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  Characterization of electroosmotic flow in rectangular microchannels Cheng Wang, Teck Neng Wong * , Chun Yang, Kim Tiow Ooi School of Mechanical and Aerospace Engineering, Nanyang Technological University, Nanyang Avenue 50, Singapore 639798, Singapore Received 5 August 2005; received in revised form 18 July 2006Available online 22 February 2007 Abstract In this paper, the electroosmotic displacing process between two solutions (namely the same electrolyte of different concentrations) ina rectangular microchannel is studied theoretically and experimentally. Firstly, the electric potential and velocity field in a rectangularmicrochannel are obtained by solving the governing equations. Fourier transform method is used to solve the electrolyte concentrationprofile equation. The electric current versus time curve through the microchannel is predicted based on the concentration profileobtained. The current monitoring technique is then used to study the electroosmotic displacing process. The results from the measuredcurrent–time relations agree well with those from the prediction, suggesting a reliable theoretical model developed in this study.   2007 Elsevier Ltd. All rights reserved. Keywords:  Theoretical model; Electroosmotic; Rectangular microchannel; Solution displacement 1. Introduction The study in microfluidics is rapidly becoming a veryimportant area of research due to numerous potentialapplications in separation and analysis. Performing biolog-ical analysis on a microfluidics based bioMEMS or lab-on-a-chip usually involves sample preparation, treatment,injection, delivery, separation and detection. Most sub-stances acquire surface electric charges when in contactwith an aqueous (polar) medium. The rearrangement of the charges on the solid surface and the balancing chargesin the liquid results in the formation of the electrical doublelayer (EDL) [1]. If an electric field is applied tangentiallyalong a charged surface, the electric field will exert a bodyforce on the ions in the diffuse layer, resulting in electroos-motic flow (EOF). EOF can provide a very flat velocityprofile, thus avoiding smearing [2]. Besides, other advanta-ges of electroosmotic pumping such as valve-less switching,accurate control of transportation and manipulation of liquid sample by an electrical field and with no solidmoving parts make electroosmosis a preferred method fortransporting liquids in microfluidics.Burgreen and Nakache [3] studied the effect of the sur-face potential on liquid transport through ultrafine capil-lary slits assuming Debye–Hu¨ckel linear approximationfor the electrical potential distribution under an imposedelectrical field. Rice and White [4] discussed the same prob-lem in narrow cylindrical capillaries. Levine et al. [5]extended the Rice and Whitehead’s model to cases involv-ing high zeta potentials. Yang and Li [6] studied the elec-trokinetic effects of pressure-driven flow in rectangularmicrochannels using a finite difference scheme. In Yang’sstudy, the motion of the liquid with electrokinetic effectswas analytically solved by employing the Green functionformulation. Mala et al. [7] reported a study of microchan-nel flow and heat transfer in two parallel plates. Tsao [8]studied the electroosmosis through an annulus withDebye–Hu¨ckel linear approximation and Kang et al. [9]studied the electroosmosis in annuli with high zeta poten-tials by introducing a correction to account for the finitethickness of the EDL and geometrically related factors.More recently, the alternating current (AC) electroosmosisin rectangular microchannels has been studied by Yanget al. [10], Erickson and Li [11], and Marcos et al. [12]. 0017-9310/$ - see front matter    2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2006.11.035 * Corresponding author. Tel.: +65 67905877; fax: +65 67911895. E-mail address:  mtnwong@ntu.edu.sg (T.N. Wong).www.elsevier.com/locate/ijhmtInternational Journal of Heat and Mass Transfer 50 (2007) 3115–3121  In applications, electroosmotic pumps have been fabricatedby packing micron sized particles into fused silica capillar-ies [13] and by using standard micromachining techniques[14].In studying the electrokinetic phenomena, one of themost important characterization parameters is zeta poten-tial. In many practical situations, it is difficult to obtain areliable estimate of zeta potential, because zeta potentialcannot be measured directly. Different ways to measurethe value of zeta potential have been proposed, theseinclude a conventional microelectrophoresis method [15]and the micro particle image velocimetry (micro-PIV) tech-nique [16]. In the latter, Yan et al. [16] formulated an expression to determine the electrophoretic velocity of thetracer particles and the EOF field of the microchannelthrough the measurement of the steady velocity distribu-tions of the tracer particles in both open- and closed-endrectangular microchannels under the same water chemicalconditions. Another simple experimental method, termedthe current monitoring method, was first suggested byHuang et al. [17] in which the electroosmotic flow ratewas monitored using the electric current change, whenone solution is electrokinetically displaced by the samesolution with a slight difference in ionic concentration.Arulanandam and Li [18] employed this method to evalu-ate the zeta potential and the surface conductance. In theseexperiments, the average velocity of electroosmosis flowwas determined by measuring the time required to com-pletely displace a solution by another similar solution inthe capillary tube. The zeta potential value was determinedby fitting the measured average velocity to the theoreticalmodel developed in the study [18]. Ren et al. [19] improved the current monitoring method by using the slope of themeasured correlation of the current versus time to achievea better accuracy. The electroosmotic displacing process ina circular capillary was later studied numerically by Renet al. [20] using a finite control volume method to solve amathematical model accounting for three zones of solution,namely solution 1 zone, mixing zone and solution 2 zone.Analytically, Ren et al. [21] developed mathematicalformulations for the displacing process in the cylindricalcapillaries based on two models: the sharp interface andthe mixing zone models.The above mentioned cases studied the electroosmoticflow displacing processes in cylindrical capillaries. How-ever, in practice, the microchannel networks in lab-on-a-chip platforms are usually fabricated by microelectronicfabrication techniques [22,23], and these channels are rect-angular rather than cylindrical in cross section. This paperpresents an in-depth analysis of electroosmotic displacingprocess of one solution with another in rectangular micro-channels. A mathematical model is developed to describethe electroosmotic flow in a rectangular microchannel.The model includes solving the Poisson–Boltzmann’s equa-tion for EDL potential distribution and the modifiedNavier–Stokes equation for the electroosmotic flow fieldby the separation of variables method, and the mass trans-port equation using Fourier transform method. Based onthe mass transport equation, the average electroosmoticvelocity is determined by measuring the time needed totransport the electrolyte across the microchannel. The zetapotential is also obtained. The theoretical predictions forthe current–time relationship are compared with measuredresults. Good agreement is obtained. 2. Mathematical model  2.1. Electrical double layer potential distribution A straight rectangular microchannel of width 2 W  ,height 2 H   and length  L  is shown in Fig. 1. In this theoret-ical model, the channel wall is assumed to be uniformlycharged so that the electrical potential in the EDL variesin the  x  and  y  directions only. Due to the symmetry of the potential and velocity fields, the solution domain canbe reduced to a quarter cross section of the channel. It isfurther assumed that the electric charge density is uninflu-enced by the external electric field due to thin EDLs; there-fore the charge convection can be ignored and thus theelectric field equation and the fluid flow equation aredecoupled [24]. Introducing the dimensionless parameters:  X   ¼  x D h ; Y   ¼  y  D h ; W ¼  z  0 ekT   w  and  D h  ¼  4  HW   H  þ W   , and assuming asmall zeta potential, the electric potential due to chargedwall can be described by the linearized Poisson–Boltz-mann’s equation which can be written in terms of dimen-sionless variables [12] as r 2 W ¼  K  2 W ;  ð 1 Þ where  K   ¼ j  D h  is the ratio of the length scale  D h  tothe characteristic double layer thickness 1/ j . Here  j  isthe Debye–Hu¨ckel parameter, given by j ¼  e r e 0 k  b T  2  z  20 e 2 n 0   1 = 2 ; where  z 0  is the valence,  e  is the fundamental electric charge, w  is the electric potential,  e r  is the relative permittivity, and e 0  is the permittivity in vacuum. Fig. 1. Schematic of the rectangular microchannel and the coordinatesystem used for modeling.3116  C. Wang et al./International Journal of Heat and Mass Transfer 50 (2007) 3115–3121  The boundary conditions along the symmetrical lines is o W o  X   ¼ 0 at  X   = 0 and  o W o Y   ¼ 0 at  Y   = 0. At the channel wall, W ¼  f  at  X   ¼  W   D h and  Y   ¼  H  D h . Using the separation of vari-able method, the solution to the linearized Poisson–Boltz-mann equation gives W ð  X  ; Y  Þ¼ 4  f X 1 n ¼ 1 ð 1 Þ n þ 1 cosh  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ð 2 n  1 Þ 2 p 2  D 2h 4  K  2 W    2 q     KY    ð 2 n  1 Þ p cosh  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ð 2 n  1 Þ 2 p 2  D 2h 4  K  2 W    2 q     KH  D h    cos  ð 2 n  1 Þ  D h p 2 W   X    þ 4  f X 1 m ¼ 1 ð 1 Þ m þ 1 cosh  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ð 2 m  1 Þ 2 p 2  D 2h 4  K  2  H  2 q     KX    ð 2 m  1 Þ p cosh  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ð 2 m  1 Þ 2 p 2  D 2h 4  K  2  H  2 q     KW   D h    cos  ð 2 m  1 Þ  D h p 2  H  Y    :  ð 2 Þ The ionic net charge density in the EDL can be expressedby [25]  q e  ¼ 2 W :  ð 3 Þ  2.2. Electroosmotic flow field  The motion of an incompressible fluid is governed bythe Navier–Stokes equation, which is expressed as q o V o t   þ q ð V    rÞ V ¼r  p  þ F þ l r 2 V    ;  ð 4 Þ where  V  is the velocity vector,  p  is the pressure,  F  is theexternal force,  q  and  l  are the density and dynamics viscos-ity of the fluid [26].Using the following assumptions(1) The fluid is Newtonian;(2) the properties of the fluid are independent of localelectric field, thus only diluted solutions are consid-ered in this study;(3) the fluid’s properties are temperature independent.Joule heating effect could increase the temperature,but it can be negligible for diluted solution or underlow electric field strength;(4) the flow field is steady, fully developed and obeyingno-slip conditions at the channel wall, and(5) there is no pressure gradient along the microchannel,and the two reservoirs are large enough to maintainthe same pressure level.Eq. (4) is thus reduced to l r 2 u ¼ q e  E  ;  ð 5 Þ where  E   is the applied electric field strength,  q e  is the ionicnet charge density and  u  is the velocity [24].The boundary conditions are described by u j  x ¼ W    ¼ 0 ;  u j  y  ¼  H   ¼ 0 ; o u o  x   x ¼ 0 ¼ 0 ;  o u o  y    y  ¼ 0 ¼ 0 :  ð 6 Þ Introducing dimensionless parameters,  U   ¼  uU  0 and usingthe separation of variables method, the analytical velocityfield inside the quarter domain is obtained as [25,27] U  ð  X  ; Y  Þ¼ 4  E  e r e 0 kT  l  zeU  0  f X 1 m ¼ 1 ð 1 Þ m þ 1 cos  ð 2 m  1 Þ  D h p 2  H   Y  h i ð 2 m  1 Þ p  cosh  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ð 2 m  1 Þ 2 p 2  D 2h 4  K  2  H  2 q     KX    cosh  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ð 2 m  1 Þ 2 p 2  D 2h 4  K  2  H  2 q     KW   D h     cosh  ð 2 m  1 Þ  D h p 2  H   X  h i cosh  ð 2 m  1 Þ  D h p 2  H  h i8>><>>:9>>=>>;  4  E  e r e 0 kT  l  zeU  0  f X 1 n ¼ 1 ð 1 Þ n þ 1 cos  ð 2 n  1 Þ  D h p 2 W    Y  h i ð 2 n  1 Þ p  cosh  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ð 2 n  1 Þ 2 p 2  D 2h 4  K  2 W    2 q     KY    cosh  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ð 2 n  1 Þ 2 p 2  D 2h 4  K  2 W    2 q     KH  D h     cosh  ð 2 n  1 Þ  D h p 2 W    Y  h i cosh  ð 2 n  1 Þ  D h p 2 W   h i8>><>>:9>>=>>; : ð 7 Þ The average electroosmotic velocity can be found as U   ¼  1  A Z   U  ð  X  ; Y  Þ d  A ;  ð 8 Þ where the integration is over the quarter domain of themicrochannel.  2.3. Solution displacement in the microchannel  The solution displacing model studied is shown in Fig. 2.There are two reservoirs containing the same electrolytesolution but with two different concentrations,  c 1  and  c 2 ,where  c 1  and  c 2  are not significantly different, (e.g., c 1  = 70%  c 2 ). A rectangular microchannel connects thetwo reservoirs. Initially, the connecting channel is filledwith solution of concentration  c 1 . Immediately after anelectric field is applied along the channel, electroosmoticflow is generated. Gradually, the solution of higher concen-tration from the reservoir 2 displaces the solution of lowerconcentration towards the reservoir 1. Fig. 2. Schematic of the solution displacement inside the microchannel. C. Wang et al./International Journal of Heat and Mass Transfer 50 (2007) 3115–3121  3117  The solution displacement along  z  direction is governedby the mass transport equation [28] o c o t   þ  u o c o  z   ¼  D o 2 c o  z  2  ;  ð 9 Þ where  c  is the bulk concentration,  t  is the time,   u  is theaverage velocity and  D  is the diffusion coefficient. By intro-ducing non-dimensional parameters:  Pe ¼  uD h  D  ,   t  ¼  tD D 2h , Z   ¼  z  D h and  C   ¼  c  c 1 c 2  c 1 , Eq. (9) in the dimensionless formcan be written as o C  o  t   þ  Pe o C  o Z   ¼ o 2 C  o Z  2  ;  ð 10 Þ where  Pe  is Peclect number. Eq. (10) is subjected to the ini-tial and boundary conditions shown below: C  j  t  ¼ 0  ¼ 0 ; C  j  z  ¼ 0  ¼ 1 ; where  L ¼  l D h is the dimensionless channel length.Introducing new variables  g ¼ Z    Pe  t   and  s ¼  t  , themass transport equation can be simplified to that shownin Eq. (11) o C  o s  ¼ o C  2 o g 2  :  ð 11 Þ Using the Fourier transform theorem, the Fourier trans-form [29] of a function  f  ( x ) is  F  ð a Þ¼  1  ffiffiffiffiffiffi 2 p p  Z   11  f  ð t  Þ e i a t  d t  ; and the inverse Fourier transform of   F  ( a ) is  f  ð  x Þ¼  1  ffiffiffiffiffiffi 2 p p  Z   11  F  ð a Þ e  i  xt  d a : Then, it can be shown that the solution to Eq. (11) can beexpressed as C  ð g ; s Þ¼ 12  1 þ erf    g 2  ffiffi s p   h i  for  g  <  0 ; 12  1  erf   g 2  ffiffi s p   h i  for  g  >  0 : 8><>: ð 12 Þ In the  z  and  t  coordinates, Eq. (12) can be rewritten as c ð  z  ; t  Þ¼ 12 ð c 2  c 1 Þ  1 þ erf    ut    z  2  ffiffiffiffi  Dt  p   h i þ c 1  for   ut   >  z  ; 12 ð c 2  c 1 Þ  1  erf   z    ut  2  ffiffiffiffi  Dt  p   h i þ c 1  for   ut   <  z  : 8><>:  ð 13 Þ From the current-monitoring method, the total displace-ment time  t max  can be obtained from measurementvia the recorded current–time relationship, and theoreti-cally it also satisfies  c ð l ; t  max Þ¼ c 2 . Together with Eq.(13), the average velocity can be determined. Finally, thecorresponding zeta potential can be found from Eqs. (7)and (8).  2.4. Current prediction Once the concentration profile of the solution in thechannel is known, the resistance of the solution and thecurrent  I  ( t ) through the channel can be found. The resis-tance of the solution is then given by Ohm’s Law  R ¼  l k  A total , in which  R  is the resistance,  l   is the length,  A total is the total cross section area of the conductor (equals thechannel cross section area), and  k  is the conductivity of the electrolyte solution. Noted that  k  depends on the con-centration of the electrolyte solution during the solutiondisplacement, the total resistance of the electrolyte in thechannel can be written as  R total ð t  Þ¼ Z   l 0 d  x k i ð c Þ  A total :  ð 14 Þ The current is given by  I  ð t  Þ¼  El R total ð t  Þ :  ð 15 Þ 3. Experimental setup and discussion In the measurement setup, the current monitoringmethod is used to study the characteristics of the electroos-motic flow in a rectangular microchannel. The experimen-tal setup consists of a high voltage power supply (CZE1000R, Spellman, USA), a personal computer (PC), a dataacquisition system (BNC 2110 unit, National InstrumentsCorporation), flow reservoirs (manufactured using Teflonmaterial), and polyimide-fused silica capillaries (PolymicroTechnologies Incorporated, USA).The microcapillaries with square cross section of 100  l m  100  l m and 75  l m  75  l m were cut to 5 cm inlength and used to connect reservoir 2 (higher concentra-tion  c 2 ) and reservoir 1 (lower concentration  c 1 ). Initially,the microcapillary was filled with NaCl (Sigma–Aldrich)electrolyte solution of a lower concentration  c 1 . Platinumelectrodes were inserted in both reservoirs with the ground-ing to reservoir 1 and high voltage power supply unit toreservoir 2. Measurements were conducted using electro-lyte solution NaCl of different concentrations, 10  2 M,10  3 M, in microchannel of different dimensions, 75  l mand 100  l m, under different applied electric fields of 200 V/cm, 400 V/cm, 600 V/cm and 700 V/cm.The current–time curves from both measured and pre-dicted are shown in Fig. 3. The electric current increaseswith time after an electric field is applied. An EOF isinduced, whereby the higher concentration solution dis-places the lower concentration one causing the variationin electrical current along the flow direction. The currentreaches a constant and maximum value when the higherconcentration solution completely displaces the lower con-centration solution. The stable current level indicates thecompletion of the displacement process. The good agree-ment between the prediction and the measured values sug-gests the validation of the theoretical model. 3118  C. Wang et al./International Journal of Heat and Mass Transfer 50 (2007) 3115–3121

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