The effect of particle–substrate adsorption on the deposition of particles from a thin evaporating sessile droplet

A mathematical model for the evaporation of, the flow within, and the deposition from, a thin, pinned sessile droplet undergoing either spatially uniform or diffusion-limited evaporation is formulated and analysed. Specifically, we obtain explicit expressions for the concentration of particles within the bulk of the droplet, and describe the behaviour of the concentration of particles adsorbed onto the substrate as well as the evolution of the masses within the bulk of the droplet, adsorbed onto the substrate, and in the ring deposit that can form at the contact line. In particular, we show that the presence of particle–substrate adsorption suppresses the formation of a ring deposit at the contact line for spatially uniform, but not for diffusion-limited, evaporation. However, in both scenarios, the final adsorbed deposit is more concentrated near to the contact line of the droplet when radial advection due to evaporation dominates particle–substrate adsorption, but is more concentrated near to the centre of the droplet when particle–substrate adsorption dominates radial advection due to evaporation. In addition, in an appendix, we investigate the formation of a ring deposit at the contact line for a rather general form of the local evaporative flux, and show that the presence of particle–substrate adsorption suppresses the formation of the ring deposit that can otherwise occur when the local evaporative flux is non-singular at the contact line.