Onset of Area-Dependent Dissipation in Droplet Spreading

We probe the viscous relaxation of structured liquid droplets in the partial wetting regime using a diblock copolymer system. The relaxation time of the droplets is measured after a step change in temperature as a function of three tunable parameters: droplet size, equilibrium contact angle, and the viscosity of the fluid. Contrary to what is typically observed, the late-stage relaxation time does not scale with the radius of the droplet—rather, relaxation scales with the radius squared. Thus, the energy dissipation depends on the contact area of the droplet, rather than the contact line.

Figure 1

(a) Schematic of a mostly disordered droplet, coexisting with an ordered wetting layer of the same liquid. (b) Interference optical microscopy images showing the relaxation of a droplet as it spreads. (c) Best fit spherical cap to the data points from the interference fringes of the droplet in (b) at $t=0$ (solid line) and at equilibrium as $t\to \infty$ (dashed line). The height scale is exaggerated as compared to the lateral scale for clarity. (d) Normalized relaxation of the contact angle of 15 different representative droplets with different $\mathrm{\Omega }$ and ${\theta }_e$, which undergo both advancing and receding motion. The solid line is the function $\exp (-t/\tau )$. The inset shows the relaxation of the contact angle of the droplet from (b) and (c) as a function of time with an exponential fit (black line).

Figure 2

The relaxation time $\tau$ as a function of ${\mathrm{\Omega }}^{2/3}$ for different values of ($T_f$, $h_e$) (with correspondingly different ${\theta }_e$) as indicated in the legend, for receding and advancing droplet motion. Solid lines are the best fit to a $\tau \propto {\mathrm{\Omega }}^{2/3}$ scaling with zero intercept.

Figure 3

The area-dependent relaxation prefactor $\kappa$ plotted as a function of ${\theta }_e$ (shown on a double logarithmic plot for clarity of the data). $\kappa$ is not a universal constant for the system, and depends on $T$ as well as ${\theta }_e$. The lines are fit to $\kappa \propto {\theta }_e$.

Figure 4

The normalized relaxation time as a function of the final contact angle for advancing (circles) and receding (squares) contact line motion. The lines are the best fit of Eq. (4) to the data, with ${\mathrm{\Lambda }}_a=22\pm 1\mu \mathrm{m}$ and ${\mathrm{\Lambda }}_r=16\pm 1\mu \mathrm{m}$.