Countering capillary pressure with electroosmotic pressure at small scales

Electroosmosis, originating in the double-layer of a small liquid-filled pore (size $R$) and driven by a voltage $V$, is shown to be effective in pumping liquid against the capillary pressure of a larger liquid droplet (size $B$) provided the dimensionless parameter $R^2 \sigma /\varepsilon | \zeta | V B$ is small enough. Here $\sigma$ is surface tension of the droplet liquid/gas interface, $\varepsilon$ the liquid dielectric constant, and $\zeta$ the zeta potential of the solid/liquid pair. As droplet size diminishes, the voltage required to pump eletroosmotically scales as $V \sim R^2/B$. Accordingly, the voltage needed to pump against smaller higher-pressure droplets can actually {\em decrease} provided the pump pore-size scales down with droplet size appropriately. In this talk, we shall focus on the electroosmotic droplet-switch, two droplets coupled by an electroosmotic pump. For millimeter-size droplets and micron-size pores, 5 volts yields switching times under 5 seconds in experiment. The down-scaling of this voltage and switching-time are of interest.