Pumping of liquids with traveling-wave electroosmosis

J. Appl. Phys. 97, 084906 (2005); https://doi.org/10.1063/1.1873034 97, 084906 © 2005 American Institute of Physics. Pumping of liquids with traveling-wave electroosmosis Cite as: J. Appl. Phys. 97, 084906 (2005); https://doi.org/10.1063/1.1873034 Submitted: 22 September 2004 . Accepted: 27 January 2005 . Published Online: 04 April 2005 A. Ramos, H. Morgan, N. G. Green, A. González, and A. Castellanos ARTICLES YOU MAY BE INTERESTED IN Fast ac electro-osmotic micropumps with nonplanar electrodes Applied Physics Letters 89, 143508 (2006); https://doi.org/10.1063/1.2358823 Electrokinetic micropump and micromixer design based on faradaic polarization Journal of Applied Physics 96, 1730 (2004); https://doi.org/10.1063/1.1767286 ac electro-osmotic micropump by asymmetric electrode polarization Journal of Applied Physics 103, 024907 (2008); https://doi.org/10.1063/1.2832624 Pumping of liquids with traveling-wave electroosmosis A. Ramosa兲 Departamento Electrónica y Electromagnetismo, University of Seville, Avenida Reina Mercedes s/n. 41012 Sevilla, Spain H. Morgan and N. G. Green School of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, United Kingdom A. González Departamento Física Aplicada, Escuela Superior de Ingenieros (E.S.I), University of Seville, Camino de los Descubrimientos s/n. 41092 Sevilla, Spain A. Castellanos Departamento Electrónica y Electromagnetismo, University of Seville, Avenida Reina Mercedes s/n. 41012 Sevilla, Spain 共Received 22 September 2004; accepted 27 January 2005; published online 4 April 2005兲 Net flow of electrolyte induced by a traveling-wave electric potential applied to an array of microelectrodes is reported. Two fluid flow regimes have been observed: at small-voltage amplitudes the fluid flow follows the direction of the traveling wave, and at higher-voltage amplitudes the fluid flow is reversed. In both cases, the flow seems to be driven at the level of the electrodes. The experiments have been analyzed with a linear electroosmotic model based upon the Debye–Huckel theory of the double layer. The electrical problem for the experimental interdigitated electrode array is solved numerically using a truncated Fourier series. The observations at low voltages are in qualitative accordance with the electroosmotic model. © 2005 American Institute of Physics.关DOI: 10.1063/1.1873034兴 I. INTRODUCTION The precise control of small masses of liquids is an important requirement in the development of “lab-on-a-chip” devices or micrototal analysis systems 共 ␮ TAS兲.1,2 Many techniques have been developed to pump liquids in microdevices, including micromechanical methods,3electroosmosis,4 electrowetting,5thermocapillary pumping,6electrohydrody- namic 共EHD兲pumping,7and ac electroosmosis.8In the latter, electrolytic solutions are pumped using asymmetric pairs of microelectrodes subjected to a low potential ac signal of the order of kilohertz. The physical mechanism that causes the electrolyte to move in ac electroosmosis is as follows. If the ac electric field is nonuniform, the component of the ac field normal to the electrode surface induces charge in the double layer, while the tangential component produces a body force on the induced charge, thus moving the fluid.9–14 The generated velocity field has a steady-state component which is observed experimentally. Each asymmetric pair of electrodes produces a nonsymmetric local flow that eventually generates a global flow in the direction of broken symmetry.8,10,15 Instead of using an asymmetric array and a single-phase ac signal, it is possible to use a traveling-wave signal applied to a symmetric array of electrodes.10,16,17 Contrary to EHD pumping,7which also employs a traveling-wave signal, traveling-wave electroosmotic 共TWEO兲pumping occurs because of electrical forces on the induced charge in the diffuse double layer, while EHD pumping occurs because of electrical forces on induced charges that appear in the liquid bulk due to gradients of conductivity and permittivity.18,19 In this paper we present some experimental observations of fluid pumping above an electrode array subjected to a travelingwave signal, and a theoretical, linear study of the TWEO pumping mechanism for such array. The traveling-wave electrode array is also used to induce dielectrophoretic motion of particles 共traveling-wave dielectrophoresis.20–23兲The mechanism responsible for particle motion can therefore on occasion be ambiguous. The analysis presented in this paper will aid in discriminating between particle motion induced by dielectrophoresis and particle motion due to fluid flow in traveling-wave- dielectrophoretic experiments. The mechanism responsible for the fluid pumping at low voltages is as follows. Above the electrode array there is an electrolyte, such as KCl, with conductivity ␴ and permittivity ␧. At the interfaces between the metal and electrolyte, and glass and electrolyte, double layers are formed. The characteristic thickness of a double layer is given by the Debye length, ␭D, which is much smaller than typical electrode dimensions. A traveling-wave potential is applied to the electrodes as shown in Fig. 1. Ions accumulate in the double layer in response to the applied signal. Due to the finite time during which the double layer is charged, a delay exists between the time of maximum-induced charge and the maximum applied signal. The electric field acts on the charge pulling the fluid in the direction of the traveling wave. The charging of the double layer is analogous to the charging of a capacitor 共the double layer兲through a resistor 共the ohmic a兲Author to whom correspondence should be addressed; electronic mail:[email protected] JOURNAL OF APPLIED PHYSICS 97, 084906 共2005兲 0021-8979/2005/97共8兲/084906/8/$22.50 © 2005 American Institute of Physics97, 084906-1 liquid兲.9,11–13 The charging time is therefore of order of ␶ c =共␧/ ␴ 兲共ᐉ/␭D兲, where ᐉis a typical length of the system and ␧/␭Dis the surface capacitance of the double layer. When the frequency of the applied signal is low ␻ ␶ cⰆ1, most of the applied voltage is dropped across the double layer, and the tangential electric field is small. Conversely, when the frequency is high ␻ ␶ cⰇ1, most of the applied voltage is dropped across the bulk electrolyte and the charge induced in the double layer is small. The frequency of maximum velocity is, therefore, of the order of ␻ c⬃1/ ␶ c, which is several orders of magnitude smaller than the charge relaxation frequency ␴ /␧. Fluid flow due to ac electroosmosis is observed in the region of this characteristic frequency ␻ c.8,9,11,13 Ehrlich and Melcher studied theoretically the fluid motion induced by traveling wave in a semi-insulating bipolar liquid.16 As in the present work, they took into account the charge induced in the double layer. For small field excitations, they obtained as the characteristic frequency of maximum velocity ␻ ⬃ ␴ /␧. However, they analyzed a system with a thick dielectric layer between the electrodes and the liquid, which is different from the system presented in this paper where the electrodes and the electrolyte are in contact. In the process of finalizing this manuscript, a paper on electroosmotic flows driven by traveling electric fields has appeared.17 These authors studied a traveling-wave device with a dielectric layer between the electrodes and the electrolyte, and obtained experimental frequencies of the order of ␴ /␧, as in the case studied by Ehrlich and Melcher. Their theoretical, linear analysis employs a single Fourier mode to describe the electrical potential at any frequency. Here we will present a Fourier series analysis in order to deal with the electrical boundary conditions, that is valid at frequencies ␻ Ⰶ ␴ /␧. In this paper we present experimental observations of unidirectional fluid flow due to the traveling electric field. We then describe a theoretical, linear analysis of travelingwave electroosmosis and compare this theory with the experimental results. II. EXPERIMENTAL OBSERVATION OF FLUID MOTION Observations of fluid motion were made on an array of N=20 interdigitated microelectrodes fabricated on a planar glass slide using photolithography. The microelectrodes were made from a metal trilayer of 10-nm titanium, 100-nm gold, and 20-nm titanium. The purpose of the top layer of titanium was to reduce the effects of corrosion at low frequencies. The array consisted of 20- ␮ m-wide electrodes separated by 20- ␮ m gaps. A diagram of the system used to generate the fluid flow is shown in Fig. 2. The electrodes were connected to an ac voltage of angular frequency ␻ and amplitude V0, i.e., V共t兲=V0cos ␻ t. The voltage on consecutive electrodes was phase shifted by 90°, so that the traveling wave had a wavelength L=160 ␮ m. A square glass chamber was constructed around the electrode array and filled with electrolyte. The electrode/electrolyte system could be observed both from above and from the side. A microscope objective and camera were pointed horizontally along the electrodes, so that the electrodes could be imaged in cross section.13 A solution of KCl in water of conductivity 2 mS/m was used for the electrolytic solution. The voltage amplitude V0ranged from 0 to 3 V, and the frequency f= ␻ /2 ␲ from 100 Hz to 100 kHz. Electrolysis was not observed at the electrodes because the capping layer of titanium forms a very thin insulating oxide layer. Fluorescent latex particles of 557 nm in diameter were used as tracers to observe the fluid flow. The movement of the particles was recorded on video and transferred to a computer. After applying the traveling-wave voltage to the microelectrode array, the fluid moved across the electrode array as shown by the composite image of particle tracks in Fig. 3. It was observed that the fluid flow was driven at the electrode FIG. 1. Diagram illustrating the physical mechanism of traveling-wave electroosmosis FIG. 2. Diagram showing the experimental arrangement of an electrode array used for traveling-wave electroosmosis. FIG. 3. Superimposed successive video frames showing the experimental streamlines for the case V0=1.2 V and f=1 kHz. 084906-2 Ramos et al. J. Appl. Phys. 97, 084906 共2005兲 surface, with highest velocities observed close to the electrode. For V0⬍2 V, a net flow was observed which was in the direction of the traveling wave, to the left in Fig. 3, as expected by the qualitative picture of TWEO described in the introduction. Because the chamber is closed at the right and left ends, the fluid recirculates, going from left to right at the top. Some small flow recirculation was also observed at the front and rear walls. In order to determine if the particle movement was indeed due to fluid flow, the expected dielectrophoretic velocity was calculated. For particles of 557 nm in diameter in this array, the horizontal dielectrophoretic velocity was estimated to be less than 0.5 ␮ m/satV0=2 V and at heights greater than 20 ␮ m.21,22 Therefore, the effect of dielectrophoresis on particle behavior can be neglected, except very close to the electrode edges, where positive dielectrophoresis can become important. For the images shown in Fig. 3, the frequency and voltage amplitude of the applied signal were 1 kHz and 1.2 V, respectively. At a height of 30 ␮ m from the surface, velocities of approximately 30 ␮ m/s were observed. Latex particles were seen to accumulate at the left edge of each electrode, probably due to dielectrophoresis. Figure 4 is a plot of the fluid velocity for different applied frequencies, for a height of approximately 64 ␮ m, and a voltage amplitude of 1 V. The figure shows that the fluid flow occurs in a frequency window centered at approximately 2 kHz. As expected from ac electroosmosis, this frequency is much smaller than the charge relaxation frequency ␴ /2 ␲ ␧共⯝0.5 MHz for the conductivity used in the experiments兲. The characteristic frequency of ac electroosmosis fc= ␻ c/2 ␲ ⬃680 Hz 共ᐉ=20 ␮ m兲is of the order of the observed frequency 2 kHz for maximum velocity. The fluid velocity was also measured at three different heights 共29, 42, and 54 ␮ m兲as a function of voltage, and this is shown in Fig. 5. The frequency of the applied signal was 1 kHz. The estimated averaged velocity at the level of the electrodes is also shown. This was estimated by assuming the velocity profile to be parabolic 共as a function of height兲, and extrapolating back to the level of the electrodes 共0 ␮ m兲. The measured velocities can be approximated to a parabolic profile by setting the velocity to zero at h=97 ␮ m due to the recirculating flow and at h=219 ␮ m due to the nonslip condition at the upper wall. Theoretically, a parabolic function is obtained from the Stokes equation with no volumetric force and assuming a unidirectional flow. Although there is scatter in the data, this is not significant, and saturation is observed above a voltage amplitude of V0=1 V. For voltage amplitudes V0艌2 V and at 1 kHz, the fluid flow changed direction, a phenomenon that is not expected from our understanding of ac electroosmosis. The observed recirculation of the tracking particles was an indication of fluid motion. Figure 6 shows the velocity of the fluid as a function of height in the channel at three different voltages for a frequency of 1 kHz. It can be seen that the velocity is nearly zero at the midpoint of the channel for the lower voltages 共1.2 and 1.4 V兲, but that the change in behavior at 2.4 V is noticeable, both in terms of the flow direction and the large jump in fluid velocity, particularly close to the electrodes. In this case, the velocities at a height of 20 ␮ m above the electrodes were around 150 ␮ m/s for a voltage amplitude of 2.4 V and frequency of 1 kHz. In general, higher voltages were FIG. 4. Measured velocity as a function of frequency for V0=1 V and at a height of 65 ␮ m from the electrodes. FIG. 5. Measured velocity as a function of voltage at f=1 kHz for three different heights, and estimated velocity at the level of the electrodes. FIG. 6. Measured velocity as a function of height for three different voltage amplitudes and f=1 kHz. 084906-3 Ramos et al. J. Appl. Phys. 97, 084906 共2005兲 required in order to observe this kind of reversal flow at higher frequencies. These voltages are too low to ascribe the flow to the electrothermal mechanism for this conductivity.19 Reversal of the fluid direction was also observed above a similar threshold voltage when gold electrodes were used. Reversals of fluid flow with increasing voltage have also been observed using arrays of asymmetric pairs of electrodes,24,25 which could be related to our observations for traveling-wave arrays. Although the effect is reproducible, the origin of this phenomenon is not clear. The numerical solutions given by Ehrlich and Melcher16 showed some cases of reversed pumping, although at angular frequencies of the order of ␴ /␧, much higher than in our experiments. Their numerical and analytical studies always indicated forward pumping at frequencies ␻ ␧/ ␴ Ⰶ1. At voltages much greater than the thermal voltage kBT/e, concentration polarization should become important and lateral concentration gradients could produce diffusio-osmosis.26,27 Also, Faradaic charge injection from the electrodes to the electrolyte could be possible without observable electrolysis. Washabaugh et al.28 observed forward and backward pumping of semi-insulating liquids pumped by a traveling-wave potential. They assumed that charge injection was responsible for the fluid motion. As the charge is injected, the sign of the charge layer close to the electrodes could change and, therefore, the direction of electrical force and slip velocity. However, their liquids are quite different to ours. For semi-insulating liquids, the applied electric field is usually much greater than the typical electric field in the double layer kBT/e␭D. For our system this electric field would be around 106V/m, greater than typical applied field required for reverse pumping, which is of the order of 105V/m 共2 V/20 ␮ m兲. Very recently, fluid flow of electrolyte subjected to ac voltages with velocities of the order of 1 mm/s has been reported,29 and the mechanism responsible for the fluid flow has been ascribed to Faradaic charging. Although, the electrode geometry is quite different to the traveling-wave array, the underlying mechanism could be the same. III. THE LINEAR ELECTROKINETIC PROBLEM Linear electrokinetic theory was used to study the system shown in Fig. 2. As shown in the figure, the system is modeled as an insulating substrate 共glass兲onto which an infinite periodic array of microelectrodes is fabricated. The electrodes are considered to be infinitely long and thin with width d1and spacing d2. Four ac signals of the same amplitude V0and angular frequency ␻ , and relative phases 0°, 90°, 180°, and 270° are applied to the array, so that the voltage applied to consecutive electrodes is phase shifted by 90°. The spatial period of the system is L=4d1+4d2. A symmetrical electrolyte, such as KCl, is placed on top of the array. In most cases, the Debye length ␭Dis of the order of 10 nm and is negligibly small compared to the other lengths in the system, which are of the order of 10 ␮ m. To a first approximation we assume that the voltage applied to the electrodes is low enough such that electrolysis of the electrolyte does not occur. If the frequency of the applied signal is also low enough, i.e., ␻ Ⰶ ␴ /␧, the double layer can be assumed to be in quasiequilibrium.30 Under these conditions the bulk electrolyte behaves in a resistive manner and the double layer considered to be an ideal capacitor. Therefore, the electrical potential in the bulk electrolyte satisfies Laplace’s equation ⵜ2⌽=0. 共1兲 The boundary condition on the electrode surface describes the charging of the double layer due to the current in the bulk, ⳵ qs/ ⳵ t=− ␴ Ey, with qsthe charge per unit area in the double layer. For sufficiently low voltage across the diffuse layer 共⌬⌽⬍kBT/e兲, there is a linear relationship between the surface charge and the voltage drop across the double layer. In this case, the surface charge conservation equation can be written using phasors as ␴⳵ ⌽ ⳵ y=i ␻ CDL共⌽−Vj兲,共2兲 where i=冑−1, CDL is the capacitance per unit of area of the whole double layer 共diffuse plus compact or Stern layers兲,⌽ is the potential just outside the double layer, and Vjis the potential applied to electrode j. At the interface between the electrolyte and the glass a similar boundary condition holds. For frequencies ␻ Ⰶ ␴ /␧, this boundary condition can be simplified to13,15 ⳵ ⌽ ⳵ y=0. 共3兲 In defining both boundary conditions we have implicitly ignored any lateral conduction current along the double layer. Also, any current due to convection in the system has been neglected. The boundary condition at y→⬁is the potential tends to zero. Once the electric potential has been solved, the electroosmotic fluid velocity at the surface of the electrodes can be calculated. For diffuse double layers in quasiequilibrium on perfectly polarizable metal surfaces, the electroosmotic slip velocity is given by the Helmholtz–Smoluchowski formula,13,31 u=␧⌬⌽ ␩ Ex.共4兲 Here ␩ is the viscosity of the fluid, ⌬⌽=⌽−⌽0is the potential drop across the diffuse double layer 共⌽0is the potential at the nonslip plane兲, and Exis the tangential electric field outside the double layer. For our problem, both ⌬⌽ and Exare oscillating functions of time, of frequency ␻ . The time-averaged slip fluid velocity at the interface between the double layer and the bulk is12,13 具u典=− ␧ 4 ␩ ⌳ ⳵ ⳵ x关共⌽−Vj兲共⌽−Vj兲*兴,共5兲 where ⴱindicates complex conjugate and ⌳is the ratio of the diffuse double-layer voltage to the total double-layer voltage. The parameter ⌳is given by ⌳=Cs/共Cs+Cd兲, where Cdand Csare the capacitances of the diffuse and Stern or compact layers, respectively. For the glass/electrolyte interface, an estimate of the potential drop across the diffuse double layer shows that this is negligibly small and, from Eq. 共4兲, the 084906-4 Ramos et al. J. Appl. Phys. 97, 084906 共2005兲 electroosmotic velocity on the glass is negligible.15 To obtain the bulk velocity, the Navier–Stokes equations are solved in the limit of small Reynolds number, usually very small in microsystems. In the steady state, and in the absence of externally applied body forces, the equations reduce to ␩ ⵜ2u−ⵜp=0, ⵜu=0, 共6兲 where pis the pressure and uthe velocity, u=uex+vey. The boundary conditions for these equations at y=0 are 共a兲the tangential velocity is equal to the slip velocity on the electrodes, given by Eq. 共5兲; 共b兲the tangential velocity is zero at the glass; 共c兲the normal velocity is zero for any xat y=0共electrodes and glass兲. Far from the electrodes in the normal direction, the fluid can be considered to be free of stress, ⳵ u/ ⳵ y=0 and v=0. Given the periodicity of the electrode array we look for periodic solutions in the xdirection. The velocity field can be obtained from an integration of the slip velocity calculated from the electrical problem.15 Defining the stream function ⌿by the pair of derivatives ⳵ ⌿/ ⳵ y=uand ⳵ ⌿/ ⳵ x=−v, the function ⌿that is solution of the Stokes problem in the bulk and satisfies the abovementioned boundary conditions can be written as15 ⌿=1 L 冕 0 L dx⬘u共x⬘,0兲ysinh共k0y兲 cosh共k0y兲− cos 关k0共x−x⬘兲兴,共7兲 where u共x⬘,0兲is the slip velocity calculated from the electrical problem and k0=2 ␲ /L. If a net fluid flow occurs, the velocity tends to u=Uexfor y→⬁, where Uis a constant. This condition presumes that the upper boundary of any actual device is very much higher than the wavelength of the problem. The value of the pumping velocity Uis equal to the average value of the slip velocity over a wavelength, i.e., U=共1/L兲兰0 Ludx. This equality can be seen, for instance, from expression 共7兲taking the leading terms when k0yⰇ1. IV. MATHEMATICAL ANALYSIS A. Single potential wave We first present the analysis for the case of a single potential wave which is simpler than the potential wave for the four-phase electrode array. If the potential applied at the level of the electrodes is approximated by a single potential wave of angular frequency ␻ and wave number k0=2 ␲ /L,of the form V0ei共 ␻ t−k0x兲, the solution for the electrical potential in the bulk is of the form ⌽=Ae−k0yei共 ␻ t−k0x兲. Applying the charge conservation boundary condition at y=0 ␴⳵ ⌽ ⳵ y=i ␻ CDL共⌽−V0e−ik0x兲,共8兲 we obtain A=i⍀V0 1+i⍀,共9兲 where ⍀= ␻ CDL/共 ␴ k0兲. The generated electroosmotic slip velocity is u=− ␧ 2 ␩ Re 冋 ⌬⌽ ⳵ ⌽* ⳵ x 册 =⌳␧k0V0 2 2 ␩ ⍀ 1+⍀2.共10兲 The velocity uis independent of the coordinate x, then the pumping velocity U=u. The maximum velocity as a function of frequency is obtained for ⍀=1 and is U=⌳␧k0V0 2 4 ␩ .共11兲 Equation 共10兲is in complete agreement with the velocity expression for low frequencies given in Ref. 17. Ehrlich and Melcher studied traveling-wave-induced motion in a bipolar semi-insulating liquid theoretically.16 Their model described the generation and recombination of bipolar carriers. Unlike our case, their system had a thick dielectric solid, which acted as a conduit wall, between the electrodes and the liquid. They studied the case of a pure traveling wave at different ranges of electric-field amplitude and signal frequencies. For small field amplitudes, they obtained a maximum velocity of 0.2␧E0 2␭D/ ␩ for a frequency ␻ = ␴ /␧. Here E0is the normal electric field inside the double layer at the nonslip plane. This frequency of maximum velocity is much higher than for our case, ␻ = ␴ k0/CDL ⬃k0␭D ␴ /␧. Analyzing their expression of the slip velocity for small field and low frequency 共 ␻ Ⰶ ␴ /␧兲 u=␧E0 2␭D 2 ␩ ␻ ␧ ␴ ,共12兲 we can reconcile their results with ours. In this expression, E0is independent of frequency. However, if the electrodes are in contact with the electrolyte 共or separated by a very thin insulator, e.g., oxide layer as for Ti electrodes兲then the field is a function of frequency. In this case, the charging time of the double layer ␶ c=␧L/ ␴ ␭Dbecomes crucial, as explained in the Introduction. The voltage across the diffuse double layer and, therefore, the field inside the double layer E0are maximum at ␻ ␶ cⰆ1, and decrease with increasing frequency. Equating E0 2to 兩⌬⌽兩2/␭D 2in Eq. 共12兲, and taking into account the frequency dependency of ⌬⌽, we recover the slip velocity expression 共10兲. Equation 共11兲indicates that the maximum slip velocity decreases as the thickness of the dielectric layer increases, through the scaling factor ⌳. With a thick dielectric conduit, as in the case studied by Ehrlich and Melcher, the slip velocity becomes inversely proportional to the thickness of the dielectric layer. Therefore, larger voltages are needed to produce fluid velocities of the same order of those obtained working with electrodes in contact with the electrolyte. B. The traveling-wave array The electrical potential for the four-phase traveling-wave electrode array can be solved numerically using Fourier series. The electrical potential can be written as ⌽=⌽1+⌽2, 084906-5 Ramos et al. J. Appl. Phys. 97, 084906 共2005兲 where ⌽1is the solution for a problem with boundary conditions as shown in Fig. 7共a兲and ⌽2as shown in Fig. 7共b兲. Furthermore, the boundary conditions for ⌽2are those for ⌽1shifted a distance 共d1+d2兲with V0changed to −iV0. Therefore, ⌽2can be written as ⌽2共x,y兲=−i⌽1关x−共d1+d2兲,y兴.共13兲 For ⌽1,x=0 is a line of even symmetry, i.e., ⳵ ⌽1/ ⳵ x =0, while x=d1+d2=L/4 is a line of odd symmetry, i.e., ⌽1=0. Taking these into account, the general form of ⌽1 using Fourier series is ⌽1=兺 n=0 ⬁ Ancos共knx兲e−kny,共14兲 with kn=共2n+1兲2 ␲ /L. The boundary conditions at y=0 for ⌽1impose −兺 n=0 ⬁ knAncos共knx兲=−iKV0+iK兺 n=0 ⬁ Ancos共knx兲 0⬍x⬍d1/2, 共15兲 −兺 n=0 ⬁ knAncos共knx兲=0 d1/2 ⬍x⬍d1/2 + d2,共16兲 −兺 n=0 ⬁ knAncos共knx兲=iK兺 n=0 ⬁ Ancos共knx兲 d1/2 + d2⬍x⬍d1+d2,共17兲 where K=CDL ␻ / ␴ . Multiplying the Fourier representation of ⳵ ⌽1/ ⳵ yby cos共kmx兲and integrating in the interval from 0 to d1+d2we arrive at −kmdAm=−iKV0cm+iK兺 n=0 ⬁ Anamn,共18兲 where d=共d1+d2兲/2 and cm= 冕 0 d1/2 cos共kmx兲dx,共19兲 amn = ␦ mnd− 冕 d1/2 d1/2+d2cos共knx兲cos共kmx兲dx .共20兲 We solve the linear system 共18兲numerically by truncating the series. The potential of the original problem is ⌽=⌽1共x,y兲−i⌽1关x−共d1+d2兲,y兴 =兺 n=0 ⬁ An兵cos共knx兲−icos关kn共x−2d兲兴其e−kny, =兺 n=0 ⬁ An关cos共knx兲−i共−1兲nsin共knx兲兴e−kny =兺 n=0 ⬁ Ane−knye−iknx共−1兲n.共21兲 The potential is the superposition of waves propagating alternately in the positive or negative xdirection depending on whether nis even or odd, respectively. From the potential, the electroosmotic slip velocity can be calculated and therefore the average velocity in a wavelength, the pumping velocity U. Because of the symmetry the average velocity on each electrode should be equal, U=1 L 冕 0 L udx =4 L 冕 −d1/2 d1/2 udx =⌳␧ L ␩ 共兩⌽−V0兩x=−d1/2 2−兩⌽−V0兩x=d1/2 2兲.共22兲 Another expression can be obtained from Eq. 共2兲 u=− ␧ 2 ␩ Re 冋 ⌬⌽ ⳵ ⌽* ⳵ x 册 =⌳␧ 2 ␩ Re 冋 i ␴ ␻ CDL ⳵ ⌽ ⳵ y ⳵ ⌽* ⳵ x 册 . 共23兲 This last expression is also valid on the gaps between electrodes, where ⳵ ⌽/ ⳵ y=0 and u=0. The pumping velocity is, therefore, U=⌳␧ 2 ␩ LRe 冋 i ␴ ␻ CDL 冕 0 L ⳵ ⌽ ⳵ y ⳵ ⌽* ⳵ xdx 册 .共24兲 Taking into account the Fourier expression of the potential U=⌳␧ 2 ␩ ␴ ␻ CDL 兺 n=0 ⬁ 共−1兲nkn 2兩An兩2.共25兲 Sometimes in experiments, the traveling wave is generated using square-wave signals applied to the electrodes instead of sinusoidal signals.7,17 We extend the present analysis to traveling-square-wave signals as follows. In order to compute the slip velocity, the voltage applied to the electrodes 共V0in a half period and −V0in the other兲should be written as a superposition of sinusoidal waves: electrodes 1 and 3 are subjected to V1=−V3=兺p=0 ⬁Bpcos关共2p+1兲 ␻ t兴with Bp =4V0共−1兲p/ ␲ 共2p+1兲, while electrodes 2 and 4 are subjected to V2=−V4=兺p=0 ⬁Bp ⬘sin关共2p+1兲 ␻ t兴with Bp ⬘=4V0/ ␲ 共2p+1兲. As a consequence, the potential in the system is a superposition of potentials with frequencies ␻ ,3 ␻ ,5 ␻ ,共2p+1兲 ␻ , FIG. 7. Boundary conditions for ⌽1and ⌽2. 084906-6 Ramos et al. J. Appl. Phys. 97, 084906 共2005兲 i.e., the complex potential is ⌽=兺p=0 ⬁⌽pei共2p+1兲 ␻ t. It can be seen that the potentials ⌽pwith p=even are traveling waves going to the right in Fig. 2 while the potentials with p =odd go to the left. The time-averaged slip velocity generated is u=−␧ ␩ 冓 ⌬⌽ ⳵ ⌽ ⳵ x 冔 =− ␧ 2 ␩ 兺 p=0 ⬁ Re 冋 ⌬⌽p ⳵ ⌽p * ⳵ x 册 .共26兲 Each term of the series can be computed using the numerical scheme of the present section, taking into account the corresponding frequency 共2p+1兲 ␻ , voltage amplitude 4V0/ ␲ 共2p +1兲, and direction of the wave, i.e., u=兺p=0 ⬁共−1兲pup. V. NUMERICAL RESULTS AND DISCUSSION AT LOW VOLTAGE The numerical convergence was verified through multiple runs of increasing number of functions. The estimated error is less than 0.1% using n=64 functions. For the case d1=d2=d, the maximum pumping velocity as a function of frequency is U=0.0502⌳␧k0V0 2/ ␩ obtained at a frequency ␻ =1.4492 ␴ k0/CDL. A plot of the pumping velocity 共in units of ⌳␧k0V0 2/4 ␩ 兲as a function of frequency 共in units of ␴ k0/CDL兲is shown in Fig. 8. For comparison, the pumping velocity obtained from a pure traveling wave, Eq. 共10兲, and from a traveling square wave applied to the electrode array is also presented. The computations show that the maximum slip velocity in a four-phase microelectrode array with d1 =d2is around five times smaller than the maximum velocity obtained by a pure potential traveling wave of voltage amplitude V0and wave number k0=2 ␲ /L. The main reason for this difference is that the slip velocity is zero on the gaps between electrodes, the resulting pumping velocity being the average over a wavelength. The pumping velocity generated by a four-phase square wave of maximum voltage V0shows a maximum pumping velocity as a function of frequency of U=0.0762⌳␧k0V0 2/ ␩ . If only the first harmonic of the Fourier series in ␻ is employed 关curve 共c兲in Fig. 8 multiplied by 16/ ␲ 2兴, the maximum value is around 7% greater than the one obtained using the complete Fourier series. The streamlines generated by the slip velocity, Eq. 共7兲, are shown in Fig. 9 for two frequencies: ⍀=⍀M/5 and ⍀ =5⍀M, with ⍀M=1.45 the nondimensional frequency of maximum pumping velocity. In this figure, the traveling wave goes from right to left in order to compare with experimental streamlines. As the frequency increases the slip velocity is less uniform on top of each electrode. The experimental streamlines show a qualitative agreement with the computed ones. These computed streamlines can be compared with those obtained numerically15 and experimentally32 for asymmetric pairs of electrodes. The fluid flow for TWEO is more uniform, and this can be an advantage in circular chromatographic applications.33 We can also compare the theoretical slip velocity for TWEO and for ac electroosmosis in asymmetric electrodes in a representative case. The predicted pumping velocity for the asymmetric array of Brown et al.8 is given in Ref. 15, U=0.52⌳V0 2mm/s. Here V0is the amplitude of the ac voltage applied to each electrode, i.e., the maximum potential difference between electrodes is 2V0. The theoretical velocity for a TWEO experiment with an array of the same wavelength 共L=50 ␮ m兲is U =4.5⌳V0 2mm/s. The TWEO velocity is a factor of the order of 10 greater for the same voltage amplitude and, in conclusion, it suggests that the TWEO is more efficient. The predicted frequency for maximum velocity in our experiments is f=776/⌳Hz. A value of ⌳⬃0.4 is needed in order to obtain the experimental value around 2 kHz. The existence of a layer of oxide that naturally forms on top of the titanium can account for this factor. Also, in Ref. 13 we had a better agreement with experiments if we used a measured double-layer impedance in boundary condition 共2兲, rather than the impedance given by perfect capacitors, i.e., 1/共i ␻ CDL兲. It is clear that the experimental velocities presented in Fig. 5 do not show a quadratic dependence on voltage. This is not completely unexpected since the Debye–Huckel approximation of the double layer 共linear theory兲can only be used tentatively for voltages across the diffuse layer greater than the thermal voltage 共kBT/e=0.025 V兲. For our system, the linear theory predicts a maximum pumping velocity U =⌳1.38 V0 2mm/s 共V0in volts兲. A factor ⌳⬃0.05 is needed to fit the experimental velocities for low voltages 共V0 FIG. 8. Pumping velocity U共in units of ⌳␧k0V0 2/4 ␩ 兲vs nondimensional frequency ⍀in TWEO: 共a兲single mode, 共b兲square waves applied to electrode array, 共c兲and sinusoidal waves applied to electrode array. FIG. 9. Streamlines for the cases: 共a兲⍀=⍀M/5, 共b兲⍀=5⍀M. Here ⍀M =1.45, the nondimensional frequency of maximum velocity. 084906-7 Ramos et al. J. Appl. Phys. 97, 084906 共2005兲 ⬃0.5 V兲. The oxide layer on top of the titanium electrode can account in part for the reduction of the experimental velocities. Other factors that can reduce the maximum achievable velocity are the three-dimensional nature of the flow, together with the front window that will increase the viscous friction. The traveling wave experiments presented in Ref. 17 show a frequency of maximum velocity of the order of ␴ /␧ because of the existence of an insulating layer between the electrodes and the electrolyte. Although those results cannot be analyzed with the present theory, valid for ␻ ␧/ ␴ Ⰶ1, we can try to draw some conclusions. Cahill et al. obtained a quantitative difference between their predicted and experimental velocities. They simulated the imposed voltage as a pure sinusoidal traveling wave. When we take into account the actual geometry of our electrode array 共very thin electrodes separated by gaps, mounted on an insulating glass substrate兲, the calculated velocity is around five times smaller than the one expected from a single mode of frequency ␻ , wave number k, and voltage amplitude V0. Our results indicate that a quantitative comparison between theoretical and experimental slip velocities should take into account the actual geometry and material properties of the system. We would expect the theoretical velocities calculated by Cahill et al. to be smaller when the actual shape of the electrodes is included in the computations. With respect to the slip velocity distribution, we would expect that this is much smaller on the gaps than on the electrodes. VI. CONCLUSIONS Net flow of electrolyte induced by a traveling-wave potential applied to an array of microelectrodes has been reported. The observed flows seem to be driven at the level of the electrodes. The results have been analyzed with a linear electroosmotic model based upon the Debye–Huckel theory. The electrical problem for the experimental interdigitated electrodes was solved numerically using a truncated Fourier series. The linear analysis shows qualitative accordance with the observations at low voltages. At applied voltage amplitudes greater than 2 V and at frequency of 1 kHz, the fluid flow changed direction. By increasing the voltage, this reversal flow can easily be ten times faster than the one at low voltages. More experimental and theoretical research is needed to clarify the origin of this kind of flow. The possibility of switching the fluid flow direction with increasing voltage may be of interest in the lab-on-a-chip technology.25 We have analyzed a problem with LⰆhⰆw, where his the height of the upper wall and wthe depth of the array. In this situation, since the electroosmotic slip velocity is U, the maximum flow rate that can be generated by the array of electrodes is F=Uhw/2, taking into account that the upper wall boundary condition is of nonslip. The pressure drop that can be generated in a unit wavelength is given by pb =12 ␩ LF/wh3=6 ␩ UL/h2. For the experimental array and a slip velocity of 50 ␮ m/s, this gives pb⬃10−3 Pa per wavelength. In order to increase the pressure drop, the number of cells should be increased but even to generate 1 Pa would require a pump with 1000 unit cells; 160 mm for this design of pump. Clearly, the height of a device and the wavelength could be reduced in order to increase the pressure drop. 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