Variational modeling and numerical simulations for evaporating thin droplets and coffee-ring effect.

Sessile liquid droplets on solid surfaces are ubiquitously found in nature and engineered applications. They exhibit many intriguing properties and phenomena because of the three phases, i.e. liquid, gas, and solid, interacting with each other. Many physical processes are involved in droplet dynamics, presenting a complex problem for fundamental understanding and practical applications. Among those processes participating in the dynamics of evaporating sessile droplets, two have been of continuous interest. The first is the moving contact line at which the evolving liquid-gas interface intersects the solid surface, and the second is the evaporation at liquid-gas interface. Coupled to the moving contact line on the substrate and the liquid flow in the droplet, the interfacial evaporation plays a key role in the evolution of evaporating sessile droplets. Based on Onsager's variational principle, we derive a continuum model for evaporating thin droplets. The variational approach ensures thermodynamical consistency in describing the coupling of multiple dissipative processes, including viscous momentum transport, contact line motion, evaporation, and vapor diffusion. A characteristic length scale is introduced by considering the competition between the liquid evaporation at liquid-gas interface and the vapor diffusion in gas space. Comparing this intrinsic length scale with that of the confinement geometry, a dimensionless parameter is identified that varies across two distinct dynamic regimes, namely the diffusion-limited regime and the transition-limited regime. Numerical results are presented for the evaporation flux and liquid flow, showing distinct characteristics in these two regimes. The coffee-ring effect is also numerically investigated for drying particle-laden droplets across different dynamic regimes with pinned and depinned contact lines.