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The role of solute diffusion in the early stages of coffee ring formation in a droplet with arbitrary smooth contact set

We study the initial evolution of the coffee ring that is formed by the evaporation of a thin, surface tension-dominated droplet with a smooth, pinned contact line. When the solutal P├ęclet number is large, we show that the competing effects of solute advection towards the contact line and solute diffusion away from the contact line controls the coffee-ring structure in the initial stages of evaporation. Utilizing a local coordinate system and a novel integrated mass formulation, we are able to construct an asymptotic prediction of the solute concentration profile valid during the early stages in which the solute remains dilute. We call this the nascent coffee ring and show that it has an approximate similarity profile depending on the evaporation model and the droplet shape. We are able to predict properties of the nascent coffee ring, such as its height and thickness.

For an axisymmetric droplet, we demonstrate the accuracy of the asymptotic predictions by comparing our results to numerical simulations of the full advection-diffusion equation. We then use the asymptotic analysis to investigate the validity of the diffusive regime: in particular, we illustrate the limited validity of the model in the diffusive evaporative flux regime. For more complicated droplet profiles, we explore the effects on the local coffee ring structure of both a geometry-dependent evaporative flux and the droplet free surface profile. In particular, by considering a constant evaporative flux, we show that the droplet geometry alone is sufficient to enhance the nascent coffee ring along parts of the contact line with high curvature.