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JOURNAL OF APPLIED PHYSICS 102, 1 共2007兲

Simultaneous mixing and pumping using asymmetric microelectrodes Byoung Jae Kim, Sang Youl Yoon, and Hyung Jin Sunga兲

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Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu, Daejeon, 305-701, Republic of Korea

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Charles G. Smith

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共Received 29 June 2007; accepted 8 August 2007兲

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This study proposes ideas for simultaneous mixing and pumping using asymmetric microelectrode arrays. The driving force of the mixing and pumping was based on electroosmotic flows induced by alternating current 共ac兲 electric fields on asymmetric microelectrodes. The key idea was to bend/ incline the microelectrodes like diagonal/herringbone shapes. Four patterns of the asymmetric electrode arrays were considered depending on the shape of electrode arrays. For the diagonal shape, repeated and staggered patterns of the electrode arrays were studied. For the herringbone shape, diverging and converging patterns were examined. These microelectrode patterns forced fluid flows in the lateral direction leading to mixing and in the channel direction leading to pumping. Three-dimensional numerical simulations were carried out using the linear theories of ac electro-osmosis. The performances of the mixing and pumping were assessed in terms of the mixing efficiency and the pumping flow rate. The results indicated that the helical flow motions induced by the electrode arrays play a significant role in the mixing enhancement. The pumping performance was influenced by the slip velocity at the center region of the channel compared to that near the side walls. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2794375兴

Semiconductor Physics Group, Cavendish Laboratory, Madingley Road, Cambridge, CB3 0HE, United Kingdom

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24 I. INTRODUCTION

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Alternating current 共ac兲 electro-osmosis has recently re26 ceived great attention due to its remarkably low working 27 voltages for moving liquid in a microchannel. While direct 28 current 共dc兲 electro-osmosis, a traditional method, mostly 29 needs several kilovolts at electrodes, ac electro-osmosis re1–8,10,11 30 quires generally very low voltage, less than about 5 V. 31 The operation under such low voltage has many important 32 advantages: 共a兲 reducing the electrode degradation caused by 33 Faradaic reactions, which lengthens the life span of a micro34 fluidic device; 共b兲 preventing air bubble generation; and 共c兲 35 not changing significantly the properties of working fluid. ac 36 electro-osmotic flow was explored by Green’s group and 37 their earlier studies were focused on the symmetric 1–4 Adjari5 predicted a net flow by introducing 38 electrodes. 39 asymmetric arrays of electrodes. Thereafter, ac electro40 osmosis was developed into a microfluidic pump. Brown et 6 7 41 al. verified the net flow experimentally. Ramos et al. con42 ducted numerical simulations using the analytical theory es3 43 tablished by González et al. and searched for the optimal 44 geometry of electrodes for maximal pumping. Moreover, 8 45 Olesen et al. found the global optimum condition for maxi46 mal pumping and investigated the effects of the channel 47 height, the Faradaic reaction in the vicinity of electrodes, and 48 nonlinear surface capacitance of the Debye layer. Most pre49 vious studies have been for pumping based only on a two1–8 50 dimensional model. Most of the previous microfluidic pumps using ac 51

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electro-osmosis have generally straight electrodes perpendicular to the channel direction. Each electrode pair consists of a narrow electrode and a wide electrode, and the pairs are periodically displaced on the bottom wall. When an ac voltage is applied to each electrode pair, a slip velocity occurs at the electrode surface. The direction of the resulting net flow is toward the wide electrode. In the meantime, the work by Stroock et al.9 has been recognized as a breakthrough in the field of micromixers, thanks to its good mixing performance. The fluids inside the channel globally rotate along the channel wall while going downstream, by the help of grooves with diagonal/herringbone shapes on the bottom wall. However, there is a limitation as a passive mixer, i.e., it requires an additional pump to work. In fact, micropumps and micromixers have been independently developed and their performances are considered to be sufficient. The purpose of this study is to realize a microfluidic device for simultaneous mixing and pumping, making use of the principles of the ac electro-osmosis pump and the staggered herringbone mixer. The key idea is to bend/incline the microelectrode strips like diagonal/herringbone shapes. Consequently, such an arrangement of electrodes produces a slip velocity with the lateral direction for mixing and with the channel direction for pumping. This kind of multifunctional device 共simultaneous mixing and pumping兲 is expected to be useful in microfluidic systems such as lab-on-a-chip. Most importantly, the number of small devices on a microfluidic system can be reduced. This means that the microfluidic system can be miniaturized further or can work with more functions by adding other devices. In addition, the possibility of malfunction of the system can be reduced because of the smaller number of parts on the microfluidic system.

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Author to whom correspondence should be addressed. Tel.: 82-42-8693027; FAX: 82-42-869-5027. Electronic mail: [email protected]

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© 2007 American Institute of Physics

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J. Appl. Phys. 102, 1 共2007兲

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Numerical simulations were performed to test the effects of the proposed ideas. In contrast to the previous studies, the 86 present study employed three-dimensional simulations. The 87 electrode geometries were the same as those adopted by 6 88 Brown et al., and the optimal frequency for the maximal 89 pumping corresponding to the dimensions followed Olesen 8 90 et al. The channel height was determined such that a large 91 recirculating flow can be generated above the leading edges 92 of the wide electrodes. The slip velocity for fluid flow was 93 obtained by solving the Laplace equation, the flow motion 94 was simulated by using the Stokes equation, and then the 95 concentration distribution was obtained through the 96 convection-diffusion equation. Four patterns of the asymmet97 ric electrode arrays were chosen depending on the shape of 98 electrode arrays. Concerning the diagonal shape, repeated 99 and staggered patterns of the electrode arrays were studied. 100 For the herringbone shape, diverging and converging pat101 terns were examined. The pumping ability was assessed by 102 the flow rate and the mixing degree was quantified by the 103 mixing efficiency. 85

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105 A. Theory of ac electro-osmosis

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104 II. NUMERICAL METHODS

The present study uses the linear theory established by 3 107 González et al. 共2000兲, which has been a basis of the works 7 8 108 of Ramos et al. 共2003兲, Olesen et al. 共2006兲, and Urbanski 10 109 et al. 共2007兲. The previous studies were made using the 110 following assumptions for symmetric electrolyte: 共a兲 the 111 electrical double layer in the vicinity of electrodes can be 112 approximated as a simple capacitor showing linear behav113 iors. This assumption is theoretically valid for low voltages 114 applied to the electrodes; 共b兲 the electrolysis of electrolyte 115 does not occur; 共c兲 the electrodes are ideally polarizable. In 116 other words, the free charge flow between electrolyte and 117 electrodes is negligible; 共d兲 the applied frequency is even 118 lower than the charge relaxation frequency of electrolyte dur119 ing whose period the electrical double layer forms 共␻ 120 = 2␲ f ␴ / ␧兲, where ␴ and ␧ are the electrical conductivity 121 and the permittivity of electrolyte, respectively. Under this 122 condition, the double layer can be assumed to be in quasi123 equilibrium; 共e兲 the ions within the double layer are trans124 ported mainly by the electrical field rather than fluid flow. 125 This assumption enables us to solve the fluid velocity inde126 pendently of the electrical potential problem. A schematic diagram of the geometrical layout of the 127 128 channel used in the present work is given in Fig. 1. The 129 electrolyte motion is governed by the Stoke equation, 106

FIG. 1. Cross-sectional view of the asymmetric electrodes displaced in parallel on the insulating wall. The pairs of asymmetric electrodes are positioned periodically.

具Uslip典 = −

␧ ⵜs兩Vext − ␾兩2 , 4␮共1 + ␦兲

共2兲 137

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where ⵜs stands for a gradient along the electrode surface and ␦ is the capacitance ratio of the diffuse layer to the Stern layer when the layers are modeled as capacitances, Vext the external voltage applied to electrodes, ␾ the electrical potential of electrolyte. Since 兩Vext − ␾兩 is the potential drop across the double layer, we have to know the potential distribution inside the electrolyte, specifically at the boundaries where the diffuse layer ends. The electrical problem associated with 兩Vext − ␾兩 shows a harmonic motion synchronized with ac. It is therefore convenient to deal with Eq. 共2兲 in a phase mode, so that Vext and ␾ are complex variables circulating with identical frequency in the phase domain. The electrolyte outside the double layer is assumed to be electrically neutral. Accordingly, the potential of the electrolyte is governed by the Laplace equation,


n · ⵜ␾ = 0,

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n · ⵜ␾ −

␧␻ ␧␻ i␾ = − Vext , ␴␭D共1 + ␦兲 ␴␭D共1 + ␦兲

共5兲

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where p, ␮, and U are the static pressure, the absolute viscosity, and the velocity vector of fluid, respectively. This equation is appropriate for microfluidics where the inertial term is negligible compared with the viscous term. No-slip velocity is imposed on the insulating walls, whereas timeaveraged slip velocity is set on the electrode surface,7

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The boundary conditions at the insulating walls and the elec- 154 trodes are given as follows, respectively: 155

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共1兲

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131 132 133 134 135 136

0 = − ⵜp + ␮ⵜ2U,

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ⵜ2␾ = 0.

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where n denotes the normal vector to the surface, ␭D the Debye length of the diffuse layer, and i the unit imaginary value.7 Equations 共1兲–共5兲 can be rewritten with the dimensionless parameters listed in Table I. The characteristic TABLE I. Parameters used in the present study. The parameters with the superscript “*” are dimensionless. Parameter

Definition

* Vext *

=Vext / V0 =␾ / V0 =W1 / G1 =W2 / G1 =G2 / G1 =h / G1 =G1␧␻ / ␴␭D共1 + ␦兲 =␧V20 / ␮G1共1 + ␦兲

␾ W*1 W*2 G*2 H* ␻* U0

Electrical potential applied to electrodes Electrical potential of electrolyte Width of the narrow electrode Width of the wide electrode Distance between electrode pairs Channel height ac frequency Characteristic velocity 关m/s兴

157 158 159 160 161


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TABLE II. Detailed conditions for three-dimensional simulations.

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FIG. 2. Comparison of the slip velocity.

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n · ⵜ ␾ − ␻ i␾ = − *

*

*

*

* ␻*Vext .

共6兲共7兲

In order to validate the numerical scheme developed, we compared it with the previous studies.7,8 The dimensions of * * * 170 the geometry are W1 = 1, W2 = G2 = 10/ 3, and the frequency is * 171 ␻ = 1.24. The channel width 共lateral length兲 is much longer 172 than the channel height. In addition, the channel height is 173 sufficiently high such that the top wall does not affect the 174 flow structure anymore. The simulation domain included one 175 pair having narrow and wide electrodes. Periodic boundary 176 conditions were imposed on the inlet and the outlet of the 177 domain. Figure 2 shows the slip velocity profile along the 178 electrode surface. The narrow electrode is located from x* 179 = 0 to 1. Only one reference is plotted in Fig. 2, since the 180 results for other two references are the same. Our data are in 8 181 excellent agreement with those of Olesen et al., which 182 verify the accuracy of our simulations. 168 169

Value

Parameters

Value

V0 G1 W*1 W*2 G*2 H* ␻*

3V 5 ␮m 1 5 3 5 1.12

U0 ␧ ␮ ␴ ␭D

0.2805 m/s 6.943⫻ 10−10 C2 / N m2 0.000 891 Pa s 0.014 68 S/m 10 nm 4 5160

␦ Pe

electrical double layer. In addition, the flow structure is not affected by the applied voltage provided that the frequency remains unchanged from the linearity of the Stokes equation. Thus, if the flow rate is obtained for one applied voltage 共V0 = 3 V兲, the flow rates can be readily calculated for other applied external voltages. The mixing performance is evaluated by observing how the concentration distribution of a chemical reagent changes along the downstream direction. The concentration distribution is obtained by the convectiondiffusion equation, U * · ⵜ *c * =

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1 * * 典 = − ⵜs*兩Vext − ␾ *兩 2 , 具Uslip 4

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length 共G1兲 is the distance between the narrow and wide 163 electrodes, and the characteristic voltage 共V0兲 is the peak-to164 peak voltage applied to the electrodes. Equations 共2兲 and 共5兲 165 are converted into a simple dimensionless form as follows: 162

Parameters

1 *2 * ⵜ c , Pe

共8兲

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where the Peclet number is Pe= U0G1 / D = 5610. The corresponding diffusion coefficient is D = 2.5⫻ 10−10 m2 / s, which is a general value for commonly used chemical reagents. Figures 3 and 4 depict a variety of electrode patterns on the bottom wall for simultaneous mixing and pumping. Hereafter, the dimensionless parameters will be expressed without the superscript “*.” Figure 3 shows two different patterns of electrode pairs having diagonal shapes: a repeated pattern 关Fig. 3共a兲—type A兴 and a staggered pattern 关Fig. 3共b兲—type B兴. The lengths of two patterns are L = 39.04 and 51.04, respectively, and the widths are the same as l = 12. Each pattern is repeated with a period of L on the bottom wall. The first electrode pair for each type starts from the point 共1,0兲. The arrows indicate the overall direction of fluid flow, predicted by the fact that the net flow is from the narrow electrode to the wide electrode in the case of two-dimensional flow. Accordingly, simultaneous mixing and pumping can be

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In this study, the simulations are extended into threedimensional geometries, because the electrodes are inclined against the channel direction. The widths of electrodes and their gaps are the same as those of Brown et al.6 The detailed simulation conditions are given in Table II. The applied frequency 共␻* = 1.12兲 is optimal for maximal pumping in two190 dimensional simulations with the present geometrical 8 191 dimensions. The potential applied to the electrodes is V0 192 = 3 V, which is free of bubble generation at the present elec11 193 trode gap 共5 ␮m兲. The working fluid is potassium chloride 194 solution with 10−3 M, whose properties are almost the same 195 as those of water, except for the electrical conductivity. The 196 temperature is 25 ° C. The capacitance ratio ␦ = 4 is taken 2,10 197 from the previous studies. 198 The slip velocity is simply proportional to the square of 199 the applied voltage, according to the linear theory of the 184 185 186 187 188 189

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183 B. Simulations

FIG. 3. Two patterns of the electrode pairs with a diagonal shape 共top view兲. Each pattern is repeated with period of L on the bottom wall. 共a兲 Repeated shape 共type A兲; 共b兲 staggered shape 共type B兲.

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J. Appl. Phys. 102, 1 共2007兲

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FIG. 4. Two patterns of the electrode pairs with a staggered herringbone shape. Each pattern is repeated with period of L on the bottom wall. 共a兲 Diverging shape 共type C兲; 共b兲 converging shape 共type D兲. 228

achieved through the diagonal flows. Because the driving force caused by the slip velocity is divided into the channel and lateral directions, the pumping performance is then ex231 pected to be reduced to some extent, compared with the gen232 eral pump having electrode lines perpendicular to the chan233 nel direction. The pumping may be useful for short channels, 234 but not for long ones, because the latter may need a high 235 pressure gradient to propel liquid. Figure 4 shows two stag236 gered herringbone patterns: a diverging case of fluid flow 237 关Fig. 4共a兲—type C兴 and a converging case of fluid flow 关Fig. 238 4共b兲—type D兴. The starting point of the first electrode is 239 共1,0兲. As shown, the difference between types C and D is 240 whether the diagonal flows converge to the center region or 241 diverge to the side walls. The staggered patterns facilitate the 242 mixing enhancement further. The patterns in Figs. 3 and 4 243 are periodically positioned on the bottom wall. 244 In order to determine the channel height for the above 245 four types, we conducted preliminary two-dimensional simu246 lations. Two different flow structures are illustrated in Fig. 5. 247 The main difference is whether a recirculating flow is made 248 in the region at a distance vertically from the leading edge of 249 the wide electrode. The channel height 共H = 5兲 was selected 250 for the present three-dimensional simulations because of the 251 relatively complicated flows for mixing. 229 230

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FIG. 6. 共Color online兲 Type A 共top view兲. 共a兲 Streamlines; 共b兲 slip velocity in the x direction. Symbols + and − indicate some nearby points where the slip velocity is high and low, respectively. 共c兲 Slip velocity in the y direction.

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repeated pattern of electrode pairs with a diagonal shape. The streamlines can be divided into two kinds. One is that the fluid in the vicinity of the starting point of each electrode pair flows along the leading edge of the wide electrode, showing helical motions. This unexpected flow may be attributed to the height effect explained in Fig. 5共a兲 and the fluid flow diagonally approaches the side wall. There are of course helical fluid motions along the narrow electrodes. However, their sizes are so small, compared to the helical motions relating to the wide electrodes, that they are not drawn in Fig. 6共a兲. The main spiral motions in Fig. 6 are initiated mostly from the left and lower region of the first wide electrode 共x = 3 , y = 0兲 and terminated at the right and upper region 共x = 15, y = 12兲. The other is that the fluid being at a distance from the starting point of each electrode pair tends to gather in the middle region of the electrodes and pass through the electrodes. This flow is similar to the expected flow in Fig. 3共a兲. The flow direction is just perpendicular to the electrode lines, having the cross-sectional flow structure shown in Fig. 5共a兲. These two representative flow patterns appear again in the region of the second electrode pair. In this manner, the upper and lower fluids in Fig. 6 exchange places on going downstream. In the meantime, the slip velocity is almost symmetric throughout the narrow electrodes, but it is asymmetric throughout the wide electrodes. Such an asymmetrical slip velocity distribution plays a primary role in inducing a net flow, which is consistent with previous studies. As for the wide electrodes, the slip velocity in the x direction 共uslip兲 is strong at the points where the leading edges contact with the side walls 关some nearby points are marked with the symbol + in Fig. 6共b兲兴, while the slip velocity in the y direction 共vslip兲 is dominant in the middle region. The flow streamlines for the staggered pattern of electrode pairs with diagonal shapes 共type B兲 are displayed in Fig. 7共a兲. Similar streamlines are observed on the electrodes, due to the same diagonal shape as type A. As mentioned


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FIG. 5. Flow structures for two channel heights.

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The numerical results are analyzed in terms of the flow streamlines, slip velocity, and concentration distribution in the center. Figure 6 displays the representative flow streamlines and the slip velocity on the electrode for type A, the

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252 III. RESULTS AND DISCUSSION

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J. Appl. Phys. 102, 1 共2007兲

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FIG. 7. 共Color online兲 Type B 共top view兲. 共a兲 Streamlines; 共b兲 slip velocity in the x direction; 共c兲 slip velocity in the y direction.

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兩c − c⬁兩dA

A

e共x兲 = 1 − 303

共9兲

, 兩c0 − c⬁兩dA

A

305

311

1, 6 ⱕ y ⱕ 12

冎

.

共10兲

The outlet condition is n · ⵜc = 0.

共11兲

This means that the dominant mass transfer is made by fluid convection. The mixing efficiency e共x兲 was calculated at the outlet 315 共x = L兲. The concentration distributions 共z = 2.5兲 for types A 316 and B are shown in Fig. 8. One interesting point here is that 317 the concentration distribution shows a cross pattern in the 318 region above the wide electrode. More precisely, the concen319 tration 共6 ⱕ y ⱕ 12兲 is going down and up again after the 320 region above the wide electrode, and the concentration 共0 321 ⱕ y ⱕ 6兲 is moving toward the direction parallel to the wide 322 electrode. These behaviors can be easily imagined according 323 to the flow streamlines in Figs. 5共a兲, 6共a兲, and 7共a兲, and by 324 the fact that the concentration distribution is closely related

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310

0, 0 ⱕ y ⬍ 6

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where c = c共y , z兲 is the cross-sectional concentration profile at a distance x downstream, and c0 and c⬁ are the concentration 306 profile and the concentration associated with an initial state 12 307 and a completely mixed state, respectively. In this study, 308 the inlet concentration profile 共x = 0兲 is specified by 304

c共y,z兲 =

with fluid convection. Types A and B show good mixing; however, type A is better than type B. The reason is, as mentioned above, that type B has a larger space between two electrode pairs so that the fluid flows without any disturbance in the space, as ascertained in Fig. 8共b兲. Analysis is extended to see the relation between the mixing enhancement and the helical motions observed. Toward this end, the fluid velocity field and the concentration distribution are put in the same figure. Figure 9 is a series of cross-sectional views of the velocity field 共v , w兲 and the concentration distribution, going downstream for type A. A small vortical structure appears first near the narrow electrode 关Figs. 9共a兲 and 9共b兲兴, and then a large vortical structure occurs by the presence of the wide electrode 关Figs.

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FIG. 8. 共Color online兲 Mixing patterns. 共a兲 Type A; 共b兲 type B.

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earlier, the fluid near the starting point of each electrode pair 295 flows along the leading edge of the wide electrodes, showing 296 spiral motions. A difference from type A is that there is a 297 larger space between the two electrode pairs, where the fluid 298 flows without any disturbances. Such a large space would 299 make the mixing performance of type B inferior to that of 300 type A. 301 To assess the mixing degree quantitatively, we intro302 duced a mixing efficiency, defined by

312 313 314


FIG. 9. 共Color online兲 Series of cross-sectional views of the concentration distribution and the velocity field 共v , w兲 along the downstream for type A. 共a兲 x = 1; 共b兲 x = 2; 共c兲 x = 4; 共d兲 x = 6; 共e兲 x = 10; 共f兲 x = 13; 共g兲 x = 17; 共h兲 x = 21.

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J. Appl. Phys. 102, 1 共2007兲

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FIG. 12. 共Color online兲 Mixing patterns. 共a兲 Type C; 共b兲 type D.

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FIG. 10. 共Color online兲 Type C 共top view兲. 共a兲 Streamlines; 共b兲 slip velocity in the x direction; 共c兲 slip velocity in the y direction. 339

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9共c兲–9共f兲兴. We can see clearly that the large vortical structure 340 moves in the right direction; at the same time, a large con341 centration packet is delivered by the fluid convection. A simi342 lar progress through Figs. 9共a兲–9共f兲 appears again from Fig. 343 9共g兲 at the appearance of the next electrode pairs. In view of 344 the results so far achieved, one can say that the helical mo345 tions along the leading edge of the wide electrode play a 346 significant role in the mixing enhancement. For the two arrangements of electrode pairs with a her347 348 ringbone shape, the flow streamlines and concentration dis349 tributions are exhibited in Figs. 10–12. As expected, two 350 cases in Figs. 10 and 11 show spiral streamlines along the 351 leading edge of the wide electrodes. Inspection of Figs. 10 352 and 11 indicates that the spiral motions of type C are more 353 activated than those of type D. This can be explained by the 354 slip velocity distribution. The slip velocity 共uslip兲 for type C 355 is high at the points where the leading edge of the electrodes

contacts with the side walls, while it is high at the bending points of the leading edges of the electrodes for type D. Consequently, the helical motions for type D are initiated from the middle region of the channel 共the bending points of the electrodes兲, differently from the other three cases. In the meantime, the helical motions are induced from the fluid flow diagonally approaching the side wall. Therefore, type D has few opportunities to activate the spiral motions because the direction of diagonal flows is toward the center of the channel, instead of the fixed side walls. Such a weak spiral fluid motion affects the mixing performance. Figure 12 verifies that the mixing performance of type D is inferior to that of type C. For a direct comparison among the four types, the flow rate and mixing efficiency are calculated and summarized in Table III. The flow rate 共Q = 兰udydz兲 was calculated at the outlet 共x = L兲. A reference is shown for comparison, which is the typical pump case whose electrode lines are straight and perpendicular to the channel direction. The dimensions of the electrodes are W1 = 1, G1 = 1, W2 = 5, and G2 = 3. The total channel length is L = 40 and the channel width is l = 12. First, the mixing efficiencies 共e兲 are greatly enhanced by the present four types, compared with the reference in which mixing is made only by diffusion. As mentioned above in qualitative terms, the mixing efficiencies of types A and C are higher than those of types B and D, respectively. Closer inspection of the data in Table III indicates that the mixing efficiency is increased at a cost of the flow rate. The flow rates of types A, B, and C are reduced to the extent of

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TABLE III. Pumping performance 共Q: flow rate兲 and mixing efficiency 共e兲. c⬁ is the concentration when completely mixed.

FIG. 11. 共Color online兲 Type D 共top view兲. 共a兲 Streamlines; 共b兲 slip velocity in the x direction; 共c兲 slip velocity in the y direction.


Type

Q

e

c⬁

Reference A B C D

0.708 0.461 0.422 0.274 0.733

0.127 0.948 0.863 0.949 0.769

0.501 0.411 0.447 0.412 0.594

356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384


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TABLE IV. 兰uslipdA is the integration value of the x-direction slip velocity. Type

兰uslipdA

uslip,max

uslip,min

A B C D

9.174 10.93 10.07 9.978

1.81 1.85 1.87 1.76

−2.60 −2.66 −2.64 −2.63

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40– 65% of the reference. One point that should be noted 386 here for type D is that, even though the mixing degree is 387 comparatively reduced to those of the other three cases, the 388 pumping and mixing are increased at the same time to the 389 reference value. Such a desirable behavior may be explained 390 by the slip velocity distribution. Recall that the electrodes of 391 type D are arranged to lead to the higher slip velocity 共uslip兲 392 at the center region of the channel and the slower slip veloc393 ity at the electrode edges near the side walls. The energy 394 going into fluid flow from the electrodes is not directed to the 395 side walls where the viscous damping slows the fluid down. 396 On the other hand, the slip velocities in the other three types 397 are higher at the edges close to the side walls. 398 In order to validate the above analysis, we integrated the 399 slip velocity 共uslip兲 throughout the electrodes, which is asso400 ciated with the pumping ability. In Table IV, 兰uslipdA is the 401 integration of the x-direction slip velocity over two electrode 402 pairs. As shown, the values of 兰uslipdA are almost the same. 403 This suggests that different flow rates in Table III are caused 404 by different slip velocity distributions. The distributions of 405 slip velocity for the reference and type D are shown in Fig. 406 13. The reference profile corresponds to the slip velocity 407 profile shown in Fig. 5共a兲. The slip velocity of the reference 408 does not vary in the y direction. However, the slip velocity 409 for type D is mostly positive in the center region of the 410 channel. Moreover, its magnitude is higher than the refer411 ence. This supports that the higher pumping is induced by 412 the contribution of higher slip velocity in the center region of 413 the channel.

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nel using asymmetric electrode arrays. The asymmetric electrode arrays were arranged to enforce fluid flows to the lateral direction for mixing and to the channel direction for pumping. Four patterns of the electrode pairs were chosen with diagonal/herringbone shapes. Concerning the diagonal shape, repeated and staggered patterns of the electrode arrays were studied. For the herringbone shape, diverging and converging patterns were examined. The pumping ability was assessed using the flow rate, and the mixing degree was quantified by a mixing efficiency. As a result, the spiral motion of flow along the leading edges of the wide electrodes played a major role in the mixing enhancement. The mixing performance was greatly increased for all four cases, compared with the reference case in which mixing occurs only by diffusion. The mixing efficiencies for types A, B, and C were increased at a cost of the pumping performances. Their pumping flow rates were reduced to the extent of 40– 65% of the reference one. However, for type D, even though the mixing degree was comparatively inferior to those of the other three cases, the pumping and mixing were increased at the same time compared with the reference. Such a desirable behavior was attributed to the slip velocity distribution. The slip velocity is higher at the center region of the channel and slower at the region near the side walls, so that the energy going into fluid flow from the electrodes is not directed to the side walls where the viscous damping slows the fluid down. On the other hand, the slip velocity for types A, B, and C is low in the middle region of the channel and high near the side walls, so that the slip velocity does not make significant contributions to the pumping because of the viscous damping effect of the side walls.

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N. G. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, Phys. Rev. E 61, 4011 共2000兲. 2 N. G. Green, A. Ramos, A. González, H. Morgan, and A. Castellanos, Phys. Rev. E 66, 206305 共2002兲. 3 A. González, A. Ramos, N. G. Green, A. Castellanos, and H. Morgan, Phys. Rev. E 61, 4019 共2000兲. 4 A. Ramos, H. Morgan, N. G. Green, and A. Castellanos, J. Colloid Interface Sci. 217, 420 共1999兲. 5 A. Ajdari, Phys. Rev. E 61, R45 共2000兲. 6 A. B. D. Brown, C. G. Smith, and A. R. Rennie, Phys. Rev. E 63, 016305 共2000兲. 7 A. Ramos, A. González, A. Castellanos, N. G. Green, and H. Morgan, Phys. Rev. E 67, 056302 共2003兲. 8 L. H. Olesen, H. Bruus, and A. Ajdari, Phys. Rev. E 73, 056313 共2006兲. 9 A. D. Stroock, S. K. W. Dertinger, A. Ajdari, I. Mezić, H. A. Stone, and G. M. Whiteside, Science 295, 647 共2002兲. 10 J. P. Urbanski, J. A. Levitan, D. N. Bruch, T. Thorsen, and M. Z. Bazant, J. Colloid Interface Sci. 309, 332 共2007兲. 11 M. Mpholo, C. G. Smith, and A. B. D. Brown, Sens. Actuators B 92, 262 共2003兲. 12 D. Li, Electrokinetics in Microfluidics 共Elsevier Academic, New York, 2004兲.

P JA

FIG. 13. Comparison of the slip velocity.


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This work was supported by the Creative Research Ini- 449 tiatives of the Korean Science & Engineering Foundation. 450

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A detailed numerical analysis has been performed to elucidate the simultaneous pumping and mixing in a microchan-

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414 IV. CONCLUSIONS

ACKNOWLEDGMENT

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