Moving contact line of a volatile fluid

Interfacial flows close to a moving contact line are inherently multiscale. The shape of the interface and the flow at meso- and macroscopic scales inherit an apparent interface slope and a regularization length, both named after Voinov, from the microscopic inner region. Here, we solve the inner problem associated with the contact line motion for a volatile fluid at equilibrium with its vapor. The evaporation or condensation flux is then controlled by the dependence of the saturation temperature on interface curvature—the so-called Kelvin effect. We derive the dependencies of the Voinov angle and of the Voinov length as functions of the parameters of the problem. We then identify the conditions under which the Kelvin effect is indeed the mechanism regularizing the contact line motion.

Figure 1

Schematic showing a liquid vapor interface $h\left(x\right)$—here, a spreading drop ($U>0$) on a cooled plate ($\Delta T<0$)—at different scales. The inner region, close to the moving contact line, is controlled by evaporation or condensation (top right). The slope changes from the Young angle ${\theta }_Y$ to the Voinov angle ${\theta }_V$ across a scale ${\ell }_V$ given by the Kelvin length ${\ell }_K$. In a mesoscopic range of scales, the shape of the interface results from the balance between viscous friction and the Laplace pressure gradient, resulting in a slope $h^{\prime }\left(x\right)$ which varies logarithmically in scale [Eq. (2)].

Figure 2

Solution of the equations for a receding contact line ($\delta =-0.02$), with an overheating $\epsilon =0.1$. The dotted line corresponds to the static case ($\delta =0$). The dashed line corresponds to the Voinov outer asymptotics (2). These solutions allow one to obtain the Voinov length ${\ell }_V$ and angle ${\theta }_V$.

Figure 3

(a) Ratio of the Young angle to the Voinov angle as a function of overheating parameter $\epsilon$, determined numerically (solid line). The dashed line is the analytical expansion (10). (b) Voinov length ${\ell }_V$ rescaled by the Kelvin length ${\ell }_K$ as a function of $\epsilon$.

Figure 4

Voinov length ${\ell }_V$ as a function of the ratio ${\ell }_{\beta }/{\ell }_K$ for $\epsilon =0$, as predicted by Eq. (13). Inset: Schematic of the kinetically limited regime. The desorption or adsorption takes place over a scale given by the mean free path $\overline{\ell }$.