Tradeoff between Mixing and Transport for ElectroosmoticFlow in Heterogeneous Microchannels with Nonuniform

Surface Potentials

Fuzhi Tian,† Baoming Li,†,‡ and Daniel Y. Kwok*,†

Nanoscale Technology and Engineering Laboratory, Department of Mechanical Engineering,University of Alberta, Edmonton, Alberta T6G 2G8, Canada, and National Key Laboratory of

Transient Physics, Nanjing University of Science & Technology, Nanjing, China

Received July 17, 2004. In Final Form: October 29, 2004

Electroosmotic flow (EOF) is a phenomenon associated with the movement of an aqueous solution inducedby the application of an electric field in microchannels. The characteristics of EOF depend on the natureof the surface potential, i.e., whether it is uniform or nonuniform. In this paper, a lattice Boltzmann model(LBM) combined with the Poisson-Boltzmann equation is used to simulate flow field in a rectangularmicrochannel with nonuniform (step change) surface potentials. The simulation results indicate that localcirculations can occur near a heterogeneous region with nonuniform surface potentials, in agreement withthose by other authors. Largest circulations, which imply a highest mixing efficiency due to convectionand short-range diffusion, were found when the average surface potential is zero, regardless of whetherthe distribution of the heterogeneous patches is symmetric or asymmetric. In this work, we have illustratedthat there is a tradeoff between the mixing and liquid transport in EOF microfluidics. One should notsimply focus on mixing and neglect liquid transport, as performed in the literature. Excellent mixing couldlead to a poor transport of electroosmotic flow in microchannels.

I. IntroductionMicrochannel network devices have been widely used

in various areas of chemistry and biochemistry. In suchmicrofluidic devices, electroosmosis is often utilized asthe tool for fluid transport and mixing simultaneously.The Reynolds number of electroosmotic flow in microfluidicdevices is usually very small, and quite often to achievingsufficient mixing in electroosmotic microchannel flow canbe a challenge. Obtaining complete mixing in eitherpressure or electrokinetically driven microfluidic devicesrequires both a long mixing channel and an extendedretention time to attain a homogeneous solution.

In general, devices to enhance mixing in microchannelsare classified into two categories: passive and activemixers. Complex specific channel geometry is used inpassive mixers to increase the interfacial area betweenthe mixing liquids. For example, Liu et al.1 proposed athree-dimensional (3D) serpentine channel which en-hances mixing via chaotic advection. Stroock et al.2,3

presented a mixing channel with patterned grooves bycreating spiral circulation around the flow axis at lowReynolds number. These methods can achieve completemixing within a short channel length. However, difficultyremains in the fabrication of complicated geometries forpractical application. Compared with passive mixers,active mixers introduce moving parts inside microchannelsby applying either an external unsteady pressure per-turbation4 or a sinusoidally alternating electric field5 tostir the flow stream.

Recently, electroosmotic flow provides an attractivemeans for controlling fluid motion in microfluidic devices.The characteristics of electroosmotic flow in a microchan-nel depend on the nature of the surface potential of thechannel wall, i.e., whether it is uniform or nonuniform.Most previous studies have focused on electroosmotic flowwith uniform surface electric potential, ψ, or zeta potential,ú.6,7 Ajdari8,9 investigated electroosmosis with nonuniformsurface potential and found circulation regions generatedby application of oppositely charged surface heterogene-ities to the microchannel wall. Ren et al.10 numericallystudied electroosmotic flow in heterogeneous circularmicrochannels with axial step change of surface potentialand illustrated the distorted velocity profiles. Keely etal.11 theoretically presented the flow profiles in capillarieswith a linear variation of surface potential in which theHelmholtz-Smoluchowski equation was employed to de-scribe the electroosmotic velocity. Herr et al.12 utilized anonintrusive, caged-fluorescence, technique to image theelectroosmotic flow in cylindrical capillaries with non-uniform surface potentials and observed a parabolicvelocity profile induced by a pressure gradient due to thesurface heterogeneity. Fu et al.13 found that a step changein zeta potential causes a significant variation in thevelocity profile and in the pressure distribution by solving

* To whom correspondence should be addressed: tel, (780) 492-2791; fax, (780) 492-2200; e-mail, daniel.y.kwok@ualberta.ca.

† University of Alberta.‡ Nanjing University of Science & Technology.(1) Liu, R. H.; Stremler, M. A.; Sharp, K. V.; Olsen, M. G.; Santiago,

J. G.; Adrian, R. J.; Aref, H.; Beebe, D. J. J. Microelectromech. Syst.2000, 9, 190.

(2) Stroock, A. D.; Dertinger, S. K. W.; Ajdari, A.; Mezic, I.; Stone,H. A.; Whitesides, G. M. Science 2002, 295, 647.

(3) Stroock, A. D.; Dertinger, S. K.; Whitesides, G. M.; Ajdari, A.Anal. Chem. 2002, 74, 5306.

(4) Lee, Y. K.; Deval, J.; Tabeling, P.; Ho, C. M. Proc. 14th IEEEWorkshop on Micro Electro Mechanical Systems (Interlaken, Switzer-land, Jan.) 2001; pp 483-6.

(5) Oddy, M. H.; Santiago, J. G.; Mikkelsen, J. C. Anal. Chem. 2001,73, 5822.

(6) Patankar, N. A.; Hu, H. H. Anal. Chem. 1998, 70, 1870.(7) Yang, R. J.; Fu, L. M.; Lin, Y. C. J. Colloid Interface Sci. 2001,

239, 98.(8) Ajdari, A. Phys. Rev. Lett. 1995, 75, 755.(9) Ajdari, A. Phys. Rev. E 1996, 53, 4996.(10) Ren, L.; Li, D. J. Colloid Interface Sci. 2001, 243, 255.(11) Keely, C. A.; van de Goor, T. A. A. M.; McManigill, D. Anal.

Chem. 1994, 66, 4236.(12) Herr, A. E.; Molho, J. I.; Santiago, J. G.; Mungal, M. G.; Kenny,

T. W. Anal. Chem. 2000, 72, 1053.(13) Fu, L. M.; Lin, J. Y.; Yang, R. J. J. Colloid Interface Sci. 2003,

258, 266.

1126 Langmuir 2005, 21, 1126-1131

10.1021/la048203e CCC: $30.25 © 2005 American Chemical SocietyPublished on Web 12/23/2004

the Navier-Stokes equation and the Nernst-Planckequation. The phenomena were also observed experimen-tally by Stroock et al.14 Erickson et al.15 used a traditionalcomputational fluiddynamics (CFD)methodtostudythesecirculation regions and exploited them as a method ofenhancing mixing in T-shaped mixers. Chang et al.16

demonstrated a new method to enhance mixing byintroducing rectangular blocks with patterned heteroge-neous upper surfaces.

From the literature mentioned above, electroosmoticflow with nonuniform surface potential appears to be anexcellent alternative to enhance mixing in microfluidicsvia manipulation of electrical charge distribution in theelectric double layer (EDL). We note, however, that thereis a tradeoff between such mixing and liquid transporta-tion, and this important factor has often been ignored.That is, excellent mixing could imply poor liquid transport.Therefore, it is the purpose of this study to investigatethis tradeoff for electroosmotic flow in heterogeneousmicrochannels. The results from our simulation can beused as guidelines for the optimization and design ofmicrodevices in terms of mixing and species transport.

II. The Model of Electric Double Layerin Microchannels with Nonuniform

Surface PotentialElectroosmosis is a phenomenon associated with the

movement of the bulk electrolyte solution or a liquidcarrying a free charge relative to a stationary chargedsurface under the influence of an imposed electric field.17

The EDL theory to model ion distributions and electricalpotential near a solid surface are briefly discussed below.We consider a straight rectangular microchannel whosegeometry is depicted in Figure 1. This rectangularmicrochannel has a length L, width W, and height H. Theheight-to-width ratio in the channel is ,1, allowing us toassume a two-dimensional flow and neglect any influenceof the side walls on the polarization of the electrolyte andthe flow field. In a straight channel, the governingequations for ion distribution and flow field essentiallyreduce to a one-dimensional problem.

The EDL theory17 relates the surface electrostaticpotential and the distribution of counterions and co-ionsby the Poisson equation

where ψ is the electrical potential, Fe is the net chargedensity, ε is the dimensionless dielectric constant of thesolution, and ε0 is the permittivity of vacuum. Under theabove assumptions, the Poisson equation can be simplifiedto one-dimensional form as

Assuming that the Boltzmann distribution applies, theequilibrium Boltzmann distribution equation can be usedto describe the ionic concentration as follows

where i is the ionic number concentration of the ith species,zi the valence of type i ions, n∞ the ionic numberconcentration in the bulk solution, e the elementaryelectrical charge, kb the Boltzmann constant, and T thetemperature. The charge density can be expressed in termsof Boltzmann distribution. For a symmetric electrolyte,i.e., z+ ) z- ) z, it is given by

Substituting the above expression into eq 2, we obtain thePoisson-Boltzmann equation

If zeψ/kbT is small (i.e., ψ e 25 mV), sinh(zeψ/kbT) ≈ zeψ/kbT, whereby the Debye-Huckel approximation can beinvoked to obtain

and

where κ2 ) (2n∞z2e2/εε0kbT) and (1/κ) is the Debye lengthfor a z: z electrolyte. The boundary conditions for solvingthe Possion-Boltzmann eq 7 are given by

As we consider heterogeneous channels, ψ1(x) and ψ2(x)are both functions of x and represent the surface potentialat the lower and upper walls, respectively. Equation 7 isa linear one-dimensional ordinary differential equationthat can be solved analytically, subject to the boundaryconditions given in eqs 8 and 9. Then the solution is

The net charge density at any point in the channel canbe obtained using eq 4 or 6 after the electrical potentialdistribution ψ has been found.

The surface potential of microchannel walls can behomogeneous or heterogeneous by implementing ψ1(x) andψ2(x) according to real applications. In practice, there aretwo methods to achieve a heterogeneous surface. One isto coat the microchannel walls with different materials;14

(14) Stroock, A. D.; Weck, M.; Chiu, D. T.; Huck, W. T. S.; Kenis, P.J. A.; Ismagilov, R. F.; Whitesides, G. M. Phys. Rev. Lett. 2000, 84, 3314.

(15) Erickson, D.; Li, D. Langmuir 2002, 18, 1883.(16) Chang, C. C.; Yang, R. J. J. Micromech. Microeng. 2004, 14, 550.(17) Masliyah, J. H. Electrokinetic transport phenomena, OSTRA

technical publication series; no. 12 (Edmonton, Alta.: Alberta Oil SandsTechnology and Research Authority, 1994).

Figure 1. Geometry of the microchannel.

∇2ψ ) -Fe

εε0(1)

d2ψdy2

) -Fe(y)εε0

(2)

ni ) n∞ exp(-ziekbT

ψ) (3)

Fe ) -2n∞ze sinh( zekbT

ψ) (4)

d2ψdy2

)2n∞zeεε0

sinh( zekbT

ψ) (5)

Fe )2n∞z2e2

kbTψ (6)

d2ψdy2

)2n∞z2e2

εε0kbTψ ) κ

2ψ (7)

ψ ) ψ1(x) at y ) 0 (8)

ψ ) ψ2(x) at y ) H (9)

ψ(x,y) )ψ2(x) - ψ1(x) cosh(κH)

sinh(κH)sinh(κy) +

ψ1(x) cosh(κy) (10)

Electroosmotic Flow Langmuir, Vol. 21, No. 3, 2005 1127

the other is to establish an electrical field perpendicularto the microchannel wall via microelectrodes which areembedded inside the solid near the solid-liquid interface.18

It has been shown that the surface potential can becontrolled both spatially and temporally. In this simula-tion, the Poisson-Boltzmann equation which governs theEDL field is solved to describe the potential and iondistribution19 with the real dimension of EDL. In the LBMalgorithm,20-22 the well-known modified bounceback bound-ary condition at the wall is applied to have a nonslipboundary condition. A periodic boundary condition in thex-direction was employed to represent a periodicallychanging nonuniform surface along the length so that theflow field of the whole channel can be obtained. In thiscase, the length of the channel can be infinite. Thedrawback of such a periodic boundary condition is theinability to simulate the inlet and outlet effect.

III. Results and DiscussionA. Description of the Physical System. In this paper,

we employ a stepwise surface potential for our hetero-geneous microchannel system shown in Figure 2. Themicrochannel has a width of 500 nm. The patterned surfacein Figure 2a is heterogeneous with symmetrically dis-tributed patches for the lower and upper channel walls,whereas Figure 2b represents a heterogeneous case wherethe patches on the lower and upper walls are asymmetric.ψn and ψp are defined as the surface potentials for thenegatively and positively charged patches, respectively.The average surface potential ψavg of the channel can becalculated from

where Lp and Ln are the lengths of the positively andnegatively charged patches, respectively. In our simula-tion, we set ψn to be -25 mV while ψp is varied from 0 to100 mV; the lengths Ln and Lp are also adjusted in termsof the ratio Ln/Lp to be either 1 or 4. A symmetric 1:1electrolyte solution is selected which has the followingphysical properties similar to those of water at 298 K: F∼ 103 kg/m3, ε ∼ 6.95 × 10-10 C2/J‚m and µ ∼ 10-3 N‚s/m2.

A schematic illustrating how circulation can occur inheterogeneous microchannels is shown in Figure 3, wherethe electric double layer thickness is shown as 1/κ. InFigure 3a, an electroosmotically driven flow over a

homogeneous surface with a surface potential ψ ) -ψ0(ψ0 > 0) is presented. The excess positive ions of the EDLare driven by the electroosmotic body force when anexternal electric field (negative downstream) is appliedacross two ends of the microchannel. This drags the bulkfluid due to viscous effect. The same effect also happensfor a heterogeneous surface with a step change nonuniformsurface potential shown in Figure 3b; in this case, theelectroosmotic body force is applied to both regions ofexcess negative and positive ions. As a result, the flownear the positively charged patch with ψ ) ψ0 (ψ0 > 0) isin the opposite direction to that of the homogeneous regions(ψ ) -ψ0). The interaction of these local flow fields withthe bulk results in a regional circulation zone as shownin Figure 3b.8,9,14

B. Influence of the Nonuniform Surface Potentialon the Flow Fields.Our simulation results are presentedhere for a 1:1 electrolyte with an ionic molar concentrationof 0.1 mM having an EDL thickness of 30.4 nm. Theexternally applied electric field was set to be 1000 V/m forthe following four cases.

1. Symmetric with Ln/Lp ) 1. The surface potentialsψp are set to vary from 0 to 25 mV in a 5 mV incrementwhile ψn remains a constant value of -25 mV for Ln/Lp )1. Parts a-c of Figure 4 show a series of velocity fieldgenerated by modeling the heterogeneous surface withsymmetrically distributed patches; symmetric implies thatthe surface potentials on the upper and lower channelsare exactly the same in terms of both magnitude and sign.As expected, the flow fields presented in these figuresexhibit local circulations near the heterogeneous regionwith a positive surface potential. No circulation was foundin Figure 4a when ψp ) 0 mV. The largest circulationresults from setting ψp ) |ψn| in Figure 4c where theaverage surface potential ψavg is zero. From Figure 4c, thesizes of the vortex appear to be the same when ψp ) |ψn|) 25 mV. We found that the absolute values of themaximum velocities in each vortex are the same. The flowfields for ψp ) 5, 10, and 15 mV are similar to that shownin Figure 4b and are not shown here.

2. Symmetric with Ln/Lp ) 4. Here, the ratio of Ln/Lpis set to be 4 while ψp varies from 0 to 100 mV in a 25 mVincrement and ψn remains a constant value of -25 mV.As shown in parts d-f of Figure 4, the magnitude of thecirculation increases as ψp increases from 0 (Figure 4d)to 100 mV (Figure 4f). Contrary to the results in Figure4c, the sizes of the vortex in Figure 4f are not the samefor a different Ln/Lp ratio; that is, the larger the dimensionLn, the larger is the circulation size. However, the absolutevalue of the maximum velocity, in the region where ψ ispositive (ψp), is larger than that with a negative ψ value(ψn). Thus, the patterns of the flow fields are dramaticallydifferent due to the magnitude and size of the positivelycharged heterogeneous regions. The circulation regionsexpand as the magnitude of the heterogeneous surfacepotential ψp increases. These expanded circulation regionsforce the bulk to flow through a narrower channel crosssection (cf. parts c and f of Figure 4), resulting in a shorterlocal diffusion length which enhances mixing.

3. Asymmetric with Ln/Lp ) 1. This section is similarto those studied above, except that the heterogeneouspatches here are arranged asymmetrically for the upperand lower channel walls. The velocity field generated fromthe asymmetric heterogeneous surfaces is shown in partsa-c of Figure 5 where ψp increases from 0 to 25 mV in 5mV increments while ψn is set to be a constant value of-25 mV. Other flow fields with ψp ) 5, 10, and 15 mV arenot shown. Similar to the symmetric case given above,local circulation regions can be obtained near the het-

(18) Schasfoort, R. B. M.; Schlautmann, S.; Hendrikse, J.; van derBerg, A. Science 1999, 286, 942.

(19) Li, B.; Kwok, D. Y. J. Colloid Interface Sci. 2003, 263, 144.(20) Wolf-Gladrow, D. A. Lattice-Gas Cellular Automata and Lattice

Boltzmann Models; Springer: Berlin, 1991.(21) Succi, S. The Lattice Boltzmann Equation for Fluid Dynamics

and Beyond; Clarendon Press: Oxford, 2001.(22) Chen, S.; Doolen, G. D. Annu. Rev. Fluid Mech. 1998, 30, 329.

Figure 2. Schematic of the patterned surfaces with (a)symmetric and (b) asymmetric stepwise variation of surfacepotential. ψ and L are the surface potential and length of theheterogeneous patches, respectively. The subscripts n and prepresent the negatively and positively charged surfaces,respectively.

ψavg )(ψpLp + ψnLn)

Lp + Ln(11)

1128 Langmuir, Vol. 21, No. 3, 2005 Tian et al.

erogeneous patch with a positive surface potential ψp. Thedifferent sizes of circulation regions are due to both themagnitude (ψp) and dimension (Lp) of the heterogeneouspatches. It is also apparent that the flow field patternsare different from the symmetric case above. Comparingthe flow fields in Figures 4c and 5c, we see that thedirection of the vortex has changed as a result ofrearrangement (whether symmetric or asymmetric) of theheterogeneous patches.

4. Asymmetric with Ln/Lp ) 4. The ratio of Ln/Lp isset to 4 here for an asymmetric arrangement of theheterogeneous patches. The simulation was based on ψn) -25 mV while ψp increases from 0 to 100 mV in 25 mVincrements; the results are shown in parts d-f in Figure5, where the circulation magnitude increases from zero(Figure 5d) to the largest in Figure 5f when ψp is 100 mV(i.e., ψavg ) 0 mV). The flow field in Figure 5f shows thatthe sizes of the vortex are not same when Ln * Lp; thatis, the larger the dimension Ln, the bigger is the circulationsize. However, the absolute value of the maximum velocityin the ψp region is larger than that of the ψn region. Becauseof the asymmetric distribution and dimension of thecharged patches, the flow field in Figure 5f is the mosttortuous. For example, the bulk flows in parts b and e ofFigure 4 are forced to converge into a narrow stream bythe symmetric circulation region while the flow fieldsshown in parts b and e of Figures 5 are more tortuous as

a result of the offset, asymmetric circulation region. Inaddition, we see by comparing parts c and f of Figures 4and 5 that movement perpendicular to the applied fieldcan be generated in the circulation regions. Such a motioncould be particularly useful in system mixing. On the otherhand, near the junction of positively and negativelycharged regions is a stagnation point which may be usefulfor manipulation of macromolecules or cells in a fluidicenvironment.

C. Influence of Nonuniform Surface Potential onthe Volumetric Flow Rate. According to the masscontinuity condition, the volumetric flow rate at any crosssection of the channel is the same and can be calculatedby

where Q is the volumetric flow rate and u(y,z) is the localvelocity in the x direction. For a 2D simulation, eq 12 canbe simplified as the velocity u is independent of the widthz. To study the systematic effect of ψp on the volumetricflow rate, we normalize Q by the maximum flow rate Qmax,i.e., the flow rate of the same channel having a uniformsurface potential of ψ ) -25 mV. As discussed before, weanticipate that nonuniform surface potential affects notonly the flow field but also the flow rate for liquid transport.

Figure 3. Schematic of an electroosmotic flow near the double layer region for (a) a homogeneous surface and (b) a homogeneoussurface with a heterogeneous patch. 1/κ is the double layer thickness.

Figure 4. Streamlines for a symmetrically arranged nonuniform surface potential when ψn ) -25 mV and (a) ψp ) 0 mV andLn/Lp ) 1; (b) ψp ) 20 mV and Ln/Lp ) 1; (c) ψp ) 25 mV and Ln/Lp ) 1; (d) ψp ) 0 mV and Ln/Lp ) 4; (e) ψp ) 75 mV and Ln/Lp) 4; (f) ψp ) 100 mV and Ln/Lp ) 4.

Q ) ∫u(y,z)W dy dz (12)

Electroosmotic Flow Langmuir, Vol. 21, No. 3, 2005 1129

The dependence of electroosmotic flow rate on the non-uniform surface potential is illustrated below:

1. Effect of ψp on Q/Qmax. The effect of ψp on thenormalized flow rate Q/Qmax is shown in Figure 6 usingthe results discussed above. In Figure 6, filled diamondsand squares represent the normalized volumetric flowrate versus the values of ψp for symmetrically distributedcharged patches; while circles and triangles representsimilar results for asymmetrically arranged patches. Thenormalized flow rates Q/Qmax for Ln/Lp ) 4 (as ψp increasesfrom 0 to 100 mV in 25 mV increments) are shown ascurve A. Those of Ln/Lp ) 1 (as ψp increases from 0 to 25mV in 5 mV increments) are shown as curve B. From thisfigure, we see that Q/Qmax decreases linearly as ψpincreases for both curves A and B. From each line, we

conclude that arrangement of nonuniform surface poten-tial, i.e., either symmetric or asymmetric, does not affectthis linear relation. Comparing with curves A and B, wesee that, for the same surface potential ψp, Q/Qmax forLn/Lp ) 1 is smaller than that when Ln/Lp ) 4.

2. Effect of Lp/(Lp + Ln) on Q/Qmax. Using our previoussimulation results, we show in Figure 7 the dependenceof the normalized flow rate Q/Qmax on the ratio R ) Lp/(Ln+ Lp). These results were obtained by adjusting Lp for R) 0, 0.2, 0.4 and 0.5 while the values of ψn, ψp, and Ln areset to be constant. In Figure 7, the circles and filled squaresrepresent the normalized volumetric flow rate versus thelength of the positively charged region Lp in the micro-channel with symmetrically and asymmetrically distrib-uted charged patches, respectively. It is again apparentthat Q/Qmax decreases linearly as the ratio Lp/(Lp + Ln)increases for both symmetric and asymmetric cases. Thus,

Figure 5. Streamlines for an asymmetrically arranged nonuniform surface potential when ψn ) -25 mV and (a) ψp ) 0 mV andLn/Lp ) 1; (b) ψp ) 20 mV and Ln/Lp ) 1; (c) ψp ) 25 mV and Ln/Lp ) 1; (d) ψp ) 0 mV and Ln/Lp ) 4; (e) ψp ) 75 mV and Ln/Lp) 4; and (f) ψp ) 100 mV and Ln/Lp ) 4.

Figure 6. Normalized volumetric flow rate versus surfacepotential of the positively charged region ψp when ψn ) -25mV. Curve A represents the case of Ln/Lp ) 4 and ψp ) 0, 25,50, 75, and 100 mV. Curve B represents the case of Ln/Lp ) 1and ψp ) 0, 5, 10, 15, 20, and 25 mV.

Figure 7. Normalized volumetric flow rate versus length ratioof the positively charged region Lp/(Lp + Ln) for ψp ) |ψn| + 25mV.

1130 Langmuir, Vol. 21, No. 3, 2005 Tian et al.

arrangement of the nonuniform surface potential, i.e.,symmetric or asymmetric, will have no effect on thenormalized flow rate.

D. Tradeoff between Mixing and Transport ofElectroosmotic Flow with Heterogeneous Surfaces.It is then instructive to study how the normalized flowrate Q/Qmax changes with the heterogeneous surfacepotentials. From eq 11, an average surface potential ψavghas been defined and can be calculated in terms of boththe magnitude of the surface potential ψp and length Lpof the heterogeneous patch. Figure 8 shows the results forthe normalized flow rate with the average surfacepotential, using the data from Figures 6 and 7. We seethat the normalized flow rate Q/Qmax indeed depends onlyon the average surface potential ψavg even though theindividual flow field could have been different. That is tosay, once an average surface potential is known, thenormalized flow rate remains the same and does notdepend on the heterogeneous pattern of the surface.Therefore, we conclude that arrangement of nonuniform

surface potential, i.e., symmetric or asymmetric, has noeffect on the flow rate. The average surface potentialdefined in eq 11 is a means by which the electroosmoticflow rate can be evaluated in a microchannel withnonuniform surface potentials to maximize mixing.

According to results in Figures 4 and 5, we see that thecomplex pattern of the heterogeneous surface potentialwill enhance mixing due to circulation and tortuosity andshould result in a higher mixing efficiency. To obtain largercirculation and tortuosity, the average surface potentialψavg calculated by eq 11 should be as close to zero aspossible. However, from the flow rate discussion above inFigure 8, as ψavg approaches zero, the normalized flowrate Q/Qmax also decreases to zero. Indeed, there is atradeoff between mixing and transport via circulationinduced by heterogeneous surface potential. Thus, designand applications of such phenomena for mixing should becarefully considered. On one hand, mixing is required; onthe other, liquid transport is also an important issue forelectroosmotic flow.

IV. SummaryA lattice Boltzmann method with the Poisson-Boltz-

mann equation has been used to simulate electroosmoticflow in a 0.5 µm microchannel with heterogeneous surfacepotential. The simulation results indicate that localcirculations can be obtained near the heterogeneousregion, in agreement with those obtained by other authorsusing the traditional CFD method. Different patterns ofthe heterogeneous surface potential cause different mag-nitudes of circulation. The largest circulation, whichimplies the highest mixing efficiency due to convectionand short-range diffusion, was found when the averagesurface potential is zero. A zero average surface potential,however, implies zero flow rate. We have shown that thereis a clear tradeoff between mixing and transport inmicrofluidics with heterogeneous surfaces.

Acknowledgment. We gratefully acknowledge finan-cial support from the Canada Research Chair (CRC)Program and Natural Sciences and Engineering ResearchCouncil of Canada (NSERC) in support of this research.

LA048203E

Figure 8. Normalized volumetric flow rate versus averagesurface potential ψavg when ψn ) -25 mV and (0) Ln/Lp ) 1 andψp ) 0, 5, 10, 15, 20, and 25 mV; ([) ψp ) 25 mV and Lp/(Lp+ Ln) ) 0, 0.2, 0.4, and 0.5; and (O) Ln/Lp ) 4 and ψp ) 0, 25,50, 75, and 100 mV.

Electroosmotic Flow Langmuir, Vol. 21, No. 3, 2005 1131