. However, the only available analytical solutions are for circular or elliptical droplets. We improve this state of affairs by finding asymptotic solutions for the evaporative flux for nearly-circular droplets. Via matched asymptotic expansions, we show that this allows for the determination of the resultant profiles, and corresponding coffee-stain residue, in large droplets. This is then extended to polygonal drops, before the framework is extended to allow for the complete calculations to be performed for arbitrary arrays of multiple droplets.