Dispersion Reduction in Open-Channel Liquid

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Anal. Chem. 2003, 75, 3352-3359

Dispersion Reduction in Open-Channel Liquid Electrochromatographic Columns via Pressure-Driven Back Flow Debashis Dutta and David T. Leighton, Jr.*

Department of Chemical Engineering, University of Notre Dame, Notre Dame, Indiana 46556

Application of electrokinetic forces to drive the mobile phase diminishes analyte dispersion in open-channel liquid chromatographic columns due to minimization of shear in the flow field. However, the retentive layer coating the inner walls of such devices slows down the average convective velocity of solute molecules in its vicinity, inherently causing dispersion of analyte bands. In this article, we explore the possibility of reducing such dispersion in electrochromatographic columns by imposing a pressure-driven back flow in the system. Analysis shows that although such a strategy introduces shear in the flow field, the overall dispersion in the mobile phase is reduced. This occurs as the streamline velocity in such a system is greater near the channel walls than that in the center of the conduit, thereby allowing fluid dispersion to counteract wall retention effects. For an optimally chosen magnitude of the back flow, hydrodynamic dispersion of any target species in the mobile phase may be shown to diminish by a factor of 3 and 10/3 in a circular tube and a parallel-plate geometry, respectively. A similar reduction in slug dispersion is also realized in rectangular conduits for all aspect ratios. In trapezoidal geometries with large wedge angles or isotropically etched profiles, this reduction factor may attain values of 10 or greater. Chromatography has long been used as a powerful analytical tool for separating neutral solute species.1 While conventionally practiced in packed-bed columns, its implementation in openchannel devices has gained considerable attention recently due to advances in miniaturization technology.2,3 In analytical applications, miniaturization of chromatographic devices is often desired to minimize the reagent consumption in the system, thereby reducing the cost involved in carrying out an experimental protocol. Moreover, theory predicts that a reduction in the length scale also enhances the separation performance of these devices.4 Although such a prediction holds for all chromatographic systems, the improvement in performance is most easily realized in open* Corresponding author. E-mail: [email protected]. Fax: 574-631-8366. (1) Giddings, J. C. Dynamics of Chromatography: Part 1; Marcel Dekker: New York, 1965. (2) Martin, M.; Guiochon, G. Anal. Chem. 1984, 56, 614-620. (3) Jinno, K.; Sawada, H. Trends Anal. Chem. 2000, 19, 352-363. (4) Reyes, D. R.; Iossifidis, D.; Auroux, P. A.; Manz, A. Anal. Chem. 2002, 74, 2623-2636.

3352 Analytical Chemistry, Vol. 75, No. 14, July 15, 2003

channel microcolumns. In liquid chromatography, for example, solutal spreading dominated by hydrodynamic dispersion in the mobile phase is significantly diminished in open-channel devices due to a reduction in the transverse diffusional length scale. In packed-bed columns, however, realizing a similar benefit by scaling down the size of the packing particles is more challenging. This is because uniformity in particle packing becomes increasingly difficult to accomplish with a decrease in the particle diameter.2,3 Any such nonuniformity in the packing density automatically leads to large variations in the fluid velocity convecting the solute molecules, thereby significantly increasing slug dispersion in the system. While the open-channel design offers a convenient and efficient choice for microchromatography, its separation performance is often limited due to convective dispersion of solute slugs in the system.5,6 In gas chromatographic applications, such dispersion is dominated by the contribution arising from transverse diffusion limitations in the stationary phase.7,8 The efficiency of liquid chromatographic columns, however, is mostly determined by the hydrodynamic dispersion component in the mobile phase, which is the focus of this work. This component arises from variations in the axial solute velocity across the channel cross section coupled with slow transverse diffusion in the mobile phase. Nonuniformities in the flow as well as a reduction in the solute velocity due to retention at the channel walls automatically leads to a stretching of the analyte bands as convection proceeds in the axial direction. This convective stretching is ultimately limited by diffusion across streamlines and is often described in terms of the Taylor-Aris dispersivity in the mobile phase, Km. This quantity in a pressure-driven parallel-plate chromatographic device, for example, is given by9-12

(5) Jacobson, S. C.; Hergenro¨der, R.; Koutny, L. B.; Ramsey, J. M. Anal. Chem. 1994, 66, 2369-2373. (6) Kutter, J. P.; Jacobson, S. C.; Matsubara, N.; Ramsey, J. M. Anal. Chem. 1998, 70, 3291-3297. (7) Knox, J. H. J. Chromatogr. Sci. 1980, 18, 453-461. (8) Knox, J. H. J. Chromatogr., A 1999, 831, 3-15. (9) Golay, M. J. E. In Gas Chromatography; Desty, D. H., Ed.; Buttersworth: London, 1958. (10) Aris, R. Proc. R. Soc. London 1959, 252A, 538-550. 10.1021/ac0207933 CCC: $25.00

© 2003 American Chemical Society Published on Web 06/06/2003

Here, Dm is the diffusivity of the solute molecules in the mobile phase, U is the average mobile-phase velocity, and d is the separation distance between the parallel plates. The symbol R in the above expression denotes the retention ratio of the analyte species in the system and is defined to be the fraction of solute in the mobile phase at equilibrium. The dimensionless group (Ud/ Dm) is the characteristic Peclet number of the device and is also sometimes referred to as the “reduced flow velocity” ν in the system. Note that the expression given in eq 1 may be related to the reduced plate height contribution in the Golay equation9 arising from slow mass transfer in the mobile phase (Cmν) as

Cmν )

( )

( )

2 Km 2 Km w Cm ) 2 νR Dm ν R Dm

(2)

As may be seen from eq 1, the overall convective dispersion in the mobile phase in a pressure-driven parallel-plate device may be decomposed into three contributions. While the first term in the expression quantifies the effect of wall retention with an uniform flow U in the system, the second term determines the effect of shear in the channel with no stationary phase. Because Taylor-Aris dispersion is essentially proportional to a weighted average of the deviation of the square of the solute velocity from its mean, the two effects also give rise to an interaction term in the system, the third term in eq 1. It is important to note that solute dispersion arising due to wall retention effects is an inherent consequence of the open-channel chromatographic technique. However, the contribution to dispersion arising from the shear in the flow field depends on the choice of the driving force for the mobile phase. Clearly, this source of dispersion may be diminished by minimizing variations in the streamline velocity across the channel cross section. In an electrokinetically driven system, for example, the flow through a conduit is essentially uniform, thereby eliminating the contributions from the “flow term” and the “cross-term” to the overall hydrodynamic dispersion in the mobile phase. In this article, we propose the use of a pressure-driven back flow to reduce solute dispersion in open-channel electrochromatographic systems. Imposition of such a back flow on a uniform electrokinetic velocity profile automatically leads to a greater fluid velocity near the channel walls than that in the center of the conduit. As a consequence, fluid dispersion in the system tends to counteract solutal dispersion arising from a reduction in the axial velocity of the analyte molecules due to wall retention. However, the imposition of a pressure gradient also introduces shear in the flow field. Here, we quantify the competing effects of additional dispersion resulting from an increased shear in the system and the reduction in solute dispersivity obtained due to the counteraction of fluid dispersion and wall retention effects. These effects are examined for a number of different channel geometries. SCALING ANALYSIS It was shown by Aris10 that the overall analyte dispersion in an open-channel chromatographic device may be expressed as (11) Giddings, J. C. J. Chromatogr. 1961, 5, 4-60. (12) Dutta, D.; Leighton, D. T. Anal. Chem. 2003, 75, 57-70.

the sum of the contributions arising from simple axial diffusion and convective dispersion in the system. While the first contribution results from an unbounded random walk motion of the solute molecules in the flow direction, the second contribution arises from variations in the axial solute velocity across the channel cross section coupled with slow transverse diffusion. Further, the effect of solute diffusivity in the mobile and the stationary phases leading to each of these contributions is also additive.10 In this situation, the various contributions leading to the overall slug dispersivity in the system may be evaluated independently as implied by the principle of additivity. Here, we investigate the performance of open-channel electrochromatographic systems upon introduction of a pressure-driven back flow in the column. Since a modification in the flow field only alters the convective dispersion component in the mobile phase,10 we focus on this contribution, ignoring all other sources of dispersion in the system. The effect of different channel cross sections on the convective dispersion component in the mobile phase was recently investigated by our research group12 for pressure-driven open-channel chromatographic systems. By decomposing the effect of shear in the flow field and solute retention at the channel walls on this contribution, it was demonstrated that the influence of any channel profile may be characterized using three functions g1, g2, and g3 as shown below12

where

g1)

g2 )

g3 )

30 P*

12 P*

210 A/m

∫f

D 1

dA/m -

∫ v(x*, y*)f D

∫ [v(x*, y*) - 1] f D

30 A/m

2

12A/m P*2

dA/m -

/ 1 dAm

∫f

f| ∂D 1 ∂D

dl*

I

210 A/m

∫f

D 2

dA/m

+

dA/m D 2

-

30 P*

f | dl* ∂D 2 ∂D

I

(3)

Here, δ denotes the ratio of the stationary-phase thickness to the narrower channel dimension (d), γ is the partition coefficient in the system governing the distribution of solute molecules between the mobile and the stationary phases at equilibrium, v(x*, y*) is the solute velocity profile in the mobile phase normalized with respect to its average value U, Am is the cross-sectional area of the mobile phase (domain D), and P is the length of the interfacial boundary (∂D) between the stationary and the mobile phases in the system. In deriving the above expressions, the retentive layer coating the inner walls of the conduit was assumed to be thin, i.e., δ , 1, and the cross-sectional area of the stationary phase was approximated by the product As ≈ δdP. The functions f1 and Analytical Chemistry, Vol. 75, No. 14, July 15, 2003

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f2 appearing in eq 3 were obtained by solving the following set of differential equations:

∫f

∂2f1

∂2f1

∂x*

2

+ 2

∂y*

∂2f2

∂2f2

∂x*

∂y*2

dA/m D 1

+ 2

+ γδ

)-

∇ B * f1‚n b|∂D ) -1

) 1 - v (x*, y*); ∇ B *f2‚n b|∂D ) 0

f| ∂D 1 ∂D

I

P* ; A/m

dl* ) 0;

∫f D

/ 2 dAm + γδ

f | dl* ) 0 (4) ∂D 2 ∂D

I

Note that all lengths in eqs 3 and 4 have been normalized with respect to the narrower channel dimension d. Also, the functions g1, g2, and g3 have been defined in this formulation such that they assume a value of unity in a pressure-driven parallel-plate geometry. It is interesting to note that the functions g1, g2, and g3 evaluated for the simple pressure-driven chromatography case12 may also be used to quantify the performance of an electrochromatographic system with a pressure-driven back flow as investigated here. To this end, consider a flow profile given by u(x*, y*) ) U[(1 + R) - Rh(x*, y*)] in an open-channel chromatographic column. This velocity field may be viewed as that arising from two independent driving forces acting on the system. While the component U(1 + R) represents an uniform electrokinetic flow in the channel due to an applied electric field, the remaining contribution -URh(x*, y*) arises from an imposed back pressure in the system. Thus, the function h defined above determines the shear in the flow field with RU being the appropriate scaling factor for the shear velocity. Further, to fairly quantify the effect of a back pressure on the performance of an electrochromatographic system, the average flow through the device is held constant (U) in all our comparisons. Therefore, the function h in this formulation has an average value of unity in the conduit and thus is identically equal to the velocity function (v(x*, y*) in eq 3) in a simple pressuredriven system for a given channel profile. In the case of simple pressure-driven chromatography, an interesting way to view the velocity function in the mobile phase, v(x*, y*), is to decompose it into two components as 1 + [v(x*, y*) - 1]. The first term (1) here denotes the difference in the average velocities of the mobile and stationary phases and, therefore, determines the effect of wall retention in the system. The second term, [v(x*, y*) - 1], has a zero mean velocity and governs the effect of shear in the flow field. As such, the TaylorAris dispersivity arising from each of these sourcessthe “wall retention term” and the “flow term” in eq 3sscales with the square of these two flow components that produces them, i.e., 1 and [v(x*, y*) - 1]2, respectively. The cross-term quantifying the interaction of the two effects varies with the product of the two flow components. Likewise, the velocity function in an electrochromatographic column with a pressure-driven back flow may also be rewritten as 1 + (- R)[h(x*, y*) - 1]. Note that this flow profile is identical to that in a simple pressure-driven system except for the fact that the shear velocity component in this case has an additional scaling factor of -R. Thus, the dispersivity in the mobile 3354

phase for the modified electrochromatographic systems may be expressed as

Analytical Chemistry, Vol. 75, No. 14, July 15, 2003

where the functions g1, g2, and g3 are evaluated for a simple pressure-driven open-channel chromatographic system. As may be seen from eq 5, the interaction of fluid shear and analyte retention at the channel walls leads to a negative crossterm in electrochromatographic devices with a pressure-driven back flow. The reduction in solute dispersivity due to this crossterm thus quantifies the counteracting effect of fluid dispersion and wall retention in such systems. The convective dispersion component in the mobile phase quantified by the “flow term” in eq 5 in this case scales with R2. It is important to note that expressing solute dispersivity in electrochromatographic columns with a back pressure in terms of the functions g1, g2, and g3 evaluated for the simple pressure-driven chromatography case leaves these quantities independent of the magnitude of the back flow (R) in the system. In this situation, solute dispersion in the modified electrochromatographic systems may be minimized by optimizing eq 5 with respect to R to yield

( ) ( ) Km Dm

)R

min

[ ( )]

2 2 2 7 g3 Ud (1-R) g1 1; Dm 12 10 g1g2

7 (1-R)g3 Ropt ) (6) 2 g2 The above expression suggests that the optimum value of R is a function of the retention ratio of the dispersing solute species. Upon introducing this optimum back flow, solute dispersion of any target species in electrochromatographic columns may be reduced by a factor of

(

reduction factor, RF ) 1 -

2 7 g3 10 g1g2

)

-1

(7)

The dispersion of any other analyte species in this case is governed by

( ) ( ) Km Dm

R

)

modifiedEC

Ud Dm

2

[ ( )(

)(

)]

2 1 - Rt (1-R)2g1 7 g3 1 - R t 1212 10 g1g2 1 - R 1-R

(8)

where Rt and R denote the retention ratios of the target and the other analyte species in the system, respectively. In Figure 1, we have plotted this dependence of Km in an optimally modified parallel-plate electrochromatographic column for different choices of the target species. Recall that for this geometry the three

this situation, however, the operating conditions may be further retuned to enhance the separations in the system. For example, in the modified electrochromatographic devices where both axial diffusion and convective dispersion in the mobile phase dominate the overall broadening of the analyte bands (e.g., low Peclet number), the separation resolution (SR) is maximized only for an appropriate choice of the average mobile-phase velocity U. In this case, the separation resolution of any two closely related solute slugs under optimum back flow conditions, i.e., R given by eq 6 for a choice of Rt set equal to the average of the retention ratios of the two species, is given by

Figure 1. Dispersion of various solute species in a parallel-plate electrochromatographic device with a pressure-driven back flow that minimizes the band broadening of a target species. Here Rt and R refer to the retention ratios of the target and other analyte species in the system, respectively.

dispersivity functions g1, g2, and g3 assume a value of unity. As may be seen from the figure, the reduction in the convective dispersion component in the mobile phase is maximized for a retention ratio equal to that of the target species, i.e., R ) Rt. For retention ratios smaller than this value, the reduction in dispersion diminishes gradually with decrease in R. In the opposite range, i.e., R > Rt, however, the optimum back flow is beneficial only for analyte species with retention ratios R < (1 + Rt)/2. For solute molecules with retention ratios greater than this value, the effect of fluid shear dominates the dispersivity in the mobile phase. Clearly, in this limit, solute dispersion in the modified electrochromatographic devices is smaller for larger choice of Rt as Ropt ∼ (1 - Rt). Examination of Figure 1 suggests that the overall benefit yielded by the modified electrochromatographic systems in this situation is maximized for a choice of Rt near 0.5. In cases where the performance of the chromatographic column is completely dictated by the separation of two closely related species, Rt may be chosen as the average of the retention ratios of these two analytes. While the magnitude of the optimum back flow given by eq 6 has been derived to minimize the dispersion component arising due to transverse diffusion limitations in the mobile-phase Km, note that the same choice also minimizes the overall dispersivity in open-channel chromatographic systems where other sources of dispersion may be important. This is because with the dispersion contributions arising due to ordinary axial diffusion in the mobile and the stationary phases, any mass-transfer resistance between the two and limited transverse diffusion in the stationary phase are all additive to Km and do not depend on the flow profile in the mobile phase. However, the minimum overall dispersivity in such a system is modified from (Km)min in eq 6 by these additive contributions whose magnitudes may be estimated based on the work of Aris.10 It is also important to note that as the contribution to the overall solutal spreading due to the other above-mentioned sources of dispersion becomes significant, the benefit yielded upon imposing an optimum pressure-driven back flow is reduced. In

Here, L is the length of the separation column and K is the mean overall dispersivity of the two solute slugs in the system whose retention ratios differ by a magnitude of ∆R. The symbol RF denotes the dispersion reduction factor as defined in eq 8. The above expression may be then optimized with respect to the average mobile-phase velocity (U) to yield

SRmax )

( )x [ ] [ ]

1 ∆R 2 Rt

Uopt )

2 L 1 (1-Rt) g1 d RF 12

Dm 1 (1-Rt)2g1 d RF 12

-1/4

-1/2

(10)

Inspection of eq 10 suggests that, for a given electrochromatographic separation resolution in the system, the imposition of a pressure-driven back flow reduces the separation length by as much as factor of (RF)1/2 when both axial diffusion and convective dispersion in the mobile phase are important. The effect of the back flow, however, is more enhanced in the absence of any axial diffusion in which case the maximum reduction factor in the length of the separation column approaches a value of RF for any given average mobile-phase velocity as suggested by eq 9. It is important to note that the above analysis strictly holds for open-channel devices and is not relevant to packed-bed electrochromatographic systems with an imposed back pressure. This is because such pressure gradients in a randomly packed column tend to induce a greater back flow (channeling) in regions with a larger interparticle pore area (hydraulic flow velocity ∼ pore area). The electroosmotic flow velocity, however, is independent of the size of these interstices and thus is nearly uniform across the packed column. The flow profiles are also complicated by the three-dimensional nature of a packed bed. As a result, it is difficult to see how dispersion can be reduced by back flow in such a geometry. Moreover, even in the absence of any channeling effects, the overall dispersion in such a system is likely to be dominated by the contribution arising from slow diffusion in to and out of the pores within the packing particles (stationary phase). This can occur as the characteristic diffusional length scale Analytical Chemistry, Vol. 75, No. 14, July 15, 2003

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Figure 2. Effect of different channel geometries on the dispersivity of a target solute species in an electrochromatographic system with an optimum pressure-driven back flow.

determining the stationary-phase dispersion component is the diameter of the packing particles while the corresponding length scale governing the mobile-phase contribution is a small fraction of this particle diameter. In this situation, the effect of the flow profile in the mobile phase, and therefore the pressure-driven back flow, is insignificant in determining the performance of the packedbed chromatographic device. This is consistent with the experimental observation of Rebscher and Pyell,13 who found that back flow either had no effect or increased dispersion for packed-bed electrochromatography. Effect of Channel Geometry. Inspection of eq 8 shows that the reduction in analyte dispersion in modified electrochromatographic devices also depends on the cross-sectional geometry of the separation column. Specifically, this improvement in performance is governed by the ratio g23/(g1g2) in the system. In this section, we investigate the dependence of this ratio on the various channel profiles often employed in chromatographic applications. Circular Tubes. The effects of fluid shear and wall retention on the solute dispersivity in open-tubular chromatographic columns have been investigated in the past by Golay,9 Aris,10 and Giddings.11 Their work on simple pressure-driven chromatography shows that the contribution to convective dispersion due to slow transverse diffusion in the mobile phase may be expressed as9-11

where d is the diameter of the separation tube. Thus, upon comparing eqs 11 and 3, the values of the dispersivity functions g1, g2, and g3 may be shown to be given by 3/8, 35/32, and 5/8, respectively, for the tubular geometry. In this situation, the ratio g23/(g1g2) assumes a value of 20/21, which upon substituting in eq 7 suggests that dispersion of any target species may be reduced by as much as a factor of 3 in open-tubular electrochromatographic systems using an optimally chosen pressure-driven back flow. (13) Rebscher, H.; Pyell, U. Chromatographia 1996, 42, 171-176.

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Figure 3. Cross section of an isotropically etched channel geometry with a rectangular central region flanked by two quarter-circular disks. The radius of the disks is the same as the channel depth due to the undercutting of the mask used in the wet etching process.

Rectangular Conduits. Analyte dispersion in open-channel rectangular chromatographic conduits has been investigated by our research group12 among others.14-16 In our previous work,12 it was shown that limited mass transfer across both the narrower and the wider dimensions of this geometry contribute to the overall convective dispersion in the mobile phase.17 While the contribution due to diffusion across the narrower dimension may be estimated using the parallel-plate result given in eq 1, the effect of slow diffusion in the wider direction is more complex. As has been previously recognized,12,16-22 the side walls of the rectangular profile tend to increase the Taylor-Aris dispersivity in the system by additionally retarding solute motion in their vicinity. This reduction in solute velocity in a simple pressure-driven chromatographic column occurs due to both the no-slip flow condition at the channel side walls and an increased analyte retention in these regions due to a greater wall perimeter. While both these contribu(14) Zhang, X.; Regnier, F. E. J. Chromatogr., A 2000, 869 (1-2), 319-328. (15) Blackburn, H. M. Comput. Chem. Eng. 2001, 25 (2-3), 313-322. (16) Poppe, H. J. Chromatogr., A 2002, 948, 3-17. (17) Desmet, G.; Baron, G. V. J. Chromatogr., A 2002, 946 (1-2), 51-58. (18) Doshi, M. R.; Daiya, P. M.; Gill, W. N. Chem. Eng. Sci. 1978, 33, 795-804. (19) Golay, M. J. E. J. Chromatogr. 1981, 216, 1-8. (20) Chatwin, P. C.; Sullivan, P. J. J. Fluid Mech. 1982, 120, 347-358. (21) Giddings, J. C.; Schure, M. R. Chem. Eng. Sci. 1987, 42 (6), 1471-1479. (22) Dutta, D.; Leighton, D. T. Anal. Chem. 2001, 73, 504-513.

Figure 4. Trapezoidal channel geometry.

tions scale with the square of the Peclet number based on the narrower dimension of the conduit (usually the channel depth d), their coefficients are functions of the aspect ratio of the geometry W/d, where W is the channel width. This dependence was derived in our previous work12 using the dispersivity functions g1, g2, and g3 defined in eq 3 for simple pressure-driven chromatography. Analysis showed that these functions may assume values as large as 2, 4.1, and 7.95, respectively in the limit of wide channel systems, i.e., d/W f 0. In this work, we have used these results to evaluate the ratio g23/(g1g2) that determines the effect of the rectangular geometry on the performance of an electrochromato-

graphic device with an optimally chosen back flow. The dependence of this ratio on the geometric parameter of the rectangular channel d /W, has been depicted in Figure 2. As may be seen, upon introduction of a pressure-driven back flow the greatest benefit is obtained in large aspect ratio geometries, yielding a reduction in solute dispersion by as much as a factor of 4. For small aspect ratio geometries, this reduction is diminished to a factor of ∼3 for any target species. Isotropic Profiles. In microchip applications, the cross section of the chromatographic column is often not rectangular. Isotropic chemical wet etching techniques employed in the fabrication of these devices, for example, result in channel profiles close to the idealization depicted in Figure 3. In this design, the side regions may be modeled as quarter-disks of radius d alongside a central rectangular section of depth d and length l. As in rectangular channels, these quarter circular end regions retard the motion of the solute molecules in the conduit, thereby affecting the dispersivity in the system. Again, the effect of these side regions were quantified in our previous work12 for the simple pressure-driven chromatography case using the dispersivity functions g1, g2, and

Figure 5. Solute dispersion in a pressure-driven chromatographic column with a trapezoidal cross section (a) effect of wall retention (b) effect of shear (c) effect of the interaction between wall retention and shear flow.

Analytical Chemistry, Vol. 75, No. 14, July 15, 2003

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Figure 6. Effect of a pressure-driven back flow on the dispersivity of solute slugs in a trapezoidal electrochromatographic column.

g3. As described earlier, the improvement yielded due to the presssure-driven back flow component is determined by the ratio g23/(g1g2) for the isotropic geometry. In Figure 2 we have depicted the dependence of this ratio for various isotropic designs at a fixed depth. As may be seen, the above ratio may attain values as large as 1.25 in the narrower isotropic profiles yielding a reduction in electrochromatographic dispersion by a factor of 8. For large aspect ratio geometries, this ratio approaches a value 1.06 yielding an improvement of a factor of ∼4. Trapezoidal Geometries. Another common channel profile used in miniaturized chromatographic devices is the trapezoidal geometry as depicted in Figure 4. Such cross sections result from anisotropic etching techniques also employed in the fabrication of chemical microchips. Upon scaling the dimensions of the trapezoidal channel with respect to its depth in the central section d, this geometry may be completely specified in terms of the parameters l/d and θ0, where l is the width of the central rectangular section and θ0 is the wedge angle of the side wall (see Figure 4). The case l ) 0 corresponds to a triangular channel. Here, we have investigated the effect of the trapezoidal cross section on solute dispersion in electrochromatographic systems using the dispersivity functions g1, g2, and g3. In Figure 5, the dependence of these functions on the various geometric parameters of the trapezoidal profile has been shown. As may be seen, solute dispersion in this geometry is significantly increased for large wedge angles, i.e., θ ∼ π/2. This occurs as the contribution to slug dispersion from the channel side walls scales with the square of the Peclet number based on the width of the side regions. In the limit θ f π/2, this width diverges, yielding large dispersion in such geometries. In the opposite limit, i.e., θ f 0, the dispersivity in the trapezoidal channel approaches that in a rectangular geometry of identical depth and width. In Figure 6, we have shown the effect of an optimally chosen pressure-driven back flow on the solutal spreading in a trapezoidal electrochromatographic column. As may be seen, for smaller wedge angles, dispersion of any target species in this geometry may be reduced by a factor of ∼3, similar to that in the rectangular conduits. For larger wedge angles, the reduction factor increases and attains a maximum value when the disparities in the average solute velocity 3358

Analytical Chemistry, Vol. 75, No. 14, July 15, 2003

in the central and the side regions of the channel is minimized. Upon increasing the wedge angle any further, the dispersion reduction factor diminishes again as the width of the side regions approaches that of the central section. Note that the use of a pressure-driven back flow also enhances the performance of the trapezoidal profiles with an increase in the width of the central section l by minimizing the side wall contribution to slug dispersion in the mobile phase. CONCLUSIONS The above analysis shows that introduction of a pressure-driven back flow reduces solutal dispersion in electrochromatographic columns by counteracting wall retention effects with fluid shear. This occurs as the axial velocity of the analyte molecules is increased near the channel walls in such a system over the motion of the solute slug. As a result, the cross-term quantifying the interaction of fluid shear and wall retention assumes negative values. Although the back pressure also introduces an additional dispersion component in the mobile phase due to fluid shear, the effect of the cross-term dominates for small back flows. The optimum magnitude of the back flow that minimizes the overall dispersion in the mobile phase is a function of the retention of the dispersing solute species. For a chosen target species with retention ratio R ) Rt, dispersion may be reduced by as much as a factor of [1 - 0.7g23/(g1g2)]-1 using an optimum ratio of a pressure-driven back flow to electroosmotic flow Ropt/(1 + Ropt) ) [3.5(1 - Rt)g3]/[g2 + 3.5(1 - Rt)g3] in the chromatographic column. This corresponds to a reduction in dispersion (or alternatively a reduction in the separation length for a given separation resolution) by a factor of ∼3 in a tube, parallel-plate device and a rectangular conduit. The reduction factor may attain values as large as 8-10 for small aspect ratio isotropic designs and wide trapezoidal geometries with large wedge angles. Note that dispersion is also reduced in such a system for other species with retention ratios R < (1 + Rt)/2. For analytes with retention ratios greater than this value, however, solutal spreading is increased due to the shear introduced by the back flow. The variation in the dispersion reduction factor with the solute retention ratio about its optimum value is gradual. This implies

that nearly optimum benefit of the pressure-driven back flow may be realized even for analytes with retention ratios significantly separated from that of the target species. In a parallel-plate device with a choice of Rt ) 0.5, for example, the yielded benefit is within 10% of that for the target species for analytes with retention ratios in the range 0.36 < R < 0.58, i.e., ∆R/Rt ≈ 45%. As a final note, it is useful to compare the performance of an open-channel electrochromatographic column with an equivalent packed-bed electrochromatographic device. Under optimal conditions, a packed column can be operated with dispersivities close to that of molecular diffusion by reducing packing size to O (3 µm) and velocities to O (1 mm/s). Similar dispersivities in openchannel electrochromatography may be achieved by employing a Peclet number of O (5), corresponding to gap widths of O (1 µm) or less. Use of optimally determined pressure-driven back

flows allows somewhat larger Peclet numbers (a factor of ∼2) to be employed before the onset of significant Taylor-Aris dispersion. The approach described here is of greatest benefit for liquid chromatographic columns chosen to be operated at high Peclet numbers where Taylor-Aris dispersion in the mobile phase dominates the overall dispersion in the system. ACKNOWLEDGMENT This research work was supported by the National Science Foundation Grant CTS-9980745.

Received for review December 31, 2002. Accepted April 7, 2003. AC0207933

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