A model for pattern deposition from an evaporating solution subject to contact angle hysteresis and finite solubility

We propose a model for the pattern deposition of the solute from an evaporating drop of a dilute solution on a horizontal substrate. In the model we take into account the three-phase contact angle hysteresis and the deposition of the solute whenever its concentration exceeds the solubility limit. The evaporating drop is governed by a film equation. We show that unless for a very small three-phase contact angle or a very rapid evaporation rate the film adopts a quasi-steady geometry, satisfying the Young-Laplace equation to leading order. The concentration profile is assumed to satisfy an advection diffusion equation subject to the standard Fick's law for the diffusive flux. We further use an integral boundary condition to describe the dynamics of the concentration in the vicinity of the three-phase contact line; we replace an exact geometric description of the vicinity of the contact line, which is usually assumed such that mathematical singularities are avoided, with general insights about the concentration and its flux. We use our model to explore the relationships between a variety of deposition patterns and the governing parameters, show that the model repeats previous findings, and suggest further insights.