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PHYSICAL REVIEW E 66, 026305 ~2002!

Fluid flow induced by nonuniform ac electric fields in electrolyteson microelectrodes. III. Observation of streamlines and numerical simulation

N. G. Green,1,* A. Ramos,2 A. Gonzalez,2,3 H. Morgan,1 and A. Castellanos21Bioelectronics Research Centre, Department of Electronics and Electrical Engineering, University of Glasgow, Rankine Build

Oakfield Avenue, Glasgow G12 8LT, Scotland, United Kingdom2Departamento de Electro´nica y Electromagnetismo, Facultad de Fı´sica, Universidad de Sevilla, Reina Mercedes s/n,

41012 Sevilla, Spain3Departamento de Fı´sica Aplicada III, ESI Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain

~Received 28 January 2002; published 19 August 2002!

The application of a nonuniform ac electric field to an electrolyte using coplanar microelectrodes results insteady fluid flow. The flow has its origin in the interaction of the tangential component of the nonuniform fieldwith the induced charge in the electrical double layer on the electrode surfaces. Termed ac electro-osmosis, theflow has been studied experimentally and theoretically using linear analysis. This paper presents experimentalobservations of the fluid flow profile obtained by superimposing images of particle movement in a planenormal to the electrode surface. These experimental streamlines demonstrate that the fluid flow is driven at thesurface of the electrodes. Experimental measurements of the impedance of the electrical double layer on theelectrodes are also presented. The potential drop across the double layer at the surface of the electrodes iscalculated numerically using a linear double layer model, and also using the impedance of the double layerobtained from experimental data. The ac electro-osmotic flow at the surface of the electrodes is then calculatedusing the Helmholtz-Smoluchowski formula. The bulk fluid flow driven by this surface velocity is numericallycalculated as a function of frequency and good agreement is found between the numerical and experimentalstreamlines.

DOI: 10.1103/PhysRevE.66.026305 PACS number~s!: 47.65.1a, 82.45.2h, 82.70.Dd, 85.90.1h

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I. INTRODUCTION

It has been demonstrated that nonuniform ac elecfields generated by coplanar microelectrodes produce stfluid flow in electrolytic solutions@1–4#. The fluid movesfrom the high strength field regions on the edges of the etrodes onto the surface of the electrodes, with the highvelocities found at the edge@3#. The flow has been characterized in terms of a number of experimental variables andependent on the frequency and amplitude of the appsignal and on the electrolyte conductivity. The velocity goto zero at high and low frequency limits and is maximuma frequency that depends on the conductivity and positionthe electrode surface@3#. This characteristic frequency imuch smaller than the charge relaxation frequency ofbulk electrolytef c5s/(2p«), wheres and « are the elec-trical conductivity and permittivity of the electrolyte. Although generated by ac fields, the fluid flow, referred to aselectro-osmosis, is steady. It can be distinguished experimtally from flow of electrothermal origin because the latterof smaller magnitude and is mainly observed at frequenaroundf c and at higher applied voltages@5–7#. The mecha-nism of ac electro-osmotic driven fluid flow has recenbeen shown to be capable of producing unidirectional puming of liquid on a microscale, as demonstrated experimtally by Brown, Smith, and Rennie@8#, following generalsymmetry arguments presented by Ajdari@9#.

The mechanism responsible for the flow is the interact

*Corresponding author. Email address: n.green@elec.gla.ac.

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of the tangential component of the electric field and theduced charge in the diffuse double layer on the electrsurface@1,4#. Consider two coplanar electrodes separateda thin gap, subjected to an ac potential difference and cered in an electrolyte as shown in Fig. 1. At a certain timthe situation is as shown in the figure; induced charge acmulates in the diffuse double layer with a sign oppositethe electrode charge. This induced charge is subjected toaction of the tangential component of the electric field, ging rise to a force directed from the center of the gap ontoelectrode surface. This force drives the fluid at the levelthe electrodes and has a direction that is independent osign of the electrode potential, so that the fluid flow hasnonzero time average. It should be noted that the mechanrequires a nonuniform electric field thus ensuring that a tgential field component exists in the diffuse double layerthe electrodes.

Other authors have also observed that spatial variationthe normal current distribution on the electrodes cause latfluid motion either in ac or dc fields@10,11#. They havesuggested that concentration gradients arising from electreactions produce a distribution of free charge adjacent toelectrodes; this charge interacts with any lateral electric figenerating the electrohydrodynamic flow. Sides@12# furtherelaborated this theory in order to make predictive calcutions for ac applied voltages. In our work, we considcharge transfer reactions to be negligible and that anycharge appears from ideal ion polarization, i.e., the chargof the double layer in the manner of a capacitor in an ac fieThe electrodes used in this work were made of Titaniummetal which readily forms a thin highly resistant oxide layTherefore, any Faradaic currents are likely to be extrem

©2002 The American Physical Society05-1

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GREEN, RAMOS, GONZA´ LEZ, MORGAN, AND CASTELLANOS PHYSICAL REVIEW E66, 026305 ~2002!

small and are not considered in the analysis.The electro-osmotic velocity has been calculated usin

simple circuit model to represent the fluid/interface syst@1#, as well as the linear analysis of the electrokinetic eqtions @4#. Both models have a good degree of correlatwith experimental data, predicting that the frequency depdence of the fluid flow is governed by electrode polarizatias demonstrated experimentally@3#.

This paper presents experimental results of the flstreamlines imaged in a plane normal to the surface ofmicroelectrodes. The streamlines were obtained by a suposition of several images of latex tracer particles movwith the fluid. In order to determine the specific impedanof the double layer at the surface of the electrodes, expmental measurements were made of the frequency-depenimpedance of the electrode/electrolyte system.

Finally, the solution for the ac electro-osmotic driven fluflow is calculated numerically using the finite elememethod. First the electric potential was solved using onetwo boundary conditions at the electrode surface; eithering the double layer capacitance determined from the linDebye-Huckel model@13# or using the experimentally detemined values for the double layer impedance. The electrstress on the fluid and the fluid velocity at the surface wthen calculated and compared with experimental measments@3#. Finally, the motion of the bulk fluid was numercally calculated, using the ac electro-osmotic velocity onelectrode surface as the boundary condition. These re

FIG. 1. A schematic diagram of the mechanism of ac elecosmosis for the experimental electrode array, consisting of two lplate electrodes separated by a narrow gap.~a! shows the inducedcharge layers and the electric fieldE at a point in time, resultingfrom a potential difference applied to the two electrodes. The etric field has a tangential componentEx at the surface of the electrodes, producing a forceFc on the charges at the surface. The timaveraged value of this force for an alternating potential is nonzproducing the steady fluid flow pattern shown schematically in~b!.The fluid flow is driven at the surface of the electrode, movingacross the surface and dragging fluid down in the center of the

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were compared with the experimental observations ofstreamlines.

II. EXPERIMENTAL RESULTS

The fluid motion has been measured in detail as a funcof the applied field frequency, voltage, and medium condtivity @3#. In general, the fluid velocity increases as the voage is increased and decreases with increasing electrconductivity. The magnitude of the velocity as a functionfrequency has a bell-shaped profile and tends to zero atand low frequency limits.

A. ac electro-osmotic streamlines

The streamlines were recorded using an experimesetup that has been described previously@14#, a schematic ofwhich is shown in Fig. 2. In summary, two coplanar belectrodes, 0.5 mm wide, 2 mm long, and 120 nm thick, wtheir long edges parallel and separated by 25mm, were mi-crofabricated on a glass slide. The electrodes consistedlayer of gold sandwiched between two layers of titaniuThe electrodes were coated in a thin titanium layer becaof the well known resistance of this metal to corrosion inor in saline solutions@15#. Titanium forms a very thin oxidelayer that usually renders it inactive. Therefore, to a goapproximation, the electrodes can be considered to befectly polarizable in the range of applied voltages~0–2.5 V!.A square glass chamber was constructed around the elecarray, so that the electrode/electrolyte system could beserved both from above and from the side. A microscoobjective and camera were pointed horizontally alongelectrodes with the gap between the electrodes in the ceof the field of view, so that the electrodes could be imagedcross section@14#. An ac potential difference was applied tthe pair of electrodes with voltage amplitude from 0 to 2.5and frequencies ranging from 102 to 105 Hz. This frequencyrange is always below the charge relaxation frequencyf c5vc/2p5(1/2p)(s/«), which is from 106 to 108 Hz forthe experimental conductivities.

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FIG. 2. A schematic diagram of the experimental setup. Telectrodes were fabricated up to the edge of the substrate aglass chamber was constructed around the array, with a verglass plate at the end of the electrodes. A microscope objectivecamera were then placed horizontally looking along the gaptween the electrodes. The camera was then focused at a point ithe chamber at sufficient distance from the electrode ends sothe fluid flow was moving in the vertical plane of focus only.

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FLUID FLOW INDUCED BY . . . . III. . . . PHYSICAL REVIEW E66, 026305 ~2002!

The electrolytic solutions used, both for the study of tfluid motion and the measurement of the double layer impance, were aqueous solutions of potassium chloride~KCl!with three different conductivities: 2.1~hereafter referred toas caseA!, 8.6 ~caseB!, and 84 mS/m~caseC!. Fluorescentlatex spheres~Molecular Probes, Oregon, U.S.!, 557 nm indiameter, were used as tracer particles@3,14#. The movementof the particles was recorded on video and transferredcomputer. Successive video frames were superimposeproduce tracks of particle movement over time. In thesence of other forces, the particles are small enough tolow the fluid so that the particle tracks can be consideredbe the streamlines of the fluid.

Three composite images of particle tracks are shownFigs. 3~a!, 3~b!, and 3~c! for different frequencies~100, 300,and 1000 Hz, respectively!, for electrolyteA. There is somedielectrophoretic capture@16,17# of particles at the edges othe electrodes and some collection of particles on top ofelectrodes~white regions!. This latter effect could be causeby dipole-dipole interparticle interaction, or particle-particparticle-electrode interactions of electrohydrodynamic natnot well understood at present@10,11,18#. However, it isclear that in the bulk, the particle tracks show that the flcirculates in two symmetrical rolls with the fastest velociticlose to the electrode edges. Particle tracks indicate thafluid moves from the center top of the image, down to tcenter of the gap and out across the electrodes as shschematically in Fig. 1.

The highest velocities were observed close to the etrode surface, a good indication that the fluid flow is drivin this region. The three images were taken at frequenwhere the fluid flow can clearly be observed. The patternthe fluid flow depends on the frequency of the applied signAt 1 kHz @Fig. 3~c!# the velocity is high at the electrode edgand decreases rapidly with increasing distance from the eThis produces a flow that has a center of circulation closethe electrode edge and movement that is restricted toregion close to the gap between the electrodes. Particlesther away~at the edge of the image! move more slowly anddemonstrate Brownian motion; the broken~jiggling! tracksare an indication of this. As the frequency is decreased@Figs.3~b! and 3~a!#, the fluid velocity at the electrode edge rmains constant or decreases slightly. However, the ratdecrease of velocity with distance over the electrodes is lresulting in the center of the streamlines moving farthaway from the electrode edge over the surface.

B. Double layer impedance measurements

In order to determine the frequency dependence offraction of the total applied voltage drop across the doulayer, double layer impedance measurements were mading a specially constructed measurement cell. The measment cell consisted of two large (35.534.5 mm2) parallelplate electrodes fabricated on planar glass slides, sepausing a thin spacer of 0.7 mm thickness. The electrodes wmade of the same sandwich of titanium/gold/titaniumthose used for fluid flow observations. The impedance ofelectrode/electrolyte system was measured using a Hew

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Packard impedance analyzer 4192A with the signal appto one electrode. The second electrode was attacheground to complete the circuit. The cell was filled with KCsolutions of varying conductivities and the impedance ofcell was measured as a function of frequency and voltagthe applied signal.

The total impedance of the sampleZT can be representeas a parallel RC circuit with frequency-dependent resistaand capacitance defined by

ZT5R

11 ivCR. ~1!

Figure 4 shows a plot ofR andC as a function of frequency

FIG. 3. Composite images of particle tracks over time, obtainby superimposing successive video frames from footage of flflow observed using the experimental setup shown in Fig. 2,three different frequencies of the applied potential:~a! 100,~b! 300,and ~c! 1000 Hz. The amplitude of the applied voltage was 2 Vall cases and the suspending medium was electrolyteA. As thefrequency increased, the rolls in the fluid flow extended a smadistance over the surface of the electrodes. The center of thealso moved close to the surface and the edge of the electrode

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GREEN, RAMOS, GONZA´ LEZ, MORGAN, AND CASTELLANOS PHYSICAL REVIEW E66, 026305 ~2002!

at an applied voltage of 0.5 V for three electrolyte condutivities as used in a previous work@3#. This data shows thathe capacitance and resistance are functions of frequenchigh frequencies, the capacitance tends to a constant vequivalent to that of a parallel plate capacitor with the pmittivity of the bulk electrolyte. The resistance is also costant at high frequencies, with a value that is inversely pportional to the conductivity of the solution. These limitinvalues correspond to the impedance of the bulk electrolsince at high frequencies the double layer impedance isligible. At low frequencies, the total measured resistancecapacitance includes the impedance of the double layer.suming that the impedance of the double layer and the bare in series, the bulk impedance can be subtracted fromtotal impedance to obtain values for the frequency-dependouble layer impedanceZDL . The double layer impedancdetermined using this method is shown in Fig. 5 for electlyte A and a voltage amplitude of 0.5 V.

The results show that the double layer polarization imp

FIG. 4. The capacitance and resistance for the three electroA, B, and C measured using two parallel plate electrodes forapplied potential of amplitude 0.5 V and plotted as a functionfrequency.

FIG. 5. The real and imaginary components of the double laimpedance for electrolyteA and 0.5 V, obtained by subtracting thbulk impedance values from the measured impedance.

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ance can be fitted to the following expression:

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vb FcosS p

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2b D G , ~2!

whereA and b are constants. This type of polarization impedance is frequently found in double layer measuremeand is referred to asconstant phase angle~CPA! impedance@15,19–23#. This expression satisfies the Kramers-Kroniglations for the impedance of a linear system@24#. For a per-fect capacitor,b51 and for a perfect resistor,b50. Fordouble layer impedance measurements on solid electrodebis typically found to be between 0.7 and 0.9@20#. An analysisof the measurements performed in this work gave valueb around 0.8060.05, depending on conductivity and volage. The voltage dependence of the impedance indicatnonlinear system, as observed by other researchers@25#. Asan example, for the three different electrolyte conductivitused in this work and an applied voltage of amplitude 0.5the values ofb were 0.75, 0.78, and 0.8. The correspondivalues for the constantA were 18 000, 18 400, and 19 00with v in rads21 andZ in ohms. At the lower voltage amplitude of 0.1 V, the values ofb were 0.80, 0.80, and 0.82 foelectrolytesA,B, andC, respectively. The corresponding vaues for the constantA were 27 000, 22 000, and 23 500.

For the ac electro-osmotic flow measurements@3#, thedouble layer impedance measurements and fluid flow obvations, the frequency of the applied signal was much lowthan the charge relaxation frequency,f c . In this lower fre-quency range, the experimental results indicate thatdouble layer is not purely capacitive~shown by the deviationof b from unity! as expected from theory@26#. The nonidealbehavior of the double layer cannot be ascribed to ionicfusion effects since the standard differential capacitancethe double layer is not modified in this frequency range@26#.The explanation for the nonunity value ofb is a topic ofconsiderable current research@19,22,23#, and a number ofpossible theories have been proposed, including surfaceorder such as atomic-scale heterogeneities, or geometriregularities~roughness!.

However, even though there is no well-defined physiexplanation, the measured CPA impedance can providuseful means of modelling the impedance of the system.specific impedance of the double layer can be obtainedmultiplying the double layer impedance by the area ofelectrode (35.534.5 mm2). This figure gives the ratio between the potential drop and the current density, whichturn, can be used to define a boundary condition fornumerical simulation that may be more valid for compariswith the experimental results.

III. THEORETICAL ANALYSIS

In this section the equations and boundary conditions gerning the electrical potential and the fluid velocity will bdescribed. The subsequent section presents the results onumerical simulation of the governing equations for the eperimental electrode system.

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FLUID FLOW INDUCED BY . . . . III. . . . PHYSICAL REVIEW E66, 026305 ~2002!

A. The physical system

As described earlier, the experimental system consisttwo long thin coplanar parallel electrodes fabricated onglass slide, as shown schematically in Fig. 2. Above theface, there is a solution of KCl bounded by an upper hozontal glass cover slip. At the glass/electrolyte and meelectrolyte interfaces, double layers are formed with a typthickness given by the Debye length,

lD5AD«

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whereD is the mean diffusion coefficient of the ions. ThDebye length is negligibly small compared to the typical sof the systemL (L;10mm,s;1022 S/m,lD /L;1023).Since the electrodes are long~; 2 mm!, both compared togap width~;25 mm! and the size of the roll observed in thfluid ~; 100 mm!, the problem can be considered to be twdimensional.

When an ac voltage is applied to the electrodes, an etrical current is established in the solution. The followinanalysis assumes that electrolysis does not occur at thetrode surfaces, i.e., the electrodes are considered to befectly polarizable, and the double layer behaves in a linmanner.

B. The electrical potential

As previously stated, for sufficiently low frequencies, i.f ! f c , the double layer is in quasiequilibrium@26#. Underthese conditions, the bulk electrolyte behaves in a resismanner and the double layer in a capacitive manner. Aresult, the potential in the bulk electrolyte satisfies Laplacequation

¹2f50 ~4!

with the boundary condition just outside the double layerthe electrode surface given by

s]f

]y5

]qDL

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whereqDL is the charge per unit area in the double layer.this equation we assume that lateral currents alongdouble layer are negligible. Equation~5! describes the charging of the double layer due to the bulk current. The relatioship between the charge and the potential drop acrossdouble layer depends on the model used. If the voltage dacross the diffuse double layer is sufficiently small (Dfd,kT/e50.025 V) there is a linear relationship between tcharge and the voltage, i.e.,qDL5CDL(f2V), and the equa-tion can be written with complex amplitudes as

s]f

]y5 ivqDL5 ivCDL~f2Vj !, ~6!

where CDL is the capacitance per unit of area of the todouble layer~diffuse plus compact layers!, and Vj is thepotential applied to the electrodej. This capacitanceCDL is

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given by the series combination of the Stern or complayer capacitance,Cs , and the diffuse double layer capactance,Cd ,

CDL5CsCd

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The compact layer capacitance is attributed almost entito the oxide layer which forms on top of the titanium eletrode and is approximately independent of the electrolconcentration. The properties of the diffuse part of tdouble layer depend on electrolyte concentration as giventhe Gouy-Chapman theory@13#. Although experimentally thepotential drop across the diffuse double layer can exc0.025 V, nevertheless the linear analysis can give usefulformation into the mechanism governing fluid flow.

At the interface between the electrolyte and the glasssimilar boundary condition holds. However, the boundacondition can be simplified in this case. In the absencelateral currents through the double layer, the total normcurrent, free plus displacement, must be continuous. Thfore, the total normal current in the electrolyte and the glare equal,

~ iv«1s!]f

]y5~ iv«G1sG!

]fG

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where«G , sG , andfG are the electrical permittivity, conductivity, and potential in the glass. Since the angular fquencyv!s/«,s/«G , and since the conductivity of theglass is negligible, the boundary condition at the glass inface in the fluid, Eq.~8!, simplifies to

]f

]y50. ~9!

In deriving these boundary conditions, the presence of latcurrents along the double layer has been neglected. Intheoretical analysis, the applied and natural surface potials are assumed to be small. In this case, as discusseGonzalez et al. @4#, the intrinsic mean ion density of thdiffuse double layer is small and the lateral currents are nligible. For high ion concentrations in the double layer, sface currents might be comparable to the currents in themal direction.

Owing to electrode polarization, the electric field in thbulk electrolyte is frequency dependent. When the frequeis low, most of the applied voltage is dropped acrossdouble layer but when the frequency is high most of tapplied voltage is dropped across the bulk electrolyte. Ttypical angular frequency of transition can be estimated frsimple circuit theory asv0;(RC)215s/(CDLL), whereLis a typical size of the system, for example, the electrode@1,2#. The capacitance per unit area can be estimated fthe Debye-Hu¨ckel theory as«/lD . The approximate transition frequency is therefore,v0;(s/«)(lD /L), which isseveral orders of magnitude smaller than the charge re

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GREEN, RAMOS, GONZA´ LEZ, MORGAN, AND CASTELLANOS PHYSICAL REVIEW E66, 026305 ~2002!

ation frequency,s/«, of the electrolyte. The fluid flow isobserved at frequencies of the order of the transitionquencyv0 @1,3,8#.

C. The fluid dynamic problem

Once the potential is solved, the electro-osmotic velocjust outside the double layer at the surface of the electrocan be calculated from this solution.

1. The solution in the diffuse layer

The fluid motion is caused by electrical stresses thatnonzero only in the diffuse double layer, since the chadensity in the bulk is zero. These stresses result in a rapvarying velocity profile in the diffuse double layer, changinfrom zero at the wall to a finite value just outside the doulayer. This velocity value can be used as a boundary cotion at the electrode surface to calculate the bulk motionthe thin double layer approximation, for diffuse layersquasiequilibrium~low frequency compared tos/«! and on aperfectly polarizable metal surface, the slip electro-osmovelocity is given by the Helmholtz-Smoluchowski formu@27#,

u52«

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In this expression,Ex is the tangential field just outside thdiffuse layer,h is the fluid viscosity, andDfd5f2c rep-resents the difference between the potentialf on the outerside of the diffuse layer and the potentialc on the inner sideof this layer, at the nonslip plane. This equation givestangential velocity on the electrodes; the normal velocityzero.

In the linear regime, the time-averaged horizontal fluvelocity at the interface between the double layer andbulk is @4#

^u&52«

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]f*

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«

4hL

]

]xuf2Vj u2, ~11!

where Re@B# means real part ofB, the asterisk indicates complex conjugate,L is the ratio between the diffuse layer ptential dropDfd and the total double layer potential droDfDL , andVj is the potential applied to the electrode. TparameterL is given by the ratio between the total capatanceCDL and the diffuse double layer capacitance,

L5CDL

Cd5

Cs

Cs1Cd. ~12!

SinceL,1, the presence of the extra capacitance due tocompact layer decreases the maximum achievable slip veity. Equation~11! gives the boundary condition for the tan

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gential velocity on the electrodes. For the glass/electrointerface an estimate of the potential drop across the diffdouble layer shows that the tangential velocity here is negibly small. At the boundary

ivCDL~f2fG!5 iv«G

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]y, ~13!

so that the potential drop across the diffuse layer on the gis

f2fG5«G

CDL

]fG

]y;

«GlD

«LfG , ~14!

which is of the orderlD divided by the typical distanceL ofthe system. Therefore, from Eq.~10!, the electro-osmotic ve-locity on the glass is negligible.

2. The solution in the bulk

To obtain the velocity in the bulk, the Navier-Stokeequations must be solved. Since, for microelectrodesReynolds number is usually very small,rmuL/h<1022, weneglect the inertial terms in the Navier-Stokes equationsthe absence of externally applied body forces, these eqtions reduce to

h¹2^u&2“p50, “•^u&50. ~15!

The boundary conditions are~1! the tangential ac electroosmotic velocity on the electrodes, Eq.~11!; ~2! zero tangen-tial velocity on the glass; and~3! zero normal velocity onevery boundary.

In addition to the time-averaged velocity there will alsbe an alternating velocity, resulting from the oscillating pof the electrical force in the diffuse double layer. This osclating electro-osmotic velocity diffuses into the bulk updistances of the order@h/(rmv)#1/2 @9#. This velocity is onlyobserved in our experiments for very low frequencies~1–10Hz!, since the amplitude of the oscillation of the trackinparticles is too small at higher frequencies.

D. Calculation of ac electro-osmotic flow using theexperimental polarization impedance

The measured polarization impedance of the meelectrolyte interface provides an empirical relationship btween the potential drop across the double layer and the etrical current. The frequency dependence of theelectroosmotic fluid flow is closely correlated with thmechanism of electrode polarization@1,3#. It can be con-cluded therefore that a phenomenological model forelectro-osmosis based on themeasuredimpedance of thedouble layer will give a better description of the fluid flow

In this case the electrical potential in the bulk is a solutiof Laplace’s equation with a new boundary condition at telectrode surface given by

s]f

]y5

1

ZDL~f2Vj !, ~16!

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FLUID FLOW INDUCED BY . . . . III. . . . PHYSICAL REVIEW E66, 026305 ~2002!

whereZDL is the measured specific impedance of the doulayer ~Sec. II B!. We assume that Eq.~11! is valid even if themeasured impedance of the double layer is not purely captive. The time-averaged slip velocity is then given by

^u&52«

4hL

]

]xuf2Vj u2, ~17!

with f given by the solution of the electric potential with thnew boundary condition, Eq.~16!. This expression for thevelocity was derived from the Smoluchowski express@Eq. ~10!# for the ideal model of a compact Stern layerseries with a diffuse layer. In this ideal model,L is given byEq. ~12! and indicates how much of the potential drop acrothe double layer,DfDL , is dropped across the diffuse layeDfd , i.e.,L5Dfd /DfDL . In our phenomenological modefor a real double layer,L is assumed to be a measure of thratio.

In the numerical section of this paper~Sec. IV!, the ve-locities calculated using Eq.~17! with the experimental im-pedance boundary condition will be compared with thocalculated using the double layer capacitance given byDebye-Huckel theory.

IV. NUMERICAL RESULTS AND DISCUSSION

Numerical calculations were performed usingFLEXPDE

~PDE Solutions! @28#, a commercially available finite element partial differential equation solver. In general, a twdimensional~2D! problem space is divided into triangulaelements and the variables are approximated by seconthird order polynomials in each element. In order to solveNavier-Stokes equation, the penalty method describedRef. @29# was used.

The problem space for the numerical simulation is shoschematically in Fig. 6. Owing to the symmetry of the eletrodes, the problem space can be restricted to one electof the two with appropriate symmetry boundary conditionThe inner edge of the electrode is 12.5mm from the symme-try boundary, giving an interelectrode separation of 25mm.The height of the solution space is 200mm, corresponding tothe height of the fluid filled chamber; the length of the sotion space was much larger than the gap and the electro

A. Numerical simulation of the electrical potentialand calculation of ac electro-osmotic velocity on the surface

of the electrodes

The electric potential satisfies Laplace’s equation@Eq. ~4!#with the following boundary conditions. On the surfacethe electrodes, we have from Eq.~16!, which is an extensionof Eq. ~6!,

sZDL

]f

]y5f2V. ~18!

In the numerical solution, two versions of this boundary codition are considered. First, using the linear Debye-Hu¨ckelmodel, the impedance corresponds to a capacitor;ZDL51/ivC, with C5«/lD . In this case, the problem need

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-

only to be solved once as a function of frequency forconductivities by defining the nondimensional frequencyV5v«L/slD , with L51024 m as the typical length of thesystem. In the second case, the constant phase elemenpedance of the double layer derived from the experimeresults is used, as described in Secs. II and III;ZDL5A( iv)2b. In this case, the problem must be solved serately for each conductivity. In addition, there is a boundacondition of odd symmetryf50 on the planex50, and aNeumann boundary condition]f/]n50 everywhere else.

Owing to the presence of complex parameters, the prlem was solved numerically as a pair of coupled equatifor the real and the imaginary parts of the potential. In ordto obtain an accurate picture of the frequency-dependenthavior the problem was solved for ten frequencies percade, evenly spaced logarithmically.

As an illustration, Figs. 7~a! and 7~b! represent the contour lines for the real and imaginary parts of the potentcorresponding to a nondimensional frequency of 8 andplied voltage amplitude 0.5 V.

The electro-osmotic velocity on the surface of the eletrodes can then be calculated using Eq.~17!. In Fig. 8 theelectro-osmotic velocity, calculated numerically for the caCDL5«/lD , is plotted as a function of the distance from thelectrode edge and nondimensional frequency. Here we hassumed thatL51. It can be seen that the velocity asfunction of frequency has a maximum that depends ondistance from the electrode edge: the higher the distancelower the frequency. Also, it can be seen that the velocitynegative for values ofx close tox5100mm, correspondingto the outer edge of the electrodes, where the liquid is floing inwards. This is in perfect agreement with experime

FIG. 6. A schematic diagram of the 2D problem space fornumerical solution of the electrical potentialf and the fluid flowu5(u,v). The boundary condition for the potential on the electrowas fEP5V1sZDL]f/]y, which is the potential at the Debylength. The boundary conditions for the velocity on the electrowere normal velocity zero and horizontal velocity~in this caseu!given by the electro-osmotic velocityuACE calculated from the so-lution of the electrical potential. The boundary conditions on tsymmetry plane were electrical potential zero, normal velocityuzero, and normal derivative of tangential velocityn zero. For theremaining boundaries, the normal derivative of the electrical pottial was zero and the fluid velocity holds the no-slip condition,u5n50.

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GREEN, RAMOS, GONZA´ LEZ, MORGAN, AND CASTELLANOS PHYSICAL REVIEW E66, 026305 ~2002!

since fluid flow in the opposite direction is observed atedge of the electrodes farthest from the gap.

Figures 9~a! and 9~b! are comparisons of the experimentvelocities and the calculated velocities for electrolytesA andB as a function of frequency, for different positions acrothe electrode. The curves correspond to an applied volamplitude of 0.5 V. For this comparison, the parameterLwas chosen to be 0.25 for caseA and 0.24 for caseB. Thisfactor is the same for all positions.

For the case of the CPA impedance, the systemsolved separately for the different conductivities. Figu10~a! and 10~b! show the corresponding comparison of texperimental and calculated velocities for the same caseAand B as above, with the same voltage amplitude 0.5Again, to make the comparison, the numerically calculacurves were multiplied by a factorL smaller than unity~0.28and 0.25, respectively!.

The comparisons demonstrate that better agreementhe frequency of maximum velocity is found when the CPimpedance is used, for both values of conductivity.

FIG. 7. The electrical potential calculated for a double layimpedanceZDL51/(ivCDL) shown as separate real~a! and imagi-nary ~b! parts. The voltage applied to the electrode was 0.25corresponding to a potential difference between the electrode0.5 V, and the nondimensional frequency was 8.

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The discrepancy between the Debye-Hu¨ckel predictionand the experimental results may be explained becausecapacitance of a real double layer is not only due todiffuse double layer, but also has a component from the copact layer. The fact that matching the experimental and coputed velocities requires that the parameterL be less than 1is an indication that the voltage drop across the diffuse lais less than across the whole double layer. A factorL;0.25 suggests that the diffuse layer impedance is a quaof the total double layer impedance, and the majority ofpotential is dropped across the compact layer. This asseseems to be supported by the experimental impedanceThe measured double layer impedance was assumed tthe sum of the impedance of the diffuse layer and that ofixed layer, i.e.,ZDL5Zd1Zs . As the conductivity of thesolution increases, the impedance of the diffuse layerZd re-duces and, for a certain conductivity, should become negible compared withZs . The ratio between the absolute impedance value for the lowest conductivity solution andhighest conductivity solution is 1.2560.10 in the frequencyrange of interest. If the double layer impedance for the hiest conductivity solution is solely due to the compact lay(ZDL'Zs), a ratio of 1.25 implies that the potential dropthe diffuse double layerDfd for the lowest conductivity so-lution is around 20% of the total potential drop,DfDL , i.e.,

U Dfd

DfDLU5U Zd

Zd1ZsU;0.2. ~19!

In other words, the parameterL in the electro-osmotic ve-locity equation should be around 0.2 for the lowest condtivity, which is close to the value 0.25 determined by coparison of the experimental and numerical velocity dshown in Fig. 10. The oxide layer increases the distabetween the diffuse double layer and the electrode, redu

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FIG. 8. The horizontal ac electro-osmotic fluid flow at the suface of the electrodes calculated from the electrical potential, pted as a function of distance from the edge of the electrodenondimensional frequency. The electrode in this case was 100mmwide, corresponding to the electrodes used for the measuremethe fluid velocity@3#. The amplitude of applied voltage was 0.5 V

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FLUID FLOW INDUCED BY . . . . III. . . . PHYSICAL REVIEW E66, 026305 ~2002!

the capacitance and increasing the impedance of the dolayer. This, in turn, reduces the achievable slip velocity@Eq.~17!#.

The fact that for each conductivity and applied voltagthe same value ofL is found for all positions across thelectrodes~showing a good agreement in every case forCPA impedance! suggests that the numerical model correcpredicts the geometry dependence for the liquid velocHowever, the frequency variation of the velocities doesagree completely with the experimental data and couldimproved by numerical analysis of a more experimentarealistic three-dimensional system. In addition, in orderpredict the conductivity and frequency dependence ofelectro-osmosis from first principles, more theoretical aexperimental work is required to understand the physchemistry of the system.

B. Numerical simulation of bulk fluid flow

The time-averaged velocity was calculated from Stokeequations@Eq. ~15!# with the following boundary conditionsOn the electrodes, the tangential velocity is given by

FIG. 9. Comparison of the calculated horizontal electro-osmvelocity at the surface of the electrodes and the experimental msurements of the fluid velocity across the electrode surface.data are plotted as a function of frequency~in Hz! for several po-sitions on the electrodes and for electrolyteA ~a! and electrolyteB~b!. The impedance of the double layer was given byZDL

5lD /( iv«) and the amplitude of applied voltage was 0.5 V.

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electro-osmotic velocityu5U @from Eq. ~17!#, with L51;at the symmetry plane, the tangential velocity satisfiesNeumann boundary condition]v/]x50; for every otherboundary the tangential velocity is zero. The normal velocis zero over the whole boundary.

Figures 11~a!, 11~b!, and 11~c! show the predicted streamlines, together with the fluid velocity at each point using tCPA impedance, for conductivityA, at frequencies of 100300, and 1000 Hz and applied voltage 2 V. According to tcalculation, the drag of the liquid by the surface stresproduces a roll on top of each electrode. As the frequeincreases, the position of the center of the rolls approacthe electrode edges, while the roll size decreases. The limoves faster near the inner edges of the electrodes, wthe electric field is stronger and changes more sharply.

This velocity distribution can be compared with the eperimental streamlines. Figures 12~a!, 12~b!, and 12~c! showthe predicted streamlines superimposed on the compositeages of the particle paths, for the same cases as in the pous figures. It can be seen that the numerical computacorrectly describes the size and shape of the rolls andposition of the center, the distance of which to the edge

ca-e

FIG. 10. Comparison of the calculated horizontal electosmotic velocity at the surface of the electrodes and the experimtal measurements of the fluid velocity across the electrode surfThe data are plotted as a function of frequency~in Hz! for severalpositions on the electrodes and for electrolyteA ~a! and electrolyteB ~b!. The impedance of the double layer was given by the Cimpedance obtained from the experimental measurements anamplitude of applied voltage was 0.5 V.

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GREEN, RAMOS, GONZA´ LEZ, MORGAN, AND CASTELLANOS PHYSICAL REVIEW E66, 026305 ~2002!

FIG. 11. Numerically computed streamlines for the fluid floarising from ac electro-osmosis on the surface of 500mm wideelectrodes, corresponding to the array used for the observatiothe streamlines. The calculations were performed using theimpedance, electrolyteA and for the three frequencies used in texperimental observations:~a! 100, ~b! 300, and~c! 1000 Hz.

02630

creases as the frequency increases. This agreement isdence of the ac electro-osmotic origin of the experimenfluid motion.

A similar calculation for the case of a Debye-Hu¨ckel im-pedance predicts similar rolls, but with a less accurate depdence on frequency.

V. CONCLUSIONS

Experimental observations of the streamlines of steelectro-osmotic fluid flow driven by ac fields have been psented. The streamlines were obtained by superimposingages of particle movement in a plane normal to the electrsurface. These experimental streamlines show that theflow is driven at the surface of the electrodes. Experimenmeasurements of the impedance of the electrical double lon the electrodes were also presented. The resulting datathen used to model the double layer behavior as a consphase-angle impedance.

The electrical potential generated by the experimenelectrodes as a function of frequency has been numericcalculated using both the Debye-Hu¨ckel specific capacitanceof the double layer and the experimental CPA impedanThe ac electro-osmotic velocity at the surface of the eltrodes has been calculated using the numerical solutionthe potential for both cases. A better description of thelocity dependence on frequency was obtained by using

ofA

FIG. 12. Comparison of the numerically computed streamlinfor 500 mm wide electrodes and the experimental streamlinesthe three frequencies used in the experimental observations:~a! 100,~b! 300, and~c! 1000 Hz. The experimental images have beenverted.

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FLUID FLOW INDUCED BY . . . . III. . . . PHYSICAL REVIEW E66, 026305 ~2002!

CPA impedance. The bulk fluid flow driven by this surfavelocity was also calculated numerically as a functionfrequency and good agreement was found between themerical and experimental streamlines, demonstrating thafluid motion is caused by ac electroosmosis.

To match the numerical and experimental velocity magtudes, the value of the parameterL was'0.25. This is be-cause of the existence of a layer of oxide layer betweentitanium electrode and the electrolyte, reducing the potendrop across the diffuse layer and, in turn, reducing thepected velocity magnitude.

This fluid flow described in this paper generally occursa system involving nonuniform fields generated by mic

, J

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electrodes at frequencies of the order of kilohertz. The respresented in this paper could be relevant to the problemparticle-particle interaction seen on top of electrodes infields such as described in Ref.@30#.

ACKNOWLEDGMENTS

The authors would like to acknowledge the EuropeUnion for financial support to N.G.G.@Contract No. BIO4-CT98-5010 ~DG12-SSM!# and the Spanish governmenagency Direccio´n General de Ciencia y Tecnologia for finacial support under Contract No. BFM2000-1056.

l,

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s

v.

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