bazant 2004 induced

8/13/2019 Bazant 2004 Induced

1/36

J. Fluid Mech. (2004), vol. 509, pp. 217252. c 2004 Cambridge University PressDOI: 10.1017/S0022112004009309 Printed in the United Kingdom

217

Induced-charge electro-osmosis

B y TO D D M . S Q U I R E S 1 AND M A RT I N Z. B A Z A N T 2

1Departments of Applied and Computational Mathematics and Physics,California Institute of Technology, Pasadena, CA 91125, USA

2Department of Mathematics and Institute for Soldier Nanotechnologies,Massachusetts Institute of Technology, Cambridge, MA 02139, USA

(Received 5 May 2003 and in revised form 13 February 2004)

We describe the general phenomenon of induced-charge electro-osmosis (ICEO) the nonlinear electro-osmotic slip that occurs when an applied eld acts on the ioniccharge it induces around a polarizable surface. Motivated by a simple physical picture,we calculate ICEO ows around conducting cylinders in steady (DC), oscillatory (AC),

and suddenly applied electric elds. This picture, and these systems, represent perhapsthe clearest example of nonlinear electrokinetic phenomena. We complement andverify this physically motivated approach using a matched asymptotic expansion to theelectrokinetic equations in the thin-double-layer and low-potential limits. ICEO slipvelocities vary as us

E 20 L , where E 0 is the eld strength and L is a geometric lengthscale, and are set up on a time scale c = D L/D , where D is the screening lengthand D is the ionic diffusion constant. We propose and analyse ICEO microuidicpumps and mixers that operate without moving parts under low applied potentials.Similar ows around metallic colloids with xed total charge have been described inthe Russian literature (largely unnoticed in the West). ICEO ows around conductors

with xed potential, on the other hand, have no colloidal analogue and offer furtherpossibilities for microuidic applications.

1. IntroductionRecent developments in micro-fabrication and the technological promise of micro-

uidic labs on a chip have brought a renewed interest to the study of low-Reynolds-number ows (Stone & Kim 2001; Whitesides & Stroock 2001; Reyes et al. (2002)).The familiar techniques used in larger-scale applications for uid manipulation, whichoften exploit uid instabilities due to inertial nonlinearities, do not work on the micronscale due to the pre-eminence of viscous damping. The microscale mixing of miscibleuids must thus occur without the benet of turbulence, by molecular diffusion alone.For extremely small devices, molecular diffusion is relatively rapid; however, in typicalmicrouidic devices with 10100 m features, the mixing time can be prohibitivelylong (of order 100 s for molecules with diffusivity 10 10 m2 s1). Another limitationarises because the pressure-driven ow rate through small channels decreases withthe third or fourth power of channel size. Innovative ideas are thus being consideredfor pumping, mixing, manipulating and separating on the micron length scale (e.g.Beebe, Mensing & Walker 2002; Whitesides et al. 2001). Naturally, many focus on

the use of surface phenomena, owing to the large surface to volume ratios of typicalmicrouidic devices.Electrokinetic phenomena provide one of the most popular non-mechanical tech-

niques in microuidics. The basic idea behind electrokinetic phenomena is as follows:


2/36

218 T. M. Squires and M. Z. Bazant

locally non-neutral uid occurs adjacent to charged solid surfaces, where a diffusecloud of oppositely charged counter-ions screens the surface charge. An externallyapplied electric eld exerts a force on this charged diffuse layer, which gives rise toa uid ow relative to the particle or surface. Electrokinetic ow around stationarysurfaces is known as electro-osmotic ow, and the electrokinetic motion of freelysuspended particles is known as electrophoresis. Electro-osmosis and electrophoresisnd wide application in analytical chemistry (Bruin 2000), genomics (Landers 2003)and proteomics (Figeys & Pinto 2001; Dolnik & Hutterer 2001).

The standard picture for electrokinetic phenomena involves surfaces with xed, con-stant charge (or, equivalently, zeta-potential , dened as the potential drop acrossthe screening cloud). Recently, variants on this picture have been explored. Anderson(1985) demonstrated that interesting and counter-intuitive effects occur with spatiallyinhomogeneous zeta-potentials, and showed that the electrophoretic mobility of acolloid was sensitive to the distribution of surface charge, rather than simply the totalnet charge. Anderson & Idol (1985) explored electro-osmotic ow in inhomogeneously

charged pores, and found eddies and recirculation regions. Ajdari (1995, 2002) andStroock et al. (2000) showed that a net electro-osmotic ow could be driven eitherparallel or perpendicular to an applied eld by modulating the surface and chargedensity of a microchannel, and Gitlin et al. (2003) have implemented these ideas tomake a transverse electrokinetic pump. Such transverse pumps have the advantagethat a strong eld can be established with a low voltage applied across a narrowchannel. Long & Ajdari (1998) examined electrophoresis of patterned colloids, andfound example colloids whose electrophoretic motion is always transverse to theapplied electric eld. Finally, Long, Stone & Ajdari (1999), Ghosal (2003), and othershave studied electro-osmosis along inhomogeneously charged channel walls (dueto, e.g., adsorption of analyte molecules), which provides an additional source of dispersion that can limit resolution in capillary electrophoresis.

Other variants involve surface charge densities that are not xed, but ratherare induced (either actively or passively). For example, the effective zeta-potential of channel walls can be manipulated using an auxillary electrode to improve separa-tion efficiency in capillary electrophoresis (Lee, Blanchard & Wu 1990; Hayes &Ewing 1992) and, by analogy with the electronic eld-effect transistor, to set up eld-effect electro-osmosis (Ghowsi & Gale 1991; Gajar & Geis 1992; Schasfoort et al.1999).

Time-varying, inhomogeneous charge double layers induced around electrodes giverise to interesting effects as well. Trau, Saville & Aksay (1997) and Yeh, Seul &Shraiman (1997) demonstrated that colloidal spheres can spontaneously self-assembleinto crystalline aggregates near electrodes under AC applied elds. They proposedsomewhat similar electrohydrodynamic mechanisms for this aggregation, in which aninhomogeneous screening cloud is formed by (and interacts with) the inhomogeneousapplied electric eld (perturbed by the sphere), resulting in a rectied electro-osmoticow directed radially in toward the sphere. More recently, Nadal et al. (2002a ) perfor-med detailed measurements in order to test both the attractive (electrohydrodynamic)and repulsive (electrostatic) interactions between the spheres, and Ristenpart, Aksay &Saville (2003) explored the rich variety of patterns that form when bidisperse colloidalsuspensions self-assemble near electrodes.

A related phenomenon allows steady electro-osmotic ows to be driven using ACelectric elds. Ramos et al. (1998, 1999) and Gonzalez et al. (2000) theoreticallyand experimentally explored AC electro-osmosis, in which a pair of adjacent, atelectrodes located on a glass slide and subjected to AC driving, gives rise to a steady


3/36

Induced-charge electro-osmosis 219

electro-osmotic ow consisting of two counter-rotating rolls. Around the same time,Ajdari (2000) theoretically predicted that an asymmetric array of electrodes withapplied AC elds generally pumps uid in the direction of broken symmetry (ACpumping). Brown, Smith & Rennie (2001), Studer et al. (2002), and Mpholo, Smith &Brown (2003) have since developed AC electrokinetic micro-pumps based on thiseffect, and Ramos et al. (2003) have extended their analysis.

Both AC colloidal self-assembly and AC electro-osmosis occur around electrodeswhose potential is externally controlled. Both effects are strongest when the voltageoscillates at a special AC frequency (the inverse of the charging time discussed below),and both effects disappear in the DC limit. Furthermore, both vary with the squareof the applied voltage V 0. This nonlinear dependence can be understood qualitativelyas follows: the induced charge cloud/zeta-potential varies linearly with V 0, and theresulting ow is driven by the external eld, which also varies with V 0. On the otherhand, DC colloidal aggregation, as explored by Solomentsev, Bohmer & Anderson(1997), requires large enough voltages to pass a Faradaic current, and is driven by a

different, linear mechanism.Very recently in microuidics, a few cases of nonlinear electro-osmotic ows aroundisolated and inert (but polarizable) objects have been reported, with both AC andDC forcing. In a situation reminiscent of AC electro-osmosis, Nadal et al. (2002b)studied the micro-ow produced around a dielectric stripe on a planar blockingelectrode. In rather different situations, Thamida & Chang (2002) observed a DCnonlinear electrokinetic jet directed away from a protruding corner in a dielectricmicrochannel, far away from any electrode, and Takhistov, Duginova & Chang (2003)observed electrokinetically driven vortices near channel junctions. These studies (andthe present work) suggest that a rich variety of nonlinear electrokinetic phenomenaat polarizable surfaces remains to be exploited in microuidic devices.

In colloidal science, nonlinear electro-osmotic ows around polarizable (metallicor dielectric) particles were studied almost two decades ago in a series of Ukrainianpapers, reviewed by Murtsovkin (1996), that has gone all but unnoticed in the West.Such ows occur when the applied eld acts on the component of the double-layercharge that has been polarized by the eld itself. This idea can be traced back atleast to Levich (1962), who discussed the dipolar charge double layer (using theHelmholtz model) that is induced around a metallic colloidal particle in an externalelectric eld and touched upon the quadrupolar ow that would result. Simonov &Dukhin (1973) calculated the structure of the (polarized) dipolar charge cloud inorder to obtain the electrophoretic mobility, without concentrating on the resultingow. Gamayunov, Murtsovkin & Dukhin (1986) and Dukhin & Murtsovkin (1986)rst explicitly calculated the nonlinear electro-osmotic ow arising from double-layerpolarization around a spherical conducting particle, and Dukhin (1986) extended thiscalculation to include a dielectric surface coating (as a model of a dead biological cell).Experimentally, Gamayunov, Mantrov & Murtsovkin (1992) observed a nonlinearow around spherical metallic colloids, albeit in a direction opposite to predicitionsfor all but the smallest particles. They argued that a Faradaic current (breakdown of ideal polarizibility) was responsible for the observed ow reversal.

This work followed naturally from many earlier studies on non-equilibrium electricsurface phenomena reviewed by Dukhin (1993), especially those focusing on the

induced dipole moment (IDM) of a colloidal particle reviewed by Dukhin & Shilov(1980). Following Overbeek (1943), who was perhaps the rst to consider non-uniform polarization of the double layer in the context of electrophoresis, Dukhin(1965), Dukhin & Semenikhin (1970), and Dukhin & Shilov (1974) predicted the


4/36


electrophoretic mobility of a highly charged non-polarizable sphere in the thin-double-layer limit, in good agreement with the later numerical solutions of OBrien & White(1978) (see, e.g., Lyklema 1991). In that case, diffuse charge is redistributed by surfaceconduction, which produces an IDM aligned with the eld, and some variations inneutral bulk concentration, and secondary electro-osmotic and diffusio-osmotic owsdevelop as a result. Shilov & Dukhin (1970) extended this work to a non-polarizablesphere in an AC electric eld. Simonov & Dukhin (1973) and Simonov & Shilov(1973) performed similar calculations for an ideally polarizable, conducting sphere,which typically exhibits an IDM opposite to the applied eld. Simonov & Shilov(1977) revisited this problem in the context of dielectric dispersion and proposeda much simpler RC-circuit model to explain the sign and frequency dependence of the IDM. OBrien & White (1978) performed a numerical solution of the full ion,electrostatic, and uid equations with arbitrarily thick double layer, which naturallyincorporated the effects of double-layer polarization. OBrien & Hunter (1981) andOBrien (1983), following Dukhins approach, arrived at a simpler, approximate

expression that incorporated double-layer polarization in the thin-double-layer limit,and compared favourably with the numerical calculations of OBrien & White (1978).Nonetheless, it seems a detailed study of the associated electro-osmotic ows aroundpolarizable spheres did not appear until the paper of Gamayunov et al. (1986) citedabove.

In summary, we note that electrokinetic phenomena at polarizable surfaces sharea fundamental feature: all involve a nonlinear ow component in which double-layercharge induced by the applied eld is driven by that same eld. To emphasize thiscommon mechanism, we suggest the term induced-charge electro-osmosis (ICEO) fortheir description. Specic realizations of ICEO include AC electro-osmosis at micro-electrodes, AC pumping by asymmetric electrode arrays, DC electrokinetic jets arounddielectric structures, DC and AC ows around polarizable colloidal particles, and thesituations described below. Of course, other electrokinetic effects may also occur inaddition to ICEO in any given system, such as those related to bulk concentrationgradients produced by surface conduction or Faradaic reactions, but we ignore suchcomplications here to highlight the basic effect of ICEO.

In the present work, we build upon this foundation of induced-charge electrokineticphenomena, specically keeping in mind microuidic applications. ICEO ows aroundmetallic colloids, which have attracted little attention compared to non-polarizableobjects of xed zeta-potential, naturally lend themselves for use in microuidic devices.In that setting, the particle is replaced by a xed polarizable object which pumps theuid in response to applied elds, and a host of new possibilities arise. The ability todirectly control the position, shape, and potential of one or more inducing surfaces ina microchannel allows a rich variety of effects that do not occur in colloidal systems.

Before we begin, we note the difference between ICEO and electrokinetic pheno-mena of the second kind, reviewed by Dukhin (1991) and studied recently by Ben &Chang (2002) in the context of microuidic applications. Signicantly, second-kindelectrokinetic effects do not arise from the double layer, being instead driven byspace charge in the bulk solution. They typically occur around ion-selective porousgranules subject to applied elds large enough to generate strong currents of certainions through the liquid/solid interface. This leads to large concentration variations

and space charge in the bulk electrolyte on one side, which interact with the appliedeld to cause motion. Barany, Mishchuk & Prieve (1998) studied the analogous effectfor non-porous metallic colloids undergoing electrochemical reactions at very largeFaradaic currents (exceeding diffusion limitation). In contrast, ICEO occurs around


5/36


inert polarizable surfaces carrying no Faradaic current in contact with a homogeneous,quasi-neutral electrolyte and relies on relatively small induced double-layer charge,rather than bulk space charge.

The article is organized as follows: 2 provides a basic background on electrokineticeffects, and

3 develops a basic physical picture of induced-charge electro-osmosis viacalculations of steady ICEO ow around a conducting cylinder. Section 4 examinesthe time-dependent ICEO ow for background electric elds which are suddenlyapplied ( 4.1) or sinusoidal ( 4.2). Section 5 describes some basic issues for ICEOin microuidic devices, such as coupling to the external circuit ( 5.1) and thephenomenon of xed-potential ICEO ( 5.2). Some specic designs for microuidicpumps, junction switches, and mixers are discussed and analysed in 5.3. Section 6investigates the detrimental effect of a thin dielectric coating on a conducting surfaceand calculates the ICEO ow around non-conducting dielectric cylinders. Section 7gives a systematic derivation of ICEO in the limit of thin double layers and smallpotentials, starting with the basic electrokinetic equations and employing matched

asymptotic expansions, concluding with a set of effective equations (with approxima-tions and errors quantied) for the ICEO ow around an arbitrarily shaped particlein an arbitrary space- and time-dependent electric eld. The interesting consequencesof shape and eld asymmetries, which generally lead to electro-osmotic pumping orelectrophoretic motion in AC elds, are left for a companion paper. The reader isreferred to Bazant & Squires (2004) for an overview of our results.

2. Classical (xed-charge) electro-osmosisElectrokinetic techniques provide some of the most popular small-scale non-

mechanical strategies for manipulating particles and uids. We present here a verybrief introduction. More detailed accounts are given by Lyklema (1991), Hunter (2000)and Russel, Saville & Schowalter (1989).

2.1. Small zeta-potentialsA surface with charge density q in an aqueous solutions attracts a screening cloudof oppositely charged counter-ions to form the electrochemical double layer, whichis effectively a surface capacitor. In the DebyeH uckel limit of small surface charge,the excess diffuse ionic charge exponentially screens the electric eld set up by thesurface charge (gure 1 a ), giving an electrostatic potential

= qw e

z

ez

. (2.1)Here w 800 is the dielectric permittivity of the solvent (typically water) and 0 isthe vacuum permittivity. The zeta-potential, dened by

qw

, (2.2)

reects the electrostatic potential drop across the screening cloud, and the Debyescreening length 1 is dened for a symmetric z:z electrolyte by

1

D = w kB T

2n0(ze )2 , (2.3)

with bulk ion concentration n0, (monovalent) ion charge e, Boltzmann constant kBand temperature T . Because the ions in the diffuse part of the double layer are


6/36


+ + + + + + + + + + + + + + + + + +

x

E U

Screening cloud

u s

Surface charge density q0

z

D

++ + + +

+++

+++++

++

++

u s

Q > 0U

E

(a) (b)

Figure 1. (a ) A charged solid surface in an electrolytic solution attracts an oppositely chargedscreening cloud of width D . An electric eld applied tangent to the charged solid surfacegives rise to an electro-osmotic ow (2.4). ( b) An electric eld applied to an electrolytic solutioncontaining a suspended solid particle gives rise to particle motion called electrophoresis, withvelocity equal in magnitude and opposite in direction to (2.4).

approximately in thermal equilibrium, the condition for a small charge density (orzeta-potential) is kB T/ze .

An externally applied electric eld exerts a body force on the electrically chargeduid in this screening cloud, driving the ions and the uid into motion. The resultingelectro-osmotic uid ow (gure 1 a ) appears to slip just outside the screening layerof width D . Under a wide range of conditions, the local slip velocity is given by theHelmholtzSmoluchowski formula,

u s =

w

E , (2.4)

where is the uid viscosity and E is the tangential component of the bulk electriceld.

This basic electrokinetic phenomenon gives rise to two related effects, electro-osmosis and electrophoresis, both of which nd wide application in analyticalchemistry, microuidics, colloidal self-assembly, and other emerging technologies.Electro-osmotic ow occurs when an electric eld is applied along a channel withcharged walls, wherein the electro-osmotic slip at the channel walls gives rise to plugow in the channel, with velocity given by (2.4). Because the electro-osmotic owvelocity is independent of channel size, (in contrast to pressure-driven ow, which

depends strongly upon channel size), electro-osmotic pumping presents a natural andpopular technique for uid manipulation in small channels. On the other hand, whenthe solid/uid interface is that of a freely suspended particle, the electro-osmotic slipvelocity gives rise to motion of the particle itself (gure 1 b), termed electrophoresis.In the thin-double-layer limit, the electrophoretic velocity is given by Smoluchowskisformula,

U = w

E e E, (2.5)

where E is the externally applied eld, and e = w / is the electrophoretic mobilityof the particle.2.2. Large zeta-potentials

For large zeta-potentials, comparable to or exceeding kB T/ze , the exponential proleof the diffuse charge (2.1) and the linear chargevoltage relation (2.2) are no longer


7/36


valid, but the diffuse part of the double layer remains in thermal equilibrium. As aresult, the potential in the diffuse layer satises the PoissonBoltzmann equation. Forsymmetric, binary electrolyte, the classical GouyChapman solution yields a nonlinearchargevoltage relation for the double layer,

q ( ) = 4 n0ze D sinh ze 2kB T , (2.6)

in the thin-double-layer limit. This relation may be modied to account for micro-scopic surface properties, such as a compact Stern layer, a thin dielectric coating, orFaradaic reactions, which enter via the boundary conditions to the PoissonBoltzmannequation. The key point here is simply that q generally grows exponentially withze/k B T , which has important implications for time-dependent problems involvingdouble-layer relaxation, as reviewed by Bazant, Thornton & Ajdari (2004).

Remarkably, the HelmholtzSmoluchowski formula (2.4) for the electro-osmoticslip remains valid in the nonlinear regime, as long as

D

aexp

ze 2kB T

1, (2.7)

where a is the radius of curvature of the surface (Hunter 2000). When (2.7) is violated,ionic concentrations in the diffuse layer differ signicantly from their bulk values,and surface conduction through the diffuse layer becomes important. As a result, theelectrophoretic mobility, e , becomes a nonlinear function of the dimensionless ratio

Du = s a

, (2.8)

of the surface conductivity, s , to the bulk conductivity, . Though this ratio wasrst noted by Bikerman (1940), we follow Lyklema (1991) in referring to (2.8) asthe Dukhin number, in honor of Dukhins pioneering calculations of its effect onelectrophoresis. We note also that essentially the same number has been called , ,and by various authors in the Western literature, and Rel by Dukhin (1993).

Bikerman (1933, 1935) made the rst theoretical predictions of surface conductance, s , taking into account both electromigration and electro-osmosis in the diffuse layer.The relative importance of the latter is determined by another dimensionless number

m =kB T ze

2 2wD

. (2.9)

Using the result of Deryagin & Dukhin (1969), Bikermans formula for s takes asimple form for a symmetric binary electrolyte, yielding

Du = 2 D (1 + m)

acosh

ze 2kB T 1 = 4

D (1 + m)a

sinh 2 ze 4kB T

(2.10)

for the Dukhin number. In the limit of innitely thin double layers, where (2.7) holds,the Dukhin number vanishes, and the electrophoretic mobility tends to Smoluchow-skis value (2.4). For any nite double-layer thickness, however, highly chargedparticles (with > k B T/ze ) are generally described by a non-negligible Dukhinnumber, and surface conduction becomes important. This leads to bulk concentration

gradients and a non-uniform diffuse-charge distribution around the particle in anapplied eld, which modies its electrophoretic mobility, via diffusiophoresis andconcentration polarization. For more details, the reader is referred to Lyklema (1991)and Dukhin (1993).


8/36


+

+ + _

_

+

E

E

+ +++++

+ ++++++

_ _ _ _ _ _ _

_

_ _ _ _ _ _ _

+ + _ _

++ _ _ +

(a) (b)

Figure 2. The evolution of the electric eld around a solid, ideally polarizable conductingcylinder immersed in a liquid electrolyte, following the imposition of a background DC eldat t = 0 ( a ), where the eld lines intersect normal to the conducting surface. Over a chargingtime c = D a/D , a dipolar charge cloud forms in response to currents from the bulk, reachingsteady state ( b) when the bulk eld prole is that of an insulator. The resulting zeta-potential,however, is non-uniform.

3. Induced-charge electro-osmosis: fundamental pictureThe standard electrokinetic picture described above involves the interaction of an

applied eld and a surface of xed charge, wherein the electro-osmotic ow is linearin the applied eld. Here we focus on the nonlinear phenomenon of ICEO at apolarizable (metal or dielectric) surface. As a consequence of nonlinearity, ICEO canbe used to drive steady electro-osmotic ows using AC or DC elds. The nonlinearityalso allows larger uid velocities and a richer, geometry-dependent ow structure.These properties stand in stark contrast to classical electro-osmosis described above,which, e.g., gives zero time-averaged ow in an AC eld.

This section gives a physically clear (as well as quantitatively accurate) sense of ICEO in perhaps the simplest case: an ideally polarizable metal cylinder in a suddenlyapplied, uniform electric eld. This builds on the descriptions of double-layer polariza-tion for an uncharged metallic sphere by Levich (1962), Simonov & Shilov (1973)and Simonov & Shilov (1977) and the associated steady ICEO ow by Gamayunovet al. (1986). We postpone to 7 a more detailed and general analysis, justifying theapproximations made here for the case of thin double layers ( D a ) and weakapplied elds ( E 0a kB T/ze ). Since Du 1 in these limits, surface conduction andconcentration polarization can be safely ignored, and ICEO becomes the dominantelectrokinetic effect around any inert, highly polarizable object.

3.1. Steady ICEO around an uncharged conducting cylinderThe basic phenomenon of ICEO can be understood from gures 2 and 3. Immediatelyafter an external eld E = E 0 z is applied, an electric eld is set up so that eld lines

intersect conducting surfaces at right-angles (gure 2 a ). Although this represents thesteady-state vacuum eld conguration, mobile ions in electrolytic solutions move inresponse to applied elds. A current J = E drives positive ions into a charge cloudon one side of the conductor ( z < 0), and negative ions to the other ( z > 0), inducing


9/36


+

++

++

+ +

++

++++

Q = 0

+

E 0

u s

E 0

Q > 0+ +

+ + +++

+

+

u s

(a) (b)

Figure 3. The steady-state induced-charge electro-osmotic ow around ( a ) a conductingcylinder with zero net charge and ( b) a positively charged conducting cylinder. The ICEOslip velocity depends on the product of the steady eld and the induced zeta-potential. Theow around an uncharged conducting cylinder ( a ) can thus be understood qualitatively fromgure 2( b), whereas the charged sylinder ( b) simply involves the superposition of the standardelectro-osmotic ow.

an equal and opposite surface charge on the conducting surface. A dipolar charge

cloud grows as long as a normal eld injects ions into the induced double layer, andsteady state is achieved when no eld lines penetrate the double layer (gure 2 b). Thetangential eld E drives an electro-osmotic slip velocity (2.4) proportional to thelocal double-layer charge density, driving uid from the poles of the particle towardsthe equator (gure 3 a ). An AC eld drives an identical ow, since an oppositelydirected eld induces an oppositely charged screening cloud, giving the same net ow.The ICEO ow around a conducting cylinder with non-zero total charge, shown ingure 3(b), simply superimposes on the nonlinear ICEO ow of gure 3( a ) the usuallinear electro-osmotic ow.

As a concrete example for quantitative analysis, we consider an isolated, unchargedconducting cylinder of radius a submerged in an electrolyte solution with very smallscreening length D a . An external electric eld E 0 z is suddenly applied at t = 0,and the conducting surface forms an equipotential surface, giving a potential

0 = E 0z 1 a 2

r 2. (3.1)

Electric eld lines intersect the conducting surface at right-angles, as shown ingure 2(a ).

Due to the electrolytes conductivity , a non-zero current J = E drives ions tothe cylinder surface. In the absence of electrochemical reactions at the conductor/

electrolyte interface (i.e. at sufficiently low potentials that the cylinder is ideallypolarizable), mobile solute ions accumulate in a screening cloud adjacent to the solid/liquid surface, attracting equal and opposite image charges within the conductoritself. Thus the conductors surface charge density q induced by the growing


10/36


11/36


Cylinder Sphere

Initial potential i E 0z 1 a 2

r 2 E 0z 1 a3

r 3

Steady-state potential s E 0z 1 + a2

r 2 E 0z 1 + a3

2r 3

Steady-state zeta-potential 2E 0a cos 32 E 0a cos

Radial ow u r2a (a 2 r 2)

r 3 U 0 cos 2

9a 2(a 2 r 2)16r 4 U 0 (1 + 3 cos 2 )Azimuthal ow u

2a 3

r 3 U 0 sin 2

9a 4

8r 4 U 0 sin 2

Charging Timescale c = D a/D s = D a/D

Induced dipole strength: g(t ) for g(t ) = 1 2 et / c g (t ) = 12 1 3 e2t/ ssuddenly applied eld E 0g for AC eld Re( E 0e it ) g =

1 i c1 + i c g = 1 i s2 + i s

Table 1. Electrostatic and hydrodynamic quantities for the induced-charge electro-osmotic(ICEO) ow around conducting spheres and cylinders, each of radius a . Here U 0 = w E 20 a/is a characteristic velocity scale, and the induced dipole strength g is dened in (4.1). SeeGamayunov et al. (1986) for ows around metal colloidal spheres in steady and AC elds.

Although we focus on the limit of linear screening in this paper, (3.9) should also

hold for nonlinear screening ( kB T/ze ) in the limit of thin double layers, as longas the Dukhin number remains small and (2.7) is satised. The relevant zeta potentialin these conditions, however, is not the equilibrium zeta potential ( 0 =0) but thetypical induced zeta-potential, E 0a , which is roughly the applied voltage acrossthe particle.

Finally, although we have specically considered a conducting cylinder, a similarpicture clearly holds for more general shapes. More generally, ICEO slip velocitiesaround arbitrarily shaped inert objects in uniform applied elds are directed alongthe surface from the poles of the object (as dened by the applied eld), towards theobjects equator.

3.2. Steady ICEO around a charged conducting cylinderUntil now, we have assumed the cylinder to have zero net charge for simplicity. Acylinder with non-zero equilibrium charge density q0 = Q/ 4 a 2, or zeta-potential, 0 =w q0, in a suddenly applied eld approaches a steady-state zeta-potential distribution,

( ) = 0 + 2 E 0a cos , (3.10)

which has the induced component in (3.5) added to the constant equilibrium value.This follows from the linearity of (3.3) with the initial condition, (, t =0) = 0. As aresult, the steady-state electro-osmotic slip is simply a superposition of the standardelectro-osmotic ow due to the equilibrium zeta-potential 0,

u Qs = u s 2w 0

sin , (3.11)


12/36


where u s is the ICEO slip velocity, given in (3.7). The associated Stokes ow is asuperposition of the ICEO ow and the standard electro-osmotic ow, and is shownin gure 3( b).

The electrophoretic velocity of a charged conducting cylinder can be found usingthe results of Stone & Samuel (1996), from which it follows that the velocity of acylinder with prescribed slip velocity u s ( ), but no externally applied force, is givenby the surface-averaged velocity,

U = 12

2

0u s ( ) d. (3.12)

The ICEO component (3.7) has zero surface average, leaving only the standardelectro-osmotic slip velocity (3.11). This was pointed out by Levich (1962) using aHelmholtz model for the induced double layer, and later by Simonov & Dukhin(1973) using a double-layer structure found by solving the electrokinetic equations.The conducting cylinder thus has the same electrophoretic mobility e = w 0/ asan object of xed uniform charge density and constant zeta-potential. This extremecase illustrates the result of OBrien & White (1978) that the electrophoretic mobilitydoes not depend on electrostatic boundary conditions, even though the ow aroundthe particle clearly does.

As above, the steady-state analysis of ICEO for a charged conductor is unaffectedby nonlinear screening, as long as (2.7) is satised (and Du 1), where the relevant is the maximum value, = | 0|+ |E 0a |, including both equilibrium and inducedcomponents.

4. Time-dependent ICEOA signicant feature of ICEO ow is its dependence on the square of the electric

eld amplitude. This has important consequences for AC elds: if the direction of theelectric eld in the above picture is reversed, so are the signs of the induced surfacecharge and screening cloud. The resultant ICEO ow, however, remains unchanged:the net ow generically occurs away from the poles, and towards the equator.Therefore, induced-charge electro-osmotic ows persist even in an AC applied elds,as long as the frequency is sufficiently low that the induced-charge screening cloudshave time to form.

AC forcing is desirable in microuidic devices, so it is important to examine the

time-dependence of ICEO ows. As above, we explicitly consider a conducting (ideallypolarizable) cylinder and simply quote the analogous results for a conducting spherein table 1. Although we perform calculations for the more tractable case of linearscreening, we briey indicate how the analysis would change for large induced zetapotentials. Two situations of interest are presented: the time-dependent response of aconducting cylinder to a suddenly applied electric eld ( 4.1) and to a sinusoidal ACelectric eld (4.2). We also comment on the basic time scale for ICEO ows.

4.1. ICEO around a conducting cylinder in a suddenly applied DC eldConsider rst the time-dependent response of an uncharged conducting cylinder inan electrolyte when a uniform electric eld E = E 0 z is suddenly turned on at t = 0.The dipolar nature of the external driving suggests a bulk electric eld of the form

(r , t ) = E 0z 1 + g(t )a 2

r 2, (4.1)


13/36


so that initially g(0)= 1 (3.1), and in steady state g(t ) = 1 (3.4). The potentialof the conducting surface itself remains zero, so that the potential drop across thedouble layer is given by

(a , , t ) =

( ) =

q ( )

w . (4.2)

Here, as before, we take q to represent the induced surface charge, so that the totalcharge per unit area in the charge cloud is q . The electric eld normal to the surface,found from (4.1), drives an ionic current

J

= q ( ) = E 0 cos (1 g ), (4.3)into the charge cloud, locally injecting a surface charge density q per unit time. Weexpress the induced charge density q in terms of the induced dipole g by substituting(4.1) into (4.2), take a time derivative, and equate the result with q given by (4.3).This results in an ordinary differential equation for the dipole strength g,

g = w a

(1 g ), (4.4)whose solution is

g (t ) = 1 2 et/ c . (4.5)Here c is the characteristic time for the formation of induced-charge screening clouds,

c = a w

=

D aD

, (4.6)

where the denitions of conductivity ( = 2 n0e2D/ k B T ) and screening length (2.3)have been used.

The induced-charge screening cloud (with equivalent zeta-potential given by (4.2))is driven by the tangential eld (derived from (4.1)) in the standard way (2.4), resultingin an induced-charge electro-osmotic slip velocity

u s = 2 U 0 sin 2 1 et/ c2 . (4.7)

More generally, the time-dependent slip velocity around a cylinder with a non-zeroxed charge (or equilibrium zeta-potential 0) can be found in similar fashion, andresults in the standard ICEO slip velocity u s (4.7) with an additional term representingstandard electro-osmotic slip (3.11)

u Qs = u s 2w 0

sin 1 et / c . (4.8)

Note that (4.8) grows more quickly than (4.7) initially, but that ICEO slip eventuallydominates in strong elds, since it varies with E 20 , versus E 0 for the standard electro-osmotic slip.

4.2. ICEO around a conducting cylinder in a sinusoidal AC eldAn analogous calculation can be performed in a straightforward fashion for sinusoidalapplied elds. Representing the electric eld using complex notation, E = E 0 eit z ,where the real part is implied, we obtain a time-dependent zeta-potential

= 2 E 0a cos Re eit

1 + i c, (4.9)


14/36


giving an induced-charge electro-osmotic slip velocity

u s = 2 U 0 sin2 Re eit

1 + i c

2 , (4.10)

with time-averaged slip velocity

u s = U 0 sin2

1 + 2 2c . (4.11)

In the low-frequency limit c 1, the double layer fully develops in phase with theapplied eld. In the high-frequency limit c 1, the double layer does not have timeto charge up, attaining a maximum magnitude ( c)2 with a / 2 phase shift. Notethat this analysis assumes that the double layer changes quasi-steadily, which requires 1D .

4.3. Time scales for ICEO ows

Before continuing, it is worth emphasizing the fundamental time scales arising inICEO. The basic charging time c exceeds the Debye time for diffusion across thedouble-layer thickness, D = 2D /D = w / , by a geometry-dependent factor, a/ D ,that is typically very large. c is also much smaller than the diffusion time across theparticle, a = a 2/D . The appearance of this time scale for induced dipole moment of a metallic colloidal sphere has been explained by Simonov & Shilov (1977) using asimple RC-circuit analogy, consistent with the detailed analysis of Simonov & Shilov(1973). The same time scale, c , also arises as the inverse frequency of AC electro-osmosis (Ramos et al. 1999) or AC pumping (Ajdari 2000) at a micro-electrodearray of characteristic length a , where again an RC-circuit analogy has been invokedto explain the charging process. This simple physical picture has been criticized byScott, Kaler & Paul (2001), but Ramos et al. (2001) and Gonzalez et al. (2000) haveconvincingly defended its validity, as in the earlier Russian papers on polarizablecolloids.

Although it is apparently not well-known in microuidics and colloidal science, thetime scale for double-layer relaxation was debated and analysed in electrochemistryin the middle of the last century, after Ferry (1948) predicted that D should be thecharging time for the double layer at an electrode in a semi-innite electrochemicalcell. Buck (1969) explicitly corrected Ferrys analysis to account for bulk conduction,which introduces the macroscopic electrode separation a . The issue was denitivelysettled by Macdonald (1970) who explained the correct charging time, c , as the RCtime for the double-layer capacitor, C = w / D , coupled to a bulk resistor, R = a/ .Similiar ideas were also developed independently a decade later by Kornyshev &Vorotyntsev (1981) in the context of solid electrolytes.

Ferrys model problem of a suddenly imposed surface charge density in a semi-innite electrolyte (as opposed to a suddenly imposed voltage or background eld ina nite system) persists in recent textbooks on colloidal science, such as Hunter (2000)and Lyklema (1991), and only the time scales D and a are presented as relevantfor double-layer relaxation. This is quite reasonable for non-polarizable colloidalparticles, but we stress that the intermediate RC time scale, c = D a , plays acentral role for polarizable objects that exhibit signicant double-layer charging.

We also mention nonlinear screening effects at large applied elds or large totalcharges, where the maximum total zeta-potential, 0 + E 0a , exceeds kB T/ze . Theanalysis of this section can be generalized to account for the weakly nonlinear limitof thin double layers, where > k B T/ze , but (2.7) is still satised (and thus Du 1 as


15/36


well). In the absence of surface conduction, (3.2) still describes double-layer relaxation,but a nonlinear chargevoltage relation, such as (2.6) from GouyChapman theory,must be used. In that case, the time-dependent boundary condition (3.3) on thecylinder is replaced by

CD ( )d dt = E r , (4.12)

where

CD ( ) = w D

cosh ze 2kB T

(4.13)

is the nonlinear differential capacitance of the double layer.If the induced component of the zeta-potential is large, due to a strong applied eld,

E 0a > k B T/ze , the charging dynamics in (4.12) are no longer analytically tractable.Since the differential capacitance CD ( ) increases with | | in any thin-double-layermodel, the poles of the cylinder along the applied eld charge more slowly than thesides. However, the steady-state eld is the same as for linear screening, as long asDu 1 .

If the applied eld is weak, but the total charge is large ( 0 > k B T/ze ), (4.12) maybe linearized to obtain the same polarization dynamics and ICEO ow as for Du 1.However, the RC time,

c( 0) = RC D = c cosh ze 02kB T

, (4.14)

increases nonlinearly with 0, as shown by Simonov & Shilov (1977). The same timeconstant, with 0 replaced by 0 + E 0a , also describes the late stages of relaxation inresponse to a strong applied eld.

Some implications for ICEO at large voltages in the strongly nonlinear limit of thindouble layers, where condition (2.7) is violated and Du 1, are discussed in 7.5. Inthis regime, the simple circuit approximation breaks down due to bulk diffusion, andsecondary relaxation occurs at the slow time scale, a = a 2/D . For an interdisciplinaryreview and detailed analysis of double-layer relaxation (without surface conductionor ow), the reader is referred to Bazant et al. (2004).

Finally, we note that the oscillatory component of ICEO ows may not obey thequasi-steady Stokes equations, due to the nite time scale, v = a 2/ , for the diffusionof uid vorticity. (Here = / is the kinematic viscosity and the uid density.)

It is customary in microuidic and colloidal systems to neglect the unsteady term, u /t , in the Stokes equations, because ions diffuse more slowly than vorticity by afactor of D/ 103. However, the natural time scale for the AC component of ACelectro-osmotic ows is c = D a/D , so the importance of the unsteady term in theStokes equations is governed by the dimensionless parameter

v c

= a D

D

. (4.15)

This becomes signicant for sufficiently thin double layers, D /a 103. Therefore,in AC electro-osmosis and other ICEO phenomena with AC forcing at the chargingfrequency, c = 1c , vorticity diffusion connes the oscillating component of ICEOow to an oscillatory boundary layer of size D a/D . However, the steady com-ponent of ICEO ows is usually the most important, and obeys the steady Stokesequations.


16/36


5. ICEO in microuidic devicesWe have thus far considered isolated conductors in background elds applied

at innity, as is standard in the colloidal context. The further richness of ICEOphenomena becomes apparent in the context of microuidic devices. In this section,we consider ICEO in which the external eld is applied by electrodes with nite,rather than innite, separation. Furthermore, microuidic devices allow additionaltechniques not available for colloids: the inducing surface can be held in placeand its potential can be actively controlled. This gives rise to xed-potential ICEO,which is to be contrasted with the xed-total-charge ICEO studied above. Finally,we present a series of simple ICEO-based microuidic devices that operate withoutmoving parts in AC elds.

5.1. Double-layer relaxation at electrodesAs emphasized above, one must drive a current J 0(t ) = E 0(t ) to apply an electriceld E 0(t ) in an homogeneous electrolyte. The electrochemical reactions associated

with steady Faradaic currents may cause uid contamination by reaction products orelectrodeposits, unwanted concentration polarization, or permanent dissolution (andthus irreversible failure) of microelectrodes. Therefore, oscillating voltages and non-Faradaic displacement currents at blocking electrodes are preferable in microuidicdevices. In this case, however, one must take care that diffuse-layer charging at theelectrodes does not screen the eld.

We examine the simplest case here, which involves a device consisting of a thinconducting cylinder of radius a D placed between at, inert electrodes separatedby 2L 2a . The cylinder is electrically isolated from the rest of the system, so thatits total charge is xed. Under a suddenly applied DC voltage, 2 V 0, the bulk electriceld, E 0(t ), decays to zero as screening clouds develop at the electrodes. For weakpotentials ( V 0 kB T /e ) and thin double layers ( D a L ), the bulk eld decaysexponentially

E 0(t ) = V 0L

et/ e , (5.1)with a characteristic electrode charging time

e = D L

D, (5.2)

analogous to the cylinders charging time (4.6). This time-dependent eld E 0(t ) thenacts as the applied eld at innity in the ICEO slip formula, (4.7).

The ICEO ow around the cylinder is set into motion exponentially over thecylinder charging time, c = D a/D , but is terminated exponentially over the (longer)electrode charging time, e = D L/D as the bulk eld is screened at the electrodes.This interplay between two time scales one set by the geometry of the inducingsurface and another set by the electrode geometry is a common feature of ICEO inmicrouidic devices.

This is clearly seen in the important case of AC forcing by a voltage, V 0 cos(t ), inwhich the bulk electric eld is given by

E 0(t ) = V 0L

Re i e

1 + i e

eit . (5.3)

Electric elds persist in the bulk solution when the driving frequency is high enough( e 1) that induced double layers do not have time to develop near the electrodes.Induced-charge electro-osmotic ows driven by applied AC elds can thus persist


17/36


Material properties of aqueous solutionDielectric constant w 800 7105 gcmV2 s2Viscosity 102 g cm1 s1Ion diffusivity D i 105 cm2 s1

Experimental parametersScreening length D 10 nmCylinder radius a 10 mApplied eld E 0 100 V cm1Electrode separation L 1 cm

Characteristic scalesSlip velocity U 0 = w E 20 a/ 0.7 mm s1Cylinder charging time c = D a/D i 104 sElectrode charging time e = D L/D i 101 sDimensionless surface potential = eE 0a/ k B T 3.9Dukhin number Du 102

Table 2. Representative values for induced-charge electro-osmosis in a microuidic device.

only in a certain band of driving frequencies, 1e 1c , unless Faradaic reactionsoccur at the electrodes to maintain the bulk eld. In AC electro-osmosis at adjacentsurface electrodes (Ramos et al. 1999) or AC pumping at an asymmetric electrodearray (Ajdari 2000), the inducing surfaces are the electrodes, and so the two timescales coincide to yield a single characteristic frequency c = 1 / e . (Note that Ramoset al. (1999) and Gonzalez et al. (2000) use the equivalent form

D / w L .)

Table 2 presents typical values for ICEO ow velocities and charging time scales

for some reasonable microuidic parameters. For example, an applied electric eldof strength 100 V cm 1 across an electrolyte containing a 10 m cylindrical post givesrise to an ICEO slip velocity of order 1 mm s 1, with charging times c0.1ms and e = 0 .1 s.

5.2. Fixed-potential ICEOIn the above examples, we have assumed a conducting element which is electricallyisolated from the driving electrodes, which constrains the total charge on theconductor. Another possibility involves xing the potential of the conducting surfacewith respect to the driving electrodes, which requires charge to ow onto and off theconductor. This ability to directly control the induced charge allows a wide variety of microuidic pumping strategies exploiting ICEO.

Perhaps the simplest example of xed-potential ICEO involves a conducting cylinderof radius a which is held in place at a distance h from the nearest electrode (gure 4 a ).For simplicity, we consider a h and h L . We take the cylinder to be held at somepotential V c , the nearest electrode to be held at V 0, the other electrode at V = 0,and assume the electrode charging time e to be long. In this case, the bulk eld(unperturbed by the conducting cylinder) is given simply by E 0 = V 0/L , and thebackground potential at the cylinder location is given by (h ) = V 0(1 h/L ). Inorder to maintain a potential V c , an average zeta-potential,

i = V c V 0 1 hL , (5.4)

is induced via a net transfer of charge per unit length of i = 2 w a i , along withan equally and oppositely charged screening cloud.


18/36


~2 L

V 0ei t

a

h

V 0

V 0

F E

u s

E 0

(a ) (b)

Figure 4. Fixed-potential ICEO. ( a ) A cylinder of radius a is held a distance h a from anearby electrode and held at the same electrostatic potential V 0 as the electrode. A secondelectrode is located a distance L away and held at zero potential, so that a eld E 0 V 0/L isestablished. ( b) Leading-order xed-potential ICEO ow for the system in ( a ). Like xed-chargeICEO, a non-zero steady ow can be driven with an AC voltage.

This induced screening cloud is driven by the tangential electric eld (3.6) in thestandard way, giving a xed-potential ICEO ow with slip velocity

u FPs = u s + 2w

V 0L

sin V c V 0 + V 0h

L , (5.5)

where u s is the quadrupolar xed-total-charge ICEO ow u s from (3.7) and (3.8),with E 0 = V 0/L . Note that both the magnitude and direction of the ow can becontrolled by changing the position h or the potential V c of the inducing conductor.A freely suspended cylinder would move with an electrophoretic velocity

U E = dhdt

= w

V 0L

V c V 0 + V 0h

L =

h h ca

U 0, (5.6)

away from the position hc = L (1 V c /V 0) where its potential is equal to the (unper-turbed) background potential. The velocity scale is the same as for xed-total-chargeICEO, although the cylinderelectrode separation h, rather than the cylinder radiusa , provides the geometric length scale. Since typically h a , xed-potential ICEOvelocities are larger than xed-total-charge ICEO for the same eld.

To hold the cylinder in place against U E , however, a force is required. FollowingJeffrey & Onishi (1981), the force F E is given approximately by

F E = 4 U E

log[(h + h 2 a 2)/a ] h 2 a 2/a 4 U E

log (2h/a ) 1, (5.7)

and is directed toward hc . The xed-potential ICEO ow around a cylinder heldin place at the same potential as the nearest electrode ( V c = V 0, hc =0) is shown in

gure 4(b). The leading-order ow consists of a Stokeslet of strength F E plus itsimages, following Liron & Blake (1981). The (quadrupolar) xed-total-charge ICEOow exists in addition to the ow shown, but is smaller by a factor a/ h and is thusnot drawn.


19/36


Figure 5. ICEO micropump designs for cross-, elbow-, and T- junctions. A conductingcylinder placed in a junction of microchannels, subject to an applied AC or DC eld, drivesan ICEO ow which draws uid in along the eld axis and expels it perpendicular to theeld axis. For the four-electrode congurations, the eld axis can be switched from verticalto horizontal by switching the polarities of two diagonally opposite electrodes, reversing thepumping direction.

With the ability to actively control the potential of the inducing surface, xed-potential ICEO ows afford signicant additional exibility over their xed-total-charge (and also colloidal) counterparts. Note that in a sense, there is little distinctionbetween inducing conductors and blocking electrodes. Both impose voltages, undergotime-dependent screening, and may drive ICEO ows. Furthermore, their sensitivityto device geometry and nonlinear dependence on applied elds open intriguingmicrouidic possibilities for ICEO ows both xed total charge and xed potential.

5.3. Simple microuidic devices exploiting ICEOOwing to the rich variety of their associated phenomena, ICEO ows have the poten-tial to add a signicant new technique to the microuidic toolbox. Below, we presentseveral ideas for microuidic pumps and mixers based on simple ICEO ows aroundconducting cylinders. The devices typically consist of strategically placed metal wiresand electrodes. As such, they can be easily fabricated and operate with no movingparts under AC applied electric elds. AC elds have several advantages over DCelds: (i) electrode reactions are not required to apply AC elds, thus eliminating theconcentration and pH gradients, bubble formation and metal ion injection that canoccur with DC elds, and (ii) strong elds can be created by applying small voltagesover small distances. For ease of analysis, we assume the cylinders to be long enoughthat the ow is effectively two-dimensional.

5.3.1. Junction pumpsThe simplest ICEO-based devices follow natually from the symmetry of ICEO ows,

which generally draw uid in along the eld axis and eject it radially. This symmetrycan be exploited to drive uid around a corner where two, three, or four micro-channels converge at right-angles, as shown in gure 5. These DC or AC electro-osmotic pumps are reversible: changing the polarity of the four electrodes (shown forthe elbow- and cross-junctions in the gure) so as to change the eld direction by 90 reverses the sense of pumping.

For variety, we give an alternative design for the T-junction pump, which uses aconducting plate embedded in the channel wall between the electrodes and cannotbe reversed. A reversible T-junction, similar to the cross- and elbow-junctions, couldeasily be designed. The former design, however, can be modied to reduce the


20/36


u

u

(a) (b)

Figure 6. AC electro-osmotic mixers. Diffusive mixing in a background ow is enhanced bythe ICEO convection rolls produced by ( a ) an array of conducting posts in a transverse AC eld,and ( b) conducting objects (or coatings) embedded in channel walls between micro-electrodesapplying elds along the ow direction.

detrimental effects of viscous drag by having the metal plate wrap around to the top

and bottom walls (not shown), in addition to the sidewall. In general, placing theinducing conductor driving ICEO ow on a channel wall is advantageous becauseit eliminates an inactive surface that would otherwise contribute to viscous drag.

The pressure drop generated by an ICEO junction pump can be estimated ondimensional grounds. The natural pressure scale for ICEO is P U 0/a w E 20 , andthe pressure decays with distance like ( a/ r )2. Thus a device driven by a cylinderof radius a in a junction with channel half-width W creates a pressure head of order P w E 20 a 2/W 2. For the specications listed in table 2, this correspondsto a pressure head on the order of mPa. This rather small value suggests thatstraightforward ICEO pumps are better suited for local uid control than for drivinguids over signicant distances.

5.3.2. ICEO micro-mixersAs discussed above, rapid mixing in microuidic devices is not trivial, since inertial

effects are negligible and mixing can only occur by diffusion. Chaotic advection (Aref 1984) provides a promising strategy for mixing in Stokes ows, and various techniquesfor creating chaotic streamlines have been introduced (e.g. Liu et al. 2000; Stroocket al. 2002). ICEO ows provide a simple method to create micro-vortices, and couldtherefore be used in a pulsed fashion to create unsteady two-dimensional ows withchaotic trajectories.

In gure 6( a ), we present a design for an ICEO mixer in which a backgroundow passes through an array of transverse conducting posts. An AC eld in theappropriate frequency range ( 1W 1a ) is applied perpendicular to the posts andto the mean ow direction, which generates an array of persistent ICEO convectionrolls. Note that the radius of the posts in the mixer should be smaller than shown(a W ) to validate the simple approximations made above, where a backgroundeld is applied to each post in isolation. However, larger posts as shown could haveuseful consequences, as the nal eld is amplied by focusing into a smaller region.We leave a careful analysis of such issues for future work.

The same kind of convective mixing could also be produced by a different design,illustrated in gure 6( b), in which an AC (or DC) eld is applied along the channel

with metal stripes embedded in the channel walls. As with the posts described above,the metal stripes are isolated from the electrodes applying the driving eld. Thisdesign has the advantage that it drives ow immediately adjacent to the wall, whichreduces dead space.


21/36


(a )

U

W

L

F

(b )

h

Figure 7. Fixed-potential ICEO pumps ( a ) and mixers ( b) can be constructed by electricallycoupling the inducing conductor, which may be a post (as shown) or a surface pattern (notshown), to one set of electrodes. These devices generalize AC electro-osmosis in at-surfaceelectrodes arrays to other situations, with different ow patterns and frequency responses.

5.3.3. Devices exploiting xed-potential ICEOBy coupling the potential of the posts or plates in the devices above to the electrodes,xed-potential ICEO can be exploited to generate net pumping past the posts. Forexample, in the cross-junction pump of gure 5, if the central post were groundedto one of the pairs of electrodes, there would be an enhanced ow sucking uid infrom the channel between the electrode pair (with an AC or DC voltage). Likewise,a xed-potential ICEO linear pump can be created in the middle of a channel, asshown in gure 7( a ).

To estimate the ow generated by the single-post device, we note that the postin gure 7( a ), if freely suspended, would move with velocity U E

w V 20 h/L2 down

the channel. The post is held in place, however, which requires a force (per unitlength) F

4 U E / (log(W/a )0.9) (Happel & Brenner 1983). The resulting owrate depends on the length of the channel; however, the pressure required to stop

the ow does not. This can be estimated using a two-dimensional analogue of thecalculation of Brenner (1958) for the pressure drop due to a small particle in acylindrical tube, giving a pressure drop P 3F / 4W . This pressure drop is largerthan that for the xed-total-charge ICEO junction pumps by a factor of O (hW 3/a 2L 2).Furthermore, multiple xed-potential ICEO pumps could be placed in series. Whenthe post is small ( a W, L ), these pumps operate in the same frequency range, 1W < < 1c , as the junction pumps above.

Clearly, many other designs are possible, which could provide detailed ow controlfor pumping or micro-vortex generation. For example, xed-potential ICEO can alsobe used in a micromixer design, as shown in gure 7( b), wherein rolls the size of the channel can be established. An interesting point is that the frequency responseof the ICEO ow is sensitive to both the geometry and the electrical couplings. Weleave the design, optimization, and application of real devices for future work, bothexperimental and theoretical.

We close this section by comparing these new kinds of devices with previousexamples of ICEO in microuidic devices, which involve quasi-planar micro-electrodearrays to produce AC electro-osmosis (Ramos et al. 1999; Ajdari 2000). As notedabove, when the potential of the inducing conductor (post, stripe, etc.) is coupled to

the external circuit, it effectively behaves like an electrode, so xed-potential ICEOis closely related to AC electro-osmosis. Of course, it shares the same fundamentalphysical mechanism, which we call ICEO, as related effects in polarizable colloidsthat do not involve electrodes. Fixed-potential ICEO devices, however, represent a


22/36


E w

Conductor = 0

E d d

D Screening cloud w

Dielectric layer d

Figure 8. A dielectric layer of thickness d and permittivity d coating a conductor. Thepotential drop between the external potential and 0 at the conductor occurs in two steps: d = E d d across the dielectric and w E w D across the double layer.signicant generalization of AC electro-osmotic planar arrays, because a non-trivialdistinction arises between electrode (applying the eld) and inducing conductor(producing the primary ICEO ow) in multi-dimensional geometries. This allows a

considerable variety of ow patterns and frequency responses. In contrast, existingdevices using AC electro-osmosis peak at a single frequency and produce very similarows (Ramos et al. 1998, 1999; Brown et al. 2001; Studer et al. 2002; Mpholo et al.2003).

6. Surface contamination by a dielectric coatingThe above examples have focused on an idealized situation with a clean metal

surface. In this section, we examine the effect of a non-conducting dielectric layerwhich coats the conductor, and nd that any dielectric layer which is thicker than thescreening length D signicantly reduces the strength of the ICEO ow. Furthermore,the ICEO ow around a dielectric object, rather than a perfectly conducting object aswe have discussed so far, is presented as a limiting case of the analysis in this section.

We start with a simple physical picture to demonstrate the basic effect of a thindielectric layer. Consider a conducting cylinder of radius a coated with a dielectriclayer of thickness d a (so that the surface looks locally planar) and permeabilityd , as shown in gure 8. In steady state, the potential drop from the conductingsurface = 0 to the potential outside the double layer occurs across in two steps:across the dielectric (where E = E d ), and across the screening cloud (where E = E w ),so that

E d d + E w D =

. (6.1)The electric elds in the double layer and in the dielectric layer are related viad E d = w E w , so that

1 + wd

d D

E w D = . (6.2)Since E w D is approximately the potential drop across the double layer, and sincethe steady-state bulk potential is given by (3.4) to be = 2 E 0a cos , we nd theinduced-charge zeta-potential to be

= 2E 0a cos 1 +

w

d /

d

D

. (6.3)

Thus unless the layer thickness d is much less than d D / w , the bulk of the potentialdrop occurs across the dielectric layer, instead of the double layer, resulting in areduced electro-osmotic slip velocity.


23/36


The modication to the charging time c for the coated cylinder can likewise beunderstood from this picture. The dielectric layer represents an additional (parallel-plate) capacitor of separation d and lled with a dielectric d , in series with thecapacitive screening cloud, giving a total capacitance

CT = w D1 + w d

d D 1

, (6.4)

and a modied RC time

c = D a

D1 +

w d d D

1, (6.5)

as found by Ajdari (2000) for a similar calculation for a compact (Stern) layer. Adiscussion of double-layer capacitance in the nonlinear regime, where the compactlayer is approximated by a thin dielectric layer is given by Macdonald (1954).

A full calculation of the induced zeta-potential around a dielectric cylinder of

radius a coated by another dielectric layer with outer radius b and thickness ( b a ) isstraightforward although cumbersome. (Note that the analogous problem for a coatedconducting sphere was treated by Dukhin (1986) as a model of a dead biological cell.)The resulting induced-charge zeta-potential is

= 2bE 0(1 + c)cos

2 + b(1 c) , (6.6)

with characteristic charging time scale

c = D b

D1 +

b

2 (1

c)

1, (6.7)

where c is dened to be

c = b2 + a 2 w / d (b2 a 2)b2 + a 2 + w / d (b2 a 2)

. (6.8)

It is instructive to examine limiting cases of the induced zeta-potential (6.6). In thelimit of a conducting dielectric coating d / w , we recover the standard resultfor ICEO around a metal cylinder, as expected. In the limit of a thin dielectric layerb = a + d , where d a , the induced zeta-potential is given by

( d a ) 2bE 0 cos 1 + d w / D d , (6.9)as found in (6.3), with a charging time

c( d a ) D b

D1 +

w d d D

1, (6.10)

as found in (6.5). Therefore, the ICEO slip velocity around a coated cylinder is closeto that of a clean conducting cylinder ( = 2 bE 0 cos ) only when the dielectric layeris much thinner than the screening length times the dielectric contrast, d D d / w .

The zeta-potential induced around a conducting cylinder with a thicker dielectriclayer,

( d D ) 2bE 0d Dw d

cos , (6.11)


24/36


is smaller by a factor of O ( D / d ), and the charging time,

c( d D ) w Dd d

D bD

, (6.12)

is likewise shorter by a factor of O ( D / d ). As a result, strong ICEO ow requires arather clean and highly polarizable surface, with minimal non-conducting deposits.

Note that the limit of a pure dielectric cylinder of radius b is found by taking thelimit a 0. This gives a zeta-potential of

(a = 0) 2bE 0 cos 1 + w b/ d D 2

d w

E 0 D cos , (6.13)

and an ICEO slip velocity which is smaller than the conducting case (3.7) by O ( D /b ).The charging time for a dielectric cylinder is given by

c(a = 0)

w

d

2D

D, (6.14)

as expected from (6.5) in the limit d D .

7. Induced-charge electro-osmosis: systematic derivationIn this section, we provide a systematic derivation of induced-charge electro-

osmosis, in order to complement the physical arguments above. We derive a set of effective equations for the time-dependent ICEO ow around an arbitrarily shapedconducting object, and indicate the conditions under which the approximations madein this article are valid. Starting with the usual electrokinetic equations for the

electrostatic, uid, and ion elds (as given by, e.g., Hunter 2000), we propose anasymptotic expansion that matches an inner solution (valid within the charge cloud)with an outer region (outside the charge cloud), and that accounts for two separatetime scales the time for the charge cloud to locally equilibrate and the time scaleover which the external electric eld changes.

Although it has mainly been applied to steady-state problems, the method of matched asymptotic expansions is well-established in this setting, where it is commonlycalled the thin double layer approximation. For simplicity, like many authors, weperform our analysis for the case of a symmetric binary electrolyte with a single ionicdiffusivity in a weak applied eld, and we compute only the leading-order uniformly

valid approximation. The same simplifying assumptions were also made by Gonzalezet al. (2000) in their recent asymptotic analysis of AC electro-osmosis.In the phenomenological theory of diffuse charge in dilute electrochemical systems,

the electrostatic eld obeys Poissons equation,

2 =

(n+ n)ew

, (7.1)

where n represent the local number densities of positive and negative ions. Theseobey conservation equationsnt

+ (n

v

) = 0 , (7.2)

where v represent the velocities of the two ion species,v =be

kB T b log n + u , (7.3)


25/36


where b is the mobility of the ions in the solvent and u is the local uid velocity.Terms in (7.3) represent ion motion due to ( a ) electrostatic forcing, ( b) diffusion downdensity gradients, and ( c) advection with the local uid velocity. The uid ow obeysthe Stokes equations, with body force given by the product of the charge density andthe electric eld,

2u p = e(n+ n) , (7.4)along with incompressibility. Strictly speaking, this form does not explicitly includeosmotic pressure gradients, as detailed below. However, osmotic forces can beabsorbed into a modied pressure eld p , and the resulting ow is unaffected.

For boundary conditions on the surface of the conductor, we require the uidow to obey the no-slip condition, the electric potential to be an equipotential (withequilibrium zeta-potential 0), and the ions to obey a no-ux condition:

u ( ) = 0 , (7.5) ( ) = 0, (7.6)

n v ( ) = 0 , (7.7)where n represents the (outer) normal to the surface . Far from the object, werequire the uid ow to decay to zero, the electric eld to approach the externallyapplied electric eld, and the ion densities n to approach their constant (bulk) valuen0.

In order to simplify these equations, we insert (7.3) in (7.2), and take the sum anddifference of the resulting equations for the two ion species to obtain

c + D 2c D2c eb [ce ] + u c = 0 , (7.8)c

e

D2

ce

eb

[c

] + u

ce

= 0 , (7.9)where we have used (7.1) and where we have dened

ce = n + + n2n0, (7.10)c = n + n. (7.11)

The rst variable, ce , represents the excess total concentration of ions, while thesecond, c , is related to the charge density via c = /e . The rst three terms in(7.8) represent a (possible) transient, electrostatic transport, and diffusive transport,respectively, and will be seen to give the dominant balance in the double layer. Thenext term represents the divergence of a ux of excess ionic concentration ce (but notexcess charge) due to an electric eld. The nal term represents ion advection withthe uid ow u . The analogous (7.9) for the charge-neutral ce lacks the electrostatictransport term. Both ce and c decay away from the solid surface and obey theno-ux boundary conditions (7.7),

n c | = ekB T

(2n 0 + ce) n | , (7.12) n ce| =

ekB T

c n | , (7.13)at the surface .

In what follows, we obtain an approximate solution to the governing equations(7.1), (7.8), and (7.9), at the leading order in a matched asymptotic expansion. We rstanalyse the solution in the inner region within a distance of order D of the surface.Non-dimensionalization yields a set of approximate equations for the inner region


26/36


and a set of matching boundary conditions which depend on the outer solution,valid in the quasi-neutral bulk region (farther than D from the surface). Similarly,approximate equations and effective boundary conditions for the outer region arefound. By solving the inner problem and matching to the outer problem, a set of effective equations is derived for time-dependent, bulk ICEO ows.

7.1. Non-dimensionalization and inner regionWe begin by examining the inner region, adjacent to the conducting surface. Denotingnon-dimensional variables in the inner region with tildes, we scale variables as follows:

r = D r , t = ( 2D )1 t , = 0 , u = U 0 u , c = 2 n0 c , c e = 2 n0 2ce , E = E 0 E(7.14)

where 0 and U 0 are potential and velocity scales (left unspecied for now), andwhere we have introduced the dimensionless surface potential,

= e 0kB T

. (7.15)

Note that ce scales with 2 to satisfy the dominant balance in (7.9). Finally, we havescaled time with the Debye time, D = ( 2D )1 = 2D /D , although the analysis willdictate another time scale, as expected from the physical arguments above.

The dimensionless ion conservation equations (7.87.9) become

c

t + c

2c 2 [ce ] Pe u c = 0 , (7.16)

ce

t + 2

ce

+

[c

] Pe u

ce

= 0 , (7.17)where we have introduced the P eclet number,

Pe = U 0D

. (7.18)

The boundary conditions (7.12)(7.13) in non-dimensional form are given by

n c | = (1 + 2ce) n | , (7.19) n ce| = c n | . (7.20)In the analysis that follows, we concentrate on the simplest limiting case. Weassume the screening length to be much smaller than any length L0 associated with

the surface geometry, parametrized through

= ( L 0)1 = D /L 0 1, (7.21)so that the screening cloud looks locally planar. As mentioned above, the singularlimit of thin double layers, 1, is the usual basis for the matched asymptoticexpansion. The regular limit of small P eclet number, Pe 1, which holds in almostany situation, is easily taken by setting Pe = 0. Finally, we consider the regular limitof small (dimensionless) surface potential,

1, which is the same as the limit that

allows the PoissonBoltzmann equation to be linearized. With these approximations,the system is signicantly simplied. First, c is coupled to ce only through terms of O ( 2) in (7.16) and boundary condition (7.19). Second, ce is smaller than c by afactor , and is thus neglected: c = 0 + O ( 2).


27/36


In this limit, we obtain the linear Debye equation and boundary condition for calone:

c

t + c = 2

2c , (7.22)

n

c

| = n

| . (7.23)The potential is then recovered from (7.1) in the form,

2 = c , (7.24)

with the far-eld boundary condition,

E 0L 0

E , (7.25)

and the (equipotential) surface boundary condition

( ) = 0 0, (7.26)

where 0 is the dimensionless equilibrium zeta-potential.Since 1, we introduce a locally Cartesian coordinate system {n, l}, where n islocally normal to the surface, and l is locally tangent to the surface. The governing

equations (7.22) and (7.24) are both linear in c and , which allows the electrostaticand ion elds to be expressed as a simple superposition of the equilibrium elds,

ceq = eq = 0en , (7.27)and the time-dependent induced elds ci and i, via

c = ceq + c

i , (7.28)

= eq + i. (7.29)

The induced electrostatic and ion elds must then obey

2 i = c

i , (7.30)

ci t

+ ci 2ci = 0 , (7.31)

subject to boundary conditions

i ( ) = 0 , (7.32) n ci = n i | , (7.33)

on the surface.Another consequence of the linearity of (7.22) and (7.24), and therefore of the

decoupling of the equilibrium and induced elds, is that different characteristic scalescan be taken for the two sets of elds. We scale the potential in the equilibriumproblem with the equilibrium zeta-potential 0, and the potential in the inducedproblem with E 0L 0, the potential drop across the length scale of the object. In orderthat the total surface potential be small ( 1), however, we require that the totalzeta-potential be small,

(| 0|+ |E 0L 0|)ekB T 1. (7.34)At the end of this section, we will briey discuss the rich variety of nonlinear effectswhich generally occur, in addition to ICEO, when this condition is violated.


28/36


We expect the charge cloud c and electric potential to vary quickly with n , butslowly with l (along the surface). Furthermore, we expect the charge cloud to exhibittwo time scales: a fast (transient) time scale, over which c reaches a quasi-steadydominant balance, and a slow time scale c , over which the quasi-steady solutionchanges. The physical arguments leading to (4.6) for the charging time suggest thatthis slow time scale is given by c = 1 / , which is not obvious a priori, but which willbe conrmed by the successful asymptotic matching.

In order to focus on the long-time dynamics of the induced charge cloud, andguided by the above expectations, we attempt a quasi-steady solution to (7.30)(7.33)of the form

ci = ci (n, l, t ), (7.35)

i = i (n, l, t ). (7.36)

It can be veried that

ci = A ( l, t ) en

A(l,

t )2 n en , (7.37)

i = A( l, t ) en + A( l, t )

2 [2 en + n en ] + B ( l, t ) + C ( l, t )n, (7.38)

solve the governing equations to O ( 2).The equipotential boundary condition (7.32) is satised when

A( l, t ) = B ( l, t ) + A( l, t ), (7.39)

so that the induced double-layer charge density is proportional (to leading order in) to the potential just outside the double layer. The normal ion ux condition (7.33),

A( l, t ) = C ( l, t ), (7.40)

relates the evolution of the double-layer charge density to the electric eld normalto the double layer. Matching the inner region to the outer region provides the nalrelations. In the limit n , the electrostatic and ion elds approach their limitingbehaviour

c 0, (7.41) B ( l, t ) + C ( l, t )n. (7.42)

7.2. Outer solutionWe now turn to the region outside the double layer, and use overbars to denote outernon-dimensional variables (scaled with length scale L0 and potential scale E 0L 0).According to (7.37) and (7.41), any non-zero charge density c that exists within theinner region decays exponentially away from the surface . A homogeneous solution(co = 0) thus satises (7.22) and the decaying boundary condition at innity. Withco = 0, (7.24) for the electrostatic eld o in the outer region reduces to Laplacesequation,

2 o = 0 , (7.43)with far-eld boundary condition (7.25) given by

o E . (7.44)To determine o uniquely, one further boundary condition on the surface isrequired, which is obtained by matching to the inner solution. The limiting value of


29/36


the outer eld o , o( r ) 0( ) + E( )n, (7.45)

must match the inner solution (7.42), which gives two relations

B ( l, t ) = 0 (l), (7.46)C ( l, t ) = E

(l). (7.47)

Using (7.47) in (7.40), the fact that C is O ( ) veries that the assumed time scale c = D / for induced double-layer evolution is indeed correct, as expected fromphysical arguments (4.6).

7.3. Effective equations for ICEO in weak applied eldsUsing the above results, we present a set of effective equations for time-dependentICEO that allows the study of the large-scale ows, without requiring the detainedinner solution. What emerges is a rst-order ODE for the dimensionless total surface

charge density in the diffuse double layer, q = A , so an initial value for q must bespecied. From (7.39) and (7.46), the outer potential on the surface is given by

o(, t ) = q (, t ), (7.48)

which, along with the far-eld boundary condition (7.25), uniquely species thesolution to Laplaces equation (7.43). From this solution, the normal eld E

(, t ) is

found, which (using (7.47) and (7.40)) results in a time-dependent boundary condition,

q t

= E

(, t ), (7.49)

which is more naturally expressed as

q t

= E

(, t ), (7.50)

in terms of the dimensionless time variable, t = t = t / c .We have thus matched the bulk eld outside the charge cloud with the inner

behaviour of the charge cloud in a self-consistent manner. The inner solutions for cand equilibrate quickly in response to the (slow) charging, and affect the boundaryconditions which determine the outer solution. Matching (7.47) and (7.40) results ina relation between the charge cloud and normal ionic ux, conrming the validityof (3.3) and (4.3), which were previously argued in an intuitive, physical manner.This analysis has demonstrated that errors for this approach are of O ( 2), O (Pe )and O ( 2). In this limit, the system evolves with a single characteristic time scale, c = D L 0/D , set by asymptotic matching, which physically corresponds to an RCcoupling, as explained above.

7.4. Fluid dynamics7.4.1. Fluid body forces: electrostatic and osmotic

Thus far, we have derived the effective equations for the dynamics of the induceddouble layer. To conclude, we examine the ICEO slip velocity that results from theinteraction of the applied eld and the induced diffuse layer. We demonstrate that, in

the limits of thin double layer and small surface potential, the Smoluchowski formula(2.4) for uid slip holds.Following Levich (1962, p. 484), we consider a conducting surface immersed in

a uid, with an applied background electrostatic potential o . In addition, a thin


30/36


double layer (with potential ) exists and obeys

2 =

(n + n)ew

= n0ew

ee/k B T ee/k B T , (7.51)so that the total potential = o + . Note that this assumes that the double layer is

in quasi-equilibrium.Fluid stresses have two sources: electric and osmotic. The electric stress in the uid

is given by the Maxwell stress tensor,

T ij = w E i E j 12 E E ij , (7.52)from which straightforward manipulations yield the electrical body force on the uidto be

F E T = w 2 w (o + )2. (7.53)Osmotic stresses come from gradients in ion concentration, and exert a uid bodyforce

F O = kB T (n+ + n) = e (n+ n) = w 2. (7.54)Thus the total body force on the uid, is given by the sum of F E and F O ,

F = w o2 = E B . (7.55)

Therefore, when both electric and osmotic stresses are included, the body force onthe double layer above a conductor is given by the product of the local chargedensity and the background electric eld E B = o (which, importantly, doesnot vary across the double layer). Note also that the same uid ow would result if the double-layer forcing ( E B + E ) were to be used, since the osmotic component(7.54) is irrotational and can be absorbed in a modied uid pressure.

7.4.2. ICEO slip velocityFinally, we examine the ICEO slip velocity that results when the electric eld odrives the ions in the induced-charge screening cloud .We look rst at the ow in the inner region of size D . For the ICEO slip velocity

to reach steady state, vorticity must diffuse across the double layer, which requiresa very small time ( D ) = 2D / 1010 s. Because is so much faster than thecharging time and Debye time, we consider the ICEO slip velocity to follow changesin or instantaneously. As noted above, the unsteady term may play a role incutting off the oscillating component of the bulk ow. However, our main concern

is with the steady, time-averaged component, which simply obeys the steady, forcedStokes equation.Non-dimensionalizing as above, we re-express the forced Stokes equations (7.4),

with forcing given by (7.55), using a stream function dened so that u l = n andu n = l . The stream function obeys

4 =

c

n l

c

l n

, (7.56)

where the stream function has been scaled by = ( w E 0 0/ ) . We perform a localanalysis around the point l = 0 of the ICEO ow driven by an applied tangential

electric eld {E , E}. Using representations of c

and around l = 0,c = c ( l) en , (7.57) = E l( l, n) En( l, n ), (7.58)


31/36


it is straightforward to solve (7.56) to O ( ). The tangential and normal ows are thengiven by

u l = w

(l)E (l) + l

E

(1 en ) + En

(3 (3 + n) en ) , (7.59)u n =

w

l

( E )(1 n en ), (7.60)where (l)E contains terms of O ( ). To leading order, then, the slip ows obey

u l w

(l)E (l) + O ( ) and un = O ( ). (7.61)

The tangential ow does indeed asymptote to (2.4), with local zeta-potentials andbackground eld, and the normal ow velocity is smaller by a factor of O ( ).

Thus an ICEO slip velocity is very rapidly established in response to an inducedzeta-potential and outer tangential eld. Furthermore, despite double-layer andtangential eld gradients, the classical HelmholtzSmoluchowski formula, (2.4),correctly gives the electro-osmotic slip velocity. This may seem surprising, giventhat the tangential eld vanishes at the conducting surface.

The nal step involves nding the bulk ICEO ow, which must be found by solvingthe unsteady Stokes equations, with no forcing, but with a specied ICEO slip velocityon the boundary , given by solving the effective electrokinetic transport problemabove.

7.5. Other nonlinear phenomena at large voltagesAlthough we have performed our analysis in the linearized limit of small potentials,it can be generalized to the weakly nonlinear limit of thin double layers, where(2.7) holds and Du 1. In that case, the bulk concentration remains uniform atleading order, and the HelmholtzSmoluchowski slip formula remains valid. Themain difference involves the time-dependence of double-layer relaxation, which isslowed down by nonlinear screening once the thermal voltage is exceeded. It canbe shown that the linear time-dependent boundary condition, (3.3) or (7.50), mustbe modied to take into account the nonlinear differential capacitance, as in (4.12).Faradaic surface reactions and the capacitance of the compact Stern layer may alsobe included in such an approach, as in the recent work of Bonnefont, Argoul &Bazant (2001).

The condition (2.7) for the breakdown of Smoluchowskis theory of electrophoresis(with = 0 + E 0a ) coincides with the condition c( ) D , where c takes intoaccount the nonlinear differential capacitance (4.14). When this condition is violated(the strongly nonlinear regime), double-layer charging is slowed down so much by

bazant 2004 induced

Documents