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Tradeoff between Mixing and Transport for Electroosmotic Flow 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 induced by the application of an electric field in microchannels. The characteristics of EOF depend on the nature of 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 rectangular microchannel with nonuniform (step change) surface potentials. The simulation results indicate that local circulations can occur near a heterogeneous region with nonuniform surface potentials, in agreement with those by other authors. Largest circulations, which imply a highest mixing efficiency due to convection and short-range diffusion, were found when the average surface potential is zero, regardless of whether the distribution of the heterogeneous patches is symmetric or asymmetric. In this work, we have illustrated that there is a tradeoff between the mixing and liquid transport in EOF microfluidics. One should not simply focus on mixing and neglect liquid transport, as performed in the literature. Excellent mixing could lead to a poor transport of electroosmotic flow in microchannels.
I. Introduction Microchannel network devices have been widely used in various areas of chemistry and biochemistry. In such microfluidic devices, electroosmosis is often utilized as the tool for fluid transport and mixing simultaneously. The Reynolds number of electroosmotic flow in microfluidic devices is usually very small, and quite often to achieving sufficient mixing in electroosmotic microchannel flow can be a challenge. Obtaining complete mixing in either pressure or electrokinetically driven microfluidic devices requires both a long mixing channel and an extended retention time to attain a homogeneous solution. In general, devices to enhance mixing in microchannels are classified into two categories: passive and active mixers. Complex specific channel geometry is used in passive mixers to increase the interfacial area between the mixing liquids. For example, Liu et al.1 proposed a three-dimensional (3D) serpentine channel which enhances mixing via chaotic advection. Stroock et al.2,3 presented a mixing channel with patterned grooves by creating spiral circulation around the flow axis at low Reynolds number. These methods can achieve complete mixing within a short channel length. However, difficulty remains in the fabrication of complicated geometries for practical application. Compared with passive mixers, active mixers introduce moving parts inside microchannels by applying either an external unsteady pressure perturbation4 or a sinusoidally alternating electric field5 to stir the flow stream. * To whom correspondence should be addressed: tel, (780) 4922791; fax, (780) 492-2200; e-mail,
[email protected]. † 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.
Recently, electroosmotic flow provides an attractive means for controlling fluid motion in microfluidic devices. The characteristics of electroosmotic flow in a microchannel depend on the nature of the surface potential of the channel wall, i.e., whether it is uniform or nonuniform. Most previous studies have focused on electroosmotic flow with uniform surface electric potential, ψ, or zeta potential, ζ.6,7 Ajdari8,9 investigated electroosmosis with nonuniform surface potential and found circulation regions generated by application of oppositely charged surface heterogeneities to the microchannel wall. Ren et al.10 numerically studied electroosmotic flow in heterogeneous circular microchannels with axial step change of surface potential and illustrated the distorted velocity profiles. Keely et al.11 theoretically presented the flow profiles in capillaries with a linear variation of surface potential in which the Helmholtz-Smoluchowski equation was employed to describe the electroosmotic velocity. Herr et al.12 utilized a nonintrusive, caged-fluorescence, technique to image the electroosmotic flow in cylindrical capillaries with nonuniform surface potentials and observed a parabolic velocity profile induced by a pressure gradient due to the surface heterogeneity. Fu et al.13 found that a step change in zeta potential causes a significant variation in the velocity profile and in the pressure distribution by solving (4) Lee, Y. K.; Deval, J.; Tabeling, P.; Ho, C. M. Proc. 14th IEEE Workshop on Micro Electro Mechanical Systems (Interlaken, Switzerland, 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.
10.1021/la048203e CCC: $30.25 © 2005 American Chemical Society Published on Web 12/23/2004
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Fe(y) d2ψ )2 0 dy
(2)
Assuming that the Boltzmann distribution applies, the equilibrium Boltzmann distribution equation can be used to describe the ionic concentration as follows
(
Figure 1. Geometry of the microchannel.
the Navier-Stokes equation and the Nernst-Planck equation. The phenomena were also observed experimentally by Stroock et al.14 Erickson et al.15 used a traditional computational fluid dynamics (CFD) method to study these circulation regions and exploited them as a method of enhancing mixing in T-shaped mixers. Chang et al.16 demonstrated a new method to enhance mixing by introducing rectangular blocks with patterned heterogeneous upper surfaces. From the literature mentioned above, electroosmotic flow with nonuniform surface potential appears to be an excellent alternative to enhance mixing in microfluidics via manipulation of electrical charge distribution in the electric double layer (EDL). We note, however, that there is a tradeoff between such mixing and liquid transportation, 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 investigate this tradeoff for electroosmotic flow in heterogeneous microchannels. The results from our simulation can be used as guidelines for the optimization and design of microdevices in terms of mixing and species transport.
ni ) n∞ exp -
∇2ψ ) -
Fe 0
(
Fe ) -2n∞ze sinh
where ψ is the electrical potential, Fe is the net charge density, is the dimensionless dielectric constant of the solution, and 0 is the permittivity of vacuum. Under the above assumptions, the Poisson equation can be simplified to one-dimensional form as (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 Sands Technology and Research Authority, 1994).
)
ze ψ kbT
(4)
Substituting the above expression into eq 2, we obtain the Poisson-Boltzmann equation
(
)
d2ψ 2n∞ze ze ψ ) sinh 0 kbT dy2
(5)
If zeψ/kbT is small (i.e., ψ e 25 mV), sinh(zeψ/kbT) ≈ zeψ/ kbT, whereby the Debye-Hu¨ckel approximation can be invoked to obtain
Fe )
(1)
(3)
where i is the ionic number concentration of the ith species, zi the valence of type i ions, n∞ the ionic number concentration in the bulk solution, e the elementary electrical charge, kb the Boltzmann constant, and T the temperature. The charge density can be expressed in terms of Boltzmann distribution. For a symmetric electrolyte, i.e., z+ ) z- ) z, it is given by
II. The Model of Electric Double Layer in Microchannels with Nonuniform Surface Potential Electroosmosis is a phenomenon associated with the movement of the bulk electrolyte solution or a liquid carrying a free charge relative to a stationary charged surface under the influence of an imposed electric field.17 The EDL theory to model ion distributions and electrical potential near a solid surface are briefly discussed below. We consider a straight rectangular microchannel whose geometry is depicted in Figure 1. This rectangular microchannel has a length L, width W, and height H. The height-to-width ratio in the channel is ,1, allowing us to assume a two-dimensional flow and neglect any influence of the side walls on the polarization of the electrolyte and the flow field. In a straight channel, the governing equations for ion distribution and flow field essentially reduce to a one-dimensional problem. The EDL theory17 relates the surface electrostatic potential and the distribution of counterions and co-ions by the Poisson equation
)
zie ψ kbT
2n∞z2e2 ψ kbT
(6)
and 2 2 d2ψ 2n∞z e ) ψ ) κ2ψ 0kbT dy2
(7)
where κ2 ) (2n∞z2e2/0kbT) and (1/κ) is the Debye length for a z: z electrolyte. The boundary conditions for solving the Possion-Boltzmann eq 7 are given by
ψ ) ψ1(x)
at y ) 0
(8)
ψ ) ψ2(x)
at y ) H
(9)
As we consider heterogeneous channels, ψ1(x) and ψ2(x) are both functions of x and represent the surface potential at the lower and upper walls, respectively. Equation 7 is a linear one-dimensional ordinary differential equation that can be solved analytically, subject to the boundary conditions given in eqs 8 and 9. Then the solution is
ψ(x,y) )
ψ2(x) - ψ1(x) cosh(κH) sinh(κH)
sinh(κy) + ψ1(x) cosh(κy) (10)
The net charge density at any point in the channel can be obtained using eq 4 or 6 after the electrical potential distribution ψ has been found. The surface potential of microchannel walls can be homogeneous or heterogeneous by implementing ψ1(x) and ψ2(x) according to real applications. In practice, there are two methods to achieve a heterogeneous surface. One is to coat the microchannel walls with different materials;14
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Figure 2. Schematic of the patterned surfaces with (a) symmetric and (b) asymmetric stepwise variation of surface potential. ψ and L are the surface potential and length of the heterogeneous patches, respectively. The subscripts n and p represent the negatively and positively charged surfaces, respectively.
the other is to establish an electrical field perpendicular to the microchannel wall via microelectrodes which are embedded inside the solid near the solid-liquid interface.18 It has been shown that the surface potential can be controlled both spatially and temporally. In this simulation, the Poisson-Boltzmann equation which governs the EDL field is solved to describe the potential and ion distribution19 with the real dimension of EDL. In the LBM algorithm,20-22 the well-known modified bounceback boundary condition at the wall is applied to have a nonslip boundary condition. A periodic boundary condition in the x-direction was employed to represent a periodically changing nonuniform surface along the length so that the flow field of the whole channel can be obtained. In this case, the length of the channel can be infinite. The drawback of such a periodic boundary condition is the inability to simulate the inlet and outlet effect. III. Results and Discussion A. Description of the Physical System. In this paper, we employ a stepwise surface potential for our heterogeneous microchannel system shown in Figure 2. The microchannel has a width of 500 nm. The patterned surface in Figure 2a is heterogeneous with symmetrically distributed patches for the lower and upper channel walls, whereas Figure 2b represents a heterogeneous case where the patches on the lower and upper walls are asymmetric. ψn and ψp are defined as the surface potentials for the negatively and positively charged patches, respectively. The average surface potential ψavg of the channel can be calculated from
ψavg )
(ψpLp + ψnLn) Lp + Ln
(11)
where Lp and Ln are the lengths of the positively and negatively charged patches, respectively. In our simulation, we set ψn to be -25 mV while ψp is varied from 0 to 100 mV; the lengths Ln and Lp are also adjusted in terms of the ratio Ln/Lp to be either 1 or 4. A symmetric 1:1 electrolyte solution is selected which has the following physical properties similar to those of water at 298 K: F ∼ 103 kg/m3, ∼ 6.95 × 10-10 C2/J‚m and µ ∼ 10-3 N‚s/m2. A schematic illustrating how circulation can occur in heterogeneous microchannels is shown in Figure 3, where the electric double layer thickness is shown as 1/κ. In Figure 3a, an electroosmotically driven flow over a (18) Schasfoort, R. B. M.; Schlautmann, S.; Hendrikse, J.; van der Berg, 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.
homogeneous surface with a surface potential ψ ) -ψ0 (ψ0 > 0) is presented. The excess positive ions of the EDL are driven by the electroosmotic body force when an external electric field (negative downstream) is applied across two ends of the microchannel. This drags the bulk fluid due to viscous effect. The same effect also happens for a heterogeneous surface with a step change nonuniform surface potential shown in Figure 3b; in this case, the electroosmotic body force is applied to both regions of excess negative and positive ions. As a result, the flow near the positively charged patch with ψ ) ψ0 (ψ0 > 0) is in the opposite direction to that of the homogeneous regions (ψ ) -ψ0). The interaction of these local flow fields with the bulk results in a regional circulation zone as shown in Figure 3b.8,9,14 B. Influence of the Nonuniform Surface Potential on the Flow Fields. Our simulation results are presented here for a 1:1 electrolyte with an ionic molar concentration of 0.1 mM having an EDL thickness of 30.4 nm. The externally applied electric field was set to be 1000 V/m for the following four cases. 1. Symmetric with Ln/Lp ) 1. The surface potentials ψp are set to vary from 0 to 25 mV in a 5 mV increment while ψn remains a constant value of -25 mV for Ln/Lp ) 1. Parts a-c of Figure 4 show a series of velocity field generated by modeling the heterogeneous surface with symmetrically distributed patches; symmetric implies that the surface potentials on the upper and lower channels are exactly the same in terms of both magnitude and sign. As expected, the flow fields presented in these figures exhibit local circulations near the heterogeneous region with a positive surface potential. No circulation was found in Figure 4a when ψp ) 0 mV. The largest circulation results from setting ψp ) |ψn| in Figure 4c where the average surface potential ψavg is zero. From Figure 4c, the sizes of the vortex appear to be the same when ψp ) |ψn| ) 25 mV. We found that the absolute values of the maximum velocities in each vortex are the same. The flow fields for ψp ) 5, 10, and 15 mV are similar to that shown in Figure 4b and are not shown here. 2. Symmetric with Ln/Lp ) 4. Here, the ratio of Ln/Lp is set to be 4 while ψp varies from 0 to 100 mV in a 25 mV increment and ψn remains a constant value of -25 mV. As shown in parts d-f of Figure 4, the magnitude of the circulation increases as ψp increases from 0 (Figure 4d) to 100 mV (Figure 4f). Contrary to the results in Figure 4c, the sizes of the vortex in Figure 4f are not the same for a different Ln/Lp ratio; that is, the larger the dimension Ln, the larger is the circulation size. However, the absolute value of the maximum velocity, in the region where ψ is positive (ψp), is larger than that with a negative ψ value (ψn). Thus, the patterns of the flow fields are dramatically different due to the magnitude and size of the positively charged heterogeneous regions. The circulation regions expand as the magnitude of the heterogeneous surface potential ψp increases. These expanded circulation regions force the bulk to flow through a narrower channel cross section (cf. parts c and f of Figure 4), resulting in a shorter local diffusion length which enhances mixing. 3. Asymmetric with Ln/Lp ) 1. This section is similar to those studied above, except that the heterogeneous patches here are arranged asymmetrically for the upper and lower channel walls. The velocity field generated from the asymmetric heterogeneous surfaces is shown in parts a-c of Figure 5 where ψp increases from 0 to 25 mV in 5 mV increments while ψn is set to be a constant value of -25 mV. Other flow fields with ψp ) 5, 10, and 15 mV are not shown. Similar to the symmetric case given above, local circulation regions can be obtained near the het-
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Figure 3. Schematic of an electroosmotic flow near the double layer region for (a) a homogeneous surface and (b) a homogeneous surface 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 and Ln/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.
erogeneous patch with a positive surface potential ψp. The different sizes of circulation regions are due to both the magnitude (ψp) and dimension (Lp) of the heterogeneous patches. It is also apparent that the flow field patterns are different from the symmetric case above. Comparing the flow fields in Figures 4c and 5c, we see that the direction of the vortex has changed as a result of rearrangement (whether symmetric or asymmetric) of the heterogeneous patches. 4. Asymmetric with Ln/Lp ) 4. The ratio of Ln/Lp is set to 4 here for an asymmetric arrangement of the heterogeneous patches. The simulation was based on ψn ) -25 mV while ψp increases from 0 to 100 mV in 25 mV increments; the results are shown in parts d-f in Figure 5, 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 that the sizes of the vortex are not same when Ln * Lp; that is, the larger the dimension Ln, the bigger is the circulation size. However, the absolute value of the maximum velocity in the ψp region is larger than that of the ψn region. Because of the asymmetric distribution and dimension of the charged patches, the flow field in Figure 5f is the most tortuous. For example, the bulk flows in parts b and e of Figure 4 are forced to converge into a narrow stream by the symmetric circulation region while the flow fields shown in parts b and e of Figures 5 are more tortuous as
a result of the offset, asymmetric circulation region. In addition, we see by comparing parts c and f of Figures 4 and 5 that movement perpendicular to the applied field can be generated in the circulation regions. Such a motion could be particularly useful in system mixing. On the other hand, near the junction of positively and negatively charged regions is a stagnation point which may be useful for manipulation of macromolecules or cells in a fluidic environment. C. Influence of Nonuniform Surface Potential on the Volumetric Flow Rate. According to the mass continuity condition, the volumetric flow rate at any cross section of the channel is the same and can be calculated by
Q)
∫u(y,z)W dy dz
(12)
where Q is the volumetric flow rate and u(y,z) is the local velocity in the x direction. For a 2D simulation, eq 12 can be simplified as the velocity u is independent of the width z. To study the systematic effect of ψp on the volumetric flow rate, we normalize Q by the maximum flow rate Qmax, i.e., the flow rate of the same channel having a uniform surface potential of ψ ) -25 mV. As discussed before, we anticipate that nonuniform surface potential affects not only the flow field but also the flow rate for liquid transport.
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Figure 5. Streamlines for an asymmetrically arranged nonuniform surface potential when ψn ) -25 mV and (a) ψp ) 0 mV and Ln/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 surface potential of the positively charged region ψp when ψn ) -25 mV. 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 ) 1 and ψp ) 0, 5, 10, 15, 20, and 25 mV.
The dependence of electroosmotic flow rate on the nonuniform surface potential is illustrated below: 1. Effect of ψp on Q/Qmax. The effect of ψp on the normalized flow rate Q/Qmax is shown in Figure 6 using the results discussed above. In Figure 6, filled diamonds and squares represent the normalized volumetric flow rate versus the values of ψp for symmetrically distributed charged patches; while circles and triangles represent similar results for asymmetrically arranged patches. The normalized flow rates Q/Qmax for Ln/Lp ) 4 (as ψp increases from 0 to 100 mV in 25 mV increments) are shown as curve A. Those of Ln/Lp ) 1 (as ψp increases from 0 to 25 mV in 5 mV increments) are shown as curve B. From this figure, we see that Q/Qmax decreases linearly as ψp increases for both curves A and B. From each line, we
Figure 7. Normalized volumetric flow rate versus length ratio of the positively charged region Lp/(Lp + Ln) for ψp ) |ψn| + 25 mV.
conclude that arrangement of nonuniform surface potential, i.e., either symmetric or asymmetric, does not affect this linear relation. Comparing with curves A and B, we see that, for the same surface potential ψp, Q/Qmax for Ln/Lp ) 1 is smaller than that when Ln/Lp ) 4. 2. Effect of Lp/(Lp + Ln) on Q/Qmax. Using our previous simulation results, we show in Figure 7 the dependence of 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 are set to be constant. In Figure 7, the circles and filled squares represent the normalized volumetric flow rate versus the length of the positively charged region Lp in the microchannel with symmetrically and asymmetrically distributed charged patches, respectively. It is again apparent that Q/Qmax decreases linearly as the ratio Lp/(Lp + Ln) increases for both symmetric and asymmetric cases. Thus,
Electroosmotic Flow
Figure 8. Normalized volumetric flow rate versus average surface potential ψavg when ψn ) -25 mV and (0) Ln/Lp ) 1 and ψp ) 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.
arrangement of the nonuniform surface potential, i.e., symmetric or asymmetric, will have no effect on the normalized flow rate. D. Tradeoff between Mixing and Transport of Electroosmotic Flow with Heterogeneous Surfaces. It is then instructive to study how the normalized flow rate Q/Qmax changes with the heterogeneous surface potentials. From eq 11, an average surface potential ψavg has been defined and can be calculated in terms of both the magnitude of the surface potential ψp and length Lp of the heterogeneous patch. Figure 8 shows the results for the normalized flow rate with the average surface potential, using the data from Figures 6 and 7. We see that the normalized flow rate Q/Qmax indeed depends only on the average surface potential ψavg even though the individual flow field could have been different. That is to say, once an average surface potential is known, the normalized flow rate remains the same and does not depend on the heterogeneous pattern of the surface. Therefore, we conclude that arrangement of nonuniform
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surface potential, i.e., symmetric or asymmetric, has no effect on the flow rate. The average surface potential defined in eq 11 is a means by which the electroosmotic flow rate can be evaluated in a microchannel with nonuniform surface potentials to maximize mixing. According to results in Figures 4 and 5, we see that the complex pattern of the heterogeneous surface potential will enhance mixing due to circulation and tortuosity and should result in a higher mixing efficiency. To obtain larger circulation and tortuosity, the average surface potential ψavg calculated by eq 11 should be as close to zero as possible. However, from the flow rate discussion above in Figure 8, as ψavg approaches zero, the normalized flow rate Q/Qmax also decreases to zero. Indeed, there is a tradeoff between mixing and transport via circulation induced by heterogeneous surface potential. Thus, design and applications of such phenomena for mixing should be carefully considered. On one hand, mixing is required; on the other, liquid transport is also an important issue for electroosmotic flow. IV. Summary A lattice Boltzmann method with the Poisson-Boltzmann equation has been used to simulate electroosmotic flow in a 0.5 µm microchannel with heterogeneous surface potential. The simulation results indicate that local circulations can be obtained near the heterogeneous region, in agreement with those obtained by other authors using the traditional CFD method. Different patterns of the heterogeneous surface potential cause different magnitudes of circulation. The largest circulation, which implies the highest mixing efficiency due to convection and short-range diffusion, was found when the average surface potential is zero. A zero average surface potential, however, implies zero flow rate. We have shown that there is a clear tradeoff between mixing and transport in microfluidics with heterogeneous surfaces. Acknowledgment. We gratefully acknowledge financial support from the Canada Research Chair (CRC) Program and Natural Sciences and Engineering Research Council of Canada (NSERC) in support of this research. LA048203E
