Low-Dispersion Turns and Junctions for Microchannel Systems

Numerical methods are employed to optimize the geom- etry of two-dimensional microchannel turns such that the turn-induced spreading of a solute band ...
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Anal. Chem. 2001, 73, 272-278

Low-Dispersion Turns and Junctions for Microchannel Systems Stewart K. Griffiths* and Robert H. Nilson

Sandia National Laboratories, Livermore, California 94551-0969

Numerical methods are employed to optimize the geometry of two-dimensional microchannel turns such that the turn-induced spreading of a solute band is minimized. An inverted numerical method is first developed to compute the electric potential and local species motion in turns of arbitrary geometry. The turn geometry is then optimized by means of a nonlinear least-squares minimization algorithm using the spatial variance of the species distribution leaving the turn as the object function. This approach yields the turn geometry producing the minimum possible dispersion, subject only to prescribed constraints. The resulting low-dispersion turns provide an induced variance 2-3 orders of magnitude below that of a comparable conventional turns. Sample results are presented for 180 and 90° turns, and the use of these turns to form wyes and tees is discussed. A sample 45° wye is presented. The use of low-dispersion turns in folding separation columns is also discussed, and sample calculations are presented for folding a column 100 µm in width and up to 900 mm in length onto a region of only 10 by 10 mm. These low-dispersion geometries are applicable to electroosmosis, electrophoresis, and some pressure-driven flows. Microchannel systems, first explored only about 10 years ago,1-5 are now under development for a wide range of applications in the detection, analysis, and synthesis of chemical and biological species.6-16 Most such systems employ some combination of electrophoresis and electroosmotic flow on a single substrate. (1) Manz, A.; Fettinger, J. C.; Verpoorte, E.; Ludi, H.; Widmer, H. M.; Harrison, D. J. Trends Anal. Chem. 1991, 10, 144-149. (2) Manz, A.; Harrison, D. J.; Verpoorte, E. M. J.; Fettinger, J. C.; Paulus, A.; Ludi, H.; Widmer, H. M. J. Chromatogr. 1992, 593, 253-258. (3) Harrison, D. J.; Manz, A.; Fan, Z. H.; Ludi, H.; Widmer, H. M. Anal. Chem. 1992, 64, 1926-1932. (4) Effenhauser, C.; Manz, A.; Widmer, H. M. Anal. Chem. 1993, 65, 26372642. (5) Jacobson, S. C.; Hergenroder, R.; Koutny, L. B.; Ramsey, J. M. Anal. Chem. 1994, 66, 1114-1118. (6) Jacobson, S. C.; Culbertson, C. T.; Daler, J. E.; Ramsey, J. M. Anal. Chem. 1998, 70, 3476-3480. (7) Jacobson, S. C.; Ramsey, J. M. Anal. Chem. 1996, 68, 720-723. (8) Chiem, N.; Harrison, D. J. Anal. Chem. 1997, 69, 373-378. (9) McCormick, R. M.; Nelson, R. J.; Alonson-Amigo, M. G.; Benvegnu, D. J.; Hooper, H. H. Anal. Chem. 1997, 69, 2626-2630. (10) Effenhauser, C. S.; Bruin, G. J. M.; Paulus, A.; Ehrat, M. Anal. Chem. 1997, 69, 3451-3457. (11) Waters, L. C.; Jacobson, S. C.; Kroutchinina, N.; Khandurine, J.; Foote, R. S.; Ramsey, J. M. Anal. Chem. 1998, 70, 5172-5176.

272 Analytical Chemistry, Vol. 73, No. 2, January 15, 2001

Electrophoresis is typically used for analysis by separation, while electroosmotic flow is employed for sample preparation and injection into the separation column.1-11,14-17 Electrophoretic separations are generally performed in a fluid phase, so electroosmosis and electrophoresis occur simultaneously in the separation column. In this case, the electroosmotic transport can improve separation efficiency by increasing the mean transit time through the column.18,19 Alternatively, purely electrophoretic separations can be performed in a stationary phase by employing a gel-filled column or by coating the column walls to suppress electroosmotic motion of a fluid. Similarly, purely electroosmotic flows are sometimes used for analysis via electrochromatographic separation in either open or packed channels.12,20-24 Both electrophoresis and electroosmotic flow permit long-range species transport with little dispersion due to nonuniform speeds.25,26 However, while this is true for straight channel segments, microchannel turns can produce dramatic skewing of an otherwise flat species band due to the locally nonuniform electric field or fluid velocity. Such skewing is generally irreversible because transverse diffusion redistributes species concentrations across the channel, and the net effect on the band is a large and permanent spreading of the species distribution along the channel downstream of the turn. Compositional interfaces are similarly deformed and spread by transport through a turn. As a result, (12) Von Heeren, F.; Verpoorte, E.; Manz, A.; Thormann, W. Anal. Chem. 1996, 68, 2044-2053. (13) Northrup, M. A.; Benett, B.; Hadley, D.; Landre, P.; Lehew, S.; Richards, J.; Stratton, P. Anal. Chem. 1998, 70, 918-922. (14) Woolley, A. T.; Sensabaugh, G. F.; Mathies, R. A. Anal. Chem. 1997, 69, 2181-2186. (15) Jacobson, S. C.; Hergenroder, R.; Moore, A. W. Jr.; Ramsey, J. M. Anal. Chem. 1994, 66, 4127-4132. (16) Jacobson, S. C.; Koutny, L. B.; Hergenroder, R.; Moore, A. W., Jr.; Ramsey, J. M. Anal. Chem. 1994, 66, 3472-3476. (17) Salimi-Moosavi, H.; Tang, T.; Harrison, D. J. J. Am. Chem. Soc. 1997, 119, 8716-8717. (18) Dasgupta, P. K.; Liu, S. Anal. Chem. 1994, 66, 3060-3065. (19) Salimi-Moosavi, H.; Tang, T.; Harrison, D. J. Electrophoresis 2000, 21, 107115. (20) Jacobson, S. C.; Hergenroder, R.; Koutny, L. B.; Ramsey, J. M. Anal. Chem. 1994, 66, 2369-2373. (21) Yan, C.; Dadoo, R.; Zhao, H.; Zare, R. N.; Rakestraw, D. J. Anal. Chem. 1995, 67, 2026-2029. (22) Ericson, C.; Holm, J.; Ericson, T.; Hjerten, S. Anal. Chem. 2000, 72, 8187. (23) Kutter, J. P.; Jacobson, S. J.; Ramsey, J. M. Anal. Chem. 1997, 69, 51655171. (24) Kutter, J. P.; Jacobson, S. J.; Matsubara, N.; Ramsey, J. M. Anal. Chem. 1998, 70, 3291-3297. (25) McEldoon, J. P.; Datta, B. Anal. Chem. 1992, 64, 227-230. (26) Martin, M.; Guiochon, G. Anal. Chem. 1984, 56, 614-620, 1984. 10.1021/ac000936q CCC: $20.00

© 2001 American Chemical Society Published on Web 12/08/2000

elbows, wyes, tees, and other common fittings are often avoided in microchannel systems. This limitation constrains system design and precludes compact channel layout on a small chip area. The detrimental effects of band spreading in turns has been recognized for some time. In 1960, Giddings analyzed the excess spreading induced by capillary coiling in packed-column gas-phase chromatography.27 Kasicka et al. similarly analyzed the effects of capillary coiling on capillary zone electrophoresis.28 More recently, Culbertson et al. investigated several sources of dispersion in microchannel devices.29 As part of this study, they collected a large set of data on the increased variance of a species band downstream of a turn. Although several studies have demonstrated the importance of dispersive spreading in turns, there have been few prior attempts to solve the problem. Kopf-Sill and Parce proposed using channel turn geometries of high aspect ratio.30 The benefit of this approach is that the channels may be narrowed without sacrificing the cross-sectional area. Narrow channels do indeed reduce turninduced dispersion since the added variance due to a turn is proportional to the square of the channel width when diffusive transport is relatively unimportant. They also proposed turns of varying depth, which may provide benefit as well, but channels having nonuniform depth are difficult to manufacture. Again recognizing the advantage of narrow channels, Paegel et al. examined two-dimensional turns in which the channel width is reduced prior to the turn and then expanded back to the original size once the turn is completed.31 Their experimental investigation demonstrated that long separation columns may be folded without serious loss of performance when good turn geometries are used. Finally, Nordman devised two-dimensional turn geometries employing an expanded correction segment just beyond the turn.32 The geometry takes advantage of the fact that skewing depends on both the channel width and included angle of the turn. The correction segment thus reduces net skewing by partially reversing the initial turn in a section of increased width. In the present study, we develop a general numerical method for optimizing channel geometries that minimize the dispersive spreading induced by turns and junctions. The resulting lowdispersion geometries alter the direction of solute motion but leave bands and interfaces nearly flat and orthogonal to the direction of motion. These new geometries are applicable to electroosmotic, electrophoretic, and some pressure-driven transport and separation processes. GOVERNING EQUATIONS Consider the electroosmotic flow and species transport in the vicinity of a two-dimensional turn. The channel segments before and after the turn are presumed to be straight and of constant width, the top and bottom surfaces are everywhere planar, and all surfaces bounding the channel are impermeable and noncon(27) Giddings, J. C. J. Chromatogr. 1960, 3, 520-523. (28) Kasicka, V.; Prusik, Z.; Gas, B. Stedry, M. Electrophoresis 1995, 16, 20342038. (29) Culbertson, C. T.; Jacobson, S. C.; Ramsey, J. M. Anal. Chem. 1998, 70, 3781-3789. (30) Kopf-Sill, A. R.; Parce, J. W. Microfluidic Systems Incorporating Varied Channel Dimensions. U.S. Patent 5842787, 1998. (31) Paegel, B. M.; Hutt, L. D.; Simpson, P. C.; Mathies, R. A. Anal. Chem. 2000, 72, 3030-3037. (32) Nordman, E. S. Serpentine Electrophoresis Channel with Self-Correcting Bends. PCT Patent Application WO 99/24828, 1999.

ducting. Assuming that the fluid is incompressible and that transport properties are constant, the time-dependent concentration field is governed by

∂c + u‚∇c ) D∇2c ∂t

(1)

where c is the local species concentration, t is time, u ) ui + vj is the local fluid velocity, and D is the coefficient of diffusion. Further assuming that the flow is steady, that there are no applied pressure gradients, and that inertial effects are small, the momentum equation may be written as

µ∇2u ) Fe∇φ

(2)

where µ is the fluid viscosity, Fe is the net local charge density, and φ is the electric potential. Finally, for a dielectric constant, , that does not vary with position, the Poisson equation governing the electric potential is

∇2φ ) -Fe

(3)

The charge density, Fe, for equivalent ions may be related to the electric potential through the Boltzmann distribution. In many cases of practical interest, the local fluid velocity in electroosmotic flow is proportional to the applied electric field.33 The main conditions necessary for such similitude are a quasisteady electric field, uniform fluid density, and uniform viscosity of the neutral fluid outside the Debye layer. Further, the Debye layer thickness must be small compared to any channel dimension, and all solid surfaces bounding the fluid must be electrically nonconducting relative to the fluid and have a uniform surface charge or surface potential. All of these conditions are usually met in microchannel systems, at least over the scale of a single turn. Under these restrictions, the electric potential outside the Debye layer is governed by the Laplace equation

∇2φ ) 0

(4)

and the local fluid velocity is everywhere given by

u)-

ζ µ

∇φ

(5)

The Navier-Stokes equations presented in eq 2 thus need not be solved under the conditions outlined above. Moreover, when these conditions are satisfied, the electric potential and fluid velocity in any two-dimensional channel bounded by parallel planes is strictly two-dimensional and is independent of the channel depth normal to the two planes.33 The equations above were developed in the context of neutral species transport in electroosmotic flow. However, when the electroosmotic fluid velocity is proportional to the electric field, these governing equations are almost the same as those describ(33) Cummings, E. B.; Griffiths, S. K.; Nilson, R. H.; Paul, P. H. Anal. Chem. 2000, 72, 2526-2532.

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273

ing the electrophoretic motion of a single charged species relative to a stationary phase. That is, in both cases, the species flux varies linearly with both the concentration gradient and the electric field. For electroosmotic flow, the local species flux is J ) -D∇c ζ∇φ/µ; the purely electrophoretic flux is J ) -D∇c + vzF∇φ. As such, solutions to the problem of electroosmotic flow also apply to electrophoresis, provided that both problems are properly normalized. Further, similar equations also govern species transport in some pressure-driven flows. In both porous materials and open channels having a very small aspect ratio, the local velocity of an incompressible fluid is proportional to the pressure gradient at low Reynolds numbers, and the pressure field is governed by the Laplace equation.34,35 The solutions presented here thus also apply to pressure-driven flows in these special cases. To solve generally for the species concentration, we now introduce a set of dimensionless variables. The normalized dependent variables are taken as c* ) c/co, u* ) u/U and φ* ) φ/aE, where co is some reference concentration yet to be specified, a and E are the channel width and magnitude of the applied axial electric field far from the turn, and U ) ζE/µ is the HelmholtzSmoluchowski speed for electroosmotic flow past a planar surface. We take U ) -vzFE for electrophoretic motion. The new independent variables are x* ) x/a, y* ) y/a, and t* ) Dt/a2, where x and y are the Cartesian spatial coordinates. This normalization leads to one new parameter, the Peclet number, given by Pe ) Ua/D. For purely electroosmotic flow, the Peclet number is Pe ) aζE/µD provided that the Debye layer thickness is small compared to both lateral channel dimensions; for purely electrophoretic processes, it is Pe ) avzFE/D ≈ azFE/ RT. Introducing these normalized variables into the primitive governing eqs 4 and 5 and rearranging slightly yields

∂2φ* ∂x*2

+

∂2φ* ∂y*2

)0

(6)

and

u* ) -

(

∂φ* ∂x*

i+

∂φ* ∂y*

j

)

(7)

for eq 5. Boundary conditions for the normalized potential are ∇φ*‚n ) 0 on all channel surfaces and ∂φ*/∂x* ) -1 in the straight section ahead of the turn. NUMERICAL METHOD To address band spreading induced by turns of arbitrary geometry, we have developed a numerical algorithm for solving the Laplace equation using a novel numerical technique. This technique employs an inverted approach in which the dependent variables are the unknown values of the spatial coordinates and the independent variables are the normalized electric potential, φ*, and an associated stream function, ψ*.36 The governing (34) Darcy, H. P. G. Les Fontaines Publiques de la Ville de Dijon; Dalmont: Paris, 1856. (35) Saffman, P. G. J. Fluid. Mech. 1986, 173, 73-94. (36) Jeppson, R. W. J. Eng. Mech. Div. Am. Soc. Civ. Eng. 1969, 95, 1, 1969.

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equations for these new dependent variables are

∂2x* ∂φ*2

+

∂2x* ∂ψ*2

)0

(8)

)0

(9)

and

∂2y*

∂2y*

∂φ*

∂ψ*2

+ 2

Boundary conditions in the physical domain are readily mapped to the inverted domain using the Cauchy-Riemann compatibility relations.

∂x* ∂φ*

)

∂y* ∂ψ*

and

∂x* ∂ψ*

)-

∂y*

(10)

∂φ*

The two channel boundaries thus become lines of constant ψ*. Consistent with our normalization and with y* increasing in the upward direction, we take ψ* ) 0 along the lower channel boundary and ψ* ) 1 along the upper boundary. Similarly, lines crossing the straight channel segments well ahead of and beyond the turn become lines of constant φ*. The advantage of this inverted approach is that the computational domain is always rectangular, regardless of the turn geometry, and the curved boundaries of the channel walls appear only as boundary conditions on the rectangular φ* and ψ* domain. The analysis of transport in complex turn geometries is thus greatly simplified. Transient species transport in this numerical scheme is simulated using a Monte Carlo method. This method is advantageous in the study of band spreading because it does not introduce the artificial numerical dispersion generated by traditional finite difference and finite element approaches.37 Tracer particles are first injected into the straight channel segment several channel widths upstream of the turn. This is followed by a series of steps in which each particle is advected along local lines of constant ψ*, as it diffuses in both the φ* and ψ* directions. The length of the convective step is u*Pe∆t*S, based on the local speed, while the length of each diffusive step is R(2∆t*)1/2S, where R is a normally distributed random variable having a mean of 0 and variance of 1;38 S is the local scale factor relating spatial steps to steps in ∆φ* and ∆ψ*. All of these steps take place in the orthogonal φ* and ψ* coordinate system. Spreading of the species distribution is computed from the final positions of the tracer particles once all of the particles have traversed the turn. The streamwise mean particle position is first computed as the simple average of all positions. The variance of the distribution is then computed as the average sum of the squares of the spatial deviations from the mean. The result is a statistical estimate of the normalized band variance, (σ/a)2, induced by the turn. OPTIMIZATION OF GEOMETRY The computation mesh for x* and y* is always bounded by lines of constant φ* and constant ψ* and so remains rectangular (37) Oran, E. S.; Boris, J. P. Numerical Simulation of Reactive Flow; Elsevier: New York, 1987. (38) Fogelson, A. L. J. Comput. Phys. 1992, 100, 1-16.

regardless of the channel geometry. Thus, to solve the governing field equations for a specific geometry, we need only provide values for the channel boundary locations, x* and y*, along the boundaries ψ* ) 0 and ψ* ) 1. In practice, these locations are given in terms of the normalized distance s* ) s/a along each ψ* boundary. This permits interpolation between the physical and inverted problems so that the boundary locations can be input as simple x* and y* pairs. Boundary locations are represented parametrically by expressions describing the mean turn radius and width of the channel as a function of position in the vicinity of the turn. Many different expressions involving up to 10 parameters have been used. In general, however, we find that the optimum geometries are fairly smooth and can be described adequately using only four parameters. Further, the optimum outer boundary is usually a curve of nearly fixed radius, where this radius is equal to the mean turn radius plus half the channel width. That is, the optimum outer boundary is roughly the same as that of an equivalent conventional turn when both the inner and outer boundaries are optimized. Finally, lower dispersion can almost always be obtained from a longer turn radius. This requires more real estate, however, so the mean turn radius can also be viewed as a constraint. On the basis of these observations, the mean and outer turn radii are simply specified for the sample results shown here, and the normalized channel width is given by

w* )

w R ) 1 - (1 + cos θ*)γ a 2

(11)

where

θ* ) 0 for |s*| e δ θ* )

|s*| - δ ξ

π for δ < |s*| < δ + ξ

θ* ) π for |s*| g δ + ξ

(12) (13) (14)

Here, s* ) s/a is the normalized distance measured along the inner boundary starting at the angular position midway through the turn. These four-parameter profiles are capable of describing a very wide variety of turn geometries having a contraction of the channel just before the turn, a portion of the turn characterized by a constant inner radius, and an expansion of the channel beyond the turn that is symmetric with contraction ahead. The parameter R determines the minimum width of the channel, γ controls the contour of the contraction and expansion, δ specifies the angular extent of the segment having a fixed radius, and the parameter ξ controls the lengths of the contraction and expansion regions. To minimize dispersion, the parameters describing the geometry are selected using a nonlinear constrained minimization algorithm, TJMAR1.39 For a candidate geometry, TJMAR1 calls a routine based on the numerical procedure above. Given trial values for each parameter, this routine computes the potential field and (39) Jefferson, T. H. TJMAR1-A Fortran Subroutine for Nonlinear Least Squares Parameter Estimation; Sandia National Laboratories Report, SLL-73-0305, 1973.

the transport of tracer particles through the turn. The turn-induced variance is then computed, and this is returned to TJMAR1 as the object function to be minimized. TJMAR1 makes several calls to this routine and then computes new trial values of the turn parameters providing somewhat lower induced dispersion. This process is continued until no further improvement can be made. The result is the turn geometry providing the minimum possible dispersion, subject only to the flexibility of the parametric description and to any imposed constraints such as the mean turn radius or minimum allowable channel width. Several hundred evaluations of candidate geometries may be required to obtain a single optimized geometry, but overall execution times are still only a minute or two owing to the speed of the inverted numerical approach. SAMPLE RESULTS Sample results are shown in Figure 1 for a 180° turn having an inner radius 20% of the channel width. The series of frames, read from top to bottom along both columns, represents the history of a thin species band traversing the turn at a Peclet number of Pe ) 10 000. The time interval between frames is fixed. High Peclet numbers represent a worst-case condition for turninduced spreading since the spreading due to the turn geometry is maximum and is independent of the Peclet number, when the Peclet number is large, but falls in proportion to the Peclet number when the Peclet number is small.40 For a 180° turn having a mean turn radius comparable to the channel width, the transition between these behaviors occurs at a Peclet number of ∼100. The results in Figure 1 are thus generally representative of the expected behavior for all Peclet numbers above ∼100. Peclet numbers much smaller than this are not of much practical interest because diffusion alone at very small Peclet numbers significantly spreads species bands even in straight channel segments of moderate length. The results in Figure 1 show the normal progression of a species band through the turn. As the band approaches the turn, the variance of the band grows linearly (but slowly) in time due to diffusion in the direction of motion. This diffusive spreading is very small in this sample owing to the large Peclet number. Close to the turn, the nonuniform fluid velocity skews the distribution profile. Species traveling along the inner radius complete the turn first, followed progressively by species closer to the outer channel wall. The result following the turn is a roughly linear band spanning the channel width but no longer orthogonal to the direction of motion. The increased variance attributable to this skewing is proportional to the square of the distance between the leading and trailing edges of the band. As this band progresses further along the downstream channel segment, transverse and streamwise diffusion will eventually produce a species distribution that is again uniform across the channel and Gaussian in the direction of motion. The variance of this final distribution once more grows linearly in time. The normalized variance computed for the turn in Figure 1 is (σ/a)2 ≈ 3.9. This is consistent with the high Peclet number relation based on a large turn radius of (σ/a)2 ≈ θ2/3 ≈ 3.3, where θ ) π is the included angle of the turn.40 Note that this normalized variance depends only on the turn geometry, which in this case (40) Griffiths, S. K.; Nilson, R. H. Anal. Chem. 2000, 72, 5473-5482.

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Figure 1. Numerical simulation of the history of species transport through a conventional turn. Band spreading at high Peclet numbers is dominated by skewing induced by the turn geometry. The total turninduced normalized variance is (σ/a)2 ≈ 3.9.

Figure 2. Simulated transport through a low-dispersion turn. Band emerges from the turn with little distortion, and the total turn-induced variance is (σ/a)2 ≈ 0.0025. One-sided tapers are largely responsible for this improved performance.

is determined entirely by the included angle and the ratio of the inner and outer radii. It does not depend on the channel width. The dimensional variance induced by a turn is thus proportional the square of the channel width. The benefit of an optimized low-dispersion turn is illustrated in Figure 2. Here we see that the preferred geometry involves a gradual taper of the inner boundary toward the outer boundary prior to the turn. This taper begins about three channel widths before the turn and extends just into the turn, terminating at a segment of uniform width that is ∼28% of the original channel width. This constricted segment then spans ∼85% of the included angle of the turn at a fixed inner radius. Finally, the reduced channel expands back to the original width. One interesting feature of this optimum geometry is that the arc lengths along the inner and outer radii are nearly the same. This provides the same distance of travel along the two paths and potentially provides the same mean electric field. However, many geometries satisfying the condition of similar path lengths do not provide similar mean fields. This feature can thus be viewed as a (weakly) necessary condition for a low-dispersion turn, but it is not also sufficient. The sample band leaving the optimized turn of Figure 2 is nearly flat and orthogonal to the direction of motion. The induced variance for this low-dispersion turn is only (σ/a)2 ≈ 0.0025, more than 3 orders of magnitude lower than that of the equivalent conventional geometry shown in Figure 1. A portion of this improvement results from the fact that much of the turn is completed at a reduced channel width. All else being the same, reducing the width by a factor of 3 will reduce the variance by a factor of ∼10. This reduced width thus accounts for only a small fraction of the improved performance of this turn; the channel width would need to be reduced by a factor of 40 to obtain a comparable induced variance. The majority of the improvement actually results from the fact that both of the tapered sections rotate the band in a direction opposite to the direction of rotation in the turn. By optimizing the profiles of these regions, the skewing induced in the turn is just offset by that induced by the two tapered regions, so the resulting band that emerges from the turn is orthogonal to the channel walls.

The small induced variance that does arise in the optimized geometry is largely associated with the slight bowing of the band. This is apparent in the last frame of Figure 2. Bowing results from species transport through the tapered regions, and both the contraction and expansion produce bowing in the same direction. The extent of bowing is determined by both the length and profile of the taper as well as the minimum channel width. Competition between the conditions yielding a perfectly orthogonal band and a perfectly flat band is thus a prime consideration leading to the optimum geometry. Finally, note that the results in Figure 2 are again based on the condition Pe ) 10 000. As previously discussed, similar behavior is expected for Peclet numbers down to ∼100. Moreover, the induced variance associated with the turn geometry (excluding streamwise diffusion) should fall as the Peclet number is reduced.40 These expectations, based on conventional turn geometries, were confirmed by calculations for the optimized geometry of Figure 2 down to a Peclet number of 10. Figure 3 shows the progression of a band through conventional and low-dispersion 90° turns. The outer radius in each case is one channel width, so the conventional turn has a zero inner radius. Here, the band images corresponding to fixed time intervals are superposed on a single image of the geometry. The band enters each turn from the left. The conventional 90° turn produces band skewing analogous to that seen in Figure 1, though the magnitude is significantly reduced. The increased variance in this case is (σ/a)2 ≈ 0.99. Again, the turn-induced variance for a conventional turn is roughly proportional to the square of the included angle at high Peclet numbers, so this result is consistent with that of Figure 1. The corresponding optimum geometry for this 90° turn yields an increased variance of only (σ/a)2 ≈ 0.0028. This is a factor of ∼350 below that of the conventional turn. In this case, the minimum channel width is ∼35% of that in the straight segments. Again, this reduction in channel width provides only a small portion of the benefit. Note that, in contrast with the low-dispersion 180° turn, the reduced segment of fixed inner radius now spans only ∼12% of the included angle, and the tapered sections thus extend well into curved portions of the turn.

276 Analytical Chemistry, Vol. 73, No. 2, January 15, 2001

Figure 3. Conventional and low-dispersion 90° turns. Total turninduced variance is (σ/a)2 ≈ 0.99 for the conventional turn; the lowdispersion counterpart yields (σ/a)2 ≈ 0.0026. Bands at fixed time intervals are superposed on a single image of the turn. Bands enter from the left.

Figure 4. Conventional and low-dispersion 45° wye. Induced variance of two exit bands is (σ/a)2 ≈ 0.24 for conventional geometry. Low-dispersion geometry reduces the variance by a factor of ∼100 to (σ/a)2 ≈ 0.0023. Subsample profiles remain orthogonal to direction of motion.

Low-dispersion turn geometries such as those of Figure 3 can also be used to form low-dispersion wyes and tees. Because all boundaries are electrically insulating, such turns can be reflected about the upper boundary of the straight inlet channel to form a branching (or joining) junction. The zero flux condition on the original upper boundary then becomes the zero flux symmetry condition along the centerline of the inlet. The common boundary dividing the inlet can then be deleted, leaving a cusp formed by the original outer radius of the turn. Figure 4 illustrates a conventional 45° wye and the lowdispersion counterpart formed by two 45° low-dispersion turns. Such devices are useful in splitting samples for subsequent parallel processing and for joining subsamples in confluence for mixing or dilution. The induced variance for the conventional wye is (σ/a)2 ≈ 0.24, while that for the low-dispersion geometry is (σ/a)2 ≈ 0.0023. The optimum geometry thus reduces the turninduced spreading by just over 2 orders of magnitude. Low-dispersion wyes such as that in Figure 4 can be paired with corresponding 45° turns to redirect the two outlet channels back parallel to the inlet axis. These channels may then be split again to form four outlet channels. By repeating this configuration, a single sample may be split into many subsamples, and each subsample remains nearly flat and orthogonal to the channel walls. All subsamples prepared in this manner also progress through the network at the same speed, provided that each wye is symmetric, and so arrive at some downstream location simultaneously.

same. Following some rearrangement, this yields

DISCUSSION OF FOLDED COLUMNS One useful application for low-dispersion turns is in folding long separation columns onto a small region. Conventional turns are generally not suitable for this use since the excessive dispersion they produce impairs column performance.29 A simple basis for determining the impact of folding a column on separation performance is to compare the turn-induced spreading with that due to diffusion alone in the straight channel segments. If the turn-induced spreading is much smaller than that due to diffusion, then the turn will have little effect on column performance. The variance for the diffusion problem is σ2 ) 2Dt, where t ) L/U and L is the length of the original channel. Equating this result with the turn-induced variance gives a relation describing the conditions for which the two sources of spreading are the

L* )

L Pe σ ) a 2 a

2

()

(15)

where (σ/a)2 is the normalized turn-induced variance previously discussed. This dimensionless parameter, L*, thus indicates roughly the minimum channel length that can be folded in half while just doubling the total added variance incurred by a band in traversing the turn and channel length. Multiple turns in a folded column should therefore have negligible effect on column performance when the length of each straight segment is ∼10 times greater than the value given by L*. In dimensional terms this can be expressed as

Lmin ≈ 10aL* ) 5aPe(σ/a)2

(16)

Recall that (σ/a)2 depends only on the turn geometry and is independent of the channel width. Also, note that the Peclet number is proportional to the channel width, so the minimum channel length is always proportional to the width squared. Now recall that the Peclet number for a purely electrophoretic processes is Pe ) avzFE/D ≈ azFE/RT, where RT/F ≈ 25 mV. Given a charge number of unity, a channel width of 100 µm, and an applied field of 50 kV/m, the Peclet number is Pe ) 200. By eq 16, the minimum straight section length for the conventional turn shown in Figure 1 is ∼0.39 m based on (σ/a)2 ≈ 3.9. As such, this turn is not useful in microchannel systems. In contrast, however, the low-dispersion turn shown in Figure 2 gives (σ/a)2 ≈ 0.0025. In this case, the minimum length of the straight section is only ∼0.25 mm based on the same channel width and Peclet number. Using this low-dispersion turn, we thus see that a 100 µm separation column up to 900 mm in length can be folded onto a region of only 10 by 10 mm. Further, such a folded column should perform about as well as a straight column of equal length up to a Peclet number of ∼8000. This is equivalent to an applied field of ∼2000 kV/m for a channel width of 100 µm. At this value, the minimum length given by eq 16 is just 10 mm. Similar results can be obtained for electroosmotic flows. Based on a fluid speed of 1 mm/s, a channel width of 100 µm, and molecular diffusivity of 10-9 m2/s, the Peclet number is 100. For this case, eq 16 yields a minimum straight channel length of Analytical Chemistry, Vol. 73, No. 2, January 15, 2001

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0.19 m for the turn of Figure 1. This length increases to 1.9 m for a diffusivity of 10-10 m2/s. The turn of Figure 2, however, yields a minimum length of only 1.2 mm for this lower diffusivity. Again, such a 100 µm column up to 900 mm long can be folded onto a 10 by 10 mm region without detrimental influence for Peclet numbers up to ∼8000. SUMMARY To quantify and remedy the dispersion occurring in turns and junctions, we have developed a numerical model describing species transport in electroosmotic flows, electrophoretic species motion, and species transport in some pressure-driven flows. This model is not based on traditional finite-difference or finite-element methods. Instead, we solve the governing equations by an inverted approach in which the dependent variables are the unknown values of the spatial coordinates and the independent variables are the electric potential and an associated stream function describing fluid or ion motion. The advantage of this approach is that the two-dimensional computational domain is always rectangular; the irregular topology of the channel walls appears only as boundary conditions on the regular computational domain. Transport in channels of arbitrary complexity is thus easily analyzed by this approach, with no need for adaptive meshing schemes or for remeshing the domain when the channel geometry is altered. This inverted numerical model is coupled with a nonlinear least-squares minimization algorithm used to optimize the turn geometry. The minimization object function is simply the increased spatial variance of a species band accumulated in traversing the turn. To obtain a single optimum geometry often requires up to 100 realizations of species transport through candidate turns, and each computational realization is equivalent to a single experiment. As such, it is unlikely that turns and junctions matching the performance of those described here could be obtained by experimental methods alone. The resulting optimum geometries yield a turn-induced spatial variance of a sample band that is 2-3 orders of magnitude below that of equivalent conventional turns. For all turn angles, the species band emerging from the turn is nearly flat and orthogonal to the channel walls. Such turns are useful in microchannel systems for sample preparation and other routine transport processes. This dramatic improvement in turn-induced spreading also permits folding long separation columns onto small areas. Using the 180° turn presented here, a 100-µm channel up to 900 mm long can be folded onto a region covering only 10 by 10 mm. The turns in this configuration do not contribute significantly to band spreading for Peclet numbers up to ∼8000. For electrophoretic processes, this corresponds to an applied electric field up to ∼2000 kV/m. The low-dispersion turns can also be used to form wyes and tees. Such junctions permit splitting a single sample into two equal subsamples for subsequent parallel processing. Cascading a series of such wyes permits precise subdivision into numerous subsamples that are are all nearly flat and orthogonal to the channel walls. Further, all subsamples will travel through this network at the same mean speed and so arrive simultaneously downstream. Low-dispersion junctions may also find use in joining subsamples for mixing and dilution processes.

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All of these low-dispersion turns and junctions are applicable to electroosmotic and electrophoretic transport. The channels may be either open or filled with a gel or porous material. They are further applicable to pressure-driven flows in packed or shallow channels. Finally, these low-dispersion geometries are readily produced using existing manufacturing methods. The optimum geometries do not rely on features very much smaller than the nominal channel width and do not require deep narrow channels having high aspect ratios. The channel depths are also uniform, so these geometries can be produced using all existing molding, embossing, and etching techniques. ACKNOWLEDGMENT This work was funded by a Sandia Phenomenological Modeling and Engineering Simulations LDRD. Sandia National Laboratories is operated by Sandia Corp. for the United States Department of Energy under Contract DE-AC04-94AL85000. NOMENCLATURE a

mean channel width

c

species concentration

D

coefficient of diffusion

E

applied electric field

F

Faraday constant

L

column length: L ) Ut

Pe

Peclet number: Pe ) Ua/D

s

streamwise position

t

time

u

local fluid velocity

U

mean fluid speed

y

transverse position

z

charge number



dielectric constant

ζ

surface electric potential

θ

included angle of turn

λ

Debye length

µ

viscosity

v

ion mobility

Fe

charge density

σ2

increased variance of species distribution

φ

electric potential

Normalized Variables t*

time: t* ) Dt/a2

u*

local fluid speed: u* ) u/U

x*

horizontal position: x* ) x/a

y*

vertical position: y* ) y/a

Received for review August 9, 2000. Accepted October 23, 2000. AC000936Q