Strong electro-osmotic flows about dielectric surfaces of zero surface charge

We analyze electro-osmotic flow about a dielectric solid of zero surface charge, using the prototypic configurations of a spherical particle and an infinite circular cylinder. We assume that the ratio $\delta$ of Debye width to particle size is asymptotically small, and consider the flow engendered by the application of a uniform electric field; the control parameter is $E$—the voltage drop on the particle (normalized by the thermal scale) associated with this field. For moderate fields, $E=O\left(1\right)$, the induced $\zeta$ potential scales as the product of the applied-field magnitude and the Debye width; being small compared with the thermal voltage, its resolution requires addressing one higher asymptotic order than that resolved in the comparable analysis of electrophoresis of charged particles. For strong fields, $E=O\left({\delta }^{-1}\right)$, the $\zeta$ potential becomes comparable to the thermal voltage, depending nonlinearly on $\delta$ and $E$. We obtain a uniform approximation for the $\zeta$-potential distribution, valid for both moderate and strong fields; it holds even under intense fields, $E\gg {\delta }^{-1}$, where it scales as $log\left|E\right|$. The induced-flow magnitude therefore undergoes a transition from an $E^2$ dependence at moderate fields to an essentially linear variation with $\left|E\right|$ at intense fields. Remarkably, surface conduction is negligible as long as $E\ll {\delta }^{-2}$: the $\zeta$ potential, albeit induced, remains mild even under intense fields. Thus, unlike the related problem of induced-charge flow about a perfect conductor, the theoretical velocity predictions in the present problem may actually be experimentally realized.

Figure 1

Average velocity $\langle v\rangle$, calculated using (5.8) for $\alpha =0.1$, as a function of the applied field $E$. Also shown (dashed) are approximations (5.9) and (5.11), respectively corresponding to small and large $\alpha E$.

Figure 2

Average $\zeta$ potential $\langle \zeta \rangle$, calculated using (5.13), as a function of $\alpha E$. Also shown (dashed) are approximations (5.14) and (5.15), respectively corresponding to small and large $\alpha E$.