Effects of surface heterogeneity on flow circulation in electroosmotic flow in microchannels

Analytica Chimica Acta 530 (2005) 273–282

Effects of surface heterogeneity on flow circulation in electroosmotic flow in microchannels Jacky S.H. Leea , Carolyn L. Renb , Dongqing Lia,∗ a

Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ont., Canada M5S 3G8 b Department of Mechanical Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ont., Canada N2L 3G1 Received 22 May 2004; received in revised form 13 September 2004; accepted 13 September 2004 Available online 8 December 2004

Abstract The characteristics of electroosmotic flow in a cylindrical microchannel with non-uniform zeta potential distribution are investigated in this paper. Two-dimensional full Navier–Stokes equation is used to model the flow field and the pressure field. The numerical results show the distorted electroosmotic velocity profiles and various kinds of flow circulation resulting from the axial variation of the zeta potential. The influences of heterogeneous patterns of zeta potential on the velocity profile, the induced pressure distribution and the volumetric flow rate are discussed in this paper. This work shows that using either heterogeneous patterns of zeta potential or a combination of a heterogeneous zeta potential distribution and an applied pressure difference over the channel can generate local flow circulations and hence provide effective means to improve the mixing between different solutions in microchannels. © 2004 Elsevier B.V. All rights reserved. Keywords: Electroosmotic; Heterogeneous; Microchannel

1. Introduction Electroosmotic flow is commonly used as a pumping method in most lab-on-a-chip devices due to its many advantages over pressure driven flow, such as plug-like velocity profile, easy to control flow direction and no mechanical moving parts. T-shaped microchannel mixers employ electroosmotic flow to pump liquids from two reservoirs (connected to the horizontal channels) to the T-intersection and mix liquids in the vertical channel when pumped downstream. T-shaped microchannel mixers have been applied in various lab-on-a-chip devices, for example, to dilute sample in a buffer solution [1] and to generate concentration gradients [2]. Generally, the effectiveness of mixing highly depends on electroosmotic flow. It can be understood from an analysis of the following con-

∗

Corresponding author. Tel.: +1 416 978 1282; fax: +1 416 978 7753. E-mail address: [email protected] (D. Li).

0003-2670/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.aca.2004.09.026

servation equation for species ∂C  = D∇ 2 C +u  · ∇C ∂t

(1)

where C is the concentration of species, u  is the vector of the bulk electroosmotic velocity and D is the diffusion coefficient of the species. In most microfluidic applications, electroosmotic velocity is on the order of 1 mm/s and the diffusion coefficient of most samples is approximately 1.0 × 10−10 m2 /s. From the order of magnitude analysis of the convection and diffusion terms in Eq. (1), [ux ]([C]/[x]) and [D]([C]/[x2 ]), it is clear that the order of magnitude of the convection term is approximately 1000 times of the diffusion term. Here the characteristic length, x, is chosen as 100 ␮m. Therefore, the flow field should give a good indication of the concentration field, consequently, a basic understanding of the mixing effectiveness. Generally, the stronger the local flow circulation is, the better the mixing is. However, the electroosmotic flow in a homogeneous microchannel is simple laminar flow without any circulation. It has been found that heterogeneous

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patterns of zeta potential on the channel wall can alter the conventional electroosmotic flow field and generate circulations [3–9]. Generally, the effects of non-uniform zeta potential on the electroosmotic flow can be understood as follows. The driving force in electroosmotic flow depends on the externally applied electrical field strength and the local net charge density in the electrical double layer (EDL) [10]. The net charge density is strongly dependent on the EDL potential at the channel wall, referred to the zeta potential, ζ. A change in zeta potential will result in a change in net charge density. Consequently, the driving force and hence the flow pattern will be changed in different locations along the microchannel if the channel wall has a non-uniform zeta potential distribution. Generally, the zeta potential is a function of the ionic valence, the ionic concentration of the electrolyte solution and the surface properties of the channel wall. For a system with a simple electrolyte solution and a homogeneous channel wall, the zeta potential is considered uniform along the channel. However, in many cases, the zeta potential on the channel wall is not uniform due to manufacture defects or special design. For example, T-shaped micromixers are designed with a non-uniform distribution of zeta potential along the channel wall to enhance the mixing between two solutions [11]. In addition, the liquids involved in various bio-lab chips are solutions containing biological particles (e.g. DNA and protein). The adhesion of these particles to channel walls will change the local zeta potential. Therefore, a better understanding of the flow behavior over a heterogeneous microchannel is very important for manipulating flow and for improving the performance of biochip devices. Many theoretical and experimental studies have been performed to investigate the effects of surface heterogeneity on electroosmotic flow in microchannels [4–9,12–18]. Ajdari [4–6] studied the effects of sinusoidally varying surface heterogeneities on the flow field in slit microchannels and predicted that surface heterogeneities can result in circulations in electroosmotic flow. This flow behavior has been experimentally observed by Stroock et al. [8] in slit microchannels and good agreements were found between the experiments and their flow model predictions. Similar flow structures have also been predicted numerically by Erickson and Li [9] in electroosmotic flows in a slit microchannel with periodically repeating heterogeneous patches. Although the heterogeneous patterns in the above studies could result in flow circulations and provide possibilities to improve flow mixing, practically making these heterogeneous patterns in microchannels is of great difficulty. The purpose of this study is to find simple heterogeneous patterns that can generate flow circulation and improve mixing in electroosmotic flow. Fu et al. [12] investigated the effect of a simple step change in zeta potential on electroosmotic flow in a rectangular microchannel. Their study indicated that an eruptive change of velocity profile occurs at the region of step change in zeta potential. However, no circulation in the flow field was observed in this study.

For electroosmotic flow in cylindrical microchannels with non-uniform zeta potentials, analytical and one-dimensional numerical models have been developed [15–18]. However, the complexity of electroosmotic flow such as flow circulation in circular heterogeneous microchannels cannot be considered and predicted by these simple analytical and onedimensional numerical models, because two-dimensional phenomena generally exist near the region of an eruptive change in zeta potential. Therefore, it is essential to develop a two-dimensional numerical model simulating the electroosmotic flow behavior in circular microchannels with various heterogeneous patterns of zeta potential. Our goal is to search simple heterogeneous patterns in circular microchannels for the purpose of improving the flow mixing. In this work, we study electroosmotic flow in a 10-mm long cylindrical microchannel with non-uniform zeta potential distributions by examining a variety of heterogeneous patterns along the channel wall. Since efficient mixing within a short distance and a short time period is desirable in microfluidic applications [19], one of the goals in this project is to discover useful flow patterns that can enhance mixing within our computational domain that can be applied to a short microchannel. The heterogeneous patterns of zeta potential and the applied pressure difference over the channel are the major controlling parameters in this study. The two-dimensional full Navier–Stokes equation is numerically solved to obtain the electroosmotic flow field and pressure distribution. In this study, the pressure field may be either the applied pressure field or the induced pressure field. The influences of heterogeneous patches on flow rate, velocity profile, and pressure distribution are discussed with the purpose of finding the effective heterogeneous patterns of zeta potential and the applied pressure difference for enhancing mixing in electroosmotic flow.

2. Electroosmotic flow field in a cylindrical microchannel The microchannel studied here is a cylindrical microchannel with a diameter of 100 ␮m and a length of 10 mm, as shown in Fig. 1, containing a 50 mM sodium carbonate buffer with a pH = 9.0. The two ends of the microchannel connect two reservoirs containing the same buffer and electrodes. When an electrical potential difference is applied to the liquid in the microchannel through these electrodes, electrical driving forces are exerted on the net charges within the EDL. The motion of the net charges will draw the adjacent liquids to move generating electroosmotic flow. The equations governing incompressible liquids (i.e. sodium carbonate buffer) are the Navier–Stokes equation, and the continuity equation.
 ∂ u  + µ∇ 2 u  e ρ (2a) +u  · ∇u  = −∇P  + Eρ ∂t  ·u ∇ =0

(2b)

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Fig. 1. Schematic diagram of the microchannel and computational domain.

where u  is the velocity vector, µ is the viscosity, ρ is the  is the pressure gradient (either an density of the fluid, ∇P induced or an applied pressure gradient), ρe is the local net  is the electrical field strength applied charge density and E to the microchannel. Since the local net charge density is not zero only in the electrical double layer (EDL), the driving  e , exists only in EDL. force for the electroosmotic flow, Eρ The electroosmotic flow velocity changes sharply in a thin layer of liquid near the wall, from zero at the channel wall surface to an approximately constant value at the outer edge of the EDL. For most on-chip microfluidic applications, buffer solutions have a concentration of the order of mM, which results in a very thin EDL. For instance, a solution with a 10 mM concentration has a double layer thickness of approximately 10 nm, which is negligible in comparison with the microchannel diameter (e.g. 100 ␮m). Therefore, for the purpose of modeling the bulk liquid flow outside the EDL, the  e in Eq. (2a) will be dropped off and driving force term Eρ the electroosmotic effect is considered by the slip boundary condition: u  |wall =

εε0 ς  µ E

 = µeo E

(3)

where µeo is the electroosmotic mobility of the buffer solution, which will change with zeta potential. We consider that the flow is steady state, therefore, the transient term in Eq. (2a) will be dropped off and Eq. (2a) is reduced to:      ∂P 1 ∂ ∂uz ∂uz ∂uz ρ ur + uz =− +µ r ∂r ∂z ∂z r ∂r ∂r  ∂ 2 uz + 2 (4a) ∂z 

∂ur ∂ur + uz ρ ur ∂r ∂z



   1 ∂ ∂P ∂ur +µ =− r ∂r r ∂r ∂r  2 ∂ ur ur + 2 − 2 ∂z r

the velocity component in z-direction are: ∂uz =0 ∂r r = r|wall , uz = µeo Ez

r = 0,

z = 0,

∂uz =0 ∂z

z = z|total ,

∂ur =0 ∂r r = r|wall , ur = 0

r = 0,

(6a) (6b)

(7a) (7b)

Similar to the case with the z-direction boundary conditions, the velocity change along the radial direction at the two end boundaries would be zero: ∂ur z = 0, =0 (8a) ∂z z = z|total ,

∂ur =0 ∂z

(8b)

The pressures at the two ends of the microchannel are generally known, specified as: P = PBC1

z = z|total ,

The flow is axisymmetric and the computation domain as shown in Fig. 1. The corresponding boundary conditions for

(5b)

An implication in the above boundary conditions is that at the two ends of the microchannel, the flow patterns are not going through any erupt changes. This condition can be ensured by applying constant zeta potentials near the end of the microchannel, so that any changes in the electroosmotic flow field occur far away from the end boundaries. The corresponding boundary conditions for velocity component in r-direction are:

z = 0,

(4b)

∂uz =0 ∂z

(5a)

P = PBC2

(9a) (9b)

Depending on the applications, PBC1 and PBC2 may be the same to simulate an electroosmotic flow, or different to simulate a pressure driven flow or a combined electroosmotic and pressure driven flow. The pressure gradient at the center axis of the microchannel is set to be zero because of symmetric

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boundary condition, and it was set to be zero at the wall as well considering no flux cross the wall: r = 0,

∂P =0 ∂r

r = r|wall ,

∂P =0 ∂r

(10a) (10b)

In the development of the numerical simulation, the following non-dimensionalization was used. u z =

uz

uz z r P − Pa , uz = , z = , r = , P = U U D D ρU 2

(11)

where U is the reference velocity, D is the diameter of the channel, and the Reynolds number (Re) is defined as ρUD/µ.

3. Numerical scheme To solve the velocity profile and the pressure gradient of electroosmotic flow in a cylindrical microchannel, the Navier–Stokes equations must be solved in cylindrical coordinate. In order to solve the velocities, the knowledge of the pressure gradient in the microchannel is required. However, the pressure gradient is often unknown. Therefore, specialized methods have been developed to deal with this problem. SIMPLE algorithm (semi-implicit method for pressure linked equations) developed by Patankar [20] is employed in this study to deal with the pressure terms in the Navier–Stokes equations. The Gauss–Seidel iterative method is utilized to increase the speed of convergence.

4. Results and discussions The above equations and the matching boundary conditions for flow field were solved numerically. Carbon dioxide buffer solution was used as the testing liquid and the following physical properties were used: ε = 80, ε0 = 8.854 × 10−12 CV−1 m−1 and µ = 0.90 × 10−3 kg m−1 s−1 . Various cases with different heterogeneous zeta potential patterns and applied pressures are studied in this paper. All the pressure figures are plotted at the axisymetric plane (r = 0). For all of the cases, the simulations were performed with the specifications presented in Table 1. When the zeta potential is changed along the channel wall, the driving force in electroosmotic flow is changed, as discussed earlier. Consequently, the flow field is not uniform along the channel. Pressure gradients are induced to ensure Table 1 Simulation parameters Microchannel length (mm) Microchannel diameter (␮m) Reference velocity (mm/s) Applied electric field strength (V/cm) Reynolds number

10 100 1 −140 0.1

the constant flow rate between different regions with different zeta potentials on the wall. For a pure electroosmotic flow (without applied pressures) in a channel with a stepwise zeta potential change in the axial direction in the middle of the channel, Fig. 2 shows that the plug-like velocity profile is distorted. It is observed that a sudden change in velocity occurs at the region of the stepwise zeta potential change. From the velocity profile, the region with non-zero zeta potential is the active electroosmotic flow (EOF) region, the region with zero zeta potential is the EOF induced flow region. When the liquid in the active EOF region flows downstream, the liquid will pull the liquid in the back (upstream region) due to the continuity. As a result, a negative induced pressure gradient (see Fig. 2(c)) is generated to induce the flow in the zero zeta potential region, creating a parabolic velocity profile in the flow direction. Because of the same pressure in the two ends of the microchannel (i.e. open to air), a positive pressure gradient is expected in the active EOF region, as shown in Fig. 2(c). This induced positive pressure gradient tends to suppress electroosmotic flow, resulting in a distorted electroosmotic velocity profile as shown in Fig. 2(b). This similar distortion has also been observed in an experimental study [17]. However, it should be noted that the velocity variation in the radial direction is observed in this study, which cannot be predicted using 1D numerical model or analytical solutions as reported in previous works [17,18]. If the direction of the stepwise zeta potential change is switched, i.e. from −100 mV to 0 mV at the midway of the microchannel, the positions of the active EOF region and the EOF induced flow region are switched. The similar velocity profile and pressure profile are not shown here. Fig. 3 shows a more complicated zeta potential pattern and its corresponding flow field and pressure field. The zeta potential is set as 0 mV for the first quarter of the channel length, then it is linearly decreased to −100 mV over the next two quarters of the length (z = 2.5–7.5 mm), finally the zeta potential is kept as −100 mV until the end of the channel, as shown in Fig. 3(a). In this case, the induced pressure gradient is nonlinear through the microchannel, as shown in Fig. 3(b). For the regions with zero and relative low zeta potentials, negative pressure gradients exist to accelerate the flow, compensating the smaller EOF driving force in this region. For the regions with relative high zeta potentials, positive pressure gradients present to against high-speed electroosmotic flow in this region. As a result, modified positive parabolic velocity profiles in the low zeta potential regions and modified negative parabolic velocity profiles in the high zeta potential regions are observed, as shown in Fig. 3(c). Motivated by improving the flow mixing by generating strong circulations in the flow, more complicated heterogeneous zeta potential patterns are examined in this study. In the case presented in Fig. 4, the zeta potential changes stepwise from −100 mV to +50 mV at the midway of the microchannel, as shown in Fig. 4(a). The most notable feature in the flow field is that there are back flows, or circulations throughout the microchannel due to the oppositely charged channel wall

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Fig. 2. Zeta potential distribution and its corresponding flow field and pressure field in a 100 ␮m diameter microchannel under an applied electrical field strength of 140 V/cm.

Fig. 3. Zeta potential distribution and its corresponding flow field and pressure field in a 100 ␮m diameter microchannel under an applied electrical field strength of 140 V/cm.

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Fig. 4. Zeta potential distribution and its corresponding flow field and streamline pattern in a 100 ␮m diameter microchannel under an applied electrical field strength of 140 V/cm.

surfaces in the two sections. The opposite signs of zeta potentials indicate opposite EOF driving forces in the two sections of the channel, which provide possibilities of generating circulations. The circulations generated in this case can be seen from the streamline pattern shown in Fig. 4(b) and (c). In the region with a positive zeta potential, a negative net charge density exists inside the EDL generating a negative flow towards anode in the inlet of the channel (the left end). In the region with a negative zeta potential, a positive net charge density exists generating a positive flow towards cathode in the outlet of the channel (the right end). It was found that the net flow direction depends on the overall strength of the zeta potential, which is defined in this study by:  ζL ¯ζ = i i i (12) i Li where ζ i and Li are the zeta potential and channel length of the i-th section. The overall zeta potential strength is negative in this case (−25 mV) and indicates a positive net flow (in z-direction), as shown in Fig. 4(d). As a result of these

opposite migrations of ions, the magnitude of both positive and negative pressure gradients are higher than that in the previous two cases. Clearly, such flow behavior can be used to enhance the flow mixing. Convinced by the above results that the flow circulation can be induced by using oppositely charged surfaces along the channel, a different zeta potential pattern is examined. In this case, two-step changes in zeta potential (from −100 mV to 100 mV and then to −100 mV) are implemented at z = 3.75 mm and z = 6.25 mm as shown in Fig. 5(a). In this configuration, strong flow mixing can be observed from the streamline plot shown in Fig. 5(b) and (c) and the flow field shown in Fig. 5(d). Net charges in the beginning and the end sections of the channel have the same sign and are opposite to that in the middle section. When an external electric field is applied, the net charges with different signs are driven to flow in opposite directions, generating flows in different directions in the microchannel and creating strong circulations in this middle region, as seen in Fig. 5(b). The net flow direction is positive (in the z-direction) in this case because the overall

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Fig. 5. Zeta potential distribution and its corresponding flow field and streamline pattern in a 100 ␮m diameter microchannel under an applied electrical field strength of 140 V/cm.

zeta potential strength is negative. The middle regions with a positive zeta potential has a parabolic velocity profile in the main flow direction in major area of the cross section. A combination of a linear change and a step change in zeta potential is presented in Fig. 6(a). A linear variation from 0 mV to 100 mV is designed in the first half of the channel length and −100 mV is designed for the rest of the channel. Distortion observed in the flow field can be understood in the same way described above. Comparing the streamline patterns shown in Fig. 4(b), Fig. 5(b) and Fig. 6(b), we can see that in Fig. 4(b), circulations occur at the region with a stepwise change in zeta potential and fairly straight and parallel streamlines are found in the other regions. Consider two parallel streams of different solutions flow through the microchannel. This design would provide circulations within each individual stream at the region with a stepwise change in zeta potential, and the mixing between two different solutions mostly occurs in the radial direction through diffusion. Due to the presence of opposite flows through the entire channel length, the flow rate is very small in this case, which can also be seen in Fig. 7. For the case presented in Fig. 5, circulations

occur in the middle region, where the zeta potential has an opposite sign to that of the rest of the channel. A narrower channel generated between the two circulations will enhance the cross-stream diffusion because the diffusion between two different, parallel flowing solutions in this region occurs over a shorter distance. The case presented in Fig. 6 is similar to that shown in Fig. 4 in that it provides circulation within each solution stream and gives rise to a smaller flow rate due to the opposite flows through the entire channel length. In addition, there are two triangular-shape circulations close to the channel wall in the half channel with a linear variation of zeta potential. The narrower and converging flow channel formed in this region tends to enhance the cross-stream mixing between the two different, parallel flowing solutions. In practice, it is often difficult to avoid completely any external pressure difference between the two ends of a microchannel, for example, due to the uneven leveling along the channel length direction and due to the different liquid heights in the wells connected by the microchannel. EOF velocity profile is very sensitive to the external pressure effect, particularly when the channel diameter is relatively large

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Fig. 6. Zeta potential distribution and its corresponding flow field and streamline pattern in a 100 ␮m diameter microchannel under an applied electrical field strength of 140 V/cm.

Fig. 7. Effects of heterogeneous patterns of zeta potential and applied pressure differences on flow rate in a 100 ␮m diameter microchannel under an applied electrical field strength of 140 V/cm.

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Fig. 8. Zeta potential distribution and its corresponding flow field, streamline pattern and pressure field in a 100 ␮m diameter microchannel under an applied electrical field strength of 140 V/cm.

(>100 ␮m) [21]. It has been found that a combination of electroosmtoic flow and pressure driven flow can generate threedimensional flows in simple microchannels [22]. Therefore, two combined electroosmotic and pressure driven flows are investigated in this study. A small pressure head is applied in both cases with the purpose of evaluating the effects on flow mixing in microchannels. The zeta potential and the applied pressure difference for the first combination is shown in Fig. 8(a) and (b). The zeta potential distribution is the same as that described in Fig. 2 and a slightly higher gage pressure of 4 Pa is applied to the outlet of the channel as compared to that in the inlet of the channel (Gage pressure = 0 Pa). Physically, this situation can represent the difference in the liquid level in reservoir wells, or a pressure pump is placed at the outlet providing a pressure driven flow against the electroosmotic flow. The pressure gradient is found to be higher at the active EOF region because a component of the pressure gradient is the externally ap-

plied pressure gradient. Consequently, circulation occurs in the active EOF region because the electroosmotic flow originated from the channel wall is in the opposite direction to the flow generated by the externally applied pressure gradient, as shown in Fig. 8(c–e). Clearly, this small difference in applied pressures between the two ends of the microchannel can cause significant change in flow field in terms of magnitude of velocity components by comparing Fig. 8(d) and Fig. 2(b). However, when a higher pressure is applied at the inlet of the microchannel instead of the outlet, both electroosmotic and pressure driven flow are in the same direction. Therefore, there is no flow circulation and the flow patterns are similar to that shown in Fig. 2 with a higher flow rate. The results are not shown here. Since electroosmotic flow has been commonly used as a pumping method in microfluidic applications, it is desirable to know the effects of the heterogeneous zeta potential patterns on electroosmotic flow rate. Fig. 7 shows the effects

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on flow rate under the same applied electrical field. As one can see that the flow rates for case one, two, four and five are approximately the same. This can be understood as follows. The overall zeta potential strength for these cases is the same (−50 mV), which indicates that the overall driving force is the same, resulting in the approximately equal flow rates. The small discrepancies in flow rate among these four cases are due to the different extents of flow field distortion corresponding to different heterogeneous patterns of zeta potential. The flow rates in case three and six are approximately half of the other cases. This can be understood as their overall zeta potential strength is −25 mV, half of the other four cases. The effects of the applied pressure difference on flow rate depend on the direction of the applied pressure driven flow and the direction of the electroosmotic driven flow. When the applied pressure driven flow is in the same direction as that of electroosmotic driven flow, the flow rate is bigger than that of pure electroosmotic driven flow, vice versa. This can be seen by comparing cases one, seven and eight. Cases seven and eight have the same zeta potential distribution as that of case one, but both have an applied pressure difference. The applied pressure difference in case seven generates a flow in the same direction as the electroosmotic driven flow and results in a higher flow rate as compared with case one. Case eight has a lower flow rate than that of case one because the pressure driven flow is in the opposite direction to the electroosmotic driven flow.

5. Conclusion Electroosmotic flow in a microchannel with non-uniform zeta potential is studied in this paper. The results show that the different types of velocity profiles in the upstream and the downstream sections can be generated by the non-uniform zeta potential distribution. It was found that simple step change in zeta potential cannot generate flow circulation if such a zeta potential change does not involve the sign change of the surface charge. Several heterogeneous patterns involving zeta potential changes from positive to negative or negative to positive have been investigated and found to be able to generate different types of flow circulations. A combination of simple step change of zeta potential without the sign change and a small applied pressures drop in the opposite direction to the EOF driving force is found to be another way to generate flow circulation. The model predictions of the flow rate and the flow field in such a heterogeneous microchannel provide useful information about how to manipulate the electroosmotic flow mixing in lab-on-a-chip devices. In order to

increase the mixing, heterogeneous patterns that can create flow circulation are suggested in this paper. Mixing would be enhanced for a series of small heterogenous patches rather than a long single patch, due to the degree of velocity disruption over this region. Moreover, the results show that if the charge density of the heterogeneous patches increase, velocity field disruption would also increase, suggesting that large difference in charge density between the microchannel’s wall and the heterogeneous patches can enhance micro mixing.

Acknowledgments The authors gratefully acknowledge the support of a Research Grant of the Natural Sciences and Engineering Research Council (NSERC) of Canada to D. Li, and an Ontario Graduate Scholarship to C. Ren.

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