Defects at the Nanoscale Impact Contact Line Motion at all Scales

The contact angle of a liquid drop moving on a real solid surface depends on the speed and direction of motion of the three-phase contact line. Many experiments have demonstrated that pinning on surface defects, thermal activation and viscous dissipation impact contact line dynamics, but so far, efforts have failed to disentangle the role of each of these dissipation channels. Here, we propose a unifying multiscale approach that provides a single quantitative framework. We use this approach to successfully account for the dynamics measured in a classic dip-coating experiment performed over an unprecedentedly wide range of velocity. We show that the full contact line dynamics up to the liquid film entrainment threshold can be parametrized by the size, amplitude and density of nanometer-scale defects. This leads us to reinterpret the contact angle hysteresis as a dynamical crossover rather than a depinning transition.

Figure 1

(a) Height $Z$ of the contact line as a function of the capillary number Ca, in the limit $\dot{Z}\to 0$ ($Z$ is rescaled by the capillary length ${\ell }_{\gamma }=\sqrt{\gamma /\rho g}$ where $\rho$ is the liquid density and $g$ the acceleration of gravity). Circles: experimental data. Solid line: theory. Inset: schematic of the dip-coating experiment in the frame of reference of the bath. (b) Schematic of the contact line deformation over defects in the frame of reference of the plate. (c) Schematized energy landscape along the reaction path. Left: realistic. Right: idealization.

Figure 2

Cosine of the free surface angle as a function of the capillary number Ca. Circles: experimental macroscopic angle $\cos {\theta }_M=(Z/{\ell }_{\gamma })\sqrt{1-0.25(Z/{\ell }_{\gamma }{)}^2}$. Squares: experimental microscopic angle $\cos {\theta }_{\mu }$. Solid lines: theory for the microscopic angle and for the macroscopic angle. The depinning thresholds at zero temperature are indicated by arrows. Inset: close-up of the advancing branch ($\mathrm{Ca}<0$).