Asymmetric flows over symmetric surfaces: capacitive coupling in induced-charge electro-osmosis

We report curious asymmetric induced-charge electro-osmotic (ICEO) flows over a symmetric, planar gate electrode under applied ac electric fields, whereas symmetric, counter-rotating rolls are expected. Furthermore, the asymmetric component of the flow is consistently directed towards the grounded electrode. We propose that capacitive coupling of the gate electrode to the microscope stage—a comparatively large equipotential surface that acts effectively as a ground—is responsible for this symmetry breaking. This stray capacitance drives the formation of a double layer whose zeta potential is proportional to the potential drop from the electrolyte directly above the gate electrode to the external stage. Therefore, the charge in this ‘stray’ double layer varies in phase with the driving field, resulting in a rectified, steady flow as with standard ICEO. We experimentally vary the stray capacitance, the electric potential of the stage and the location of the gate electrode, and find that the effect on the stray flow is qualitatively consistent with the predictions of the proposed mechanism. In the process, we demonstrate that capacitive coupling offers an additional means of manipulating fluid flow over a polarizable surface.


Introduction
Recent years have seen intense interest in microfluidic systems [1]- [3], which, in turn, has spurred a resurgence in electrokinetics for micron-scale flow manipulation. In particular, electroosmotic flows occur when the thin (typically ∼1-10 nm) ionic clouds that screen charged surfaces are forced into motion by an applied electric field E. The surrounding fluid is dragged along with the ions, establishing an electro-osmotic 'slip' velocity given by Smoluchowski's classic formula where and η are the permittivity and viscosity of the electrolyte solution, and ζ is the electrostatic potential drop across the diffuse part of the ionic screening cloud [4,5]. Recent interest has been especially keen in the development of nonlinear electrokinetic phenomena, such as ac electrokinetics (ACEK) [6,7], induced charge electrokinetics (ICEK) [8,9], nonlinear electrokinetic instabilities [10]- [12], and ACEK micropumps [13]- [15]. The nonlinearity of these phenomena arises when the (applied) electric field induces an ionic charge cloud around a polarizable surface, then subsequently drives the induced charge into motion. Under ac applied fields at sufficiently low frequencies ω, the electric field and induced zeta potential ζ i vary in phase, giving a nonzero, rectified steady flow. By contrast, the standard equilibrium double layer is presumed to be static in time, so that the standard electro-osmotic flow that results from an oscillatory applied field simply time averages to zero. In particular, if one assumes that the induced double layer (ζ i sin ωt) and equilibrium double layer (ζ eq ) simply superpose, which is reasonable at least in the limit |ζ i + ζ eq | k B T /e, then u s = − η < (ζ i sin(ωt) + ζ eq )E sin(ωt) >= − In other words, only the induced double layer contributes to steady flow. Most ICEK studies [8,9,16,17], as well as earlier work involving metal colloids (e.g. [18,19]) have involved cylinders and spheres. Instead, we use a planar system containing the channels is bonded to the glass slide, with the driving electrodes in each end of the channel. The gate electrode is connected to the metal pads, whose area is given by A P = 2s 2 . (b) Channel cross-section. The coordinate x originates at the center of the channel, and the gate electrode is centered at x G . The length of the channel is 2L and the width of the gate electrode is 2w. The gate electrode and pads are separated from the microscope stage by the glass slide of thickness d. The powered electrode is taken to be on the left, and the grounded electrode on the right. (figure 1) that allows the induced-charge electro-osmotic (ICEO) slip velocity to be measured directly [20]. A thin, metal 'gate' electrode is deposited on the glass surface at the bottom of a microfluidic channel. The electric field in the channel is applied through two driving electrodes: the 'powered' electrode carries the applied voltage and the 'grounded' electrode is held at electrical ground. Figure 2 illustrates the basic physics behind ICEO. An electric field that is applied along the channel polarizes the gate electrode. Ions in solution are driven along the electric field 4 lines, with positive ions driven towards one-half of the gate electrode and negative ions driven along the other half. Under 'ideally polarizable' conditions, no current passes through the electrolyte/metal surface, and instead the ions that arrive form an induced double layer, whose zeta potential varies with position along the gate electrode. Since the gate electrode-a floating conductor-is electrically isolated, its total charge is fixed. The induced zeta potential, which is the only component that drives a nonzero flow in an ac field, must therefore contain zero net charge and the spatially averaged flow over the gate electrode, should vanish. Given that the electrode and the solution are symmetric, and that an ac field is applied, one should expect quite generally that the time-averaged ICEO flow should be symmetric, with slip velocities directed towards the center of the electrode, driving two rolls of fluid in the bulk above.
Herein, we investigate a curious observation in the planar ICEO system. During experiments, we frequently observe a time-averaged asymmetric flow over the gate electrode, giving a net flow over the gate electrode in a particular direction. This stands at odds with the symmetric flow one would naturally expect from this symmetric system. Moreover, the net drift velocity always appears to be directed towards the grounded electrode: switching the leads (i.e. switching which electrode is powered, and which is grounded) reverses the direction of the flow asymmetry. Similar asymmetries have been observed in other ICEO experiments [20], and in ac electrohydrodynamic experiments [21].
We propose that the mechanism responsible for these asymmetric flows involves a capacitive coupling between the gate electrode and the 'external' experimental apparatus. For example, the experiments described here are performed atop a metal microscope stage and objective, which behave effectively as an electrical ground. In our original description, charge flows left-right only; in the full system, however, charge can flow along the electrode, from the portion of the gate that contacts the electrolyte (and which thus forms a double layer) to the portion whose capacitive coupling with the outside world is strongest (figure 3(a)). In the above discussions, only the potential gradient along the channel has been considered, and therefore the ICEO slip velocity is assumed to be independent of the absolute potential above the gate electrode. Given that the apparatus is an effective ground, the potential must drop from its value in the bulk electrolyte φ B to zero at the microscope stage. In response to this voltage drop, the solution forms a uniform 'stray' double layer over the gate electrode. Because this stray double layer is induced by the field, it varies in phase at sufficiently low frequencies, and it also gives rise to a nonzero, time-averaged flow just like ICEO ( figure 3(b)). When all double layers are accounted for, the result is a uniform flow superposed on a symmetric ICEO flow (figure 3(c) and (d)), which breaks the expected symmetry. In particular, as detailed below, one expects the stray flow to be directed towards the grounded electrode quite generally, as observed in experiments.
The effect we report here involves a modulation of the ζ -potential above the gate electrode via capacitive coupling with the external apparatus (here passively, later through actively controlling the potential of the stage). An analogous 'field-effect' flow control via external modulation of ζ has been demonstrated in the so-called flow-FET, both dc [22]- [25] and ac [26]. Below we discuss similarities and differences between these two systems.
In what follows, we develop a simple, predictive theory for the effect of stray capacitance upon ICEO flows, and the asymmetric flows that one should expect. The velocity of the stray flow is shown to depend upon the magnitude of the stray capacitance and the potential drop between the electrolyte directly above the gate and the microscope stage. We then test our theory by experimentally varying the following three quantities and measuring the resulting asymmetric flow: (i) We vary the electrostatic potential of the electrolyte (but not its gradient) above the gate by changing the relative position of the gate electrode between the powered and driving electrodes. (ii) We control the geometric size of the stray capacitor-and thus its capacitance-by adding metal pads to the gate electrode. (iii) We directly vary the potential of the microscope stage itself, demonstrating that the asymmetry can be enhanced, reversed or eliminated with an electrified stage. In particular, setting the potential of the microscope stage equal to the potential in the bulk solution over the gate electrode eliminates the asymmetric flow completely, regardless of the magnitude of the stray capacitance. There is a sizeable quantitative disparity between the predicted and measured stray flow velocities; however, such discrepancies are ubiquitous in ICEO experiments [20]. By normalizing our measured ICEO velocities relative to a reference measurement, however, we scale out any constant discrepancy, and find the relative flows show strong quantitative agreement with the theory.

Theory
ICEK phenomena result when a charge cloud is induced on a polarizable surface by an applied electric field, and is subsequently driven by that same field [8,9]. In the discussion that follows, all ICEO equations and diagrams will be pertinent to the planar geometry, shown in figure 1, that was used in experiments. The channel, of length 2L, is filled with an electrolyte solution.
The coordinate x describes the position in the channel, where the origin is chosen to be the midpoint of the channel. The gate electrode is a thin layer of metal of width 2w attached to the floor of the channel, whose center is in general located at x G (in the figure, x G = 0). The theory is developed with the convention that the powered electrode is on the left, but the section concludes with remarks on the effect of switching the powered and grounded electrodes. When a voltage is applied across the channel (figure 2), the resulting electric field forces electrons to migrate in the gate electrode, which causes the external field lines to perpendicularly intersect the surface of the gate electrode. While this is the steady state solution in vacuum, in an electrolyte these field lines drive positive and negative ions toward the gate electrode surface, thereby charging the induced double layer. Steady state is reached when enough ions have been driven into the double layer to completely screen the gate electrode from the applied electric field. At this point, only the electric field along the channel (parallel to the electrode surface) remains in the bulk solution.
This simplified picture describes the response of the double layer to a dc field. A very important characteristic of ICEO is that it generates a steady flow even in an ac field. Since the electric field induces the double layer charge, the sign of the charge varies in phase with the direction of the electric field. There is thus a nonzero net force, (which depends on the product of the two), which drives a steady flow. However, it takes time for the double layer to form. For conducting spheres and cylinders, this charging time τ c is given exactly by where D is the ionic diffusivity, and λ D is the Debye length of the double layer, where w is the permittivity of the water, k B is Boltzmann's constant, T is the temperature, n 0 is the concentration (of a symmetric electrolyte), and e is the electron charge. For a planar gate electrode, there is a spectrum of charging modes, so that equation (4) represents a characteristic charging time of the system. When the frequency of the applied potential is much smaller than the charging frequency 1/τ c , the double layer evolves quasisteadily: the bulk electric field and the double layer depend on the instantaneous value of the applied voltage and not on its prior evolution.
The voltage applied to the powered electrode is given by so the electric field in the channel is approximately uniform and is given by 7 so the potential in the bulk of the solution drops linearly from the powered electrode to the grounded electrode, and is given explicitly by In the quasi-steady limit, where the potential of the gate electrode φ G (t) can only be a function of time, since a conductor is a spatial equipotential. To determine φ G (t), we use a global charge constraint: the fieldinduced double layer cannot change the total charge in the double layer, only its distribution. Therefore, if we integrate over the entire gate electrode, we require 1 2w x G +w where ζ eq is the equilibrium zeta potential, which is present even in the absence of an applied electric field. Applying this condition to equation (9) and solving for φ G (t), we find The zeta potential over the gate electrode can now be written as The corresponding time-averaged flow, given by the Helmholz-Smoluchowski slip velocity, is Note that ζ eq does not affect the flow, since it does not vary in phase with the field. The slip velocity depends on the difference x − x G : no matter where the gate electrode is located, the ICEO slip velocity will look the same. In addition, the flow is an odd function of x − x G , which means that there should be a perfectly symmetric flow converging on the middle of the gate electrode. Finally, the flow is independent of the bulk potential directly above the gate electrode-and therefore, one would expect the ICEO flow not to depend on which electrode is powered and which is grounded. The above picture, however, takes into account only the potential gradient along the channel. We have neglected a rather large conductor that has close proximity to our system: the metallic microscope stage. When we impose the boundary condition that the potential must vary from φ B (x G , t) in the channel to φ S (t), the potential of the stage, we immediately break the symmetry of the system. The stage is typically grounded, and we now have to consider the effects of this vertical (i.e. perpendicular to the surface of the gate electrode) potential gradient (figure 3). The gate electrode interacts capacitively with the stage: when the potential in the bulk φ B (x G , t) is positive (figure 3(a)), the bottom surface of the gate electrode becomes negatively charged. Because the area of the gate electrode is so small, one would expect the capacitive interaction with the stage to be inconsequential, and not yield observable consequences. However, the gate electrode often extends past the walls of the channel (figure 1(a)), which increases the capacitive coupling to the stage, and thereby causes a significant amount of charge to migrate to the bottom surface of the gate electrode. Because the global charge balance still applies to the gate electrode, this vertical charge migration will cause negative charges to accumulate at the top surface of the gate that is in contact with the electrolyte. In response, a positively charged 'stray' double layer forms in the fluid above the gate, which is then driven towards the grounded driving electrode. When the electric field switches direction, the stray double layer becomes negatively charged, so the fluid continues to flow towards the grounded driving electrode ( figure 3(b)).
To determine the amount of charge in the stray double layer, we model the potential drop from the bulk solution to the stage as taking place over two capacitors in series. The first capacitor is the stray double layer itself, which consists of a layer of ionic charge and the top surface of the gate electrode. The capacitance of the double layer, in the Debye-Huckel (ζ 1) limit, is where A G is the area of the gate electrode (in contact with the electrolyte), and w is the electric permittivity of the electrolyte. (In the experiments, λ D ∼ 30 nm and A G = 1.5 × 10 4 µm 2 .) The voltage drop across the stray double layer, from the bulk to the gate electrode, is given by −ζ S by definition. The second capacitor is modeled as a parallel plate capacitor between the bottom surface of the gate electrode (including any area that extends beyond the channel walls) and the microscope stage. The glass slide occupies the space between the two plates. Thus, the capacitance of this stray capacitor is where A P is the area of the metal pads (plus the area of the gate, which is a negligible correction), g = 4.6 is the electric permittivity of glass (pyrex, Corning), and d = 1 mm is the thickness of the glass slide. The potential drop across both capacitors is the difference between the bulk potential directly above the gate electrode φ B (x G , t) and the potential of the stage φ S . Although typically the stage is grounded, in the experiments we allow it to take on any voltage with the same frequency and phase as the driving potential, so Imposing the condition that the total charge on both of the capacitors in series is equal yields The electric field drives this double layer just like any other, generating a 'stray' fluid slip velocity given by the Helmholz-Smoluchowski relation Only the time average of the flow is experimentally measurable, and it is given by In what follows, we describe experiments that test equation (19) in three ways: controlling V S , changing C S (by varying the area of the metal pads), and moving x G . Importantly, the standard ICEO characteristics of the system are theoretically unaffected by these parameters. 9 Finally, we have assumed in the above derivation that the powered electrode was on the left, and the grounded electrode was on the right. Switching the leads to the powered and grounded electrodes reverses the direction of the stray flow, although this is not obvious from equation (19). To make this clear, let us choose x G = 0 for simplicity, so that φ B (x G , t) = φ D (t)/2 remains unchanged by switching the leads. In the experiments, one of the electrodes is powered and the other is grounded, but it is instructive to consider the left electrode at a potential φ L (t) = V L sin(ωt) (20) and the right electrode at a potential In this case, the electric field is and the potential above the gate is The stray slip velocity is then given by Equation (24) shows that switching the powered and grounded electrode will reverse the direction of the asymmetry; if the stage is grounded (V S = 0), then the stray flow will always be directed towards the grounded electrode (see video 1, available at stacks.iop.org/NJP/11/075030/mmedia, where we do this). If x G = 0, then the stray flow still switches directions upon flipping the leads, but the magnitude also changes since x G effectively becomes −x G , so φ B (x G , t) takes on a different value (see video 2).

Device fabrication
The gate electrode is patterned onto the glass slide using standard soft lithography procedures. The glass slide (Corning 2947-75 × 50) is first cleaned in piranha solution (3 : 1 H 2 SO 4 : H 2 O 2 ), followed by a dehydration bake at 200 • C for 5 min. The slide is then first spin coated with hexamethyldisilazane (HMDS) (Microchem), followed by AZ5214 photoresist (Microchem), both at 4000 rpm for 30 s. This is followed by a soft bake at 95 • C for 1 min. The slide is exposed to UV at 7.5 mW cm −2 for 10 s, under a high-resolution (20 000 dpi) transparency mask (CAD/Art Services, Inc.), after which it must be reverse baked at 110 • C for 1 min. Finally, the slide is flood exposed for 1 min, then developed in MF-701 developer for 45 s with agitation. A 5 nm layer of titanium was deposited by e-beam assisted Joule evaporation at 0.5 Å s −1 , followed by a 20 nm layer of gold at 1 Å s −1 . The resist was removed by lift-off in acetone in an ultrasonic bath. The slide was stored in a container (Fluoroware) for not more than two days before running experiments, in order to prevent significant amounts of contaminant from building up on the gate electrode over time.
The channels are fabricated by casting PDMS against a master mold in a procedure very similar to that described in [27]. The mold starts as a 3" silicon wafer with a thick oxide (∼1 µm) grown on it. The channels are patterned onto the wafer with the same lithography procedure described above (except only a 7 s exposure is needed), up to and including the develop step. The wafer is then dipped in HF for 3 min, followed by an acetone/isopropanol lift-off. This is followed by an 80 min Bosch etch (∼80 µm) and another 3 min HF rinse. The mold is finished after an 8 min piranha etch.
The PDMS (Motive RTV615) is poured in the ratio 10A:1B, and placed in a planetary mixer (Thinky ARE-250) for 1 min at 2000 rpm, followed by a 2 min defoaming at 2200 rpm. The mold is exposed to tridecafluoro-1,1,2,2-tetrahydrooctyltrichlorosilane (Gelest) for 30 min under a vacuum desiccator to make the surface hydrophobic (this step is only necessary the first time a mold is used). The mold is placed in a Petri dish covered in aluminum foil. The PDMS is then poured over the mold, placed under vacuum for 1 h to remove bubbles, then baked at 80 • C for 1 h. The PDMS is then cut to size, and access holes to the channels are punched using gauge 26 luer stub adapters (McMasterCarr). The surfaces are cleaned with scotch tape.
To bond PDMS to glass, the PDMS is placed in a photo-ozone reactor for 5 min in order to oxidize the surface. Also, the glass slide is cleaned with acetone followed by isopropanol in an ultrasonic bath for 2 min each, after which it is placed under oxygen plasma for 30 s. The PDMS is aligned over and pressed onto the glass slide, then baked at 120 • C for 20 min. Hollow stainless steel pins (New England Small Tube) are inserted into the access holes, and 15 cm segments of microline tubing (Silastic) are connected to the pins. All experiments were run immediately following the fabrication of the device.

Data collection
The device is first taped securely onto the stage of an inverted epifluorescence microscope (Nikon Eclipe TE2000-U). A 100 µM KCl solution is prepared from Milli-Q ® water (Millipore), and 0.02%wt. 500 nm diameter fluorescent beads (Bangs Laboratories FS03F) are added for flow visualization. Electrolyte is introduced into the channel by hand using a luer-lock syringe (BD) attached to the microline tubing. Once the electrolyte fills the channel, the two pieces of microtubing attached to each of the pins are submerged in a small beaker filled with the electrolyte. While submerged, the microtubing is carefully removed from the syringe using tweezers, preventing stray pressure gradients and air bubbles from entering the tubing, thereby halting any pressure driven flow. The tracer particles are illuminated with light from a mercury lamp (Nikon C-SHG1) that passes through a B-2E/C dichroic fluorescence filter. A highly sensitive, backlit EMCCD fluorescence video camera (Andor iXon, Model DV885KCS-VP) is used to record the image, and all image capturing was handled with the accompanying Andor Solis software.
To generate the electric field in the channel, a function generator (Agilent 3320A) was run through a 100× voltage amplifier (Trek PZD350 M/S), and the leads were connected to the driving electrodes. To control the stage voltage V S , the driving potential was additionally applied to nineteen resistive voltage dividers (each consisting of a 4.7 k resistor and a 10 k potentiometer). The dividers could be quickly switched between using a manual rotary switch. The stage and objective lens casings were powered from these voltage dividers, but caution is needed to make sure the microscope is not connected to electrical ground. Additionally, the stage was electrically isolated from the objective casing by placing scotch tape over the metal extensions on which the stage rests. In the majority of experiments, when x G = 0, the one-half voltage divider was connected to the objective lens in order to isolate the stage as the only contributor to the stray capacitance. A sequence of simple steps was followed to record a video of the fluid flow. First, the optical shutter was opened (the shutter was closed between videos to prevent photobleaching of tracer beads). The electric field was then turned on, and a video was immediately recorded. All videos were captured in frame transfer mode for 130 frames, with an exposure time of 0.03441 s (26.199 frames per second), the baseline clamp on, and no gain. The electric field was turned off and the shutter was closed immediately following video capture.

Velocity measurements
The velocity of the fluid flow was determined by microparticle image velocimetry (µPIV), which uses a correlation algorithm to analyze the motion of the fluorescent tracer particles in two successive frames of a video, and thereby deduce a two-dimensional (2D) velocity field [28,29]. The algorithm divides the images into 'interrogation regions' and determines a velocity vector for each such region. In our case, the frame was 512 × 256 pixels, and the interrogation regions were 128 × 32 with 75% overlap. The µPIV algorithm ensemble averages over the 129 frame pairs of each video to produce the final vector field (figure 4).
The slip velocity at a given x-position of the gate electrode is calculated by averaging over the x-components of velocity at each position of y. The error bars in figure 5(a) reflect the standard deviation of this data set. The y-component of velocity, in the plane of the gate electrode and perpendicular to the electric field, can be attributed to the Brownian motion of the tracer particles between video frames and is neglected. Because the ICEO flow creates two counter-rotating rolls of fluid above the gate electrode, there is a nontrivial velocity component in the z-direction, out of the plane of the gate electrode. This velocity component is greatest at the location where the ICEO flow converges, which, for a symmetric flow, is near the middle of The error in the x-component of the velocity due to Brownian motion of the tracer particles can be estimated by calculating the root-mean-square displacement of a tracer particle divided by the time between consecutive frames [29]. Thus, the absolute error in velocity associated with Brownian motion of a single tracer particle is estimated by where it is assumed that the diffusion coefficient of the tracer particle is given by the Stokes-Einstein relation, where k B is the Boltzmann constant, T is the temperature of the electrolyte (assumed 300 K), η is the dynamic viscosity of the electrolyte (assumed to be that of pure water, 0.89 cP [30]), d is the diameter of the tracer particles (500 nm), and t is the time between successive frames (0.038 s). Since the Brownian displacement is randomly distributed over all directions, it can be reduced by averaging over many tracer particle images. The Brownian error then goes like ε B / √ N , where N is the number of particle images in the average. In the experiments, the velocity data are ensemble averaged over 129 frame pairs, and there are at least 10 tracer particles in each interrogation region, so N = 1290. The error in the x-component of the velocity due to the Brownian motion of the tracer particles is estimated to be 0.2 µm s −1 .
Additional sources of uncertainty include the precise location of the focal plane, and the 4.3 µm depth of focus of the optical system (Objective, Nikon Plan Fluor 20×, NA = 0.45; CCD, 8 × 8 µm pixel size). Using bright-field illumination, we initially focus on the edge of the electrode, locating the focal plane to within ±4 µm of the electrode surface. Furthermore, 13 particles that are out of focus contribute to the PIV correlation via a weight (z) that is sharply peaked at the focal plane, and falls to 1% over a distance z corr , which for our experiment is approximately 4.5 µm [20,31]. In order to ensure a sufficient number of tracer particles appear within the focal plane, we use the fine focus to raise the focus 4 µm above the electrode. Captured images thus represent a focal volume approximately 4 µm thick, centered 4 ± 4 µm above the electrode, and velocities measured with µPIV represent weighted averages over this volume. A slip velocity u s (x), over an electrode of half-width L, drives counterrotating vortices whose vertical scale is (roughly) set by L. Velocity gradients thus scale like ∂u x /∂z ∼ u s (x)/L, so that the velocity a distance z above the electrode is approximately given by u x (z)/u s (x) ∼ O(z/L). Significantly, because z L (i.e. the distance between the focal plane and the electrode is small relative to the gradient length scale L), we argue that these uncertainties affect the magnitude of the measured velocity profile via a multiplicative constant, but not the features we report here. (Note variations across the correlation depth are approximately linear, and since z corr L, z corr does not appreciably affect the average.) That is, a linear slip velocity profile at the electrode will be measured by µPIV as a linear velocity profile above the electrode. Furthermore, the location of the stagnation point (where u x = 0) should be correctly recovered with the µPIV algorithm.
The gate electrode on the photolithography mask is 50 µm wide, yet the photomask resolution results in an electrode whose width in practice ranges from 41 to 46 µm. The velocity data are typically only reliable at least one interrogation region width (32 pixels ∼11.44 µm) from the edge of the gate electrode; therefore, velocity data are only presented for the central 23 µm of the gate. Given that our focus is z ∼ 4 ± 4 µm above the electrode, and L ≈ 20 µm, giving measured velocities that are ≈20-40% lower than the true 'slip' velocity u s (x). Our goal in this work is not quantitative agreement between theory and experiment. Rather, we seek to measure how the stray velocity due to capacitive coupling with the microscope stage depends upon V S , C S and x G . Our central conclusions rest upon comparisons of measured velocities relative to each other, which effectively normalizes away any systematic error that occurs in the absolute velocity measurement.

Effect of stage potential
The stray flow can be robustly controlled by varying the potential of the microscope stage. Figure 5(a) shows the fluid velocity (due to both ICEO and stray effects) as a function of position along the gate electrode, for various values of V S . The device used in these experiments had pad area A P = 10.0352 mm 2 . It is convenient to quote the pad area in terms of its direct effect on ζ S , and for that purpose we refer to a scaled pad areaÃ P ≡ (1 + C DL /C S ) −1 , here 0.001 15. The powered electrode is driven with a signal amplitude of 50 V, resulting in an electric field in the channel of 50 V cm −1 . The gate electrode is located at x G = 0, and therefore the potential in the bulk directly above the gate has amplitude V B = 25 V. As V S increases from 0 to 50 V, the velocity of the fluid over the gate decreases uniformly (that is, the decrease in velocity is independent of position). This strongly supports the notion that the stray flow is superposed on the symmetric, standard ICEO flow. The net fluid velocity over the gate electrode can be written as 14 The effect of varying V S is only to alter the stray component u S . The slope of the curves, given theoretically by − w | E| 2 /η, is determined only by the standard ICEO characteristics of the system, and the experiments confirm that this slope is unaffected by V S . This allows us to easily infer the stray flow component by In other words, the stray flow is equal to the flow at the center of the gate electrode, because the standard ICEO flow vanishes there. In this manner, the stray flow was deduced from the linear regressions of plots like those in figure 5(a), and the results are shown in figure 5(b). The error bars in figure 5(b) represent the uncertainty in the flow velocity due only to the uncertainty in the location of the center of the gate electrode (∼1 µm). This demonstrates the linear dependence of u S upon V S , as predicted by equation (19). Notably, the stray flow vanishes when the stage voltage is equal to the bulk voltage above the gate (V S = 25). This demonstrates an effective method of restoring the symmetry. As V S increases past this value, the stray velocity begins to grow in magnitude, but now it is directed toward the powered electrode. The qualitative features of the observed behavior perfectly match the predicted consequences of the stray capacitance theory.

Effect of location of gate electrode
The effect of moving x G is to change the potential drop φ B (x G , t) − φ S (t), and it therefore has similar consequences as changing V S , described above. Figure 6 shows the fluid velocity over the gate electrode for x G = 0.33 and −0.33 cm. In this experiment,Ã P = 0.00115, V S = 0 (stage grounded), and E = 50 V cm −1 . Again, the two curves have the same slope (to within experimental error). The stray flow is deduced from the linear regression to be u S = 22.5 µm s −1 when x G = −0.33 cm, and u S = 4.7 µm s −1 when x G = 0.33 cm. The ratio of the stray flow speeds is then 4.8, in excellent agreement with the predicted value of 4.9, which comes from equation (19).

Effect of pad size
We have shown thus far that some electrical interaction with the microscope stage is certainly responsible for the stray flow, but not that it is necessarily mediated by the pads attached to the gate electrode. In figure 7, the stray flow velocity is plotted against the scaled pad area, with V S = 0, x G = 0, and E = 50 V cm −1 . The error bars represent the combined uncertainty in the stray flow due to the uncertainty in the location of the center of the gate electrode (∼1 µm), and the deviation in the data collected from five identical experiments. Importantly, the stray flow is eliminated when the pads are removed (whereuponÃ P 1). The stray flow tends to increase as A P increases, though the linearity of this relationship is not clear.

Discussion
The qualitative results of our experiments strongly support the simple theory we developed. We have demonstrated that the magnitude and direction of the stray flow can be easily manipulated by controlling the potential of the microscope stage. In addition, we have shown that the stray flow increases with increasing pad size, and that removing the pads eliminates the stray flow completely. Furthermore, we have shown that the magnitude of the stray flow depends on the location of the gate electrode in the channel. The effect of mutual capacitive coupling plays a role analogous to fixed-potential ICEO [9], where the potential of the (metal) inducing surface is forced to vary in phase with the field.
Here, however, the gate voltage varies naturally due to the capacitive coupling, rather than due to active control. The stray capacitance, nonetheless, causes a uniform potential on the gate to vary in phase with the driving field. The flow-FET [22,25,26] also involves a ζ -potential that is modulated using an external potential, with an external gate electrode playing the role of the stage, and no internal gate electrode. Several differences then arise: modern flow-FETs employ a thin oxide layer upon the electrode, whose acid sites give rise to a high buffer capacitance [32] in parallel with the double-layer capacitance, and thus reduce ζ . Second, the internal gate electrode in our system allows greater geometric variation (e.g. connecting to pads outside of electrical contact), giving greater control over C S and thus flow. A high stray capacitance C S can be achieved with a large pad area A P , whereas the double-layer capacitance exists only over the gate/electrolyte contact area A G . Because the total charge on the gate is constant, ζ S is geometrically amplified (C DL /C S ∝ A G /A P ).
The absolute magnitudes of both the standard and stray slip velocities are much smaller than their respective theories predict. This is a typical, though poorly understood feature of ICEO flows, and warrants further investigation [20]. In all of the experiments, the applied electric field was 50 V cm −1 at 100 Hz. At x = −11.5 µm (the greatest distance from the center of the gate for which velocities can be reliably calculated), the standard induced zeta potential amplitude is ζ i = 57.5 mV, and should generate a slip velocity u = 102 µm s −1 . If we extrapolate the standard ICEO flow component from the data (by ignoring u S ), we typically measure slip velocities in the range of 3.7-9.2 µm s −1 . The standard ICEO theory overpredicts the experimental data by a factor between 11 and 28.
For pad sizeÃ P = 0.00115, the stray double layer has zeta potential amplitude ζ S = 28.75 mV, and the expected stray slip velocity is 51 µm s −1 . The experimentally determined stray flow is u S = 3 µm s −1 . Similar calculations for the other pad sizes shows that our theory overshoots the experimental data by a factor between 8 and 17.
Of course, approximations are made in the theory that are not necessarily compatible with the conditions of the experiment. For one, the linear Debye-Hückel theory is only valid for ζ kT /e ≈ 26 mV, and in our experiments both the standard and stray zeta potentials frequently exceed this value, although not dramatically. Other assumptions made in our theory are fairly standard (although perhaps incorrect): (i) the gate electrode is clean and smooth, (ii) the double layer forms fully and quasi-steadily, (iii) Stern layers do not form, (iv) no faradaic reactions occur, (v) ions are point-like and respond to the mean electrostatic field (i.e. Poisson-Boltzmann equation), and (vi) the fluid behaves as a homogeneous continuum.
Because this quantitative discrepancy has been observed frequently in other ICEO experiments [33], we believe that the missing bit of physics responsible for explaining the discrepancy has more to do with general features of electrokinetic and ICEK systems than our specific geometry. To account for this (uniform) quantitative discrepancy between our data and theory, we have normalized our measurements by a common relative velocity. With this normalization, our theory correctly predicts how the stray flow should scale as various parameters-stray capacitance (via pad size), electrode location, stage potential, which electrode is powered-are varied. However, the fact that our theory shares the same margin of error in predicting the results of experiments as a previously existing theory is not a rigorous proof that the same missing bit of physics, once discovered, will correct both of the theories. For example, it has been argued that any contaminant on the gate electrode acts as an additional capacitor C d in series with the double layer, reducing the induced zeta potential [9] and thus the slip velocity. Such a contaminant reduces the (symmetric) ICEO flow much more strongly than the stray flow discussed here. The induced ζ driving the symmetric component of the ICEO flow depends exclusively upon V B − V G ; any dielectric capacitor thus reduces the potential drop across the diffuse layer. By contrast, the induced ζ driving the stray flow depends upon V B − V S , which already includes the glass slide as capacitor. Any additional nanometer-scale contaminant has a negligible effect upon the total dielectric capacitance, and thus upon the stray flow. Therefore, the relatively small magnitude of the measured stray slip velocity cannot be ascribed to a contaminant layer on the gate. Other mechanisms-including steric effects within highly concentrated double layers-may play a role [34]- [36].
The frequency of the electric field was not varied in these experiments, but rather kept low (100 Hz) to allow the double layer time to fully charge. In standard ICEO, theory predicts that the slip velocity falls with increasing frequency once the 'RC' charging time for the surface is exceeded [9], and intuition suggests that the stray flow should behave similarly. The charging time for the stray double layer may differ, however, since ions are brought in from the bulk rather than from one side of the surface to the other. A study into the frequency dependence of standard ICEO and capacitive ICEO could reveal interesting differences. Similarly, the electric field was held at 50 V cm −1 for all of these experiments. At higher field strengths (e.g. 100 V cm −1 ), we observed the slip velocity profile to saturate towards the edges of the gate electrode, where the ζ i is highest. We believe that this reflects the onset of more significant nonlinearity in the double layer, where other effects-e.g. steric repulsion between ions, noncontinuum effectsmay become important [33]. It is interesting to note, however, that even at a field strength of 50 V cm −1 , the amplitude of the standard ICEO-induced zeta potential at x = −11.5 µm, neglecting the stray double layer, is ζ ≈ 50 mV, or approximately twice the thermal voltage. Nevertheless, the slip velocity was measured to vary linearly with position along the gate electrode. This linearity was also observed for the variation of stray flow with stage voltage ( figure 5(b)), for which the stray zeta potential, withÃ P = 0.00115, reaches ζ S = 28.75 mV (when the stage is grounded). We chose to gather data using the smallest electric field possible for which the slip velocity was still easily observable.
The asymmetric flow that we have described was a surprise when it was first observed in experiments intended to characterize other facets of ICEO. The fact that it results directly from the experimental apparatus used to measure the flow can render it a troublesome experimental quirk. In fact, mutual capacitive coupling may also be responsible for unexpected observations in electrohydrodynamic experiments [21,37,38], in which asymmetric behavior was observed to consistently occur around either a grounded or powered electrode. Understanding this mechanism provides a number of routes to restore the (expected) symmetry. The most obvious solution is to remove any unnecessary 'external' conductors: we found that replacing the metal stage with a plastic one drastically reduces the asymmetry, but the contribution of the metal objective casing to the asymmetry cannot be similarly removed. In our case, stray flows can be most conveniently eliminated by ensuring the gate electrode remains mostly confined to the electrolyte, and particularly by not attaching the electrode pads (that are the source of the greatest capacitive coupling to the device's surroundings). In other experiments, where the source of the stray capacitance cannot be removed as such, any surrounding conductors should be held at the correct voltage to restore the symmetry in the system. Using a transformer as a phase splitter, the potentials of both electrodes can be driven symmetrically about ground, rather than one electrode being powered and the other grounded, to give φ B (x = 0, t) = 0. We tested this method on our system and showed that it successfully eliminated the stray flow.
Finally, the capacitive coupling mechanism described need not be exclusively an experimental nuisance, but can in fact be exploited to manipulate double layers and electrokinetic flows over polarizable surfaces. These double layers can be controlled in several simple ways, as we demonstrated, and can generate a robust variety of fluid flows.