Pore-Scale Dispersion in Electrokinetic Flow through a Random

Anal. Chem. 2007, 79, 113-121

Pore-Scale Dispersion in Electrokinetic Flow through a Random Sphere Packing Dzmitry Hlushkou,† Siarhei Khirevich,†,‡ Vladimir Apanasovich,‡ Andreas Seidel-Morgenstern,†,§ and Ulrich Tallarek*,†

Institut fu¨r Verfahrenstechnik, Otto-von-Guericke-Universita¨t Magdeburg, Universita¨tsplatz 2, 39106 Magdeburg, Germany, Department of Systems Analysis, Belarusian State University, Francysk Scoryna Avenue 4, Minsk, 220050, Belarus, and Max-Planck-Institut fu¨r Dynamik komplexer technischer Systeme, Sandtorstrasse 1, 39106 Magdeburg, Germany

The three-dimensional velocity field and corresponding hydrodynamic dispersion in electrokinetic flow through a random bulk packing of impermeable, nonconducting spheres are studied by quantitative numerical analysis. First, a fixed bed with interparticle porosity of 0.38 is generated using a parallel collective-rearrangement algorithm. Then, the interparticle velocity field is calculated using the lattice-Boltzmann (LB) method, and a randomwalk particle-tracking method is finally employed to model advection-diffusion of an inert tracer in the LB velocity field. We demonstrate that the pore-scale velocity profile for electroosmotic flow (EOF) is nonuniform even under most ideal conditions, including a negligible thickness of the electrical double layer compared to the mean pore size, a uniform distribution of the electrokinetic potential at the solid-liquid interface, and the absence of applied pressure gradients. This EOF dynamics is caused by a nonuniform distribution of the local electrical field strength in the sphere packing and engenders significant hydrodynamic dispersion compared to pluglike EOF through a single straight channel. Both transient and asymptotic dispersion behaviors are analyzed for EOF in the context of packing microstructure and are compared to pressuredriven flow in dependence of the average velocity through the bed. A better hydrodynamic performance of EOF originates in a still much smaller amplitude of velocity fluctuations on a mesoscopic scale (covering several particle diameters), as well as on the microscopic scale of an individual pore. Electrokinetic flow and transport play a central role in several analytical, technological, and environmental processes, including water removal from industrial slurries and natural porous media, soil remediation, and electrochromatography in monoliths or particulate beds, as well as electrophoretic separations and sample focusing in microfluidic devices.1-15 Electroosmotic flow (EOF) along a solid-liquid interface is generated by interaction of the * To whom correspondence should be addressed. Fax: +49-(0)391-67-12028. E-mail: [email protected]. † Otto-von-Guericke-Universita¨t. ‡ Belarusian State University. § Max-Planck-Institut. (1) Deyl, Z.; Svec, F. Eds. Capillary Electrochromatography; Elsevier: Amsterdam, 2001. 10.1021/ac061168r CCC: $37.00 Published on Web 11/22/2006

© 2007 American Chemical Society

local tangential component of an applied electrical field with the net volume charge in the fluid-side electrical double layer (EDL).2 Transient behavior, as well as the magnitude, stability, and uniformity EOF in porous media, is inherently related to the physicochemical nature of the surface and associated quasiequilibrium interface dynamics, the pore space morphology, and properties of the liquid electrolyte. A careful analysis of these aspects is important as it critically guides the performance and design strategy of electrokinetic phenomena compared to alternative diffusive-convective transport schemes. In capillary electrochromatography, for example, EOF is utilized for higher separation efficiencies than in liquid chromatography (LC) with pressuredriven flow (PDF).16,17 Relatively high electroosmotic permeabilities allow us to pack small porous particles as stationary phase, which further improves column efficiency and alleviates the need for prohibitively high operating pressures in LC. The permeability criterion and straightforward implementation also make the EOF very attractive for electrokinetic pumping in lab-on-a-chip devices and the actuation of microscale mechanical components.18-20 (2) Delgado, A. V., Ed. Interfacial Electrokinetics and Electrophoresis; Marcel Dekker: New York, 2002. (3) Ista, L. K.; Lopez, G. P.; Ivory, C. F.; Ortiz, M. J.; Schifani, T. A.; Schwappach, C. D.; Sibbett, S. S. Lab Chip 2003, 3, 266-272. (4) Stachowiak, T. B.; Svec, F.; Frećhet, J. M. J. J. Chromatogr., A 2004, 1044, 97-111. (5) Li, D. Electrokinetics in Microfluidics; Elsevier Academic Press: Burlington, MA, 2004. (6) Wong, P. K.; Wang, T. H.; Deval, J. H.; Ho, C. M. IEEE-ASME Trans. Mechatron. 2004, 9, 366-376. (7) Stone, H. A.; Stroock, A. D.; Ajdari, A. Annu. Rev. Fluid Mech. 2004, 36, 381-411. (8) Kirby, B. J.; Hasselbrink, E. F., Jr. Electrophoresis 2004, 25, 187-202. (9) Saichek, R. E.; Reddy, K. R. Crit. Rev. Environ. Sci. Technol. 2005, 35, 115-192. (10) Leinweber, F. C.; Tallarek, U. J. Phys. Chem. B 2005, 109, 21481-21485. (11) Pumera, M. Talanta 2005, 66, 1048-1062. (12) Cui, H. C.; Horiuchi, K.; Dutta, P.; Ivory, C. F. Anal. Chem. 2005, 77, 78787886. (13) Nischang, I.; Chen, G.; Tallarek, U. J. Chromatogr., A 2006, 1109, 32-50. (14) Ghosal, S. Annu. Rev. Fluid Mech. 2006, 38, 309-338. (15) Jung, B.; Bharadwaj, R.; Santiago, J. G. Anal. Chem. 2006, 78, 2319-2327. (16) Dittmann, M. M.; Wienand, K.; Bek, F.; Rozing, G. P. LC-GC 1995, 13, 800-814. (17) Crego, A. L.; Gonza´lez, A.; Marina, M. L. Crit. Rev. Anal. Chem. 1996, 26, 261-304. (18) Paul, P. H.; Arnold, D. W.; Rakestraw, D. J. In Micro Total Analysis Systems ’98; Harrison, D. J., Van den Berg, A., Eds.; Kluwer: Dordrecht, The Netherlands, 1998; pp 49-52. (19) Kirby, B. J.; Shepodd, T. J.; Hasselbrink, E. F., Jr. J. Chromatogr., A 2002, 979, 147-154.

Analytical Chemistry, Vol. 79, No. 1, January 1, 2007 113

For process intensification and the design of new stationaryphase materials, it is important to understand the pore-scale mechanisms of flow and transport. In particular, EOF velocity fields need to be evaluated a priori and optimized by adjustment of the operating conditions, pore space morphology, and surface properties. This is crucial for realizing more efficient transport of chemical and biological species over long distances, that is, at the expense of relatively little hydrodynamic dispersion. Consequently, theories and experimental approaches must resolve the pore-level EOF and transport in three-dimensional porous media, e.g., random close-sphere packings or monoliths, in order to address microscopic and mesoscopic details of the velocity field underlying flow and dispersion on the macroscopic scale. A number of theoretical studies have been reported for EOF in open-channel systems21-31 and packed beds.32-35 However, since the EOF problem has no general solution, it can be solved analytically only for a limited set of simple configurations like a flat solid-liquid interface or a single straight channel. Most theoretical approaches to the understanding of EOF in random porous media like a packed bed are based on capillary models, which represent the material as an assembly of individual capillaries. Since this ignores important details of the real morphology, the capillary models are limited to express an equivalence only in terms of the mean velocity and become unrealistic if, for instance, dispersion in packed beds is analyzed.36 TheEOFproblemalsohasbeenintenselystudiednumerically,37-50 but the majority of studies is concerned with EOF in open (20) Yao, S. H.; Hertzog, D. E.; Zeng, S. L.; Mikkelsen, J. C.; Santiago, J. G. J. Colloid Interface Sci. 2003, 268, 143-153. (21) Griffiths, S. K.; Nilson, R. H. Anal. Chem. 2000, 72, 5473-5482. (22) Cummings, E. B.; Griffiths, S. K.; Nilson, R. H.; Paul, P. H. Anal. Chem. 2000, 72, 2526-2532. (23) Santiago, J. G. Anal. Chem. 2001, 73, 2353-2365. (24) Dutta, P.; Beskok, A. Anal. Chem. 2001, 73, 1979-1986. (25) Li, D. Colloids Surf. A 2001, 195, 35-57. (26) Ghosal, S. Anal. Chem. 2002, 74, 771-775. (27) Gleeson, J. P. J. Colloid Interface Sci. 2002, 249, 217-226. (28) Qiao, R.; Aluru, N. R. J. Chem. Phys. 2003, 118, 4692-4701. (29) Yang, J.; Masliyah, J. H.; Kwok, D. Y. Langmuir 2004, 20, 3863-3871. (30) Xuan, X.; Li, D. J. Micromech. Microeng. 2004, 14, 290-298. (31) Pennathur, S.; Santiago, J. G. Anal. Chem. 2005, 77, 6772-6781. (32) Rathore, A. S.; Horva´th, Cs. J. Chromatogr., A 1997, 781, 185-195. (33) Vallano, P. T.; Remcho, V. T. Anal. Chem. 2000, 72, 4255-4265. (34) Wu, R. C.; Papadopoulos, K. D. Colloids Surf. A 2000, 161, 469-476. (35) Liapis, A. I.; Grimes, B. A. J. Chromatogr., A 2000, 877, 181-215. (36) Hlushkou, D.; Apanasovich, V.; Seidel-Morgenstern, A.; Tallarek, U. Chem. Eng. Commun. 2006, 193, 826-839. (37) Coelho, D.; Shapiro, M.; Thovert, J.-F.; Adler, P. M. J. Colloid Interface Sci. 1996, 181, 169-190. (38) Patankar, N. A.; Hu, H. H. Anal. Chem. 1998, 70, 1870-1881. (39) Ermakov, S. V.; Jacobson, S. C.; Ramsey, J. M. Anal. Chem. 1998, 70, 44944504. (40) Griffiths, S. K.; Nilson, R. H. Anal. Chem. 1999, 71, 5522-5529. (41) Arulanandam, S.; Li, D. Colloids Surf. A 2000, 161, 89-102. (42) Conlisk, A. T.; McFerran, J.; Zheng, Z.; Hansford, D. Anal. Chem. 2002, 74, 2139-2150. (43) Dutta, P.; Beskok, A.; Warburton, T. C. J. Microelectromech. Syst. 2002, 11, 36-44. (44) Erickson, D.; Li, D. J. Phys. Chem. B 2003, 107, 12212-12220. (45) Fu, L.-M.; Lin, J.-Y.; Yang, R.-J. J. Colloid Interface Sci. 2003, 258, 266275. (46) Li, B. M.; Kwok, D. Y. J. Chem. Phys. 2004, 120, 947-953. (47) Hlushkou, D.; Kandhai, D.; Tallarek, U. Int. J. Numer. Methods Fluids 2004, 46, 507-532. (48) Van Theemsche, A.; Gzil, P.; Dan, C.; Deconinck, J.; De Smet, J.; Vervoort, N.; Desmet, G. Anal. Chem. 2004, 76, 4030-4037. (49) Kirchner, J. J.; Hasselbrink, E. F., Jr. Anal. Chem. 2005, 77, 1140-1146. (50) Datta, S.; Ghosal, S.; Patankar, N. A. Electrophoresis 2006, 27, 611-619.

114

Analytical Chemistry, Vol. 79, No. 1, January 1, 2007

microchannel systems. For example, the work of Griffiths and Nilson40 demonstrates that the longitudinal dispersion coefficient of a neutral, nonreacting solute in EOF can be orders of magnitude smaller than for PDF. This is in agreement with experimental results for EOF through a single straight capillary, revealing plugflow behavior under most ideal conditions.51,52 Compared to the dynamics in single-channel geometries or two-dimensional, spatially periodic structures, the direct numerical simulation of electrokinetic flow and transport in three-dimensional random porous media is a challenging task. Apart from the necessity to resolve, in general, the coupled hydrodynamic, electrostatic, and mass transport problems subjected to complex geometrical boundary conditions (represented by the solid-liquid interface in random porous media), these simulations usually must be accomplished at length scales ranging from the EDL thickness to the characteristic dimension of the whole system, e.g., the size of the channel confining the packing. These widely disparate length scales require huge computational resources. One way to reduce this problem is to assume that the EDL thickness is negligible compared to the sphere diameter (thin-EDL limit), which allows decoupling of hydrodynamic and electrostatic problems. Although recent advances in understanding electrokinetic flow and transport in packed beds are due, in large part, to the implementation of experimental techniques like pulsed magnetic field gradient nuclear magnetic resonance53,54 and confocal laser scanning microscopy,55-57 a quite powerful complementation by high-resolution numerical simulation of the pore-level EOF dynamics comes with the development and utilization of the latticeBoltzmann (LB) method.58-62 Over the past decade, it has achieved great success as a highly efficient numerical scheme for the simulation of transport phenomena in porous media.63-70 For example, the complete velocity field can be calculated in a sufficiently large packed bed of particles, which allows to relate (51) Paul, P. H.; Garguilo, M. G.; Rakestraw, D. J. Anal. Chem. 1998, 70, 24592467. (52) Tallarek, U.; Rapp, E.; Scheenen, T.; Bayer, E.; Van As, H. Anal. Chem. 2000, 72, 2292-2301. (53) Tallarek, U.; Scheenen, T. W. J.; Van As, H. J. Phys. Chem. B 2001, 105, 8591-8599. (54) Tallarek, U.; Rapp, E.; Seidel-Morgenstern, A.; Van As, H. J. Phys. Chem. B 2002, 106, 12709-12721. (55) Tallarek, U.; Pacˇes, M.; Rapp, E. Electrophoresis 2003, 24, 4241-4253. (56) Leinweber, F. C.; Tallarek, U. Langmuir 2004, 20, 11637-11648. (57) Leinweber, F. C.; Pfafferodt, M.; Seidel-Morgenstern, A.; Tallarek, U. Anal. Chem. 2005, 77, 5839-5850. (58) Higuera, F. J.; Jimeńez, J. Europhys. Lett. 1989, 9, 663-668. (59) Benzi, R.; Succi, S.; Vergassola, M. Phys. Rep. 1992, 222, 145-197. (60) Qian, Y. H.; d’Humie`res, D.; Lallemand, P. Europhys. Lett. 1992, 17, 479484. (61) He, X.; Luo, L.-S. Phys. Rev. E 1997, 56, 6811-6817. (62) Chen, S.; Doolen, G. D. Annu. Rev. Fluid Mech. 1998, 30, 329-364. (63) Maier, R. S.; Kroll, D. M.; Kutsovsky, Y. E.; Davis, H. T.; Bernard, R. S. Phys. Fluids 1998, 10, 60-74. (64) Manz, B.; Gladden, L. F.; Warren, P. B. AIChE J. 1999, 45, 1845-1854. (65) Maier, R. S.; Kroll, D. M.; Bernard, R. S.; Howington, S. E.; Peters, J. F.; Davis, H. T. Phys. Fluids 2000, 12, 2065-2079. (66) Hill, R. J.; Koch, D. L.; Ladd, A. J. C. J. Fluid Mech. 2001, 448, 243-278. (67) Kandhai, D.; Hlushkou, D.; Hoekstra, A. G.; Sloot, P. M. A.; Van As, H.; Tallarek, U. Phys. Rev. Lett. 2002, 88, art. no. 234501. (68) Schure, M. R.; Maier, R. S.; Kroll, D. M.; Davis, H. T. Anal. Chem. 2002, 74, 6006-6016. (69) Freund, H.; Bauer, J.; Zeiser, T.; Emig, G. Ind. Eng. Chem. Res. 2005, 44, 6423-6434. (70) Sullivan, S. P.; Sani, F. M.; Johns, M. L.; Gladden, L. F. Chem. Eng. Sci. 2005, 60, 3405-3418.

the pore-scale hydrodynamics to macroscopic transport behavior. In contrast to more conventional numerical schemes based on the discretization of the macroscopic continuum equations, the LB method utilizes mesoscopic kinetic equations to recover the macroscopic Navier-Stokes equation in the long-time, large-scale limit. In this work, we utilize the LB method for resolving the mechanism of pore-scale dispersion in electrokinetic flow through a random packing of hard (impermeable, nonconducting) spheres. In contrast to a macroscopic capillary model, our approach allows us to obtain complete three-dimensional information on electrical potential, species concentration, and velocity fields in the simulated system and carry out a detailed analysis of the structure-transport relations, with a focus on pore-scale dispersion, in particular, based on these data. To our knowledge, the topic of pore-scale dispersion in EOF through fixed bedsswhich is one of the key issues, for example, in electrochromatographic separationssuntil now has neither been resolved experimentally, nor theoretically or numerically. Even in the thin-EDL limit, a theoretical description of EOF through fixed beds requires the solution of the partial differential equation subjected to the geometrically complex boundary conditions. Since this task is not amenable to analytical solution, the high-resolution numerical simulation remains as promising, if not the only strategy to understanding the pore-level dynamics of EOF in packed beds and random porous media, in general. NUMERICAL METHODS The simulation of pore-scale dispersion in electrokinetic flow through a random sphere packing performed in this work involves three distinct methods for simulating the bed structure, liquid flow, and advective-diffusive transport (Figure 1). A parallel, collectiverearrangement algorithm is employed to generate the sphere packing, the LB method is utilized to calculate the velocity fields in the interparticle pore space, and a random-walk particle-tracking method is used to model the motion of an inert tracer in the LB velocity fields. Simulation of Velocity Fields. The general mathematical formalism and all details of the simulation of electrokinetic flow through a random sphere packing can be found in a preceding article.71 Here, only the most essential elements are recalled for clarity. Solid, dielectric (impermeable, nonconducting) spheres of uniform diameter dp were randomly packed into a box with quadratic cross section. For the present study, 5920 particles were packed into a box with dimensions 10dp × 10dp × 50dp (Figure 1a). The mean interparticle porosity, or void volume fraction, is inter ) 0.38. The iterative, parallel packing algorithm is based on a technique reported by Jodrey and Tory,72 adapted to parallel computers. Periodic boundary conditions were used in the generation of the bed and flow simulations. Therefore, the present study does not involve effects manifested in porosity oscillations close to the hard wall of a confining container.53,71 The bed in Figure 1a represents the bulk part of a random close-sphere packing; i.e., this bulk packing is free of ordering effects caused by walls and is therefore useful in determining bulk or macroscopic electrokinetic transport properties of random packings. (71) Hlushkou, D.; Seidel-Morgenstern, A.; Tallarek, U. Langmuir 2005, 21, 6097-6112. (72) Jodrey, W. S.; Tory, E. M. Phys. Rev. A 1985, 32, 2347-2351.

Figure 1. Numerical approach for simulation of hydrodynamic dispersion in a random sphere packing. (a) Generation of sphere packing with periodic boundaries based on a modified Jodrey-Tory algorithm. (b) Calculation of the velocity field with the LB method. (c) Simulation of tracer motion in the velocity field using a random-walk particle-tracking method.

The velocity field was obtained using the LB method, which is a simplified kinetic approach with discrete space and time.62 The macroscopic fluid dynamics is approximated by synchronous interactions between fictitious fluid particles on a regular lattice (we employed the D3Q19 lattice60), invoking the idea that fluid flow is determined mainly by the collective behavior of many molecules and not by detailed molecular interactions. It has been shown73 that the LB method corresponds to the truncation of the Boltzmann equation in a Hermite velocity spectrum space or that, in other words, it can be viewed as a special finite-difference approximation of the Boltzmann equation. Among the advantages of the LB method are its inherent parallelism in view of computational efficiency and the capability to deal with a geometrically complex solid-liquid interface like those in random porous media, e.g., the bulk sphere packing in Figure 1a. For the study reported here, we used a lattice with dimensions of 300 × 300 × 1500 points in the x, y, and z directions, respectively (i.e., (73) He, X.; Luo, L.-S. Phys. Rev. E 1997, 56, 6811-6817.


115

spheres had a diameter of 30 lattice points), where z is the axial coordinate (direction of macroscopic flow). This large box size allowed to eliminate recorrelation that originates in the repeated experience of macroscopic flow features by tracer particles traversing the length of the packing more than once. This artifact would result in an overestimation of the dispersion coefficient.65 At the solid-liquid interface Γ the no-slip boundary condition was implemented for PDF, while the apparent-slip boundary condition was used for EOF. The latter involves the thin-EDL approximation, implying that the EDL thickness is negligibly small compared to the sphere diameter or that, in other words, dp/λD f ∞ (λD is the Debye screening length). In turn, the fluid velocity at the solid-liquid interface for EOF is determined by eq 1, where the local strength of the applied electrical field (E) was obtained by numerical solution of the Laplace equation subjected to the insulating boundary condition at this interface71

v|Γ ) -

0rζE ) vslip µ

(1)

ζ is the local electrokinetic potential (or zeta potential), µ the dynamic viscosity of the fluid, 0 the permittivity of vacuum, and r the relative permittivity of the fluid. With the thin-EDL approximation, the shear plane (characterized by ζ) and the slipping plane (characterized by vslip) coincide, and the no-slip condition at the solid-liquid interface for the EOF is replaced by the velocity boundary condition formulated at the slipping plane according to eq 1. In this picture, the EOF slips past the spheres’ surface with vslip. Simulation of Mass Transport and Dispersion. Hydrodynamic dispersion was simulated by a Lagrangian approach, a particle-tracking method. Its basic idea is to move a large number of passive tracer particles by a superposition of convective motion (along flow streamlines) and diffusive motion.65 To define the hydrodynamic dispersion coefficient, N ) 107 tracer particles were randomly distributed in the interparticle pore space of the sphere packing. During an elementary time step (∆t), the displacement of each tracer molecule (∆r) is determined as the sum of convective and diffusive contributions.

∆r ) v(r)∆t + ϑ

(2)

Here, v(r) is the fluid velocity at point r and ϑ is a vector with random orientation in space and a length governed by a Gaussian distribution with standard deviation (6Dm∆t)1/2, where Dm is the free molecular diffusion coefficient.68 If the position of a tracer was inside the solid phase, its displacement was rejected and the random-walk procedure repeated without incremental increase of the time counter until the new position was in the fluid phase. The time step was defined such that the average displacement did not exceed ∆x/2, where ∆x is the discrete space step used to calculate the velocity field by the LB method. The Cartesian components of the time-dependent hydrodynamic dispersion coefficient (DR, R ) x, y, z) were determined from the tracers’ displacements as

DR(t) )

1 d

N

∑(∆r 2N dt

Ri

- < ∆rR > )2

i)1

116


(3)

where ∆rRi and denote the corresponding Cartesian components of the displacement of the ith tracer and average displacement for the tracer ensemble, respectively. With the numerical approach used in this work, it is possible to obtain complete information on the three-dimensional velocity field and address quantitatively structure-transport relations for EOF through a random-sphere packing. It includes the analysis of time and length scales underlying transient dispersion, or the dependence of asymptotic dispersion on the average velocity through the packing. RESULTS AND DISCUSSION Figure 2 shows a typical velocity distribution (color image) for electrokinetic flow through the sphere packing. Most notably and in contrast to widespread belief,74-83 the pore-scale flow profiles extracted from an arbitrary region of the packing (here, the one indicated by the white square) demonstrate a systematic nonuniformity. The velocity profiles are curved over the whole cross section of an interstitial pore. To understand this unique behavior, it needs to be realized that, even in the thin-EDL limit, a pluglike EOF velocity profile can occur only in a single straight channel. Because of the electrical flux conservation (Gauss’s law), any variation in the channel cross-sectional area or shape results in a nonuniform distribution of the local axial component of the applied electrical field, particularly at the solid-liquid interface, which leads to local variations in the slip velocity used to formulate the velocity boundary condition. It was shown theoretically84 and experimentally22 that variations in channel cross-sectional area lead to distortions of the uniform (pluglike) velocity profile of the EOF. A nonuniform distribution of EOF velocities in fixed beds of solid, dielectric spheres originates in a nonuniformity of the local electrical field strength and should be interpreted in the context of an expected similitude for the electrical field and EOF velocity field in the thin-EDL limit.22,23,85 This similarity is evidenced by Figure 2 demonstrating that the local electrical and velocity fields are nearly identical. To facilitate a visual comparison of these fields in Figure 2, we have scaled the local electrical field strength by the factor 0rζ/µ, which relates the apparent slip velocity at the solid-liquid interface to the local electrical field strength (eq 1). To summarize, even the imposition of the slip-velocity boundary condition at the solid-liquid interface does not result in a locally flat velocity profile for electrokinetic flow through the interparticle pore space of packed beds, which are employed, e.g., for electrochromatographic separations in capillary or chip format. (74) Fanali, S.; Rudaz, S.; Veuthey, J.-L.; Desidero, C. J. Chromatogr., A 2001, 919, 195-203. (75) Xia, D.; Feng, Y.-Q.; Da, S.-L. J. Liq. Chromatogr. Relat. Technol. 2001, 24, 1881-1894. (76) Mistry, K.; Krull, I.; Grinberg, N. J. Sep. Sci. 2002, 25, 935-958. (77) Walhagen, K.; Huber, M. I.; Hennessy, T. P.; Hearn, M. T. W. Biopolymers 2003, 71, 429-453. (78) Li, Y.; Xiang, R.; Wilkins, J. A.; Horva´th, Cs. Electrophoresis 2004, 25, 22422256. (79) Bandilla, D.; Skinner, C. D. J. Chromatogr., A 2004, 1044, 113-129. (80) Simal-Gandara, J. Crit. Rev. Anal. Chem. 2004, 34, 85-94. (81) Steiner, F.; Scherer, B. Electrophoresis 2005, 26, 1996-2004. (82) Szekely, L.; Guttman, A. Electrophoresis 2005, 26, 4590-4604. (83) Progent, F.; Augustin, V.; Tran, N. T.; Descroix, S.; Taverna, M. Electrophoresis 2006, 27, 757-767.

Figure 2. Cross-sectional distribution of axial velocity in the sphere packing computed for EOF in the thin-EDL limit and with a uniform ζ-potential at the solid-liquid interface (top image), and comparison of the three-dimensional distributions of the axial component of the electrical field and velocity field in a selected region of that cross section (indicated by the white square).

An intriguing question that immediately arises from an analysis of Figure 2 is how much the velocity nonuniformity for EOF differs from that for PDF through the same packing and how the remaining differences are then manifested quantitatively in the macroscopic dispersion characteristics. To resolve this question, we have compared both types of liquid flow at the same average velocity (uav) through the bed, corresponding to a Pećlet number (Pe ) uavdp/Dm) of 10. Even though the employed numerical approach can deal with an arbitrary distribution of the ζ-potential, the EOF was simulated by assuming a uniform value of ζ at the solid-liquid interface (ζ ) -50 mV) in order to eliminate any secondary, flow-disturbing effects14 and thereby facilitate the comparison of hydrodynamic dispersion resulting from each type of liquid flow under ideal conditions. (84) Brotherton, C. M.; Davis, R. H. J. Colloid Interface Sci. 2004, 270, 242246. (85) Morrison, F. A. J. Colloid Interface Sci. 1970, 34, 210-214.

First, to validate the computed EOF velocity fields, we employed the macroscopic model of Overbeek and Wijga86 to determine the mean EOF velocity through a packing of impermeable, nonconducting particles in the thin-EDL limit as

< v > ) βE ) -

0rζ K* E interµ K∞

(4)

where β is the transport coefficient characterizing electroosmotic mobility in the packing, K∞ is the electrical conductivity of the liquid electrolyte beyond the EDL, and K* is the conductivity of the packing saturated with the same liquid. For estimating β by eq 4, we used K*/K∞ ) 0.24 determined experimentally by Zeng (86) Overbeek, J. Th. G.; Wijga, P. W. O. Recl. Trav. Chim. Pays-Bas 1946, 65, 556-563.


117

Figure 3. Probability distribution functions (histograms) of the axial interparticle velocity for EOF and PDF through the bulk sphere packing at the same average velocity (Pe ) 10).

et al.87 for a packed capillary with inter ) 0.37 and a column-toparticle diameter ratio of 150. Because the wall effect manifests itself only in a near-wall annulus of ∼4-5 particle diameters,71 this aspect ratio is high enough to assume the wall effect as being negligible and apply this measured value also to the bulk packing in Figure 1a. The calculation by eq 4 yields βtheor ) 2.237 × 10-8 m2 V-1 s-1 using inter ) 0.38, r ) 80, µ ) 1.0 × 10-3 kg m-1 s-1, and ζ ) -50 mV. The value of βsim obtained by our simulation of EOF through the random sphere packing results in a relative difference (1 - βsim/βtheor) of less than 2%. This good agreement between results obtained by simulation and a macroscopic theoretical model indicates that the numerical model employed in this work allows us to arrive at an adequate description of EOF through the bulk packing on the macroscopic (bed) scale, as well as on a microscopic (pore) level. Figure 3 compares the probability distribution functions P(vz) of the local axial velocity at Pe ) 10 for EOF and PDF from the complete packing. For PDF, the velocity distribution function is dominated by slow flow near the solid surfaces. With a uniform pressure gradient in the longitudinal direction, P(vz/) ≈ exp(-vz/) for vz > 0.63 Most of the flow is slower than , but there is a long high-velocity tail. In contrast to the sharp peak in P(vz) for PDF near vz ) 0, P(vz) for EOF is centered close to and demonstrates a much narrower, more symmetric distribution resembling a skewn Gaussian. At this point, it should be mentioned that the imposition of the slip-velocity boundary condition, in the thin-EDL limit, does not significantly change the pore-scale EOF velocity distribution compared to the use of the no-slip boundary condition. This has been verified by analyzing the probability distribution functions of the local EOF velocity in a close simple-cubic array of hard spheres, obtained with the apparent-slip condition (dp/λD f ∞), as well as with the no-slip condition (using dp/λD ) 100).71 In particular, the statistical relative frequency corresponding to vz ) 0 did not depend markedly on the velocity boundary condition. It can be concluded that the distinctions in the probability distribution functions of the axial velocity component for PDF and EOF revealed by Figure 3 originate in the local (pore-scale) velocity profile and are not an artifact caused by the different velocity boundary conditions (“no-slip” for PDF vs “slip” for EOF). (87) Zeng, S.; Chen, C.-H.; Mikkelsen, J. C.; Santiago, J. G. Sens. Actuators, A 2001, 79, 107-114.

118 Analytical Chemistry, Vol. 79, No. 1, January 1, 2007

Next, we analyzed the pore-scale velocity fields, which are manifested in the overall velocity distribution functions for EOF and PDF seen in Figure 3. As shown in Figure 4, the computed velocity fields demonstrate significant differences for both types of liquid flow also at the microscopic level in realizing identical macroscopic velocity (Pe ) 10). In particular, the location of the (light) yellow regions, which correspond to areas with high fluid velocity, can be easily discriminated for EOF and PDF. While for PDF the high-velocity regions are basically located in the largest pores, farthest away from the solid surface, the EOF exhibits highvelocity regions in relatively narrow pores where electromotive forces (concentrated near the solid surface) are stronger due to a locally increased field strength. In addition, the color map for PDF in Figure 4 demonstrates the existence of a significant amount of interparticle pores in which the velocity is much higher than the highest velocity for EOF. This reflects the high-velocity tail in P(vz) observed for PDF (cf. Figure 3). Thus, the analysis by Figure 4 complements and consistently explains, from the miscroscopic point of view, the fundamental differences in P(vz) for PDF and EOF illustrated by Figure 3. After a detailed analysis of the velocity fields for EOF and PDF in the sphere packing, we recorded the resulting longitudinal hydrodynamic dispersion. In this study, we covered a range of velocities with Pe up to 25, which reflects the potential operational domain in capillary and chip (electro)chromatography. For this purpose, passive tracer particles initially were uniformly distributed in the interparticle pore space as a δ-pulse. Then, they were subjected to diffusion and a particular type and strength of convection. Axial spreading of tracer zones could be followed quantitatively from injection, via the transient, toward the asymptotic dispersion regime. This tracer dynamics is illustrated in Figure 5 with Pe ) 20. It is obvious that this approach for studying zone spreading in porous media benefits from the fact that significantly different injection histories and postcolumn effects in nanoLC and chip (electro)chromatography practice are simply absent. These contributions have often blurred a clear experimental analysis and straightforward comparison of the structure-flow-related dispersion. By contrast, the theoretical approach used in this work allows one to relate zone spreading on both transient and asymptotic time and length scales exclusively to the packing microstructure and type of liquid flow. In Figure 5, we have defined a dimensionless diffusive time (tD ) 2Dmt/dp2) to better analyze the time scale of dispersion. First, Figure 5a demonstrates that to reach the asymptotic dispersion regime, reflected by the tracer zones for PDF and EOF at tD ) 30, the length of the packing (Lbed ) 50dp; Figure 1a) is sufficient to avoid a re-entrance of tracer and related artifacts even for the highest macroscopic velocity. Second, it is evident from Figure 5b, which reflects the evolving tracer dynamics (tD ) 0 f tD ) 30) in Figure 5a from just another point of view, that not only is the dispersion coefficient reached for EOF at Pe ) 20 significantly smaller than for PDF (by a factor of nearly 3.5) but that the time needed for attainment of asymptotic behavior is also much shorter. While the asymptotic behavior for PDF is reached at tD ≈ 5, we find 0.5 < tD < 1 for EOF. These time scales demonstrate that lateral, diffusion-limited exchange between velocity extremes in the interparticle flow field

Figure 4. (a) Distribution of the axial velocity with EOF and PDF through the same cross section of the sphere packing (Pe ) 10). EOF was computed in the thin-EDL limit and with a uniform ζ-potential at the solid-liquid interface. (b) Comparison of velocity distributions for EOF and PDF in a selected region of that cross section (indicated by the white square).

for EOF is dominated by pore-level equilibration (tD < 1), while for PDF the exchange is dominated by distances on the order of a few particle diameters (tD ≈ 5) in reaching the asymptotic regime. In other words, for EOF through the packing, the velocity heterogeneity and required equilibration dominates at the microscopic scale (pore level), while for PDF, it extends significantly toward mesoscopic scale. This fundamental difference becomes clear after reinspecting the color maps in Figure 4. The highvelocity regions for PDF (located in the larger pores) span a mesoscale heterogeneity through the velocity field covering several particle diameters. By contrast, velocity nonuniformities on this length scale are not readily discernible for EOF. Although they exist (the EOF demonstrates high-velocity regions in narrow pores), they are less intense than for PDF. After having analyzed the transient dispersion regime, we recorded the effective longitudinal dispersivity for PDF and EOF (88) van Deemter, J. J.; Zuiderweg, F. J.; Klinkenberg, A. Chem. Eng. Sci. 1956, 5, 271-289.

as a function of average velocity through the bed up to Pe ) 25 (Figure 6a). It should be pointed out that our results for PDF are similar to those of Maier et al.65 obtained also for a random packing of spherical particles (Figure 9 in ref 65). A remaining discrepancy can be explained by the different porosities (inter ) 0.38 in our work vs inter ) 0.44 in ref 65). Compared to PDF, the EOF reveals a weak, almost linear dependence on Pe. At the highest Pe, Dz/ Dm for EOF is a factor of 4 lower than for PDF. It should be cautioned that the transient behavior, as well as the value of the asymptotic dispersion coefficient for flow through random media, e.g., a disordered sphere packing, differs significantly from those in regular, homogeneous media, e.g., a periodic array of spheres. Therefore, the structure-transport relations for random media cannot be deduced from those for spatially periodic structures. To facilitate the chromatographic analysis of the longitudinal dispersion, we converted the effective dispersivities into reduced plate heights (h) and subjected the dependence on Pe (Figure Analytical Chemistry, Vol. 79, No. 1, January 1, 2007

119

Figure 5. (a) Zone spreading for an inert tracer transported through the packing by PDF or EOF (Pe ) 20). Initial tracer distributions at tD ) 0 were δ-pulses. (b) Corresponding transient and asymptotic behavior of the longitudinal dispersion coefficient (Dz) normalized by the free molecular diffusion coefficient (Dm) for PDF and EOF as a function of dimensionless diffusive time (tD). The dispersion coefficient was determined by ensemble averaging according to eq 3.

6b) to the van Deemter equation88

2Dz b + cPe )h)a+ PeDm Pe

(5)

This simple three-parameter equation has been derived from an analysis of the analytical solution derived by Lapidus and Amundson89 to the general rate model of chromatography. The van Deemter equation is successfully applied in LC to plate height data acquired over a narrow range of velocities, as in this work with Pe e 25.90 The b-term in eq 5 accounts for longitudinal diffusion in the packing. It is related to the tortuosity factor of the interconnected pore space (τ) by b/2 ) τ-1 ) γ ) Deff/Dm (where Deff is the effective diffusion coefficient in the packing). τ is the inverse of the obstruction factor γ, which is often used in the chromatographic literature.91 Naturally, the b-term is identical for EOF and PDF. We have preferred to determine Deff independently (with greater accuracy than by fitting eq 5 to the limited set of data in Figure 6b) by recording the long-time, asymptotic limit of the timedependent, apparent diffusion coefficient in the packing, without superimposed convective motion (data not shown). The result is (89) Lapidus, L.; Amundson, N. R. J. Phys. Chem. 1952, 56, 984-988. (90) Tallarek, U.; Bayer, E.; Guiochon, G. J. Am. Chem. Soc. 1998, 120, 14941505. (91) Giddings, J. C. Dynamics of Chromatography, Part I: Principles and Theory; Marcel Dekker: New York, 1965.

120 Analytical Chemistry, Vol. 79, No. 1, January 1, 2007

Figure 6. Dependence of (a) effective longitudinal dispersivity and (b) reduced plate height on Pe for PDF and EOF through the random sphere packing (cf. Figure 1a).

b ) 1.292 and is typical for a bulk packing of spherical particles.90,92 In turn, this value was fixed during the analysis of plate height data (Figure 6b) by means of eq 5. The a-term in eq 5 accounts for the convectively driven contribution to the dispersion, which originates in the stochastic velocity fluctuations induced in the mobile phase by the randomly distributed bed particles. The value obtained for EOF (a ) 0.086) is significantly smaller than for PDF (a ) 0.216) and can be explained by the smaller amplitude of velocity fluctuations among different pores of the packing, on a mesoscopic scale, as already discussed in the context of Figures 4 and 5. Thus, EOF engenders significantly less dispersion due to the multiple flow paths between different particles, often referred to as eddy diffusion, than PDF. Finally, the c-term in eq 5 accounts for mass-transfer resistances occurring, in general, in the mobile and stationary phases. Under the present conditions (solid particles, no retention), it is, however, limited to the mobile-phase contribution, which originates in the velocity heterogeneity on the scale of an individual pore in the packing. Equilibration occurs by lateral diffusion, and the resulting dispersion is proportional to the pore-scale velocity spread. A much smaller value for EOF (c ) 2.6 × 10-4) compared with PDF (c ) 0.013) indicates that interparticle mass-transfer resistance becomes practically negligible for electrokinetic flow through the sphere packing. It is in agreement with flow profiles in Figure 4 demonstrating a much higher pore-scale velocity spread for PDF than for EOF. This is due, not least, to the possibility of realizing an apparent slip velocity at the solid-liquid (92) Knox, J. H. J. Chromatogr., A 1999, 831, 3-15.

interface with electrokinetic flow, not only in numerical simulations carried out in this work but also over a substantial range of experimental conditions under which the EOF can be used for transporting bulk liquid and solutes through porous media. SUMMARY The presented numerical approach allows us to simulate electrokinetic transport in three-dimensional, random porous media under most general conditions, including arbitrary value and distribution of the electrokinetic potential at the solid-liquid interface, electrolyte composition, and pore space morphology. It provides quantitative information on the spatial distribution of the simulated velocities. This feature was utilized to analyze porescale dispersion in electrokinetic flow through a microscopically disordered, macroscopically homogeneous bed of solid, dielectric spheres with interparticle porosity of 0.38. Simulations for EOF, carried out in the thin-EDL limit, reveal a generally nonuniform distribution of the axial velocity component, which demonstrates that, even when electrokinetic volume forces are concentrated in the negligibly thin EDL at the solid-liquid interface, the EOF velocity profile in the individual pores of the packing deviates fundamentally from the ideal plug-flow profile in a single straight channel. While for PDF a parabolic pore-level velocity profile results from the radial distribution of the shear stress over the whole pore cross section, a nonuniform pore-level profile for EOF in the thin-EDL limit originates in a nonuniformity of the local electrical field strength (Figure 2), particularly at the solid-liquid interface, which can be directly related to local variations in cross-sectional area of the pores in a bed of solid, dielectric spheres. This dynamics of the local EOF in packed beds compared with the much simpler and more readily anticipated plug-flow profile in a

single straight channel has a purely morphological origin. In addition, a mesoscale velocity heterogeneity in packed beds is caused by variations in surface-to-volume ratio of the pores. For example, for PDF, larger pores give rise to higher velocities. By contrast, higher velocities for EOF are observed in smaller pores. The flow field heterogeneities for PDF and EOF (Figures 3 and 4) are consistently retrieved in the transient and asymptotic dispersion behavior (Figure 5), as well as in a scaling of dispersion with the average velocity (Figure 6). While the velocity heterogeneities for EOF through a bed of solid, dielectric spheres clearly contradict widespread anticipation,74-83 they are less intense compared to PDF. This results in significantly reduced hydrodynamic dispersion depending on the achievable values of Pe. Future studies will be analyzing the electrokinetics and hydrodynamics in fixed beds of porous (permeable and conducting) spheres. In contrast to nonporous spheres used in this work, porous spheres exhibit intraparticle EOF55 butsdue to the EDL overlap in nanometer-sized intraparticle pores (∼ 10 nm)s demonstrate ion-permselective behavior as well. This phenomenon makes the coupled mass and charge transport scheme far more complex and results in electrical field-induced concentration polarization around the charge-selective particles.57 ACKNOWLEDGMENT This work was supported by the Deutsche Forschungsgemeinschaft (Bonn, Germany) under grants TA 268/1, TA 268/2, and HL 56/1, as well as by the Fonds der Chemischen Industrie (Frankfurt a.M., Germany). Received for review June 28, 2006. Accepted October 17, 2006. AC061168R


121

Pore-Scale Dispersion in Electrokinetic Flow through a Random

Recommend Documents