On the Surface Conductance, Flow Rate, and Current Continuities of

Feb 12, 2005 - The characteristics of electrokinetic flow in a microchannel depend on both the nature of surface potentials, that is, whether it is un...
0 downloads 0 Views 159KB Size
Download PDF
2192

Langmuir 2005, 21, 2192-2198

On the Surface Conductance, Flow Rate, and Current Continuities of Microfluidics with Nonuniform Surface Potentials Fuzhi Tian and Daniel Y. Kwok* Nanoscale Technology and Engineering Laboratory, Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta T6G 2G8, Canada Received October 22, 2004. In Final Form: January 7, 2005 The characteristics of electrokinetic flow in a microchannel depend on both the nature of surface potentials, that is, whether it is uniform or nonuniform, and the electrical potential distribution along the channel. In this paper, the nonlinear Poisson-Boltzmann equation is used to model the electrical double layer and the lattice Boltzmann model coupled with the constraint of current continuity is used to simulate the microfluidic flow field in a rectangular microchannel with a step variation of surface potentials. This current continuity, including surface conduction, convection, and bulk conduction currents, has often been neglected in the literature for electroosmotic flow with nonuniform (heterogeneous) microchannels. Results show that step variation of ion distribution caused by step variation surface potential will influence significantly the electrical potential distribution along the channel and volumetric flow rate. For the system considered, we showed that the volumetric flow rate could have been overestimated by as much as 70% without consideration of the current continuity constraint.

I. Introduction Liquid flow in microchannels is often influenced by a double layer of electrical charges (also known as the electrical double layer, or EDL) near the solid-liquid interface. The counterions and co-ions in an aqueous solution are preferentially distributed near the solid surface so that the net charge density becomes nonzero. When a pressure gradient is applied, a streaming current is established due to the movement of counterions in the EDL. Corresponding to this streaming current is an electrokinetic potential, namely, the streaming potential that builds up along the channel. This streaming potential will drive the counterions in the EDL in the direction opposite to the streaming current, that is, opposite to the pressure-driven flow direction, generating an electrical current known as a conduction current. As the motion of ions causes bulk movement downstream, the conduction current acts in an opposite direction against the pressuredriven flow. Therefore, the predicted flow rate by traditional fluid mechanics theory is higher than that of the experimental value. This results in an apparently higher liquid viscosity, and the phenomenon is typically referred to as an electroviscous effect. Alternatively, when an electric field is applied between two ends of the channel, the excess counterions in the EDL will be driven by an electric body force which drags the liquid due to viscosity, resulting in an electroosmotic flow.1-3 The pressure-driven and electroosmotic flows mentioned above are the frequently employed methods for liquid transport in most microfluidics applications. For electrokinetic flows in microchannels with a homogeneous surface potential, analytical solutions are available in cylindrical capillaries,1,4,5 slit channels,2,6 and straight rectangular * To whom correspondence should be addressed. Phone: (780) 492-2791. Fax: (780) 492-2200. E-mail: [email protected]. (1) Rice, C. L.; Whitehad, R. J. Phys. Chem. 1965, 69, 4017. (2) Burgreen, D.; Nakache, F. R. J. Phys. Chem. 1963, 67, 1084. (3) Levine, S.; Marriott, J. R.; Neale, G.; Epstein, N. J. Colloid Interface Sci. 1975, 52, 136. (4) Yang, J.; Kwok, D. Y. Langmuir 2003, 19, 1047.

microchannels.7-9 Such analytical solutions are often restricted to low surface potential. If the surface potentials of these channels are heterogeneous, the electrokinetic problems become more difficult and would have to be solved numerically. The characteristics of such a flow would depend on both the pattern of the surface potential (i.e., whether the surface charge is a step or continuous change) and the potential distribution along the channel. For electroosmotic flows, several authors have shown that the flow field could vary from laminar to multidirection with circulations. For example, Ajdari10,11 theoretically investigated electroosmosis with a nonuniform surface potential and found circulation regions generated by the application of oppositely charged surface heterogeneities on the microchannel wall. The phenomena were also observed experimentally by Strook et al.12-14 Ren and Li15 numerically studied electroosmotic flow in heterogeneous circular microchannels with an axial step change of surface potential and illustrated the distorted velocity profiles. Keely et al.16 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.17 (5) Masliyah, J. H. Electrokinetic transport phenomena; OSTRA technical publication series, no. 12; Alberta Oil Sands Technology and Research Authority: Edmonton, AB, Canada, 1994. (6) Yang, J.; Kwok, D. Y. J. Colloid Interface Sci. 2003, 260, 265. (7) Yang, J.; Bhattacharyya, A.; Masliyah, J. H.; Kwok, D. Y. J. Colloid Interface Sci. 2003, 261, 21. (8) Yang, C.; Li, D. Colloids Surf., A 1998, 143, 339. (9) Arulanandam, S.; Li, D. Colloids Surf., A 2000, 161, 89. (10) Ajdari, A. Phys. Rev. Lett. 1995, 75, 755. (11) Ajdari, A. Phys. Rev. E 1996, 53, 4996. (12) Stroock, A. D.; Weck, M.; Chiu, D. T.; Huck, W. T. S.; Kennis, P. J. A.; Ismagilov, R. F.; Whitesides, G. M. Phys. Rev. Lett. 2000, 84, 3314. (13) Stroock, A. D.; Dertinger, S. K.; Ajdari, A.; Mezic, I.; Stone, H. A.; Whitesides, G. M. Science 2002, 295, 647. (14) Stroock, A. D.; Dertinger, S. K.; Whitesides, G. M.; Ajdari, A. Anal. Chem. 2002, 74, 5306. (15) Ren, L.; Li, D. J. Colloid Interface Sci. 2001, 243, 255. (16) Keely, C. A.; van de Goor, T. A. A. M.; McManigill, D. Anal. Chem. 1994, 66, 4236. (17) Herr, A. E.; Molho, J. I.; Santiago, J. G.; Mungal, M. G.; Kenny, T. W. Anal. Chem. 2000, 72, 1053.

10.1021/la0473862 CCC: $30.25 © 2005 American Chemical Society Published on Web 02/12/2005

Microfluidics with Nonuniform Surface Potentials

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 the pressure gradient due to the surface heterogeneity. Erickson and Li18,19 studied electrokinetic flows with patchwise surface potentials. In all of the above studies, however, the continuity of electrical current across sections of nonuniform microchannels was neglected and only the continuity of flow rate was automatically satisfied. As a matter of fact, a typically employed assumption is the constancy of potential distribution along such channels which obviously does not reflect physical reality for electroosmotic flows. Until recently, Yang et al.20 examined both flow rate and electrical current continuities for electroosmotic flow in circular microchannels with nonuniform surface potentials. However, their analytical solution was solved on the basis of a Debye-Hu¨ckel approximation and the applicability is limited to low surface potential. In addition, the effect of surface conductance induced by the accumulated charge in the EDL at the solid-liquid interfaces was also neglected. As the concept of surface conductance has been around since the 1930s,21 it would be instructive to examine how important is such an effect. In 1939, Rutgers22 proposed an indirect method to determine the surface conductivity and derived a relationship between the surface, bulk, and total conductivity of a solid-liquid flow system. The importance of surface conductance has been quantified by Lyklema et al.23,24 in terms of a Dukhin number. As for experiments, Gu et al.25 and Erickson et al.26 utilized streaming potential measurements to determine the ζ potential and surface conductivity. In general, the effect of surface conductance is comparable to the bulk conductance when the channel height or radius is small ( 10) and the Reynolds number Re is low. A lattice Boltzmann method is employed to solve the nonlinear PoissonBoltzmann equation numerically. B. Constraint of Current Continuity. From the electrokinetic theory, the total current in the steady state consists of three components: bulk conduction current Icb, surface conduction current Ics, and convection current Is.5,21,22 Thus, the total current It can be calculated by

It ) Icb + Ics + Is

(9)

The convection current Is for the slit rectangular channel shown in Figure 1 is defined by

Is )

∫0Hux(x, y) Fe(x, y)W dy

(10)

In the above equation, ux(x, y) is the velocity profile in the x direction, which varies in the y direction with homogeneous surface potentials; however, ux(x, y) will change in both the x and y directions with nonuniform heterogeneous surface potentials. Thus, with the knowledge of net ion distribution and velocity profile, the convection current can be determined. For a pressure-driven flow, this convection current is also called the streaming current due to the motion of net charges in the EDL region. The conduction current Ic includes two parts, Icb and Ics, and can be calculated by

(

Ic )

(6)

)

Pw φ φ φ Ic ) Icb + Ics ) Acλb + Pwλs ) Ac λb + λs L L L Ac

(11)

where Ac and Pw are respectively the cross-sectional area and wetted perimeter of the channel, λb is the bulk conductivity, φ is the streaming potential in pressuredriven flow or the imposed electric potential in an electroosmotic flow by setting the potential at the outlet to be zero, and λs is the surface conductivity. For the rectangular cross-sectional channel with H/W , 1 shown in Figure 1, Pw/Ac ≈ 2/H; hence, eq 11 can be simplified to

(

)

WHλbφ 2λs WHλbφ (1 + Du) 1+ ) L λbH L

(12)

where Du ) 2λs/λbH is the Dukhin number.24 It should be pointed out that the original definition of the Dukhin number is Du ) λs/λbl, where l is a characteristic dimension. For a spherical particle or cylindrical pore with a radius of a, the characteristic dimension l becomes a.23,24 However, in our study, the cross section of the channel is a rectangular shape with a height of H; thus, it is logical to select H/2 as the characteristic dimension so that Du ) 2λs/λbH. This representation of Du is consistent with that given elsewhere.25 The surface conductance effects can be neglected when the channel is larger, for example, H > 200 µm. 1. Pressure-Driven Flow. For steady-state pressuredriven electrokinetic flow in microchannels, the total electric current should be zero so that the constraint of current continuity on the flow system is

It ) Icb + Ics + Is ) 0

(13)

Once the flow field is known, the convection current or the streaming current can be evaluated by eq 10. Combining eqs 11 and 13, we can determine the streaming potential from

φ ) -L

∫0Hux(x, y) Fe(x, y) dy

(

)

Pw λb + λs H Ac

(14)

The conduction current can then be obtained by eq 11 or 12. The strength of the induced electric field E is φ/L. As the induced streaming potential will produce an electroosmotic effect, an additional body force caused by the streaming potential acting on the EDL will be applied to the LBM equation as well as pressure body force; the velocity will then be recalculated in order to obtain a more accurate streaming potential. 2. Electroosmotic Flow. In electroosmotic flow, an externally applied electric field is utilized to drive the liquid and two ends of the channel are exposed to an atmospheric pressure. In this case, the total current can be calculated by substituting eqs 10 and 11 into eq 9. For the heterogeneous microchannel with nonuniform surface potentials (ψ1 and ψ2) represented as the two sections in Figure 2, the constraint of current continuity is It1 ) It2 and hence

It1 )

∫0Hux1(x, y) Fe1(x, y)W dy + AcλbE1 + Pwλs1E1

)

∫0Hux2(x, y) Fe2(x, y)W dy + AcλbE2 + Pwλs2E2

) It2 E1 )

φ1 - φ2 L1

E2 )

φ2 - φ3 L2

(15)

The subscripts 1 and 2 represent the properties of the left and right sections in Figure 2, respectively. By considering this constraint, the electric potential φ2 at the junction between the two different surface potentials (ψ1 and ψ2) can be evaluated according to the velocity profile and the

Microfluidics with Nonuniform Surface Potentials

Langmuir, Vol. 21, No. 6, 2005 2195

Figure 2. Geometry of a 2D heterogeneous microchannel, consisting of two different sections. ψ1 and ψ2 are the two stepwise variations of surface potentials for the heterogeneous channel.

externally applied electric potential φ1 when φ3 ) 0 V. For instance, if the surface potential of the left section ψ1 is set to be zero (i.e., no EDL effect), the net charge Fe1 and surface conductivity λs1 are also zero; hence, eq 15 can be simplified to

It1 ) AcλbE1 )

∫0Hux2(x, y) Fe2(x, y)W dy + AcλbE2 + Pwλs2E2 ) It2

E1 )

φ1 - φ2 L1

E2 )

φ2 L2

(16)

Figure 3. Streaming potential along a 1 cm rectangular microchannel (H ) 500 nm) with a uniform surface potential for ψ ) -25 and -75 mV. The filled and open symbols represent the cases with (λs ) 1.64 nS) and without (λs ) 0 nS) consideration of surface conductance, respectively; the pressure gradient is dP/dx ) 10 MPa/m.

In this case, φ2 can be calculated by

Acλb H - 0 ux2(x, y) Fe2(x, y)W dy L1 φ2 ) Acλb Acλb Pwλs2 + + L1 L2 L2

∫

(17)

III. Results and Discussion As mentioned earlier, the purpose of this study is to investigate the effect of a nonuniform surface potential on the potential distribution along the channel and volumetric flow rate, by considering current continuity using a LBM method in which flow rate continuity is satisfied automatically. Surface conductance, which was neglected in the analytical solution in ref 20, is also considered here. A stepwise variation of surface potential is selected and shown in Figure 2, where ψ1 and ψ2 are the surface potentials of the left and right sections with a length of L1 and L2, respectively. A symmetric electrolyte solution (KCl) with z:z ) 1:1 was employed, and its physical properties are 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 In this simulation, we assume that the variation of the surface potential will not influence the bulk and surface conductance for the KCl electrolyte solution. The ionic molar concentration of the solution is set to be 10-1 mM with an EDL thickness 1/κ of 30.4 nm. The bulk conductivity λb of the solution is 1.42 × 10-3 S/m. The height of the channel H is assumed to be 500 nm for κH ) 16. At this ratio, the EDL will not overlap and a pluglike electroosmotic flow can be obtained. The boundary condition at the wall is nonslip and was implemented in the LBM as a bounce-back boundary condition; a periodic boundary condition was employed at the outlet and inlet and can represent an infinitely long channel. Validation

Figure 4. Streaming potential along a 1 cm rectangular microchannel (H ) 500 nm) with a nonuniform surface potential for ψ1 ) 0 and ψ2 ) -25 and -75 mV. The filled and open symbols represent the cases with (λs1 ) 0 nS and λs2 ) 1.64 nS) and without (λs1 ) 0 nS and λs2 ) 0 nS) consideration of surface conductance, respectively; the pressure gradient is dP/dx ) 10 MPa/m.

of our LBM to simulate the electrokinetic flow can be found elsewhere.27-29 A. Pressure-Driven Flow. It is well-known that a streaming potential sets up when a liquid is forced through a microchannel under an applied pressure gradient. Obviously, the streaming potential can be obtained by considering the constraint of current continuity, as shown in eq 13. For a channel with a uniform surface potential, the streaming potential is proportional to the surface potential and the distribution along the channel is linear, as shown in Figure 3 for two different ψ values: -25 and -75 mV. Here, the Du number is 4.62 when the surface conductance λs is 1.64 nS. For a channel with a nonuniform surface potential such as a stepwise variation of surface potential, the streaming potential φ will be different. In our simulation, we set the surface potential ψ1 in Figure 2 as zero while ψ2 as (27) Tian, F.; Li, B.; Kwok, D. Y. J. Comput. Theor. Nanosci., in press. (28) Li, B.; Kwok, D. Y. Langmuir 2003, 19, 3041. (29) Li, B.; Kwok, D. Y. Phys. Rev. Lett. 2003, 90.

2196

Langmuir, Vol. 21, No. 6, 2005

Tian and Kwok

Figure 5. Potential distribution along a 1 cm rectangular microchannel (H ) 500 nm) with a stepwise variation of surface potential when the constraint of current continuity is considered in electroosmotic flow: (a) ψ1 ) 0 mV and ψ2 ) -50 mV; (b) ψ1 ) -50 mV and ψ2 ) 0 mV. The filled and open circle symbols represent the cases with (λs1 ) 0 nS and λs2 ) 1.64 nS) and without (λs1 ) 0 nS and λs2 ) 0 nS) consideration of surface conductance, respectively.

beingnegatively charged so that λs1 ) 0 nS and λs1 * 0 nS. The junction of two sections is at the midpoint of a 1 cm channel (L ) 2L1 ) 2L2 ) 1.0 cm). The convection current Is in the left section will be zero because of the zero net charge in the solution (no EDL), so that the streaming potential is also zero. For the section on the right, a streaming potential will be set up and the results are presented in Figure 4 for ψ2 ) -25 and -75 mV. This result indicates that the streaming potential relates the surface potential and ion distribution associated with the surface potential, in agreement with intuition. In both Figures 3 and 4, the filled and open symbols represent the cases with (λs2 ) 1.64 nS) and without (λs2 ) 0 nS) surface conductance, respectively. From these figures, we see that the magnitude of the streaming potential depends on whether surface conductance is neglected in the current continuity. If the effect of surface conductance is completely neglected (λs2 ) 0 nS), the streaming potential will be overestimated. This phenomenon is consistent with that studied elsewhere.25 B. Electroosmotic Flow. The constraint of current continuity is, however, not apparent in electroosmotic flow. In this section, the volumetric flow rate and potential distribution for electroosmotic flow in a channel are calculated with a stepwise variation of surface potential. The constraint of current continuity is also considered for two cases: λs2 ) 0 nS (Du ) 0) and λs2 ) 1.64 nS (Du ) 4.62), both with λs1 ) 0 nS. The results are compared with those without such a constraint. We set the externally applied potentials at the inlet as φ1 ) 100 V and those at the outlet as φ3 ) 0 V. The surface potential of the left section ψ1 in Figure 2 is set as 0, while that of the right section ψ2 is set as being negatively charged. 1. Potential Distribution along the Channel. For a channel with a uniform surface potential, the electric potential distribution along the channel governed by the Laplace equation (eq 3) is linear and the electric field strength E is constant based on the assumptions of a thin EDL and constant permittivity . Thus, the potential φ2 at the junction equals (φ1 - φ3)/2 and hence φ2 ) 50 V. When the surface potential is nonuniform, such as that shown in Figure 2, the potential distribution along the channel depends on whether current continuity is considered. If the constraint of current continuity It1 ) It2 is not involved, the potential distribution is usually regarded as being identical to the case with a uniform surface potential, that is, φ2 ) (φ1 - φ3)/2. However, the realistic situation is that current continuity should also be considered in electroosmotic flow for a nonuniform surface

Figure 6. Electrical circuit analogy to electroosmotic flow in the heterogeneous microchannel in Figure 2.

potential and the potential distribution along the channel between φ1 and φ3 will not be linear. For a stepwise variation in surface potential, there is a step change in the potential distribution, as given in Figure 5. From Figure 5a with ψ1 ) 0 mV and ψ2 ) -50 mV, we see that the potential at junction φ2 becomes 15 V for λs2 ) 1.64 nS and 49 V for λs2 ) 0 nS, rather than 50 V. This represents a 70% (35 V) and 2% (1 V) decrease from the linear assumption [φ2 ) (100 - 0)/2 ) 50 V] when the constraint of current continuity is considered. The potential reduction for λs2 ) 0 nS is in agreement with the results in ref 20 without consideration of surface conductance. However, if surface conductance is considered (e.g., λs2 ) 1.64 nS), a large reduction in φ is obtained due to the effect of the surface conductance which is comparable to the bulk conductance in the simulated system. To illustrate the symmetry of our system, Figure 5b shows the results for the case where the surface potentials for two sections are interchanged, that is, ψ1 ) -50 mV and ψ2 ) 0 mV. As the electric strengths E1 and E2 in Figure 5b equal E2 and E1 in Figure 5a, respectively, the effective driving force on the flow remains unchanged and hence the flow rate remains unchanged as well. Furthermore, as a periodic boundary condition at the inlet and outlet was applied, it is sufficient to illustrate the problems of interest by the case shown in Figure 5a. The above phenomenon can be explained by the analogy of an equivalent electrical circuit shown in Figure 6. In this circuit, the conductances λ1 and λ2 correspond to the conductances for the left and right sections in Figure 2, respectively. Compared to the electroosmotic system mentioned earlier, λ1 ) λb and λ2 is the sum of the bulk conductance λb, the surface conductance λs2, and the equivalent conductance due to convective current Is. Obviously, the value of λ1 is smaller than λ2. Thus, the resistance R1 should be higher than R2. According to Ohm’s law and the current continuity I1 ) I2 in a series circuit, we have (φ1 - φ2)/R1 ) (φ2 - φ3)/R2 so that the φ2 value for R1 > R2 should be smaller than that for R1 ) R2, given

Microfluidics with Nonuniform Surface Potentials

Langmuir, Vol. 21, No. 6, 2005 2197

Figure 7. Normalized potential at the junction of the two heterogeneous sections in Figure 2 versus the surface potential of the right section ψ2 with current continuity when ψ1 ) 0 mV, φ1 ) 100 V, and φ3 ) 0 V: (a) λs1 ) 0 nS and λs2 ) 0 nS; (b) λs1 ) 0 nS and λs2 ) 1.64 nS.

that φ1 and φ3 are fixed at the same value. This analogy applies to our electroosmotic system when nonuniform surface potential and current continuity are involved; the potential φ2 at the junction between the two heterogeneous sections will decrease significantly from the case with uniform surface potential or without current constraint. Nevertheless, the potential distribution in each section remains linear, as shown in Figure 5. Figure 7 presents the ratio of the potential φ2/φ1 for different surface potentials ψ2 of the section on the right. The potential φ2 is normalized by the potential φ1 at the inlet. In Figure 7a, surface conductivity is not considered (λs2 ) 0 nS), while λs2 is assumed to be 1.64 nS in Figure 7b. From this figure, we see that the potential φ2/φ1 or φ2 increases nonlinearly with the surface potential ψ2 for both cases. The variation of the potential ∆φ2 as a function of ψ2 is relatively small (∼1 V) and is due to the assumption of constant surface conductivity λs and bulk conductivity λb in our simulation. This small variation is induced only by the change in the convective current which is only a small fraction of the conduction current. Furthermore, we see that the variation is between 49 and 50 V when surface conductivity is neglected. When the surface conductivity of the right section λs2 is 1.64 nS, the variation is between 14 and 15 V and is much lower than that when λs2 ) 0. As a result, the apparent conductivity for the right section with an EDL effect in Figure 2 is much higher than that of the left section when surface conductance is considered. This difference is attributed to the surface conductance and convection current induced by the EDL effect. 2. Flow Rate. Figure 8 shows the results of the flow rate Q versus the variation of the surface potential ψ2 of the heterogeneous channel. To eliminate the effect of channel width W in a 2D simulation problem, the volumetric flow rate is normalized by the maximum flow rate Qmax when the surface potential is -100 mV without consideration of current continuity. In this figure, circles (filled for λs2 ) 1.64 nS and open for λs2 ) 0 nS) and open squares represent the volumetric flow rate with and without the constraint of the current continuity, respectively. As can be seen, the flow rates in all cases are inversely proportional to the negative surface potential, as is well-known. However, the flow rate (filled circles) with the constraint of the current continuity and consideration of surface conductance (λs2 ) 1.64 nS) is much smaller than those of the other two cases. As discussed above, electroosmotic flow is induced by acting the externally applied electric potential on the excess counterions in the EDL. In other words, both the

Figure 8. Normalized volumetric flow rate in electroosmotic flow versus the surface potential of the right section ψ2 with and without the constraint of current continuity when ψ1 ) 0 mV, φ1 ) 100 V, and φ3 ) 0 V. The filled and open circle symbols represent the cases with (λs1 ) 0 nS and λs2 ) 1.64 nS) and without (λs1 ) 0 nS and λs2 ) 0 nS) consideration of surface conductance, respectively.

externally applied electric potential and the EDL will influence the velocity profile and hence the volumetric flow rate. For the electroosmotic system simulated in this paper, the effective driving potential is the difference between φ2 and φ3 because the surface potential ψ1 in Figure 2 is zero. Thus, a decrease in the potential φ2 due to the constraint of current continuity will induce a smaller driving force so that the flow rate will become smaller for the same surface potential. As a result, in this specific example, the flow rate has been overestimated by as much as 70% when the constraint of current continuity and surface conductance are neglected, respectively. C. Effect of the Du Number. From the definition in eq 12, we can obtain a different Du number by changing λs, λb, and H. In this section, we study the effect of the Du number on the flow rate Q, the potential φ2, and the total current It, with λb ) 1.42 × 10-3 S/m, H ) 0.5 µm, ψ1 ) 0 mV, ψ2 ) -50 mV, φ1 ) 100 V, and φ3 ) 0 V. Figure 9 shows the results for the normalized potential at the junction of the two heterogeneous sections in Figure 2 versus the Du number for the right section with current continuity. From this figure, we see that the normalized potential φ2/φ1 decreases nonlinearly as Du increases; the variation of potential becomes more sensitive when Du is smaller than 10. Correspondingly, the normalized flow rate shown in Figure 10 has a similar trend. When the Du

2198

Langmuir, Vol. 21, No. 6, 2005

Figure 9. Normalized potential at the junction of the two heterogeneous sections in Figure 2 versus the Du number of the right section with current continuity when ψ1 ) 0 mV, φ1 ) 100 V, and φ3 ) 0 V.

Tian and Kwok

Figure 11. Total current in electroosmotic flow versus the Du number of the right section with the constraint of current continuity when ψ1 ) 0 mV, φ1 ) 100 V, and φ3 ) 0 V.

Overall, from Figures 9, 10, and 11, we see that when the Du number is small, the normalized flow rate Q/Qmax is large due to high normalized potential φ2/φ1 and small total current It, suggesting that most of the energy is used to drive the flow. However, when Du is large, the normalized flow rate Q/Qmax is quite small due to lower normalized potential φ2/φ1 and higher total current It. This implies that most energy is consumed by electric current rather than flow rate. IV. Summary

Figure 10. Normalized volumetric flow rate in electroosmotic flow versus the Du number of the right section with the constraint of current continuity when ψ1 ) 0 mV, φ1 ) 100 V, and φ3 ) 0 V.

number approaches zero, the normalized flow rate Q/Qmax is nearly 1, implying that the effect of surface conductance is negligible. If Du is large, a significant decrease in flow rate can be found when compared with the case without surface conductance, suggesting that surface conductance cannot be ignored in the constraint of current continuity. Figure 11 presents how the total current in electroosmotic flow will change as a function of the Du number for the right section with the constraint of current continuity when ψ1 ) 0 mV, φ1 ) 100 V, and φ3 ) 0 V. From this figure, we see the total current It increases nonlinearly with the Du number and essentially reaches a plateau. When the Du number decreases and approaches zero, the total current is determined mainly by the bulk conductance which is nearly zero. When the Du number is very large, the total current is dominated by surface conductance and hence results in minimal variation.

A lattice Boltzmann model has been used to simulate an electrokinetic flow field in a 500 nm nonuniform microchannel by consideration of both flow rate and electric current continuities. The electric current incorporates all surface conduction, convection, and bulk conduction currents. For pressure-driven flow, the constraint of current continuity results in the well-known streaming potential effect from the electrical double layer. For electroosmotic flow with this constraint under a step variation in surface potential, the potential distribution along the channel is also a step function. As the potential distribution varies, the effective potential to drive the flow due to an EDL effect decreases significantly. For the selected system, we have shown that the volumetric flow rate for electroosmotic flow without consideration of current continuity could have been overestimated by as much as 70 and 2% for λs ) 1.64 nS and λs ) 0 nS, respectively. In addition, when the Du number approaches zero, the effect of the surface conductance can be neglected. Acknowledgment. D.Y.K. gratefully acknowledges financial support from the Canada Research Chair Program and the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Discovery Grant. LA0473862