Steric effects on ac electro-osmosis in dilute electrolytes

The current theory of alternating-current electro-osmosis (ACEO) is unable to explain the experimentally observed flow reversal of planar ACEO pumps at high frequency (above the peak, typically 10–100 kHz), low salt concentration $\left(1–1000\mu \text{M}\right)$, and moderate voltage (2–6 V), even taking into account Faradaic surface reactions, nonlinear double-layer capacitance, and bulk electrothermal flows. We attribute this failure to the breakdown of the classical Poisson-Boltzmann model of the diffuse double layer, which assumes a dilute solution of pointlike ions. In spite of low bulk salt concentration, the large voltage induced across the double layer leads to crowding of the ions and a related decrease in surface capacitance. Using several mean-field models for finite-sized ions, we show that steric effects generally lead to high-frequency flow reversal of ACEO pumps, similar to experiments. For quantitative agreement, however, an unrealistically large effective ion size (several nanometers) must be used, which we attribute to neglected correlation effects.

Figure 1

One period of an asymmetric planar array of microelectrodes in an ac electro-osmotic (ACEO) pump. This geometry has been studied experimentally by several groups [10, 12, 14, 18] and is the subject of the present theoretical study. The dimensions are $W1=4.2\mu \text{m}$, $W2=25.7\mu \text{m}$, $G1=4.5\mu \text{m}$, and $G2=15.6\mu \text{m}$, using the notation of Ref. [19].

Figure 2

(Color online) Pumping velocity vs frequency for the planar ACEO pump in Fig. 1 from experiments and simulations. Points: experimental data at 3 Vpp in 0.03 mM KCl from Ref. [18] (taken by J. P. Urbanski). Solid curve: linear Debye-Hückel-Stern model of the diffuse layer with a constant compact layer capacitance $C_s=\epsilon /\left(\delta {\lambda }_D\right)$, where $\delta =0.5$ is chosen to fit the experimental peak frequency. Dashed curve: nonlinear Gouy-Chapman-Stern (Poisson-Boltzmann) model of the diffuse layer also with $\delta =0.5$.

Figure 3

(Color online) (a) Schematic of the equilibrium distribution of counterions near a positively charged surface taking into account a minimum ion spacing $a$ for large applied voltages. As the voltage increases, the width of the condensed layer will increase causing a decrease the capacitance. (b) The voltage dependence of the capacitance of the diffuse layer for PB and MPB at different values of $\nu$; where $\nu =2a^3{c}_0$ is the bulk volume fraction of ions.

Figure 4

(Color online) Average slip velocity as a function of frequency for three different models at 2.5 V. The solid blue curve is for $\nu =0.01$ with the complete model, the black dash-dotted line assumes the linear Debye-Hückel relationship $q=-{\Psi }_D$, and the red dashed curve is for $\nu =0$ with a Stern layer capacitance of $\delta =0.1$. Only the model with steric effects predicts flow in the negative direction. Points $A$, $B$, and $C$ will be referred to in Fig. 6.

Figure 5

(Color online) Basic mechanism for flow in ACEO devices. Here we consider a symmetric array of electrodes under a suddenly applied voltage. In (a) we show electric field lines at the instance the field is turned on. The electric field points from the positive to the negative electrode. The location on the electrodes at $y=0$ is shown. In (b) we show the charge density over the electrodes at 1/10 the charging time $t_c$ at the charging time and at 10 times the charging time. In (c) we show streamlines of the electric field at the charging time. In (d) we show electric field lines at the charging time. The arrows denote the flow direction.

Figure 6

(Color online) Streamlines (a),(c),(e) and slip velocity (b),(d),(f) for three different cases. The first row is for $V=100$, $\nu =0.01$, and $\omega =0.7$. This frequency corresponds to the peak in the forward flow for the steric model. In (a) we show the streamlines. In (b) we show the time-averaged slip velocity (solid line) and the cumulative spatial integral of this velocity (the dashed curve). The second row, (c) and (d), is for $V=100$, $\nu =0.01$, and $\omega =3$. This frequency corresponds to the peak in the reversed flow. The third row, (e) and (f), is for $V=100$ and $\omega =3$ with the linear Debye-Hückel model ${\Psi }_D=-q$.

Figure 7

(Color online) Flow response as a function of voltage for various values of $\nu$, the steric parameter. We show values of $\nu =0.01$ (a), $\nu =0.001$ (b), $\nu =0.0001$ (c), and $\nu =0$ (d). The contours show constant levels of total flow rate. The heavy line shows the $U=0$ contour, and the region above this line denotes the region where flow reversal occurs.

Figure 8

(Color online) Flow as a function of frequency at different voltages. The voltages are $V=0.5$, 1, 1.5, 2.9, 3.6, 4.5, and 6 V rms. The voltages are selected to match Fig. 6 in Studer et al. [12]. The data are presented in dimensional form in order to compare the frequency response. The flow velocity is corrected for the hydraulic resistance of the experimental flow loop. In (a), using the Bikerman model, we used $a=4.4\text{nm}$. In (b), using the Carnahan-Starling model, we used $a=2.2\text{nm}$. The general shape of both frequency response curves has many of the experimentally observed features.