Anal. Chem. 2000, 72, 1987-1993
Finite Element Simulation of an Electroosmotic-Driven Flow Division at a T-Junction of Microscale Dimensions F. Bianchi, R. Ferrigno, and H. H. Girault*
Laboratoire d’Electrochimie, Ecole Polytechnique Fe´ de´ rale de Lausanne, CH - 1015 Lausanne, Switzerland
A finite element formulation is developed for the simulation of an electroosmotic flow in rectangular microscale channel networks. The distribution of the flow at a decoupling T-junction is investigated from a hydrodynamic standpoint in the case of a pressure-driven and an electroosmotically driven flow. The calculations are carried out in two steps: first solving the potential distribution arising from the external electric field and from the inherent ζ potential. These distributions are then injected in the Navier Stokes equation for the calculation of the velocity profile. The influence of the various parameters such as the ζ potential distribution, the Reynolds number, and the relative channel widths on the flow distribution is investigated. A large amount of studies have been reported on the development of micromachined devices for micro Flow Injection Analysis applications.1-5 Electroosmotic pumping, generated by means of an external potential field, shows to be a suitable way to drive successfully nanovolumes within a capillary network.6-10 When an external electrical field is applied in a channel, the walls of which are charged, the migration of the ions present in excess in the double layer induces the motion of the bulk solution due to the viscous drag. This phenomenon is used in different microdevices and for various applications as, for example, microfractionation,11,12 electrophoresis,6,12-15 and microspray generation sys* Corresponding author: (e-mail)
[email protected] (1) Haswell, S. J. Analyst (Cambridge, U.K.) 1997, 122, R1-R10. (2) Effenhauser, C. S. Top. Curr. Chem. 1998, 194, 51-82. (3) Burggraf, N.; Manz, A.; Verpoorte, E.; Effenhauser, C. S.; Widmer, H. M.; de Rooij, N. F. Sens. Actuators, B 1994, 20, 103-110. (4) Manz, A.; Graber, N.; Widmer, H. M. Sens. Actuators, B 1990, 1, 244-248. (5) Harrison, D. J.; Glavina, P. G.; Manz, A. Sens. Actuators, B 1993, 10, 107116. (6) Harrison, D. J.; Manz, A.; Fan, Z. H.; Ludi, H.; Widmer, H. M. Anal. Chem. 1992, 64, 1926-1932. (7) Harrison, D. J.; Fluri, K.; Seiler, K.; Fan, Z. H.; Effenhauser, C. S.; Manz, A. Science (Washington, D.C.) 1993, 261, 895-897. (8) Seiler, K.; Fan, Z. H. H.; Fluri, K.; Harrison, D. J. Anal. Chem. 1994, 66, 3485-3491. (9) Ramsey, R. S.; Ramsey, J. M. Anal. Chem. 1997, 69, 1174-1178. (10) Li, P. C. H.; Harrison, D. J. Anal. Chem. 1997, 69, 1564-1568. (11) Effenhauser, C. S.; Manz, A.; Widmer, H. M. Anal. Chem. 1995, 67, 22842287. (12) Raymond, D. E.; Manz, A.; Widmer, H. M. Anal. Chem. 1996, 68, 25152522. (13) Effenhauser, C. S.; Manz, A.; Widmer, H. M. Anal. Chem. 1993, 65, 26372642. (14) Manz, A.; Harrison, D. J.; Verpoorte, E. M. J.; Fettinger, J. C.; Paulus, A.; Ludi, H.; Widmer, H. M. J. Chromotogr. 1992, 593, 253-258. 10.1021/ac991225z CCC: $19.00 Published on Web 03/28/2000
© 2000 American Chemical Society
tems.9,16 This novel fluid handling necessitates, compared with a pressure-driven flow system, the development of new devices addressing the integration of a high-voltage field without perturbing the analysis system as in Capillary Electrophoresis with Electrochemical detection (CEEC). The development of channel networks within micro-devices involves the necessity to characterize the flow behavior in different flow configurations for such small channel dimensions implying a very low Reynolds number. The T-junction is one of the first geometrical tools used for electrokinetic injection in a microchip.6,14 However, this method introduces an injection bias inherent in the difference of electrophoretic mobilities of the various compounds present in the injected sample.17 New sample injection schemes, preventing electrokinetic bias and based on time-independent injected volume,13 fall in two categories: the X-junction15,18,19 and the double T-junction.7,13,20 However, hydrodynamic phenomena still need to be understood properly to forecast flow behavior at the intersection of three channels.21 Moreover, in µ-FIA systems, the T-junction scheme has been widely introduced in microchips for others applications, where mixing between different flow streams is necessary in order to obtain a chemical reaction between various compounds.8,10 Multiple strategies and chip designs are encountered for the pre- or postcolumn labeling process, electrokinetic focusing, DNA sequencing, on-line PCR analysis, on-chip enzymatic sample digestion, fraction isolation, and immunoassays.2,22,23 Another type of application using the T-junction in a chip includes devices, where a decoupling system between the high voltage and the end column outlet of the capillary, is necessary. End- and on-column electrochemical detectors in a chip performing electrophoresis (CEEC) are good examples of this kind of (15) Jacobson, S. C.; Ramsey, J. M. Electrophoresis 1995, 16, 481-486. (16) Xue, Q. F.; Foret, F.; Dunayevskiy, Y. M.; Zavracky, P. M.; Mcgruer, N. E.; Karger, B. L. Anal. Chem. 1997, 69, 426-430. (17) Li, S. F. Y. Capillary Electrophoresis; Elsevier Science Publisher B. V.: Amsterdam, 1992. (18) Fan, Z. H.; Harrison, D. J. Anal. Chem. 1994, 66, 177-184. (19) Jacobson, S. C.; Hergenroder, R.; Koutny, L. B.; Warmack, R. J.; Ramsey, J. M. Anal. Chem. 1994, 66, 1107-1113. (20) Manz, A.; Verpoorte, E.; Effenhauser, C. S.; Burggraf, N.; Raymond, D. E.; Widmer, H. M. Fresenius, J. Anal. Chem. 1994, 348, 567-571. (21) Shultzlockyear, L. L.; Colyer, C. L.; Fan, Z. H.; Roy, K. I.; Harrison, D. J. Electrophoresis 1999, 20, 529-538. (22) Mastrangelo, C. H.; Burns, M. A.; Burke, D. T. Proc. IEEE 1998, 86, 17691787. (23) Effenhauser, C. S.; Bruin, G. J. M.; Paulus, A. Electrophoresis 1997, 18, 2203-2213.
Analytical Chemistry, Vol. 72, No. 9, May 1, 2000 1987
system.24-26 Using a T-junction structure allows a separation between the high- voltage field from the electrochemical detection field, but also the partial decoupling of the current pathway from the sample. Dasgupta et al.27,28 showed that auxiliary pressure electroosmotic pumping could be used to propel fluids toward a capillary, where no electric field was applied. Ramsey and Ramsey9 used a similar phenomenon at a T-junction to generate, at the outlet of the microdevice, an electrospray in a silicon microdevice interfaced with a Mass Spectrometer (MS). The EOF is reduced by coating the sidearm channel with linear polyacrylamide. In our laboratory, we have also developed a decoupling system in a photoablated polymer device29 used for an electrophoresis application.30,31 This system is composed of an array of microholes implemented onto the microchannel. The objective of this device is to separate the mass pathway from the high-voltage current. Other applications, which can be derived from the decoupling principle, are, for example, electroosmotic pumping in a chromatographic column with a conductivity detector or nanovolume manipulation in immunoassay applications. The aim of this paper is to describe, by the use of a numerical model, the behavior of a flow division at a T-junction from a hydrodynamic standpoint using both pressure and electroosmotic driven flows. The numerical models implemented in finite element method software are largely inspired from the recent works of Patankar et al.32 and Ermakov et al.33 In the former report, results concerning the electroosmotic flow at the intersection of a cross channel during the injection procedure are presented, whereas Ermakov et al. describe a numerical model for the simulation of the sample dilution by electrokinetic focusing. Compared with these two manuscripts, we will focus, in the present communication, on pressure effects, as dominant factors to predict the flow behavior. We report here results on the behavior of a decoupling configuration featuring a main channel with a sidearm. The influences of geometrical and physical parameters on the flow division are investigated. MATHEMATICAL MODEL Theory. The theoretical considerations are already reported in the two references listed above.32,33 Here, we briefly summarize the main assumptions made in these works and describe the equations derived in a finite element formulation in order to solve the electroosmotic flow problem. Electroosmotic flow is generated when an electrical field is imposed through an electrolyte solution parallel to the charged surfaces of a capillary. The general (24) Gavin, P. F.; Ewing, A. G. Anal. Chem. 1997, 69, 3838-3845. (25) Woolley, A. T.; Lao, K. Q.; Glazer, A. N.; Mathies, R. A. Anal. Chem. 1998, 70, 684-688. (26) Wallenborg, S. R.; Nyholm, L.; Lunte, C. E. Anal. Chem. 1999, 71, 544549. (27) Dasgupta, P. K.; Liu, S. Anal. Chem. 1994, 66, 1792-1798. (28) Dasgupta, P. K.; Liu, S. R. Anal. Chem. 1994, 66, 3060-3065. (29) Roberts, M. A.; Rossier, J. S.; Bercier, P.; Girault, H. Anal. Chem. 1997, 69, 2035-2042. (30) Rossier, J. S.; Schwarz, A.; Reymond, F.; Ferrigno, R.; Bianchi, F.; Girault, H. H. Electrophoresis 1999, 20, 727-731. (31) Schwarz, A.; Rossier, J. S.; Bianchi, F.; Reymond, F.; Ferrigno, R.; Girault, H. H. In Micro Total Analysis Systems ‘98; Harrison, D. J., van den Berg, A., Eds.; Kluvert Academic Publishers: Dordrecht, 1998; pp 241-244. (32) Patankar, N. A.; Hu, H. H. Anal. Chem. 1998, 70, 1870-1881. (33) Ermakov, S. V.; Jacobson, S. C.; Ramsey, J. M. Anal. Chem. 1998, 70, 44944504.
1988
Analytical Chemistry, Vol. 72, No. 9, May 1, 2000
equations governing these phenomena are the continuity equation (eq 1) and the momentum equation adapted for electrical external forces (eq 2)
∇v ) 0 F
(1)
[∂v∂t + v∇v] ) -∇p + µ∇ v + F E 2
e
(2)
where v is the velocity vector, F is the density, p the pressure, µ the viscosity, Fe the electric charge density, and E the electric field. The electric field density is calculated from the electric potential by
E ) -∇Φ
(3)
where Φ is the electric potential. This parameter is derived from the Poisson equation taking into account the charges at the surface
∇2Φ ) -(Fe/)
(4)
where is the electrical permittivity of the solution. As the sample is usually a dilute solution, the same properties are assumed in the whole bulk solution in the microchannels. The physical parameters such as the viscosity are taken to be equal to the solvent characteristics. The electrical potential can be divided in two terms being φ, the potential induced by the external field, and ψ, the electrical potential due to the charges at the surface. The distribution of the latter potential is determined following the Gouy-Chapman theory. Equation 4 can be split in two equations, 5 and 6
∇2φ ) 0
(5)
∇2ψ ) κ2ψ
(6)
where κ-1 is the Debye length and is relative to the thickness of the diffuse layer. Finally, the set of equations determining the velocity field induced by electroosmosis is, when appropriate dimensionless operation is performed
∇V ) 0 Re
(7)
[∂V∂t + (V∇)V] ) -∇p + ∇ V + (κ hH)ψ∇φ 2
2
(8)
∇2ψ ) (κh)2ψ
(9)
∇2φ ) 0
(10)
The Reynolds number is defined as
Re )
Fv∞d FUinζ h µ µH µ
(11)
Figure 1. Simulated geometry modeling the T-junction.
where d is the characteristic length of the capillary chosen here as the channel width h2. The same parameters are used here for the dimensionless terms except for the velocity, where V replaces v. The dimensionless process is performed using the procedure given by Patankar et al. (see Figure 1): the lengths are divided by the main channel width h; the potential φ by the value of the potential applied to inlet 3, Uin, whereas outlet 1 is grounded; the potential ψ by the Zeta potential ζ; the velocity V by v∞ ) (Uinζ/µH), where H is the distance between the inlet 3 and outlet 1; and the pressure p by (Uinζ/hH). Comparison with experimental data can be performed by introducing the dimensional parameters and by calculating the electroosmotic velocity using eq 12
v ) Vv∞ ) V
Eζ µ
(12)
Electroosmotic pumping in microchannels is characterized by a low Reynolds number, therefore inertial forces do not predominate over viscous forces. The calculations of ψ and φ can be performed independently of the velocity profile V. As demonstrated by Patankar et al. at the corners of the geometry, inertial forces cannot be neglected and result mathematically in the appearance of a singularity at the corner. Indeed, V depends on both parameters ψ and φ. In the case of a Newtonian fluid flowing in one dimension across a flat surface, Rice and Whitehead34 solved analytically the set of equations governing the electroosmotic flow for a cylindrical capillary. Similar work executed by Andreev included thermal and diffusion influences on the separation efficiency in capillary free zone electrophoresis.35 Andreev also solved the set of equations in Cartesian coordinates for rectangular channels36 considering inhomogeneous Zeta potential distribution at the opposite walls of the channel. (34) Rice, C. L.; Whitehead, R. J. Phys. Chem. 1965, 69, 4017-4024. (35) Andreev, V. P.; Lisin, E. E. Electrophoresis 1992, 13, 832-837. (36) Andreev, V. P.; Dubrovsky, S. G.; Stepanov, Y. V. J. Microcolumn Sep. 1997, 9, 443-450.
Numerical Model. The set of equations is written in a finite element formulation and implemented in numerical software Flux Expert (Flux Expert-Simulog, 60 rue Lavoisier, 38330 Montbonnot Saint Martin). Equations 9 and 10 are solved independently in order to determine the electrical potential distributions. Then, the parameters ψ and ∇φ are injected in eq 8 as source terms and both hydrodynamic equations, 7 and 8, are solved to determine the velocity field. The decoupling geometry, Figure 1, is represented as a T-intersection of rectangular microchannels where the flow inlet at the bottom and the outlet flow at the top bound the main microchannel. The auxiliary or lateral microchannel is the decoupling part of the device. The various boundary conditions applied to this geometry are illustrated in Figure 2. The external potential is imposed between the inlet and outlet 1 boundaries, whereas a ζ potential is imposed at each charged wall. For the hydrodynamic calculations, no split condition is imposed at each wall and an unperturbed velocity distribution is assumed at the inlet and outlet boundaries. A nonlinear steady-state algorithm is used for the hydrodynamic calculations, whereas a steady-state linear one is used for the potential distributions. The simulations are carried out on a UNIX workstation (Silicon Graphics Indigo 2 Solid Impact 10 000 with 640 Mb of RAM). As explained by other authors,32 the main problem here is to represent the Debye-Hu¨ckel thickness, of the order of few nanometers, whereas the geometrical parameters are in the micrometer range. The discretisation of the geometry would require an enormous memory capacity if the ratio between the two geometrical parameters was taken into account. One way to solve this problem is to increase artificially the order of magnitude of the Gouy-Chapman diffuse layer thickness and therefore this assumption gives us a qualitative and relative description of the simulated problem rather than a direct quantitative one. The second approach is to assume a split condition at the charged walls. In our work, the first approach is chosen and κ is artificially decreased to a value of κ(h/2) ) 10 (κ-1 ) 2.5 µm), if the channel width is taken as 50 µm, to be able to solve numerically the fluidic problem. This value, as described by Andreev et al., corresponds to a buffer concentration of 10-5 M and to a Zeta potential of -100 mV. Some calculations have been carried out using a value κ(h/ 2) ) 50 (κ-1 ) 500 nm) in order to evaluate the influence of κ(h/2) on the flow distribution. The three-dimensional geometry is reduced to only two dimensions. Model Validation. The integral of the simulated electroosmotic velocity profile, v∞, over a section of a normalized rectangular channel matches the theoretical value given by eq 12 within an error of less than 1%. Moreover, a comparison with the results of Rice34 and Andreev et al.35 has been carried out. Our results are in good agreement with simulations of Rice and Whitehead because we made a similar assumption on the size of the GouyChapman layer thickness. In contrast, Andreev’s studies did not take into account this approximation. The integration of our simulated velocity profile over the channel section shows 0.17% deviation with Rice’s results and 2% with Andreev’s results. The assumption of the absence of the temperature dependence on the flow profile can be considered valid as long as the plotted representation of the current as a function of the applied high Analytical Chemistry, Vol. 72, No. 9, May 1, 2000
1989
Figure 2. (a) ψ potential distribution, (b) φ potential distribution, (c) velocity distribution. ζ1/ζ2 ) 0.5, Re ) 1, h1/h2 ) 1.
voltage is linear.29 In the opposite case, a noticeable change in radial buffer viscosity due to the temperature gradient appears,37 leading to a quasi-parabolic profile. A very focused geometry around our T-junction is chosen in order to reduce calculation time and the memory space. Nevertheless, to ensure that this approximation does not interfere with the validity of our results, larger geometry has been simulated, and less than 8% deviation has been observed between both cases. Furthermore, we use a similar geometry to that shown in Figure 1 to simulate the pressure-driven flow behavior. RESULTS AND DISCUSSION The hydrodynamic behavior occurring at the intersection of three channels is investigated according to the variation of parameters such as the ζ potential distribution, the Reynolds number, and the geometry. The results are interpreted according to the final application of the T-junction: the mass decoupling from the electrical pathway. The aim is the optimization of the mass flowing toward a detector, located at the outlet 2. Influence of the Charged Walls. The Gouy-Chapman theory predicts the formation of a diffuse double layer in an electrolyte solution near a charged wall. The double layer potential distribution follows an exponential law as represented in Figure 2a. When an external electric field E (see Figure 2b) is applied perpendicularly to the Gouy-Chapman potential distribution, the ions in excess in the diffuse layer migrate and carry away the solvent molecules in the vicinity. It results, near the walls, in a motion parallel to the current lines induced by E (Figure 2c). The electroosmotic flow is clearly generated near the walls in the area, where both the electrical field E and the ψ potential are present. Out of this region, the shearing strength between the different layers of the fluid allows the bulk motion of the solution. In all the following simulations, the Reynolds number and the ζ potential ζ2 are imposed to be equal to 1. The influence of the ratio between the ζ potential at the walls of the lateral channel and the ζ potential of the main channel walls, ζ1/ζ2, is first investigated. In Figure 3, the velocity profiles at the inlet section (Figure 3a) and at the outlet sections in the straight and lateral channels (Figure 3 parts b and c, respectively) are represented for different ratios of ζ1/ζ2. When the ζ potential distribution between main and lateral channels stands in the range 0 e ζ1/ζ2 (37) Knox, J. H. Chromatographia 1988, 26, 329-337.
1990 Analytical Chemistry, Vol. 72, No. 9, May 1, 2000
Figure 3. Influence of the ratio ζ1/ζ2 on the velocity profile at a T-junction. (a) entrance, (b) outlet 2, (c) outlet 1. Re ) 1, h1/h2 ) 1. ζ1/ζ2 ) 0 (solid line). ζ1/ζ2 ) 0.5 (dotted line). ζ1/ζ2 ) 1 (dotted-dashed line). ζ1/ζ2 ) 2 (dashed line).
< 1, the flow velocity profile at the inlet section presents a concave curvature resulting from a combination between a parabolic and a flat profile, i.e., an electroosmotic profile perturbed by the pressure evolution at the intersection. The pressure drop9 or deceleration effect induced by the reduced ζ1 potential at the walls of the sidearm acts as an overpressure source generating a pressure-driven flow in the straight channel as noticed at the outlet 2 where only a parabolic velocity profile is observed (Figure 3b). In the case of the lateral channel, the flow velocity profile over the outlet section is also a combination between a flat and a parabolic profile (Figure 3c). When the ζ potential ratio (ζ1/ζ2) is increased above a value of 1, the larger electroosmotic velocity in the lateral channel compared with that in the main channel results in a flow inversion in the straight channel (Figure 3b). In this case, a depression at
Figure 4. Influence of the ratio ζ1/ζ2 on the dimensionless pressure drop Ip3, Ip2 and Ip1. Re ) 1, h1/h2 ) 1. R2 values are 0.9995, 0.9997, and 0.9998, respectively.
the intersection is created and the flow is pumped from outlet 2 toward outlet 1. In Figure 4, the dimensionless pressure-drop integrals in the channels, Ip1, Ip2, and Ip3, are represented as a function of ζ1/ζ2. The counter pressure34 per unit length, ∆p/L, related to the EOF, can be expressed by eq 13 using the relations developed by Burgreen and Nakache for rectangular section capillaries38
∆p 3ζE ) 2 [1 - G(R,κh)] L hπ
(13)
where G is a function depending on κh and on the term R ) ezψ0/ kT, where ψ0 is the potential at the capillary surface, k is the Boltzmann constant, and T is the temperature. This function, which is an integral, represents the ψ distribution and has been numerically calculated by Burgreen et al. If we consider that G is independent of the ζ potential, the pressure drop, IP2, generated at the intersection by the inhomogeneous distribution of the EOF between channel 3 and 1, can be therefore formulated using eq 13
Ip3 - Ip1 ) Ip2 )
[ ]
ζ1 3(1 - G) 1π ζ2
(14)
The eq 14 gives a linear relation between Ip2 and the ratio of ζ1/ ζ2 as observed in Figure 4. The resulting relationship calculated from the simulated data is as follows
ζ1 Ip2 ) 0.86 - 0.82 ζ2
(15)
The theoretical slope, 3(1 - G)/π, of the representation Ip2 vs ζ1/ζ2 is equal to 0.86 whereas the slope obtained with the simulated data in Figure 4 is 0.82. The value, 0.86, at the ordinate origin of the linear regression also gives a very good fit to the theoretical value. The sign inversion of the pressure drop in the straight channel, IP2, appears clearly above the limit ζ1/ζ2 ≈ 1 and corroborates (38) Burgreen, D.; Nakache, F. R. J. Phys. Chem. 1964, 68, 1084-1091.
Figure 5. Influence of the ratio ζ1/ζ2 on the normalized flow rate Iv2/Iv3 and Iv1/Iv3. Re ) 1, h1/h2 ) 1.
the reverse flow observed in Figure 3b. When ζ1/ζ2 ) 0, the same pressure drop occurs in the straight and lateral channel. The EOF is only present in the main channel, and a pressure-driven flow is observed in both lateral and straight channels (Figure 3b and 3c). For the other cases, ζ1/ζ2 * 0, the pressure in the lateral channel is always superior to the straight one, demonstrating the loss of mass toward the auxiliary channel as shown quantitatively in the next paragraph. It can be pointed out that for ζ1/ζ2 ) 1.2, no pressure drop is observed (Ip2 ) 0). All the simulations presented in Figure 3 are carried out for κ(h/2) ) 10. The influence of this parameter on the ratios of Iv1/ Iv3 and Iv2/Iv3 is presented in Figure 5, where this ratio is given as a function of ζ1/ζ2 for both parameter values κ(h/2) ) 10 and κ(h/2) ) 50. The term Iv3 is the integral of the dimensionless velocity over the inlet section and corresponds to a normalized flow rate. The terms Iv1 and Iv2 describe the normalized flow rate over the outlet sections 1 and 2, respectively. When the simulated geometry is applied to the decoupling of electrical fields, the aim is to optimize the ratio of Iv2/Iv3 compared with the ratio of Iv1/ Iv3. The former represents the mass quantity detected by a sensor at the end of the straight channel whereas the latter is the mass lost in the decoupling part (lateral channel). Even if the GouyChapman layer thickness influences the value of the velocity integral (deviation of 15%), the ratio of Iv1/Iv3 is only slightly influenced (less than 5%) by the parameter κ(h/2). In Figure 5, the ratio of Iv2/Iv3 is reduced from 49 to 4% when the ratio of ζ1/ζ2 ranges from 0 to 1. It can become even negative when ζ1/ζ2 > 1.2. It is interesting to point out that when ζ1/ζ2 ) 1, a residual flow rate is noticed at the outlet 2. The pressure drop at the intersection cannot arise from the inhomogeneous ζ potential at the walls as in the cases when ζ1/ζ2 * 1. In this case, the pressure drop can be attributed to the external field distribution at the intersection. In fact, it can be observed in Figure 2b that the equipotential lines in this area are not perfectly perpendicular to the double layer potential distribution. The influence of this distribution can depend on the size of the simulated geometry and, therefore, this effect can be damped if a larger geometry is simulated. If the optimization of a decoupling system is considered, the maximal mass transported toward outlet 2 is obtained for the case of ζ1/ζ2 ) 0, i.e., when the pressure drop is maximal in the straight channel (Figure 4). This representation shows that, for this Analytical Chemistry, Vol. 72, No. 9, May 1, 2000
1991
specific geometry, the maximum mass quantity obtained at the outlet 2 is only 50% of the initial injection. This ζ potential ratio corresponds to the case of a pressure-driven flow as demonstrated in the next paragraph. The symmetric distribution of the flow between the main and the lateral channel can be correlated to the small Reynolds number encountered in electroosmotic-driven devices and also to the small mechanical resistance of the sidearm aperture. To optimize the mass quantity flowing toward outlet 2, geometrical parameters such as the ratio of the channel widths and the Reynolds number are also studied. Influence of the Reynolds Number. In this section, the pressure-driven flow case is first considered. Numerical simulations are carried out by solving the continuity equation, eq 1, and Navier Stokes equation, eq 16, in order to characterize the division of a flow at a T-junction
F
[∂v∂t + (v∇)v] ) -∇p + µ∇ v 2
Figure 6. Influence of the Reynolds number on the flow distribution at a T-junction for a pressure-driven flow. h1/h2 ) 1.
(16)
The velocity profile at the inlet section is assumed to be parabolic and to follow the relations
vx ) 0 vy ) 4v0y
x x 1h h
(
)
(17)
vz ) 0 where v0y is the maximum velocity at the center of the parabola. The division of the flow at a T-junction in this case is described in Figure 6 as a function of the Reynolds number, Re, ranging from 0.0025 to 100. For the hydrodynamic calculations, changing v0y varies the Reynolds number. The flow is equally divided at the junction between channels 1 and 2 for Re e 1. When Re >1, the flow division between the lateral and the straight channel becomes inhomogeneous because inertial forces start to predominate over the shearing ones. When a liquid turns at 90° in a fluidic system like in a T-junction, the energy required to follow the 90° pathway depends strongly on the inertial forces of the system. The higher Reynolds number favors the straight pathway compared with the lateral one, and the increase of Iv2/Iv3 is noticed. The Re number in the case of an electroosmotic flow is governed by the external electrical field applied to the geometry. The division of the fluid at the T-junction is represented in Figure 7 as a function of the Reynolds number ranging from 0.01 to 10, for three different ζ potential ratios of 0, 0.5, and 1 (larger Re number values cannot be investigated due to numerical convergence problems). Simulated values for the ratio equal to 0 are very similar to the pressure-driven flow distribution shown in Figure 6. For ζ1/ζ2 ) 0, a division of 50% is observed at low Re number and the ratio Iv2/Iv3 increases with the Re number. The same behavior is observed for the ζ ratios of 0.5 and 1. However, at low Re number, the division is not symmetric and Iv2/Iv3 values are only 20 and 1%, respectively. The distribution reaches, at the maximum, the value obtained with a pressure-driven flow as already noticed. When Re e 1, the distribution of the fluid is constant and predominantly influenced by the ζ potential ratio. The Reynolds number can be a parameter allowing the optimization of mass transport toward outlet 2 only when Re > 1. 1992
Analytical Chemistry, Vol. 72, No. 9, May 1, 2000
Figure 7. Influence of the Re number on the flow distribution at a T-junction for an electroosmotic-driven flow. h1/h2 ) 1, ζ1/ζ2 ) 0, 0.5, and 1.
However, working with such a range of Reynolds values in capillary electrophoresis (CE) implies the use of a large electric field E (with experimental parameters expressed before, Re ) 10 requires an electric field of 30 kV/cm), which is not always compatible with CE applications. Influence of the Geometry. In the present work, we have considered different geometries as defined by the ratio h2/h1. The influence of the geometry is more complex as it can be involved in two ways: (i) it can be responsible for the variation of the external potential distribution and (ii) it can influence the mechanical resistance encountered by the fluid reaching the intersection area. To separate these two influences, the calculations are run for a ratio R1/R3 ) 1, where R1 and R3 are the electrical resistances of the lateral and main channels, respectively. In this manner, the relative potential gradient in each channel, i.e., the main and the lateral one, remains constant and only the influence of the mechanical resistance is investigated. The simulated data plotted in Figure 8 show that the decrease in the side channel width, h1, i.e., increasing the mechanical resistance, results in the linear enhancement of the ratio of Iv2/Iv3. The flow rate ratio can reach a value of 80% when the electroosmotic flow is completely removed in the sidearm. It appears that the most efficient way to maximize the quantity of sample reaching the outlet 2 after a T-junction is to reduce the dimension of the sidearm cross section. It is obvious that the maximum quantity flowing toward outlet 2
Figure 8. Influence of the ratio h1/h2 on the fluid distribution for an electroosmotic-driven flow at a T-junction. Re ) 1. R2 values are 0.992, 0.9996, and 0.9998, respectively.
(100%) will be obtained for a complete blocking of the sidearm aperture (h1 f 0), in this case new schemes must be considered to design devices, for example, integrating high-voltage electrodes in the capillary or using conductive membranes to close the sidearm channel. Another important geometrical parameter, which is not investigated here, can be the angular dependence between the two channels on the mass transport toward the outlet of the straight channel. However the ζ distribution between the main and the lateral channel still remains a determinant parameter as the increase in Iv2/Iv3 ratio is more pronounced for ζ1/ζ2 ) 0 than for ζ1/ζ2 ) 1. CONCLUSIONS The characterization of a T-junction under electroosmotic conditions, which is used as a decoupling system, is performed
using finite element simulations. A two-step method allows, first, the calculation of the ψ and φ potential distributions and then the determination of the resulting streamline flow. The influence of various parameters such as the ζ potential distribution, the Reynolds number, and the mechanical resistance of the lateral channel is emphasized. The two predominant parameters affecting the distribution of the flow at the intersection are found to be the relative ζ potential and channel widths. With ratios ζ1/ζ2 ) 0 and h1/h2 ) 0.5, 80% of the inlet flow can be separated from the electrical pathway. In particular, we have shown that pressure effects play a dominant role in the understanding of the flow behavior in microscale devices driven by electroosmotic flow. This phenomenon can be used to induce pressure-driven flow without applying any external pressure gradient. It can be pointed out that this geometry represents the half scheme of a double T-junction for capillary electrophoresis. So the same numerical method can be applied to predict the leakage effect, due to the pressure effect, during injection using a double-T structure. ACKNOWLEDGMENT One of the authors (F.B.) thanks l’Ecole Polytechnique Fe´de´rale de Lausanne (EPFL) for a doctoral fellowship.
Received for review October 29, 1999. Accepted February 1, 2000. AC991225Z
Analytical Chemistry, Vol. 72, No. 9, May 1, 2000
1993
