Coffee Stains from Drops with Receding Contact Lines

We present a framework for calculating the surface density profile of a coffee stain deposited by a drying drop with a receding contact line. For standard coffee stains, the fluid pins to the substrate, forces flow towards the exterior of the drop and deposits a thin, concentrated ring of particles. Unlike a pinned drop, a receding drop pushes fluid towards its interior and continuously deposits mass across its substrate as it evaporates. This gives rise to a new class of mountain-like morphologies that are not seen in the standard coffee ring effect but are reminiscent of recent experimental results. For a thin, circular drop with uniform evaporation, we calculate the surface density profile analytically and find that it diverges towards the center of the drop as $\eta \propto r^{-1/2}$, where $r$ is the distance from the center. We estimate how this divergence is softened due to solute interactions at the final stage of drying. Our framework can easily be extended numerically or analytically to investigate novel stain morphologies left by drying drops of different shapes and evaporation profiles.