(26d) A Criterion for “Coffee-Ring” Formation
We consider evaporation of liquid from a suspension droplet containing spherical particles resting on a wetting substrate. When the particles are small, their presence near the edge of the drop is believed to enhance the pinning of the edge of the drop, i.e. the contact line separating the air, the liquid, and the substrate. Since the evaporation rate of the liquid is greatest near the edge of the drop, the liquid from the bulk of the drop flows to the edge advecting with it the particles and depositing thereby particles near the edge of the drop, a phenomenon referred to in the literature as the ?coffee-ring? formation. We have carried out experiments with suspensions of polystyrene particles in water. While small particles indeed form rings near the drop edge, the larger particles show the opposite behavior. Evaporation of a suspension droplet containing both small and large particles is seen to segregate the particles with the larger particles accumulating near the drop center. To understand this phenomenon, we examine the problem of determining the capillary force on particles protruding from the gas-liquid interface and the effect of the particles on the contact angle at the drop edge. In order that the particles pin the contact line, the rate at which the contact angle increases with the particles accumulating near the drop edge must be equal or greater than the rate at which the contact angle decreases due to liquid evaporation. A criterion for the ring formation is derived in terms of drop and particle radius, particle volume fraction, evaporation Capillary number, drag coefficient, and particle and substrate contact angles, and shown to be in reasonable agreement with the experiments.