Electrokinetic Transport of Charged Samples through Rectangular

May 14, 2008 - conduits of all aspect ratios within an accuracy of 5% with significantly less computational effort than that required in the numerical...
0 downloads 0 Views 449KB Size
Download PDF
Anal. Chem. 2008, 80, 4723–4730

Electrokinetic Transport of Charged Samples through Rectangular Channels with Small Zeta Potentials Debashis Dutta* Department of Chemistry, University of Wyoming, Laramie, Wyoming 82071 In this article, we analyze the electrokinetic transport of charged samples through rectangular channels having a small zeta potential at their walls. Using the “method of moments” formulation, the diffusion-advection equation has been solved numerically to evaluate the mean velocity and the dispersion of analyte bands in these systems. In addition, a semianalytical theory has been presented for estimating the solutal spreading rate by decoupling the effects of vertical and horizontal velocity gradients in the channel. We demonstrate that this theory can estimate the band broadening of charged samples in rectangular conduits of all aspect ratios within an accuracy of 5% with significantly less computational effort than that required in the numerical simulations. Moreover, our analysis shows that while the side walls in a rectangular conduit modify the solute velocity only to a moderate extent, they can increase the hydrodynamic dispersion of sample slugs by as much as an order of magnitude under strong Debye layer overlap conditions. In the opposite limit of thin Debye layers, however, the increase in dispersion due to the side regions is only by a factor of 2 and remains nearly unaffected by the transverse electromigration of the solute molecules and the aspect ratio of the channel. Electrokinetic transport presents a powerful method for manipulating fluid and solute samples in microfluidic devices because of its better controllability and ease of implementation at shorter length scales.1–3 While this mode of actuation is well understood for systems with electrical double layers (EDL) much smaller than the channel dimensions, its implementation under EDL overlap conditions is an area of intense research.4–6 With the development of simple techniques for fabricating channels with lateral dimensions in the submicrometer and nanometer range (nanochannels),7–9 it has only been recently possible to observe * To whom correspondence should be addressed. E-mail: [email protected]. (1) Reyes, D. R.; Iossifidis, D.; Auroux, P. A.; Manz, A. Anal. Chem. 2002, 74, 2623–2636. (2) Squires, T. M.; Quake, S. R. Rev. Mod. Phys. 2005, 77, 977–1026. (3) Janasek, D.; Franzke, J.; Manz, A. Nature 2006, 442, 374–380. (4) Eijkel, J. C. T.; van den Berg, A. Microfluid. Nanofluid. 2005, 1, 249–267. (5) Holtzel, A.; Tallarek, U. J. Sep. Sci. 2007, 30, 1398–1419. (6) Pennathur, S.; Eijkel, J. C. T.; van den Berg, A. Lab Chip 2007, 7, 1234– 1237. (7) Mijatovic, D.; Eijkel, J. C. T.; van den Berg, A. Lab Chip 2005, 5, 492–500. (8) Bakajin, G.; Fountain, E.; Morton, K.; Chou, S. Y.; Sturm, J. C.; Austin, R. H. MRS Bull. 2006, 31, 108–113. (9) Perry, J. L.; Kandlikar, S. G. Microfluid. Nanofluid. 2006, 2, 185–193. 10.1021/ac7024927 CCC: $40.75  2008 American Chemical Society Published on Web 05/14/2008

some of the interesting electrokinetic transport phenomena under EDL overlap conditions.10–16 For example, the reduction in the electroosmotic flow velocity of electrolytic fluids with an increase in the extent of EDL overlap as predicted by classical theories has been quantitatively demonstrated by researchers only in the past few years.11,17,18 Moreover, Pennathur and Santiago11 among others have shown that the transport of charged analytes in large EDL systems is significantly modified from that in a microchannel device due to the strong influence of the lateral electric field induced by the channel surface charges. This electric field in the case of glass nanochannels (as was employed by Pennathur and Santiago) focuses positively charged analytes near the walls while it repels negatively charged species toward the channel center. The resulting nonuniformity in the analyte concentration when combined with the nonuniform electroosmotic flow profile realized under EDL overlap conditions yields transport rates for solute samples which are significantly different from that observed in the limit of thin EDLs. In general, positively charged species are slowed down and negatively charged analytes are sped up relative to the average fluid velocity when flown through glass nanochannels having a negative surface charge density. In order to quantitatively understand the experimental observations described above, there has also been some recent effort toward theoretically modeling solutal transport in nanometer scale channels. For example, Pennathur and Santiago19 mathematically described the flow of charged samples through nanochannels, capturing the leading order effects of the surface charge induced lateral electric field on analyte transport. Although their model was able to determine the effect of analyte transverse electromigration (TEM) on the average solute velocity under Debye layer overlap conditions, it failed to estimate the broadening of analyte (10) Stein, D.; Kruithof, M.; Dekker, C. Phys. Rev. Lett. 2004, 93, Art. No. 035901. (11) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6782–6789. (12) Cross, J. D.; Strychalski, E. A.; Craighead, H. G. J. Appl. Phys. 2007, 102, Art. No. 024514. (13) Karnik, R.; Duan, C. H.; Castelino, K.; Daiguji, H.; Majumdar, A. Nano Lett. 2007, 7, 547–551. (14) Kim, S. J.; Wang, Y. C.; Lee, J. H.; Jang, H.; Han, J. Phys. Rev. Lett. 2007, 99, Art. No. 044501. (15) Qiao, Y.; Cao, G. X.; Chen, X. J. Am. Chem. Soc. 2007, 129, 2355–2359. (16) van der Heyden, F. H. J.; Bonthuis, D. J.; Stein, D.; Meyer, C.; Dekker, C. Nano Lett. 2007, 7, 1022–1025. (17) Yuan, Z.; Garcia, A. L.; Lopez, G. P.; Petsev, D. N. Electrophoresis 2007, 28, 595–610. (18) Sadr, R.; Yoda, M.; Gnanaprakasam, P.; Conlisk, A. T. Appl. Phys. Lett. 2006, 89, Art. No. 044103. (19) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6772–6781.

Analytical Chemistry, Vol. 80, No. 12, June 15, 2008

4723

kinetic flow along the channel axis and a transverse electrophoretic component b ut (in the x-y plane) induced by the lateral electric field within the Debye layer. Under these conditions, one can show that the average velocity (U) and the Taylor-Aris dispersivity (K) of the sample slugs are given by eqs 1 and 2 respectively22 (see Supporting Information for derivation). U) K ) 1 + Pe2 D Figure 1. A schematic of a rectangular channel described in this article.

MATHEMATICAL FORMULATION We begin our analysis by considering the transport of nonneutral solute molecules through a rectangular channel of depth d and width W under the influence of an applied electric field Ez in the axial direction (see Figure 1). The solute velocity field in this system has an axial component uz, arising from the electro(20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35)

Xuan, X. C.; Li, D. Q. Electrophoresis 2006, 27, 5020–5031. De Leebeeck, A.; Sinton, D. Electrophoresis 2006, 27, 4999–5008. Griffiths, S. K.; Nilson, R. H. Anal. Chem. 2006, 78, 8134–8141. Giddings, J. C. J. Chem. Phys. 1968, 49, 81–85. Martin, M.; Giddings, J. C. J. Chem. Phys. 1981, 85, 727–733. Brenner, H.; Edwards, D. A. Macrotransport Processes; ButterworthHeinemann: Boston, 1993; pp 65-154. Dutta, D. Electrophoresis 2007, 28, 4552–4560. Doshi, M. R.; Daiya, P. M.; Gill, W. N. Chem. Eng. Sci. 1978, 33, 795–804. Golay, M. J. E. J. Chromatogr. 1981, 216, 1–8. Chatwin, P. C.; Sullivan, P. J. J. Fluid Mech. 1982, 120, 347–358. Dutta, D. J. Colloid Interface Sci. 2007, 315, 740–746. Dutta, D.; Ramachandran, A.; Leighton, D. T. Microfluid. Nanofluid. 2006, 2, 275–290. Dutta, D.; Leighton, D. T. Anal. Chem. 2001, 73, 504–513. Dutta, D.; Leighton, D. T. Anal. Chem. 2003, 75, 3352–3359. Dutta, D.; Leighton, D. T. Anal. Chem. 2003, 75, 57–70. Ghosal, S Electrophoresis 2004, 25, 214–228.

4724

Analytical Chemistry, Vol. 80, No. 12, June 15, 2008

/

∫u S

(1)

0 z

S

/

z

φ0h dA/ ) 1 + Pe2f

(2)

where the function h satisfies the differential equation f

bands in these systems. In a recent article, Xuan and Li20,21 studied the dispersion of charged molecules in micro- and nanochannels by solving the governing transport equations using numerical techniques. Griffiths and Nilson22 presented an alternative formulation based on previous works by Giddings23,24 and Brenner25 to simplify these computations and showed results in the limit of small zeta potentials. In a recent article by Dutta,26 this study was further extended to systems with large zeta potentials using numerical techniques. While these studies focused on the transport of charged samples through simple channel geometries, e.g., parallel-plate or a tube, actual nanochannel cross-sections etched on glass or silicon-based substrates are often close to rectangles or trapezoids. For such geometries, the presence of the side walls tends to significantly modify analyte transport by altering the fluid flow profile27–32 as well as enhancing the interaction of the solute molecules with the channel surface charges.33–35 In this article, we present a detailed analysis of the electrokinetic transport phenomena for charged samples through rectangular ducts having a small zeta potential at their walls. Using analytical and numerical techniques, the effect of the channel side walls has been quantified on the flow velocity and Taylor-Aris dispersivity of solute samples in these systems.

∫ φ u dA

f

f

f

∇/2h + βu t/ · ∇ /h ) 1 - uz/; B.C. (∇ /h) · n |∂S ) 0;

∫ φ h dA ) 0 /

S

0

(3)

The quantity φ0 ) [exp(-βψ*)]/[∫S exp(-βψ*) dA*] in the above equations is the normalized analyte concentration profile across the channel cross-sectional plane S and the parameter Pe ) [Ud]/D is the Peclet number for the analyte molecules that have a diffusion coefficient D. The symbol ∂S used in the integrals here refers to the wall of the rectangular conduit (see Figure 1) while dA* denotes a differential area on its cross-sectional plane normalized with respect to the square of the channel depth (d2). The parameter β ) [Utd]/D appearing in eq 3 is a measure of the TEM velocity for the solute particles relative to their diffusive transport rate in the system, and the symbol b n used in the same equation refers to the unit vector pointing outward normal from b* here denote the channel walls. Finally, the symbols ∇*2 and ∇ the two-dimensional Laplacian and gradient operators (only based on the x- and the y-coordinates) normalized with respect to d2 and d, respectively. Note that in a fluidic channel, the electrokinetic solute velocity in the lateral and axial directions may be evaluated bψ and uz ) -(εζ/η)(1 - ψ/ζ)Ez + µEz. Here, ψ as36 b ut ) -µ∇ denotes the electric potential induced by the channel surface charges, µ denotes the electrophoretic mobility of the analyte molecules, ζ denotes the zeta potential at the channel walls, ε denotes the electrical permittivity of the fluid and η denotes its viscosity. In the above equations, these two quantities have been rendered dimensionless as b ut* ) b ut/Ut and uz* ) uz/U, respectively, where U is the average velocity of the analyte bands as defined in eq 1. Also, since an appropriate normalization factor for ψ is ζ, we have chosen Ut ) µζ/d yielding β ) µζ/D. While all length scales have been normalized in this problem by the narrower dimension of the rectangular channel d, there exist other choices for this parameter too. One such alternative is the hydraulic diameter of the rectangular conduit. If one chooses to use this alternative however, the dispersivity function f would compare channels of fixed hydraulic diameters (for a given value of U and D) which is rather a difficult parameter to control when fabricating nanometer scale channels. In this situation, the use of channel depth for normalizing all length scales may be a more convenient choice for comparing the results presented here to experimental data. (36) Cummings, E. B.; Griffiths, S. K.; Nilson, R. H.; Paul, P. H. Anal. Chem. 2000, 72, 2526–2532.

The Taylor-Aris dispersivity of analyte samples has been evaluated in this article using the formulation presented above. In our analysis, the electric potential induced by the channel surface charges ψ* has been determined based on the GouyChapman model for the EDL after applying the Debye-Hu¨ckel approximation valid for small zeta potential systems.37 For the sake of simplicity, the case of a monovalent symmetric electrolytic medium with single anionic and cationic species was chosen for these computations. Under these conditions, the Poisson equation governing ψ* takes the form ∇*2ψ* ) λ2ψ* with boundary condition ψ* |∂S ) 1. Here λ ) d/δ denotes the ratio of the channel depth (d) to the EDL thickness (δ) in the system. For a rectangular channel shown in Figure 1, this corresponds to the following differential equation and boundary conditions: ∂2ψ/ ∂2ψ/ + /2 ) λ2ψ/ B.C. ψ/|x/)(W⁄ 2d ) ψ/|y/)(1⁄ 2 ) 1 ∂x/2 ∂y

(4)

where x* ) x/d and y* ) y/d. It has been previously shown30 that eq 4 yields an analytical solution given by ∞ ψ * ) [cosh(λy * )] ⁄ [cosh(λ ⁄ 2)] + Σn)0 An cos{(2n + 1)πy*} ×

cosh{[(2n + 1)2π2 + λ2]1⁄2x * } , where

An )

4(-1)nλ2 (2n + 1)π{(2n + 1)2π2 + λ2}cosh √(2n + 1)2π2 + λ2

(

W 2d (5)

)

In this work, we have quantified slug dispersion in a rectangular channel in terms of the function f ) ∫S uz/φ0hdA/ defined in eq 2, which compares the dispersivity for a given value of Pe in the system. This was accomplished in our work by solving eqs 1 through 3 numerically using codes written in MATLAB based on the analytical solution for ψ* given by eq 5. We would like to point out that for a spherical analyte molecule, the parameter β used here to quantify solutal TEM is equal to the product of the dimensionless zeta potential [ζ* ) eζ/(kBT)] and the valence of the analyte species (z).11,19–22 This is because the electrophoretic mobility of a sphere of radius a may be expressed as ze/(6πηa) while its molecular diffusivity may be estimated by the Stokes-Einstein value D ) kBT/(6πηa) yielding β ) [µkBT/(eD)]ζ* ) zζ*. These relationships for the electrophoretic mobility and the molecular diffusivity, however, may not be valid for solute molecules that are highly charged or nonspherical.38,39 In this situation, we find it more appropriate to use the parameter β over zζ* for quantifying the effects of solutal TEM in our analysis. Further, note that the effect of molecular diffusivity on solute dispersion in this situation, i.e., when β * zζ*, can be significantly different from that in a simple nonuniform (37) Probstein, R. F. Physicochemical Hydrodynamics, 2nd ed.; John Wiley & Sons: New York, 1994. (38) Theo van de Ven, G. M. Colloidal Hydrodynamics; Academic Press: San Diego, 1986. (39) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989.

Figure 2. Effect of channel side walls on the electroosmotic flow velocity of charged sample zones (U) in a rectangular conduit of depth d and width W. (a) Variation in the ratio U/U|| with the extent of the Debye layer overlap in the system. (b) Variation in the ratio U/U|| with channel aspect ratio. Here λ denotes the ratio of the channel depth (d) over the Debye length (δ) and U|| denotes the electroosmotic velocity of the sample zone in a rectangular duct with no side walls, i.e., a parallel-plate geometry.

axial fluid flow or chromatographic device. While the effect of the quantity D on the solutal spreading rate can be scaled out in those cases,40–42 this cannot be accomplished for analyzing the transport of charged species through micro- and nanochannels when β * zζ*. This is because the parameter β, which depends on the molecular diffusivity, strongly influences the dispersivity function f in the system. Although the mathematical formulation described above can be used to determine the increased dispersion caused by the side walls of a rectangular channel, it involves solving partial differential equations that are challenging. To reduce this computational effort, we also present a semianalytical approach that may allow us to estimate the function f with reasonable accuracy. In this approach, the effect of the channel side walls on solute dispersion may be estimated by treating the rectangular conduit as a parallel-plate geometry with a one-dimensional velocity profile obtained by averaging uz in the y-direction. The overall dispersion in the rectangular duct may then be evaluated by simply adding this (40) Aris, R. Proc. R. Soc. A 1956, 235, 67–77. (41) Aris, R. Proc. R. Soc. A 1959, 252, 538–550. (42) Taylor, G. I. Proc. R. Soc. A 1953, 219, 186–203.

Analytical Chemistry, Vol. 80, No. 12, June 15, 2008

4725

directions of a large aspect ratio channel are additive, yielding the following expression for the overall dispersivity in the limit d/W f 0: K D

( )

d⁄ Wf0

( )

( )( )

U||d 2 1 U||d f|| + D 3 D

)1+

2

γ d

2

(6)

It has been also shown that for a rectangular channel of any aspect ratio, the Taylor-Aris dispersivity may be expressed as K/D ) 1 + (Ud/D)2f where the dispersivity function f is determined only by the aspect ratio and the flow profile in the channel.26,28,31 In this situation, the quantity f for this cross-section can be evaluated as f ) f|| + (1/3)(γ/d)2 in the limit d/W f 0 based on eq 6. Here f|| denotes the dispersivity function in a parallel-plate device (rectangular channel with no side walls) and may be evaluated as22 f| )

∫

1⁄2

-1⁄2

{

φ0| (1 ⁄ φ0| )

∫

y′

-1⁄2

]

2

φ0| [1 - (uz| ⁄ U|)]dy^ dy′.

(7)

The terms uz| ) - (εζEz ⁄ η){1 - [cosh(λy^)] ⁄ [cosh(λ ⁄ 2)]}

(8)

and φ0| ) exp[-β cosh(λy^) ⁄ cosh(λ ⁄ 2)] ⁄

{∫

1⁄2

-1⁄2

exp [-βcosh(λy^) ⁄

}

cosh(λ ⁄ 2)]dy^ Figure 3. Effect of solutal transverse electromigration on the electrokinetic transport rate of a sample zone through a rectangular duct of depth d and width W (a) in the absence of axial electrophoresis, i.e., ν ) 0, and (b) in the presence of axial electrophoresis and specifically when ν ) 0.2.

contribution due to the side regions to the Taylor-Aris dispersivity in a parallel-plate device. It is important to note that this treatment is strictly valid for infinitely wide rectangular conduits and can exactly predict the Taylor-Aris dispersivity of sample slugs only in the asymptotic limit of d/W f 0. The validity of the above-described approach was first demonstrated by Doshi et al. for simple pressure-driven flow through a rectangular channel of large aspect ratio.27 In a recent work, Desmet and Barron showed that the additional band broadening due to the side regions in a rectangular conduit could also be quantified by assuming the presence of a pseudostationary layer on the channel side walls.43 In their work, the thickness of this layer γ was evaluated from the slowing down of the fluid by the channel side walls as γ⁄d ) lim (1/2)(W/d)(1 - U/U||) where d⁄Wf0 U|| denotes the average fluid velocity in a parallel-plate geometry. Based on the analysis presented by Desmet and Barron,43 the thickness of the pseudostationary layer may then be used to evaluate the additional Taylor-Aris dispersivity in a rectangular duct due to the channel side walls as (D/3)(U||d/D)2(γ/d)2. It is important to note that the average velocity in a rectangular conduit of large aspect ratio is equal to that in a parallel-plate device of identical depth, i.e., U|d/Wf0 ) U||. Moreover, the contributions to dispersion from fluid shear across the vertical and horizontal (43) Desmet, G.; Barron, G. V. J. Chromatogr. A 2002, 946, 51–58.

4726

Analytical Chemistry, Vol. 80, No. 12, June 15, 2008

(9)

here refer to the electroosmotic velocity field and the local zeroth order moment of the solute concentration in a parallel plate device under small zeta potential conditions. As mentioned earlier, the above-described approach for estimating the effect of channel side walls on solute dispersion by ignoring the velocity gradients across the narrower dimension of a rectangular duct is strictly valid in the limit of d/Wf0. In some recent works by Dutta et al.,30,31 however, it was shown that this approach can also predict band broadening in rectangular channels of all aspect ratios within an accuracy of 5% under both electroosmotically- and pressure-driven flow conditions. In order to simplify our calculations, we have taken the same approach in this article to derive a semianalytical expression that can approximate the spreading rate for charged samples in a rectangular channel under electroosmotically driven flow conditions. As in the other work,30,31 such an expression may be obtained by treating the rectangular channel ˆ ) and as a parallel-plate device with an effective fluid velocity (U solute concentration profile (φˆ0) given by the corresponding depth-averaged functions, i.e., ˆ ) - εζ Ez U η

1⁄ 2

∫

-1⁄ 2

( )

φ0 1 -

1⁄ 2

ψ/ / ˆ0 ) φ0dy/ dy and φ ζ/ -1⁄ 2

∫

(10) These 1-dimensional variations may then be used to estimate the Taylor-Aris dispersivity in the system based on the “method of moments” 40 formulation as

ˆ ( ) ( ) ∫ φˆ hˆ( UU ) d x

K Ud 2 Ud )1+ f + D D || D

2

) ( )

|

W⁄2d

/

(11)

0

-W⁄2d

where

(

ˆ d dhˆ U dhˆ ˆ0 * )φ ˆ0 1φ B.C. * * U dx dx dx

W⁄2d x/)(W⁄2d ) 0

and

∫

ˆ 0hˆ ) 0 φ

-W⁄2d

(12) The dispersivity function f in this situation may be calculated from eqs 5 and 6 as W⁄ 2d

f ) f|| +

∫

-W⁄ 2d

()

ˆ U ˆ 0hˆ φ dx/ U

(13)

through direct numerical integration using common mathematical softwares. MATLAB was applied in this work to evaluate the quantity f, requiring less than 1s of computational time for each case running on a typical desktop PC. RESULTS AND DISCUSSION Solute Velocity. For purely electrokinetically driven systems, the overall axial transport of charged samples arises from two different phenomena. The first among these is the electroosmotic advection of the analyte molecules by the fluid in the system while the second contribution originates from solutal electrophoresis along the channel axis. In this situation, the average velocity of the sample zone may be expressed as U ) ∫S (1 - ψ*) φ0 dA* (β/ζ*2)ν where the first and the second terms quantify the abovedescribed electroosmotic and electrophoretic contributions, respectively. Here, the symbol ν ) e2ηD/(εkB2T2) and therefore the term (β/ζ*2)ν ) [(µζ/D)/(eζ/kBT)2][e2ηD/(εkB2T2)] ) µ/(εζ/ η) is a measure of the analyte’s electrophoretic transport rate in the axial direction relative to the Helmholtz-Smoluchowski velocity in the channel. We begin our discussion by looking at the effect of the channel side walls on the electroosmotic component of the solute velocity. This has been quantified in our work in terms of the ratio of an analyte’s velocity in a rectangular channel (U) to that in a parallelplate geometry (U||) for ν ) 0. In Figures 2 and 3, we have depicted this ratio as a function of the operating parameters, β, λ and d/W. Figure 2(a) shows that in the limit of thin Debye lengths e.g., λ . 10, the ratio (U/U||)ν)0 approaches a value of unity as the average fluid speeds in both the rectangular and the parallel-pate geometry converge to the Helmholtz-Smoluchowski velocity, -εζEz/η. As the value of λ is reduced in the system, the ratio (U/U||)ν)0 decreases monotonically eventually reaching an asymp∞ totic value of 1 - 192(d/W){∑n)0 {tanh[(2n + 1)πW/(2d)]/[(2n 5 5 28 + 1) π ]}} in the limit of λf0. Note that this asymptotic limit is identical to the ratio of U/U|| for neutral samples in simple pressure-driven flow systems. This is because the electroosmotic flow profile approaches that in a pressure-driven flow device with an increase in the extent of EDL overlap in the channel.44 It is also interesting to note that the ratio U/U|| approaches values independent of the parameter β for both small and large Debye (44) Griffiths, S. K.; Nilson, R. H. Anal. Chem. 1999, 71, 5522–5529.

Figure 4. Effect of channel side walls on the Taylor-Aris dispersivity function (f) for charged sample zones in the absence of axial electrophoresis (ν ) 0). (a) Variation in the ratio f/f|| with channel aspect ratio. (b) Variation in the ratio f/f|| with the extent of the Debye layer overlap in the system. Here λ denotes the ratio of the channel depth (d) over the Debye length (δ) and f|| denotes the Taylor-Aris dispersivity function of the sample zone in a rectangular duct with no side walls, i.e., a parallel-plate geometry. The lines here correspond to predictions made by the semianalytical theory (eqs 10–13) and the open symbols refer to the results from numerical simulations in a 3-dimensional rectangular channel.

layers. In the limit of small EDLs, this occurs as the lateral electric field is experienced only across a distance proportional to the Debye length from the channel walls. Because the fraction of solute molecules within this region reduces with an increase in the value of λ, the effect of TEM on the average solute velocity also diminishes in the limit of λ . 1. For systems with large Debye layers on the other hand, all solute molecules in the rectangular channel reside within the EDL. Under these conditions, however, a decrease in the value of λ reduces the magnitude of the lateral electric field in the channel and therefore also diminishes the TEM of analyte molecules. Note that this reduction in lateral electric field under EDL overlap conditions occurs as the electric potential induced by the surface charges across the entire channel approaches the zeta potential in the system for λ , 1. In this situation, the mean velocity of charged samples in the conduit approaches that of the fluid (or neutral solutes) yielding an asymptotic value for U/U|| independent of the parameter β. In Figure 2(b) we have presented the variation in (U/U||)ν)0 with the aspect ratio of the rectangular channel, which shows that the side walls in this geometry tend to reduce the solute velocity Analytical Chemistry, Vol. 80, No. 12, June 15, 2008

4727

3(b) shows that the axial electrophoresis of solute molecules can contribute significantly to its overall transport rate in the system. Moreover, the ratio U/U|| in this situation varies with both β and ζ* in the system unlike in the case of ν ) 0 when the solute velocity varied independently of ζ*. In Figure 3(b), we have shown the dependence of the ratio U/U|| on the parameter β for ζ* ) 1 and ν ) 0.2. Note that while U/U|| increases monotonically with β in the absence of axial electrophoresis for a given ζ* and β > 0 (see Figure 3(a)), this dependence changes for nonzero values of ν. As may be seen from Figure 3(b), the trend actually reverses yielding a monotonic decrease in U/U|| with increase in β under conditions when axial electrophoresis is dominant. One can analyze this situation by rewriting the ratio U/U|| as

Figure 5. Effect of solutal transverse electromigration on the Taylor-Aris dispersivity of sample zones in the absence of axial electrophoresis (ν ) 0). The solid line here corresponds to the predictions made by the semianalytical theory (eqs 10 and 13), and the open symbols refer to the results from numerical simulations in a 3-dimensional rectangular channel. Here the line types - · -, - - and s correspond to the cases d/W ) 0.1, d/W ) 0.33 and d/W ) 1, respectively, while the symbols O, 3 and fcorrespond to cases λ ) 1, λ ) 3 and λ ) 10, respectively.

in the system. This reduction is more prominent in small aspect ratio designs (larger values of d/W) and under strong Debye layer overlap conditions, i.e., small values of λ. Figure 2(b) also shows that while (U/U||)ν)0 varies monotonically with the parameters d/W and λ, its variation with the quantity β can be nonmonotonous. In fact, the ratio (U/U||)ν)0 exhibits a minimum in all aspect ratio rectangular conduits as β is varied for values of λ of the order of 1 or less. To highlight this variation, we have plotted the ratio (U/U||)ν)0 against the parameter β in Figure 3(a). The figure shows that this ratio increases monotonically with increase in the magnitude of β for large positive and negative values of this parameter. In the case of species with β > 0, analyte molecules are more strongly repelled by the channel walls for larger values of β. This also implies that the side regions of the rectangular duct are depleted of the solute particles under these conditions yielding a sample zone velocity that approaches the corresponding analyte speed observed in a parallel-plate geometry. For species with β < 0, an increase in the magnitude of β leads to stronger focusing of the solute molecules toward the channel walls. As the sample gets concentrated very close to the channel surface, the effect of the actual cross-section of the duct becomes minimal. In this situation, the ratio (U/U||)ν)0 goes through a minimum as β is varied from large negative to large positive values. On closer inspection, we observe that this minimum occurs right around the conditions when the center of the channel starts getting depleted of the solute molecules due to their TEM toward the channel walls. Figure 3(a) also shows that the minimum in the ratio (U/U||)ν)0 becomes less prominent for channel geometries with large aspect ratios and thin Debye layers, i.e., λ . 1. While Figures 2(a), 2(b) and 3(a) ignore the effect of axial electrophoresis on solutal transport rates, this phenomenon has been accounted for in Figure 3(b) using a nonzero value of the dimensionless parameter ν. Note that this parameter typically assumes values of about 0.2 or higher for analyte molecules with diffusivities D g 10-6 cm2/s in water-based electrolytes (ε ) 7.08 × 10-10 C/Vm, η ) 10-3 Pa · s). For such typical values of ν, Figure 4728

Analytical Chemistry, Vol. 80, No. 12, June 15, 2008

U ) U||

( ) U U||

βν (U||)ν)0ζ *2 ν)0 βν 1(U||)ν)0ζ *2 -

(14)

Note that in the above equation, the numerator and denominator on the right-hand side assume negative values for βν/[(U||)ν)0ζ*2] > 1 as (U/U||)ν)0 is always less than unity. Moreover, because (U/U||)ν)0 increases with β for a given ζ* and λ under these conditions, the ratio U/U|| assumes values larger than 1 and decreases for larger values of β. In this situation, U/U|| approaches a value of unity as β f ∞. Equation 14 also implies that the ratio U/U|| goes through a singularity at βν/[(U||)ν)0ζ*2] ) 1 when an analyte’s electroosmotic transport rate is exactly canceled by its axial electrophoresis in the opposite direction in a parallel-plate device, i.e., U|| ) 0. Note that this singularity occurs as the electroosmotic and electrophoretic contributions to an analyte’s flow velocity vary differently with β. For example, while the electroosmotic transport rate of a species in a channel with ζ* > 0 decreases with the value of β, its electrophoretic motion increases in magnitude. In this situation, the ratio U/U|| assumes negative values when (U/U||)ν)0 < βν/[(U||)ν)0ζ*2] < 1. When β is further decreased such that βν/[(U||)ν)0ζ*2] < (U/U||)ν)0, U/U|| again assumes values greater than zero eventually approaching unity when β f -∞. Note that over this entire range, however, the ratio U/U|| increases monotonically with a reduction in β. Taylor-Aris Dispersivity. The effect of channel geometry on solute dispersion has been often quantified in the literature using the dispersivity function f as defined in the previous section. In this situation, the ratio f/f|| may be used to describe the effects of the channel side walls on band broadening in rectangular conduits where f|| denotes the dispersivity function in a parallel-plate geometry. Note that this ratio assumes values greater than unity under all conditions (see Figures 4 and 5) implying a greater dispersion in a rectangular channel compared to that in a parallel plate device for a fixed Peclet number, Pe ) Ud/D. The larger dispersion in rectangular conduits occurs due to the slowing down of the fluid and hence the solute molecules around its vertical side walls. This results in an increase in the average fluid shear and also produces velocity variations in the transverse direction, which usually, on account of its greater length scale, is subject to greater diffusional limitation. It is important to note that these two effects dominate the dispersion process in a rectangular channel in two different

scenarios. For small aspect ratio profiles (d/W ≈ 1), the characteristic length scales for diffusion along the vertical and horizontal directions of the rectangle are similar. In this case, the larger dispersion in a rectangular conduit over that between two parallel plates occurs due to a greater fluid shear introduced by the channel side walls. With a decrease in the value of d/W, however, this contribution to band broadening diminishes. This occurs as the channel side walls in a rectangular conduit slow down the fluid only over a length scale of the order of δ (EDL thickness) around them. In this situation for designs with d/W , 1, the large diffusional resistance across the channel width starts dominating the dispersion of the sample zones in the rectangular conduit. While the fraction of solute molecules that experience the effect of the side regions is only an order d/W in these geometries, the fact that these slow moving regions are widely separated by a distance W has a large effect on the overall band broadening. As a result, the additional zone dispersivity due to the channel side walls still scales with d2 in the limit d/W f 0. Moreover, the dispersion coefficient multiplying Pe2 for this contribution can be several times larger than that resulting from diffusion limitations in the narrower direction.27–30 In Figure 4(a), we have plotted the variation in f/f|| as a function of the aspect ratio in a rectangular channel. As may be seen from the figure, this quantity decreases monotonically with an increase in the ratio d/W for any given λ and β in the system. Moreover, the differential in the value of f/f|| in going from a geometry with d/W f 0 to a square cross-section, i.e., d/W ) 1, increases with increase in the extent of EDL overlap in the system. For example, in the case of neutral species while f/f|| decreases from 7.95 to 1.76 for λ,1, this ratio only varies between 5.14 and 1.85 for λ ) 3. In the limit of thin Debye layers, i.e., λ . 10, the effect of the channel aspect ratio on solute dispersion is further diminished. In fact the variation in f/f|| with d/W is completely eliminated in the limit of λ f ∞. In this limit, the dispersion in a rectangular channel is exactly twice that in a parallel-plate device for a given Pe in the system30,45–47 (see Figure 4(b)). For neutral samples, it has been previously shown by Zholkhoskij et al. that the side walls of a rectangular channel increase dispersion by a factor a two in the limit of thin Debye layers.45 Figure 4(b) demonstrates that this result is also valid for all charged analytes in any aspect ratio rectangular geometry. Figure 4(b) further shows that the variation in f/f|| with λ is maximum in the limit of d/Wf0 and diminishes significantly for a square cross-section. It is also important to note that the effect of the parameter β on f/f|| is eliminated both for systems with thin and thick Debye layers. In the limit when λ , 1, solute dispersion in electrokinetically driven devices actually becomes similar to that in a pressuredriven system eventually approaching the result for pressuredriven flows as λ f 0. In this limit, the effect of β on f/f|| is also eliminated as the lateral electrical field in the channel vanishes (see Figure 4(b)). On the other hand, for channels with thin Debye layers the fraction of solute molecules that experience the lateral electric field in the conduit becomes (45) Zholkovskij, E. K.; Masliyah, J. H.; Czarnecki, J. Anal. Chem. 2003, 75, 901–909. (46) Zholkovskij, E. K.; Masliyah, J. H. Anal. Chem. 2004, 76, 2708–2718. (47) Zholkovskij, E. K.; Masliyah, J. H. Chem. Eng. Sci. 2006, 61, 4155–4164.

negligible.44 Figure 5 shows that the ratio f/f|| tends to increase with β for most values of λ and d/W in the system. However, this trend has been observed to reverse in large aspect ratio designs and thin Debye layer conditions (see Figure 5). It is important to point out that Figures 4 and 5 in this article has been presented ignoring any axial electrophoresis in the system, i.e., for a value of ν ) 0. This is because the presence of axial electrophoresis does not alter the magnitude of the Taylor-Aris dispersivity (K) due to its spatial uniformity across the entire cross-section of the rectangular channel. However, this phenomenon does modify the axial transport rate of the analyte molecules and therefore the Peclet number (Pe) in the system. In this situation, one can evaluate the function f for nonzero values of ν using the relationship (Pe2f )ν*0 ) (Pe2f )ν)0 and the results presented in Figures 4 and 5. In Figures 4 and 5 the lines correspond to the predictions by the semianalytical model, i.e., eqs 10–13 and the open symbols correspond to results from the numerical simulations described earlier, i.e., eqs 1–3. As may be seen from the figures, the semianalytical model is capable of predicting the Taylor-Aris dispersivity in rectangular channels of all aspect ratios to within an accuracy of 5%. While the model yields the exact the value of f/f|| in the limit of d/W f 0, it is most inaccurate for a square cross-section (see Figure 4(a)). Also, the error in the predictions from this model is largest under strong Debye layer overlap conditions as may be seen from Figure 4(b) (see the case d/W ) 1). Overall, however, the semianalytical model provides a useful estimate for solute dispersion in rectangular channels with a significantly less computational effort. Finally, it is important to point out that the sign of ζ* does not alter the ratio U/U|| or the dispersivity of the analyte bands in the above-described transport process. This is because the ratio β/ζ*2 determining the relative magnitude of axial electrophoresis always assumes the same sign as β while the dimensionless electric potential ψ* governing electroosmotic flow is unaffected by the sign of the dimensionless zeta potential. CONCLUSIONS To summarize, we have presented a detailed analysis on the electrokinetic transport of charged samples through rectangular channels having a small zeta potential at their walls. Numerical techniques have been employed to solve the governing transport equations and characterize the effect of the channel side walls on this flow process. In addition, a semianalytical model has been presented for estimating the Taylor-Aris dispersivity of the sample zones in the conduit. Because the semianalytical model decouples the effect of diffusion limitations across the wider and narrower dimensions of the rectangular channel, it reduces the equations governing the moments of solute concentration to a 1-dimensional form. In this situation, the Taylor-Aris dispersivity of the sample slugs may be evaluated through integration of an ordinary differential equation significantly reducing the computational effort required for the analysis. The predictions from the semianalytical model have been also compared with simulation results in which the coupling between the diffusion limitations across the narrower and the wider channel dimensions has been accounted for. While the semianalytical model is strictly valid in the limit d/Wf0, it has been shown to predict the Analytical Chemistry, Vol. 80, No. 12, June 15, 2008

4729

Taylor-Aris dispersivity of sample slugs to within 5% accuracy for all channel aspect ratios. The analysis also shows that the side walls in a rectangular channel reduce the transport rate of sample slugs under electroosmotically driven flow conditions. Moreover, the magnitude of this reduction increases monotonically with a decrease in the aspect ratio W/d and the parameter λ in the system. This reduction in the solute velocity due to the channel side walls, however, shows a nonmonotonous dependence on the parameter β which characterizes the solutal TEM in the system. The side walls of the rectangular duct also tend to increase the dispersion of analyte bands, and the extent of this increase decreases monotonically with the ratio d/W. Further, the effect of the channel side regions on band broadening is most prominent under strong Debye layer overlap conditions (λ , 1) in which case the dispersion can be increased by as much as an order of magnitude in large aspect ratio designs. Interestingly, the effect of channel aspect ratio on solutal spreading is diminished in the limit of thin Debye layers. The dispersivity in a rectangular channel actually

4730

Analytical Chemistry, Vol. 80, No. 12, June 15, 2008

asymptotes to a value twice that in a parallel-plate device independent of the parameter β and the ratio d/W as λ f ∞. While this result has been previously derived for neutral molecules by Zholkovskij et al.,45 the present work demonstrates that it is also valid for charged species under electroosmotically driven conditions. ACKNOWLEDGMENT This research work was supported by a start-up grant from the University of Wyoming. SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

Received for review December 7, 2007. Accepted March 10, 2008. AC7024927