Oscillating electroosmotic flow in channels and capillaries with modulated wall charge distribution

TL;DR

This work analyzes AC electroosmosis in electrolyte-filled channels with spatially modulated wall charge, predicting time-dependent vortices whose sense reverses with the drive period $2\pi/\omega$. It provides analytical solutions for rectangular channels and cylindrical capillaries, revealing unidirectional transport in certain orientations and a nonzero longitudinal advective current that produces $I$-$V$ hysteresis with a defined memory time $\tau_M$ related to $\nu$ and $\kappa^2$ and a memconductance that can diverge or become negative as $V\to0$. Memory effects arise from the history-dependent advective flow, linking fluid dynamics to memristive-like conductance and suggesting potential for memory-enabled signal processing in ionically conducting systems. Overall, the results offer a reconfigurable route to control mixing, solute transport, pumping, and energy conversion in microfluidic devices, with connections to iontronic memory concepts and toroidal vortex structures in capillaries.

Abstract

Electrolyte-filled channels with modulated wall charge distribution subjected to an applied DC electric field, form time-independent vortices whose sense of circulation is determined by the field direction [Physical Review Letters $ \mathbf{75}, 755, (1995)$]. In this paper we show that an electrolyte in a channel or cylindrical capillary subjected to an external \emph{alternating} (AC) electric field gives rise to various laminar flow structures, including vortices whose sense of circulation changes with the period of oscillation of the applied AC field. The introduction of a period of oscillation lifts certain degeneracies associated with its time-independent counterpart. Although, in general, the mass flux vanishes, the charge flux is nonzero. The flow is accompanied by a longitudinal (oscillating) advective current that displays hysteresis accompanied by a diverging and negative self-similar conductance that depends on the applied voltage [Nano Letters $\mathbf{10}, 2674, (2010)$]. We show that this behavior can be interpreted with respect to a ``memory retention time'', that depends on frequency, viscosity and the Debye length and could thus form the impetus for investigating control protocols of signal carriers.

Figures (12)

• Figure 1: AC electroosmosis with charge-modulated walls in a rectangular channel: A uniform alternating electric field $\mathbf{E} \sim e^{-i\omega t}\hat{\mathbf{x}}$ of frequency $\omega$ exerts a body force on a $1:1$ electrolyte in a rectangular channel of width $2h$ with surface charge $\sigma^\pm = \sigma_0 \cos q x$ on the upper and lower insulating walls, respectively. This gives rise to a system of vortices whose sense of rotation changes periodically in time according to the external electric field, cf. Fig. \ref{['psi12']}.
• Figure 2: Oscillating vortex structure in a channel of width $h=1$, induced by a uniform alternating electric field $\mathbf{E} = E_\parallel e^{-i\omega t}\hat{\mathbf{x}}$ of frequency $\omega$ that is parallel to the wall charge variation wavevector $q \hat{\mathbf{x}}$, cf. Fig. \ref{['channelAC1']} for the coordinate system. Left: dimensionless instantaneous streamlines from (\ref{['psiphi1']}) determined for symmetric surface charges $\sigma^\pm = \sigma_0 \cos q x$ along the channel walls. Right: dimensionless instantaneous streamlines from (\ref{['psiphi2']}), determined for antisymmetric surface charges $\sigma^\pm = \pm \sigma_0 \cos q x$. Compare with the upper and lower row of Ajdari1995. It is thus clear that the pattern displayed on the left smoothly deforms and becomes the one to the right by continuously varying the relative phase of the charge distribution between the upper and lower channel walls (see discussion in the main text).
• Figure 3: AC electroosmosis in a cylindrical capillary with charge-modulated walls: A uniform alternating electric field $\mathbf{E} = Ee^{-i\omega t}\hat{\mathbf{z}}$ of frequency $\omega$ exerts a body force on a $1:1$ electrolyte in a cylindrical capillary of radius $a$ with inhomogeneous surface charges, $\sigma = \sigma_0 \cos q z$ on its insulating wall. This gives rise to a system of oscillating tori (whose cross-section are the displayed vortices) whose sense of rotation changes according to the external field's, cf. Fig. \ref{['psi12cyl']} for a calculated cross-section. Compare with figures \ref{['channelAC1']} and \ref{['psi12']} of the channel case.
• Figure 4: Oscillating toroidal vortex structure in a capillary of radius $r=1$, induced by a uniform alternating electric field $\mathbf{E} = Ee^{-i\omega t}\hat{\mathbf{z}}$ of frequency $\omega$ that is parallel to the wall charge variation wavevector $q \hat{\mathbf{z}}$ (parallel to the cylinder axis), cf. Fig. \ref{['cylinder1ac']} for the coordinate system. Here we display an $r-z$ slice of the dimensionless instantaneous streamlines $\psi(r,z,t) = \psi(r) e^{-i\omega t}\cos qz$ with $\psi(r)$ from (\ref{['psi4r']}) and (\ref{['AB']}) determined for charges $\sigma= \sigma_0 \cos q z$ on the capillary wall located at $r=1$.
• Figure 5: AC electroosmosis in a rectangular channel with applied field perpendicular to wall charge variation. Top: An alternating electric field $\mathbf{E} \sim e^{-i\omega t}\hat{\mathbf{y}}$ of frequency $\omega$ exerts a body force on a $1:1$ electrolyte in a rectangular channel of width $2h$ with symmetric inhomogeneous wall charges, $\sigma^\pm = \sigma_0 \cos q x$ on the upper and lower insulating walls. Over each tile, the charge distribution gives rise to a non-vanishing velocity averaged over the channel width (\ref{['v1']}). Bottom: Oscillating velocity profile in a channel of width $2h$, induced by a uniform alternating electric field $\mathbf{E} = E_\perp e^{-i\omega t}\hat{\mathbf{y}}$ of frequency $\omega$ that is perpendicular to the wall charge variation for the coordinate system. Snapshot of the velocity profile $v(x,z,t)\hat{\mathbf{y}}$ from (\ref{['v1']}) at time $t=1$ when the surface charges at the chanel walls $z=\pm h$ are symmetric $\sigma^\pm = \sigma_0 \cos q x$ (left). Snapshot of the velocity profile $v(x,z,t)\hat{\mathbf{y}}$ (by exchanging the cosines with sines in (\ref{['v1']}), and vice-versa) at time $t=1$ when the surface charges at the chanel walls $z=\pm h$ are antisymmetric $\sigma^\pm =\pm \sigma_0 \cos q x$ (right).