Fig. 1. Schematic of a sessile droplet drying on a substrate. Panels (a) and (b) are the top and the side views of the droplet, respectively. Relevant parameters are the radius of the contact line r, the radius of the spherical crown R, the contact angle θ, the height of the droplet h at position x.

Fig. 2. Evolution of the contact line r/r₀ of drying droplets for three values of initial contact angles, θ₀ = 0.2, 0.4, and 0.6. The time is in units of τ_re. The fitting tangent slopes k in the later evaporation processes confirm the Tanner’s law. For all calculations, θ_e = 0, k_cl = 0 and k_ev = 0.

Fig. 3. Droplet shape evolution for evaporation on frictionless substrate for three situations: (a) θ₀ < θ_e, (b) θ₀ = θ_e, and (c) θ₀ > θ_e. The corresponding evolution of the contact line r/r₀ is shown in panel (d), and the contact angle θ is shown in panel (e). For all calculations, θ_e = 0.8 and k_ev = 0.001.

Fig. 4. Evolution of (a) contact radius r/r₀ and (b) contact angle θ of a drying droplet for various evaporation rates k_ev. By increasing the evaporation rate, the evolution mode changes from CCA to a continuously decreasing one. For all cases, the initial contact angle θ₀ of the droplet is the same as its equilibrium contact angle θ_e, and k_cl is fixed as k_cl = 0.

Fig. 5. The comparison of the evolution of r calculated from the simple model and the full Onsager model. The results are in good agreement. The parameters are: θ₀ = θ_e = 0.4, k_ev = 1, k_cl = 0.

Fig. 6. Droplet shape evolution on frictional substrate. Panel (a) is for pinned contact line motion with k_cl = 1000. Panel (b) is the corresponding evolution of r/r₀ and θ. Panel (c) is the shape evolution for small k_cl = 25, and its corresponding evolution of r/r₀ and θ is shown in panel (d). The calculation parameters for panels (a) and (b) are θ₀ = 0.8, θ_e = 0.2, and k_ev = 0.01, while for panels (c) and (d), θ₀ = 1.2, θ_e = 0.6, and k_ev = 0.5.

Fig. 7. The comparison of θ evolution between the reduced model and the Onsager full model when the contact line is pinned, r˙=0. The calculation parameters are: θ₀ = 1.0, θ_e = 0, k_ev = 1, k_cl = 1000.