Ratchetlike Motion of a Shaken Drop

We study the motion of a drop lying on a plate simultaneously submitted to horizontal and vertical harmonic vibrations. The two driving vibrations are adjusted to the same frequency, and, according to their relative amplitude and phase difference $\Delta \Phi$, the drop experiences a controlled directed motion with a tunable velocity. We present a simple model putting in evidence the underlying mechanism leading to this ratchetlike motion of the drop. Our model includes the particular case $\Delta \Phi =\pi$ corresponding to the climbing of a drop on a vertically vibrated inclined substrate, as recently observed by Brunet et al. [Phys. Rev. Lett. 99, 144501 (2007)]. This study gives insights in the fundamental issue of wetting dynamics and offers new possibilities of controlled motion in droplet microfluidics applications.

Figure 1

(a) Experimental setup used to combine vertical and horizontal vibrations of the substrate. (b) Images taken from different vibration cycles at various oscillation phases. Here $f_v=f_h=70\mathrm{Hz}$, $\varphi =\pi /4$ $A_v=140\mu \mathrm{m}$, and $A_h=300\mu \mathrm{m}$. (c) Position of the triple line on the right versus time. The lines are a guide to the eyes to put in evidence the stops of the triple line.

Figure 2

Displacement of the droplet center of mass for two different frequency shifts $\Delta f=1$ and 2 Hz. Inset: Deducted drop velocity as a function of $\Delta \Phi$. The two data sets fall on the same curve.

Figure 3

(a) Droplet velocities as a function of the phase difference for various parallel vibration amplitudes $A_h$. In the inset, for a weak parallel vibration $A_h=130\mu \mathrm{m}$ the two maxima are clearly visible. (b) Velocity as a function of the $A_h$.

Figure 4

(a) Mechanical model. (b) Position of the box as a function of time. (c) Box velocity as a function of $\Delta \Phi$. (d) Box velocity as a function of $A_h$.