Capillary force on particles near a drop edge resting on a substrate and a criterion for contact line pinning

When a drop of liquid containing particles is allowed to evaporate from a substrate, the flow induced by the liquid evaporating from the drop edge carries the particles to the edge. If these particles prevent the drop edge from receding as the evaporation proceeds, then more particles will be accumulated near the drop edge resulting in the formation of a deposit that resembles coffee rings. We determine the capillary force on the particles near a drop edge and the effect of the particles on the gas-liquid-substrate contact angle to derive a condition that must be satisfied for particles to form the ringlike pattern.