Response function of a moving contact line

The hydrodynamics of a liquid-vapor interface in contact with a heterogeneous surface is largely impacted by the presence of defects at the smaller scales. Such defects introduce morphological disturbances on the contact line and ultimately determine the force exerted on the wedge of liquid in contact with the surface. From the mathematical point of view, defects introduce perturbation modes, whose space-time evolution is governed by the interfacial hydrodynamic equations of the contact line. In this paper we derive the response function of the contact line to such generic perturbations. The contact line response may be used to design simplified one-dimensional time-dependent models accounting for the complexity of interfacial flows coupled to nanoscale defects, yet offering a more tractable mathematical framework to explore contact line motion through a disordered energy landscape.

Figure 1

(a) The response function of the dynamic contact line is determined in the dip-coating geometry. A vertical plate is plunged $(U_p>0)$ or withdrawn $(U_p<0$, as in the figure) from a bath at the velocity ${\vec{U}}_p=U_p{\vec{e}}_x$. The contact line profile is denoted by $\xi$. The bath, far from the plate, is at equilibrium. Its asymptotic profile when approaching the plate resembles that of a static bath and would join the plate, if prolonged, at an angle ${\theta }_M$ called the macroscopic contact angle [see Eq. (12)]. (b) Relation between the apparent contact angle ${\theta }_M$ and the capillary number $\mathrm{Ca}=\eta U_p/\gamma$ determined using a slip length ${\ell }_s=2.5\times {10}^{-6}{\ell }_{\gamma }$ and a microscopic contact angle ${\theta }_0=0.5\mathrm{rad}$, corresponding to about $28.6^{\circ }$. Here ${\mathrm{Ca}}_{\mathrm{c}}$ is the threshold capillary number below which a liquid film is entrained on the plate and corresponds to ${\theta }_M=0$. The thin red line is the approximation by the Cox-Voinov formula (13).

Figure 2

(a) Schematic of the geometrical model relating the response function to the penetration length $L$. The length $L$ is set by the smallest of three lengths: (b) the wavelength $\lambda =2\pi /q$, (c) the dynamic length $\gamma /\eta \omega$, and (d) the capillary length ${\ell }_{\gamma }$.

Figure 3

Response function as a function of the rescaled wave number $q$, for different values of $\mathrm{\Omega }$: $\mathrm{\Omega }=0$ (solid line), $\mathrm{\Omega }=0.22$ (long dashed line), $\mathrm{\Omega }=1$ (dashed line), $\mathrm{\Omega }=4.5$ (dotted dashed line), $\mathrm{\Omega }=20$ (dotted line). The other parameters are fixed at $tan{\theta }_0=0.55,{\ell }_s=2.5\times {10}^{-6}{\ell }_{\gamma }$, and $\mathrm{Ca}={10}^{-5}$. The thin red lines are the predictions for the large $q$ asymptotics given by Eq. (32), in which a corrective factor 0.87 was applied in front of the imaginary part.

Figure 4

Response function as a function of the rescaled wave number $q$, in the limit of vanishing $\mathrm{\Omega }$, for different angles $tan{\theta }_0$: $tan{\theta }_0=0.1$ (solid line), $tan{\theta }_0=0.3$ (dashed line), $tan{\theta }_0=0.6$ (dotted dashed line), and $tan{\theta }_0=1.1$ (dotted line). The other parameters are fixed at ${\ell }_s=2.5\times {10}^{-6}{\ell }_{\gamma }$, and $\mathrm{Ca}={10}^{-5}$. The thin red lines are the predictions for the large $q$ asymptotics given by Eq. (32), in which a corrective factor 0.87 was applied in front of the imaginary part.

Figure 5

Response function as a function of the rescaled wave number $q$, in the limit of vanishing $\mathrm{\Omega }$, for different values of $\mathrm{Ca}$: $\mathrm{Ca}=-{10}^{-3}$ (dotted line), $\mathrm{Ca}=0$ (solid line), and $\mathrm{Ca}={10}^{-3}$ (dashed line). The other parameters are fixed at $tan{\theta }_0=0.55,{\ell }_s=2.5\times {10}^{-6}{\ell }_{\gamma }$. The thin red lines are the predictions for the large $q$ asymptotics given by Eq. (32), in which a corrective factor 0.87 was applied in front of the imaginary part.

Figure 6

Left: response function as a function of the rescaled frequency $\mathrm{\Omega }$, in the limit of vanishing $q$, for different values of $\mathrm{Ca}$: $\mathrm{Ca}=-{10}^{-3}$ (dotted line), $\mathrm{Ca}=0$ (solid line), and $\mathrm{Ca}={10}^{-3}$ (dashed line). The other parameters are fixed at $tan{\theta }_0=0.55,{\ell }_s=2.5\times {10}^{-6}{\ell }_{\gamma }$. The three curves almost collapse for the imaginary part of $\mathcal{C}$ (blue curves) but are split for its real part (green curves), at small $\mathrm{\Omega }$. The thin red lines are the predictions of the intermediate asymptotics limited by the dynamical length (small $q$ limit and intermediate range of $\mathrm{\Omega })$, as given by Eq. (33). Right: response function in the limit of vanishing $\mathrm{\Omega }$ and vanishing $q$, as a function of $\mathrm{Ca}$. The thin red lines are the predictions of the quasisteady asymptotics (small $q$ limit and small $\mathrm{\Omega })$, as given by Eq. (34).