Drying of droplets of colloidal suspensions on rough substrates

In many technological applications, excess solvent must be removed from liquid droplets to deposit solutes onto substrates that are rough. We present a lubrication-theory-based model of the drying of droplets of colloidal suspensions on a substrate containing a topographical defect. The model consists of a system of one-dimensional evolution equations for droplet shape and depth-averaged colloidal particle concentration, along with a precursor film, disjoining pressure, and one-sided evaporation. Finite-difference solutions reveal that the droplet contact line can pin to a defect due to a balance between capillary-pressure gradients and disjoining-pressure gradients. The time-evolution of the droplet radius and contact angle exhibits the constant-radius and constant-contact-angle stages that have been observed in experiments. When colloidal particles are present and the defect is absent, the model predicts that particles will be deposited near the center of the droplet in a cone-like pattern. However, when a defect is present, particle deposition near the droplet edge in a coffee-ring pattern is predicted. These predictions are consistent with prior experiments, and illustrate the critical role contact-line pinning plays in controlling the dynamics of drying droplets.