Droplet evaporation on inclined substrates

Drying of droplets on inclined substrates is relevant to applications such as spray coating, ink-jet printing, and crime-scene reconstruction. Recent experiments have shown that steeper substrate inclination can significantly slow down droplet evaporation due to faster droplet depinning. Motivated by these experiments, we develop a lubrication-theory-based model to study the effect of substrate inclination on the evaporation of two-dimensional pure-solvent and particle-laden droplets on smooth and rough inclined substrates. A system of partial differential equations describing the time evolution of the droplet thickness and the colloidal particle concentration is derived and then solved with a finite-difference method. The contact-line motion is described with a disjoining-pressure/precursor-film approach and the well-known one-sided model is used to describe solvent evaporation. Our results indicate that on smooth substrates steeper inclination speeds up evaporation due to larger deformation of the droplet interface, which leads to a smaller droplet thickness. On rough substrates, the effect of substrate inclination on evaporation depends on the Bond number (Bo). At lower Bo, steeper substrate inclination slows down evaporation due to faster droplet depinning, which leads to a larger droplet thickness. At higher Bo, the droplet does not pin due to stronger gravitational forces, and steeper substrate inclination speeds up evaporation, similar to smooth substrates. When colloidal particles are present, the resulting final particle deposition patterns are strongly dependent on the initial conditions (Bo, inclination angle, initial particle concentration, wettability) rather than being a function of only substrate inclination. Our model predictions qualitatively agree with previous experimental work and disentangle the effects of evaporation, substrate inclination, and surface roughness.