Thin Films in Partial Wetting: Internal Selection of Contact-Line Dynamics

When a liquid touches a solid surface, it spreads to minimize the system’s energy. The classic thin-film model describes the spreading as an interplay between gravity, capillarity, and viscous forces, but it cannot see an end to this process as it does not account for the nonhydrodynamic liquid-solid interactions. While these interactions are important only close to the contact line, where the liquid, solid, and gas meet, they have macroscopic implications: in the partial-wetting regime, a liquid puddle ultimately stops spreading. We show that by incorporating these intermolecular interactions, the free energy of the system at equilibrium can be cast in a Cahn-Hilliard framework with a height-dependent interfacial tension. Using this free energy, we derive a mesoscopic thin-film model that describes the statics and dynamics of liquid spreading in the partial-wetting regime. The height dependence of the interfacial tension introduces a localized apparent slip in the contact-line region and leads to compactly supported spreading states. In our model, the contact-line dynamics emerge naturally as part of the solution and are therefore nonlocally coupled to the bulk flow. Surprisingly, we find that even in the gravity-dominated regime, the dynamic contact angle follows the Cox-Voinov law.

Figure 1

Schematic of the tangent construction on the bulk free energy, $f(h)$, leading to the coexistence of wet, $h=h_{*}$, and dry, $h=0$, states. In the absence of intermolecular forces, the bulk free energy does not reduce to the solid-gas interfacial energy as $h\to 0$ unless $\mathcal{S}=0$, which implies complete wetting [10].

Figure 2

Comparison of the mobility with and without slip [$\mathcal{M}(h)=h^3$ , $\mathcal{M}(h)=h^3+3b_s{h}^2$, $b_s=10d_0$ , $\mathcal{M}(h)=h^3+(3/2){\lambda }^2(d\kappa /dh)h^2$, $\lambda =10d_0$, ${\theta }_Y=\pi /12$ ]. While Navier slip is global, our proposed slip model is localized to the contact-line region, where it dominates the Navier slip, consistent with molecular simulations [28, 29].

Figure 3

The rate of spreading is influenced by the volume of liquid. For small volumes ($\mathrm{Bo}⪅1$, ), capillary forces are dominant and the drop takes the shape of a spherical cap while viscosity resists the spreading, leading to Tanner’s law $A\sim t^{1/5}$. For large volumes ($\mathrm{Bo}\approx 360$, ), gravity dominates, leading to a $t^{1/4}$ scaling; the liquid puddle takes the shape of a pancake at equilibrium. $A_f$ is the final equilibrium area and ${\theta }_Y=\pi /12$.

Figure 4

(a) The dynamic contact angle is defined at the inflection point of the drop profile (top panel), where its slope (bottom panel) reaches a plateau [$\tan \theta =(h_{*}/R_f)(dh/dr)$]. (b) The dynamic contact angle, ${\theta }_d$, follows the Cox-Voinov law ${\theta }_d^3-{\theta }_Y^3=9\mathrm{Ca}\ln (l_M/l_{\mu })$, but increases with the volume of the liquid. (c) The nonlocal influence of bulk flow can be conflated into the macroscopic length scale, $l_M$, leading to a collapse of the dynamic contact angle data for the different volumes onto a single curve (the solid line represents the Cox-Voinov law). The stars represent the classic experiments of Ref. [43] corresponding to the silicone oil-air interface in a capillary tube ($l_M/l_{\mu }\approx 1.25\times {10}^3$). The rescaled contact angle data from the model accounts for the fact that the microscopic length scale $l_{\mu }$ is magnified by ${10}^4$ in the simulations: $({\theta }_d^3-{\theta }_Y^3{)}_r={\theta }_d^3-{\theta }_Y^3+(9\ln {10}^4)\mathrm{Ca}$. (Inset) An approximate fit $l_M/l_{\mu }=4.8\times {10}^4\times [1-\exp (-0.75{\mathrm{Bo}}^{1/2})]$ is shown as the dashed line.