Electrokinetic Effects on the Transport of Charged Analytes in

Anal. Chem. 2005, 77, 5839-5850

Electrokinetic Effects on the Transport of Charged Analytes in Biporous Media with Discrete Ion-Permselective Regions Felix C. Leinweber,† Matthias Pfafferodt,† Andreas Seidel-Morgenstern,†,‡ and Ulrich Tallarek*,†

Institut fu¨r Verfahrenstechnik, Otto-von-Guericke-Universita¨t Magdeburg, Universita¨tsplatz 2, 39106 Magdeburg, Germany, and Max-Planck-Institut fu¨r Dynamik komplexer technischer Systeme, Sandtorstrasse 1, 39106 Magdeburg, Germany

The influence of external electrical fields on local concentration distributions and the mass transport of ionic background (buffer) species, as well as eluting co- and counterionic tracer molecules, was investigated in a fixed bed of native glass beads by confocal laser scanning microscopy and numerical simulations. Due to the negative surface charge of the porous glass beads and significant electrical double layer overlap, the intraparticle mesopore space becomes ion-permselective. This cation selectivity and the externally superimposed electrical fields induce concentration polarization in the bulk electrolyte solution adjacent to the particles. At the anodic hemisphere of a bead, the actual interplay of convection, diffusion, and electromigration leads to the formation of a convective-diffusion boundary layer with reduced ion concentrations relative to the bulk solution. At the opposite, cathodic hemisphere where counterions leave a bead in the direction of the applied field, electrolyte concentrations increase generating an enriched concentration polarization zone. Complementary data from quantitative confocal laser scanning microscopy and numerical simulations provide insight into the spatial variations of chemical and electrical potential gradients in the hierarchically structured material, including molar flux densities of the background ionic species, and reveal the elution dynamics of co- and counterionic analytes. These results demonstrate that concentration polarization in the external fluid domain, as well as the magnitude and sign of electrophoretic with respect to electroosmotic mobility in the ion-permselective domain, are major local contributions to coupled mass and charge transport, reflecting analyte retention, migration, and dispersion on a macroscopic scale. Porous media with a tailored adsorption capacity are the basis for many industrial processes including applications in catalysis,1,2 * To whom correspondence should be addressed. Fax: +49-(0)391-67-12028, E-mail: [email protected]. † Otto-von-Guericke-Universita¨t. ‡ Max-Planck-Institut. (1) Coronas, J.; Santamaria, J. Catal. Today 1999, 51, 377-389. (2) Takahashi, R.; Sato, S.; Sodesawa, T.; Yabuki, M. J. Catal. 2001, 200, 197202. 10.1021/ac050609o CCC: $30.25 Published on Web 08/04/2005

© 2005 American Chemical Society

gas purification,3,4 or liquid chromatography.5,6 The materials typically consist of discontinuous solid phase, as in fixed particulate beds, or they have a continuous skeleton realized in a variety of monolithic devices.7-11 Due to the high mass diffusivity and low viscosity of gases at elevated temperatures, the adsorbents involving gas-phase applications are often characterized by a monomodal distribution of micropore size and high surface areas. In liquid-phase operation, by contrast, corresponding properties (bulk viscosity, analyte diffusivities) prevent the use of microporous adsorbents. Liquid-phase transport then can be tailored by introducing bimodal pore size distributions, while maintaining good adsorption or reaction capacities. In particular, a high correlation of interconnectivity between flow-through macropores and smaller mesopores allows one to improve the performance of a material in view of hydraulic permeability, hydrodynamic dispersion, and adsorption capacity.10 Diffusion-limited transport in the mesopore liquid of a hierarchically structured material, e.g., a bed of porous particles (with interparticle macropores and intraparticle mesopores), mainly depends on the morphology, i.e., geometry and topology of the pore space.12-14 In particular, the local concentration of a passive, electroneutral analyte remains unaffected by the charge density at the solid-liquid interface, which develops upon saturation of the support material with an electrolyte due to a number of possible mechanisms.15 These charges at the external and internal surfaces of support particles, however, can significantly affect the coupled mass and charge transport of ionic species such as simple ions, charged analytes, globular particles, or colloids. The geo(3) Hart, K. M.; Pankow, J. F. Environ. Sci. Technol. 1994, 28, 655-661. (4) Kruk, M.; Jaroniec, M. Chem. Mater. 2001, 13, 3169-3181. (5) Giddings, J. C. Dynamics of Chromatography. Part I: Principles and Theory; Marcel Dekker: New York, 1965. (6) Neue, U. D. HPLC Columns: Theory, Technology, and Practice; Wiley-VCH: New York, 1997. (7) Svec, F.; Fre´chet, J. M. J. Ind. Eng. Chem. Res. 1999, 38, 34-48. (8) Vergunst, T.; Linders, M. J. G.; Kapteijn, F.; Moulijn, J. A. Catal. Rev.-Sci. Eng. 2001, 43, 291-301. (9) Heck, R. M.; Gulati, S.; Farrauto, R. J. Chem. Eng. J. 2001, 82, 149-154. (10) Leinweber, F. C.; Tallarek, U. J. Chromatogr., A 2003, 1006, 207-228. (11) Jungbauer, A.; Hahn, R. J. Sep. Sci. 2004, 27, 767-777. (12) Dullien, F. A. L. Porous Media-Fluid Transport and Pore Structure; Academic Press: San Diego, CA, 1992. (13) Brenner, H.; Gaydos, L. J. J. Colloid Interface Sci. 1977, 58, 312-356. (14) Tallarek, U.; Vergeldt, F. J.; Van As, H. J. Phys. Chem. B 1999, 103, 76547664. (15) Lyklema, J. Fundamentals of Interface and Colloid Science, Vol. I: Fundamentals; Academic Press: San Diego, CA, 1991.

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Figure 1. (a) Schematic representation of equilibrium ion concentrations for a strong, symmetrical background electrolyte, transport numbers, and ratio of local pore diameter (dmacro or dmeso) to the EDL thickness (approximated by the Debye screening length, δEDL) in the interconnected macro- and mesopore spatial domains of hierarchically structured materials. No external field is applied. The concentration of mobile counterions + (cmobile ) and their transport number (z+t+; t+ is the transference number) in the pore fluid of the mesopore domain exceed those of the co-ions. (b) SEM images and pore size distributions characterizing many sphere packings and monolithic structures. As shown, these fixed bed adsorbents reveal a bimodal pore size distribution due to the intraparticle (intraskeleton) mesopores and interparticle (interskeleton) macropores.

metrical dimensions in the interparticle macropore space of a sphere packing substantially exceed the typical thickness of the quasi-equilibrium electrical double layer (EDL) at the particle’s external surface. Thus, the macropore space can be considered as quasi-electroneutral and the transport numbers of co-ions and counterions in the background electrolyte are well balanced.15 By contrast, intraparticle mesopore size is comparable to the EDL thickness. The resulting mesopore-scale EDL interaction causes counterion enrichment, but exclusion of co-ions depending on the mobile-phase ionic strength and surface electrical potential. At electrochemical equilibrium, without applied electrical field, a phase boundary potential between interconnected bulk (macroporous) and ion-permselective (mesoporous) domains balances the tendency of ions to level out the existing concentration differences.16 This potential is also known as Donnan potential.17,18 An illustration of this behavior in biporous media, e.g., a random close packing of porous particles is provided by Figure 1. The equilibrium electrolyte in the interconnected pore space is distributed in two compartments characterized by different concentrations of ionic species, thus, by different transport numbers of co- and counterions. For ideal ion permselectivity, which is approached by using electrolytes of low ionic strength and particles with small, highly charged pores, the transport numbers for co-ions are zero simply because no mobile co-ion is present then in the chargeselective domain, while this sum equals unity for the counterions.16 When an electrical field is superimposed externally on the internal potential gradients, both the mass transport and local distribution of ionic species in the hierarchically structured (16) Helfferich, F. Ion Exchange; McGraw-Hill: New York, 1962. (17) Donnan, F. G. Z. Elektrochem. 1911, 17, 572-581. (18) Donnan, F. G.; Guggenheim, E. A. Z. Phys. Chem. 1932, 162, 346-360.

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biporous material (sphere packing) become much more complicated. While a jetting effect caused by the intraparticle electroosmotic flow (EOF), i.e., forced convection in the ion-permselective region, on the electrophoretic mobility of a single porous particle,19 as well as the overall electroosmotic mobility in fixed beds of porous particles,20-22 has been recognized, the description of coupled mass and charge transport through ion-permselective materials rather has been focused on membrane systems related to electrodialysis23 and fuel cells.24 The transport of ionic as compared with electroneutral species is more complicated because electrical fields or currents, concentrations, and velocity distributions are mutually dependent. Mass and charge transport is inherently coupled,25-27 becoming particularly complex in hierarchically structured materials such as the sphere packing, which contains discrete ion-permselective regions in form of the charged, porous particles. Physicochemical models of the governing equations for ion flux densities, charge balance, and momentum conservation were established long ago during the analysis of coupled mass and charge transport through ion-exchange membranes.28-31 An (19) Chen, G.; Tallarek, U. Langmuir 2003, 19, 10901-10908. (20) Li, D.; Remcho, V. T. J. Microcolumn Sep. 1997, 9, 389-397. (21) Stol, R.; Poppe, H.; Kok, W. Th. J. Chromatogr., A 2000, 887, 199-208. (22) Tallarek, U.; Rapp, E.; Seidel-Morgenstern, A.; Van As, H. J. Phys. Chem. B 2002, 106, 12709-12721. (23) Sørensen, T. S., Ed. Surface Chemistry and Electrochemistry of Membranes; Marcel Dekker: New York, 1999. (24) Vielstich, W.; Lamm, A.; Gasteiger, H. Handbook of Fuel Cells: Fundamentals, Technology, Applications; John Wiley & Sons: New York, 2003. (25) Rastogi, R. P.; Srivastava, R. C.; Singh, S. N. Chem. Rev. 1993, 93, 19451990. (26) Saracco, G. Chem. Eng. Sci. 1997, 52, 3019-3031. (27) Re´vil, A. J. Colloid Interface Sci. 1999, 212, 503-522. (28) Morrison, F. A.; Osterle, J. F. J. Chem. Phys. 1965, 43, 2111-2118.

important result of that work is the quantitative relation between concentration polarization (CP) and limiting current density. For the presence of an electrical potential gradient over a negatively charged, ion-permselective membrane, concentrations of ionic species will decrease at the anodic bulk solution-membrane interface but increase on the cathodic side. The depleted and enriched CP zones are induced locally due to a unique interplay of electromigration, diffusion, and ion-permselective transport through the membrane. Thus, CP is a complex of effects related to the formation of concentration gradients in the bulk electrolyte solution adjacent to an ion-permselective interface upon the passage of electrical current normal to that interface.32 Caused by steep concentration gradients in the depleted CP zone, the transport of charged species through this boundary layer becomes diffusion-limited. According to the classical theory of CP, local electroneutrality is preserved in the enriched and depleted CP zones. As the applied electrical field strength is increased further, the ion concentrations in the diffusion boundary layer decrease toward zero and electrical current through the interface is expected to approach a value that, in the classical description of membrane processes, is also referred to as limiting current density.33 An increase of current density beyond this limiting value can only be explained by sophisticated electrohydrodynamic models34,35 or induced by strong electroconvection close to the membrane surface that destroys the CP zones locally, thereby removing their inherent mass-transfer resistance.34,36-38 Most of the experimental studies of CP in membranes have focused on macroscopic properties such as the current density or have analyzed the distribution of ionic species in the liquid external to a membrane.39,40 The species distributions within an ion-exchange membrane, although of key importance, remained largely out of focus. This may be caused by experimental difficulties during a spatiotemporal analysis of local concentration in extended and optically opaque membranes using electrochemical or optical methods. In contrast to flat membranes, the distribution of charged fluorescent tracer in granular porous media can be studied relatively easy by confocal laser scanning microscopy (CLSM) in combination with either low optical density polymers as stationary phase41 or by employing solid-liquid refractive index matching.42 The advantages of CLSM (high lateral resolution, dye sensitivity, fast data sampling43) optimally prepare (29) Isaev, N. I.; Shaposhnova, V. Ind. Lab. 1965, 31, 1518-1524. (30) Mandersloot, W.; Hicks, R. E. Ind. Eng. Chem. Processes Des. Dev. 1965, 4, 304-308. (31) Gross, R. J.; Osterle, J. F. J. Chem. Phys. 1968, 49, 228-234. (32) Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1962. (33) Forgacs, C.; Ishibashi, N.; Leibovitz, J.; Sinkovic, J.; Spiegler, K. S. Desalination 1972, 10, 181-214. (34) Rubinstein, I.; Zaltzman, B. Phys. Rev. E 2000, 62, 2238-2251. (35) Rubinstein, I.; Zaltzman, B. Math. Models Methods Appl. Sci. 2001, 11, 263300. (36) Rubinstein, I.; Maletzki, F. J. Chem. Soc., Faraday Trans. 2 1991, 87, 20792087. (37) Dukhin, S. S.; Mishchuk, N. A. J. Membr. Sci. 1993, 79, 199-210. (38) Mishchuk, N. A. Colloids Surf. A 1998, 140, 75-89. (39) Bath, B. D.; Lee, R. D.; White, H. S.; Scott, E. R. Anal. Chem. 1998, 70, 1047-1058. (40) Choi, J. H.; Park, J. S.; Moon, S. H. J. Colloid Interface Sci. 2002, 251, 311317. (41) Ljunglo ¨f, A.; Hjorth, R. J. Chromatogr., A 1996, 743, 75-83. (42) Tallarek, U.; Rapp, E.; Sann, H.; Reichl, U.; Seidel-Morgenstern, A. Langmuir 2003, 19, 4527-4531.

this technique for the quantitative spatiotemporal analysis of bulk and intraparticle mass transfer in sphere packings in an attempt to resolve the underlying diffusive44-48 and electrokinetic transport mechanisms.49-51 In this work, we employ quantitative CLSM and numerical simulations for an evaluation of the intraparticle and external fluidside concentration profiles of charged fluorescent tracer molecules in a bed of ion-permselective glass beads. In contrast to the membrane geometry (continuous and flat), the ion-permselective domains in a sphere packing (mesoporous particles) appear discrete and curved. Consequently, while adjacent CP zones have no direct contact for a flat membrane, although they depend on each other via the current transferred through it, the CP zones developing around a glass bead are in direct contact. Thus, the CP phenomenon appears locally on particle scale in a sphere packing, but the bed itself is macroscopically charge nonselective. Since the literature lacks the detailed description of CP in sphere packings and its influence on the steady-state distribution and transport of background ionic species, as well as the elution of co- and counterionic analytes, complementary data sets presented in this work provide a detailed, spatially resolved picture of the existing driving force gradients and resulting molar flux densities in a hierarchically structured (bi)porous medium. This includes the CP phenomenon and associated, diffusion-limited mass transfer. In addition to the steady-state distribution and transport of the background buffer components, dynamic elution behavior is investigated for both co- and counterionic analytes under the influence of external electrical fields superimposed on the internal potential gradients. Results improve our current understanding of bulk species transport and the elution dynamics in liquid-phase separation techniques using external electrical fields and fixed beds (e.g., sphere packings and monoliths, which inherently contain relatively discrete ion-permselective regions) such as capillary electrochromatography, electrical field-assisted preparative chromatography, or counteracting chromatographic electrophoresis.52-55 EXPERIMENTAL SECTION Microfluidic Device. The experimental device has been designed for acquiring reproducible data of transient and stationary tracer distributions in porous media by quantitative CLSM,49 allowing one to realize the following: (i) uniform background (43) Shotton, D.; Sheppard, C. Confocal Laser Scanning Microscopy; BIOS Scientific Publishers: Oxford, U.K., 1998. (44) Malmsten, M.; Xing, K. Z.; Ljunglo¨f, A. J. Colloid Interface Sci. 1999, 220, 436-442. (45) Linden, T.; Ljunglo ¨f, A.; Kula, M.-R.; Tho ¨mmes, J. Biotechnol. Bioeng. 1999, 65, 622-630. (46) Song, Y.; Srinivasarao, M.; Tonelli, A.; Balik, C. M.; McGregor, R. Macromolecules 2000, 33, 4478-4485. (47) Kasche, V.; de Boer, M.; Lazo, C.; Gad, M. J. Chromatogr., B 2003, 790, 115-129. (48) Hubbuch, J.; Linden, T.; Knieps, E.; Ljunglo ¨f, A.; Tho ¨mmes, J.; Kula, M.-R. J. Chromatogr., A 2003, 1021, 93-104. (49) Tallarek, U.; Pacˇes, M.; Rapp, E. Electrophoresis 2003, 24, 4241-4253. (50) Dziennik, S. R.; Belcher, E. B.; Barker, G. A.; DeBergalis, M. J.; Fernandez, S. E.; Lenhoff, A. M. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 420-425. (51) Leinweber, F. C.; Tallarek, U. Langmuir 2004, 20, 11637-11648. (52) Deyl, Z., Svec, F., Eds. Capillary Electrochromatography; Elsevier: Amsterdam, 2001. (53) Ivory, C. F.; Gobie, W. A. Biotechnol. Prog. 1990, 6, 21-32. (54) Rudge, S. R.; Basak, S. K.; Ladisch, M. R. AIChE J. 1993, 39, 797-808. (55) Keim, C.; Ladisch, M. Biotechnol. Bioeng. 2000, 70, 72-81.

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Figure 2. (a) Details of the home-built microfluidic device. (b) Two-dimensional images of the tracer concentration distribution in and around a single porous particle were generated by scanning with the laser beam at a particle’s midplane, which is parallel to the direction of the applied electrical field and pressure gradient but perpendicular to the optical path. As evidenced, the intraparticle fluorescence intensity (cintra,eff) of a neutral, nonadsorbing tracer is reduced by the effective porosity in a glass bead. For charged tracers, cintra,eff is further determined by the local electrostatics, i.e., the actual strength of the co-ion exclusion and counterion enrichment at electrochemical equilibrium.

electrolyte concentration around the glass beads; (ii) fast and reversible variation of mobile phase composition in pulse experiments; (iii) application of electrical field strengths up to 100 kV/m without complex safety precautions; (iv) efficient dissipation of Joule heat. For this purpose, a stainless steel capillary with 50µm i.d., 250-µm o.d. (Harry Rieck Edelstahl GmbH, Hilden, Germany) was inserted into a glass capillary with 300 × 300 µm quadratic cross section (Hilgenberg GmbH, Malsfeld, Germany) and fixed by two-component epoxide glue. Selected native glass beads with 100-200-µm particle diameter and 20-nm mean mesopore size (Schuller GmbH, Steinach, Germany) were vacuumaspirated into the capillary to reach a bed length of 2-3 mm. Prepunched glass-fiber filter plugs and a second stainless steel capillary terminated the sphere packing (Figure 2). The complete device was fixed overnight on a microscope slide using epoxide glue. Mobile phase in all experiments was a 90:10 mixture of dimethyl sulfoxide (DMSO; refractive index RI ) 1.479) and an aqueous sodium acetate buffer solution (pH 5.0, concentration 0.4 mol/L). The effective buffer concentration of the electrolyte consequently reduces to 40 × 10-3 mol/L (ionic strength 25.6 × 10-3 mol/L). The refractive index of the DMSO-water solution was adjusted to that of the glass beads (RI ) 1.468) to minimize loss of fluorescence light due to an aberration inside the porous medium at increasing penetration depth.42 This RI match, together with the known bulk concentration of a tracer (10-5 mol/L) allowed one to relate fluorescence intensities to concentrations within the linear range of detection. 5842 Analytical Chemistry, Vol. 77, No. 18, September 15, 2005

Hydraulic flow through the microfluidic device was generated using a dual-syringe pump (Harvard Apparatus, Holliston, MA) operating at a constant volumetric flow rate of 6.0 µL/min, which corresponds to a linear velocity of ∼1.1 mm/s through the sphere packing. Switching between pure and dyed electrolyte solutions was realized with an electrically actuated six-port valve. Each of the following dyes was used at a concentration of 10-5 mol/L: Bodipy 493/503 (electroneutral) and Bodipy-disulfonate (twice negatively charged) from Molecular Probes Europe (Leiden, The Netherlands), as well as the positively charged Rhodamine 6G (Fluka, Taufkirchen, Germany). Low dead-volume connecting units and small-diameter fused-silica capillaries were used to minimize the precolumn band dispersion. Electrical fields were generated by a medium dc voltage supply with a maximum of 1250 V (F.u.G. Elektronik GmbH, Rosenheim, Germany), while the stainless steel capillaries (cf. Figure 2) served as anode directly at the inlet and as cathode directly at the outlet of the fixed bed of porous glass beads. Confocal Laser Scanning Microscopy. Experiments were performed on an Axiovert 100 confocal laser scanning microscope (Carl Zeiss, Jena, Germany) equipped with two continuous noble gas lasers (argon ion gas laser, 488 nm, 25 mW output power; helium-neon ion gas laser, 543 nm, 1 mW) and a 40× oil immersion objective (1.3 NA). To realize a high signal-to-noise ratio, but avoid photobleaching, the laser power was adjusted in advance by acoustooptical tunable filters (Ar laser, 6.25 mW; HeNe laser, 1 mW). Glass beads were analyzed in the section-scanning mode, i.e., in the xy-plane, which is perpendicular to the optical

Figure 3. (a) Three-dimensional model with lateral rotational symmetry obtained by aligning the electrical field and macroscopic flow directions with the axial coordinate, Ltube ) 600 µm and rtube ) 150 µm. (b) Coordinate system, subdomains (external fluid domain, Ωf, and particle domain, Ωp), and boundaries of the two-dimensional model space.

axis, but parallel to the axis of the microfluidic device and macroscopic flow direction, generating slices of 230.4 µm × 230.4 µm with a voxel size of 1.80 µm in xy-directions and 3.2 µm in the z-direction. As shown in Figure 2, the focal plane was adjusted to capture the midplane through the glass bead. A complete scan with 128 × 128 pixels and 12-bit data depth was generated every 245 ms. Data Processing. Time-series raw data were exported as TIF files and processed by the software package ScionImage for Windows (Scion Corp., Frederick, MD) employing macros written in-house that allow the quantitative evaluation of arbitrarily shaped regions of interest in view of kinetic data or stationary concentration distributions. The tracer uptake and release kinetics (intraparticle mean fluorescence intensity vs time) were analyzed to extract effective intraparticle diffusion coefficients and intraparticle EOF velocities based on a differential mass balance equation accounting for both diffusion in the direction of decreasing concentration and forced electroosmotic convection in the direction of the unidirectional dc electrical field,56 following a previously published protocol.49 By a comparison with free-solution diffusivities,57,58 lumped coefficients could be derived that characterize effective diffusive transport of the co- and counterionic fluorescent tracers, as well as the buffer components inside the glass bead under a given set of conditions. Thus, in addition to the intraparticle pore space morphology, these coefficients also reflect the local electrostatics and adsorption behavior relevant for a particular ionic species. NUMERICAL MODEL The transport of charged as compared to electroneutral species is more complex because it is inherently related to the charge transfer via the local electroneutrality condition. An account of (56) Jennings, A. A.; Mansharamani, P. Environ. Modell. Software 1999, 14, 625634. (57) Handbook of Chemistry and Physics, 78th ed.; CRC Press: Boca Raton, FL, 1997-1998. (58) Koneshan, S.; Rasaiah, J. C.; Lynden-Bell, R. M.; Lee, S. H. J. Phys. Chem. B 1998, 102, 4193-4204.

coupled mass flux, charge balance, and momentum conservation is particularly important for hierarchically structured materials such as fixed beds of porous spherical particles due to the presence of discrete spatial domains (intraparticle pore space), which develop ion-permselective transport behavior over a wide range of experimental conditions. In the present work, the relevant balance equations were implemented with a simplified model geometry representing a single hemisphere of a spherical, ionpermselective particle surrounded by electrolyte solution (Figure 3). This geometry allowed us to reduce the solution domain in the numerical simulations of transient and steady-state mass transport used as complement to the quantitative CLSM measurements on a single-particle scale. The model dimensions and fluidphase properties were adjusted to those prevailing in the experimental microfluidic device. While in this work we begin to analyze the consequences of coupled mass and charge transport on the scale of a single ion-permselective particle, the actual electrohydrodynamics in a random-close sphere packing becomes more complex due to multiparticle effects, e.g., by interaction of the CP zones from neighboring spheres.59 The overall mass flux density of a charged species Jtotal,i is the result of diffusive flux caused by chemical potential gradients (Jdiff,i), electrical transference due to electrical potential gradients (Jel,i), and forced convection (Jcon,i)

Jtotal,i ) Jdiff,i + Jel,i + Jcon,i ) - Di(∇ci + ci∇(lnβi)) - ziciµep,i∇φ + ciu

(1)

where Di, ci, and µep,i are diffusivity, molar concentration, and electrophoretic mobility of the ionic species; φ and u denote local electrical potential and fluid velocity. Equation 1 represents a general description of transport in the model geometry; e.g., convective transport (Jcon,i) may originate in any combination of (59) Tallarek, U.; Leinweber, F. C.; Nischang, I. Electrophoresis 2005, 26, 391404.


5843

Table 1. Subdomain Settings and Boundary Conditions Used for Steady-State Simulations (Figures 4-6)a,b balance equation

subdomain fluid (Ωf) particle (Ωp)

momentum

uax, urad

mass flux density

Dm,i uax, urad

charge

Dintra,i ) kiDm,i uax ) ueo,exp urad ) 0 ωcfix

boundary δΩin uax(r) ) 2Fv/πrtube2 (1 - (r/rtube)2) urad ) 0

δΩout ×

cp ) cp,in ctr ) ctr,in cn,in ) (zpcp + ztrctr)/(-zn) φ ) LtubeE

δΩp

δΩwall

δΩsym,i

p)0 t‚u ) 0

uax, urad ) 0

uax, urad ) 0

n‚u ) 0

cp,out ) cp,in cn,out ) cn,in ctr,out ) ctr,in φ)0

Ji(Ωp) ) Ji(Ωf)

n‚Ji ) 0

n‚Ji ) 0

a Parameters. u and u ax rad are the local velocities in axial and radial direction; Dm,i, free-solution diffusion coefficient; ki, parameter that accounts for effective intraparticle diffusion (Dintra,i) including, if applicable, electrostatic and adsorption effects for a charged species; ueo,exp, experimentally determined intraparticle EOF velocity; cfix, fixed-charge concentration at the surface; Fv, volumetric flow rate; cp, cn, and ctr are, respectively, the molar concentrations of the positive and negative buffer ions, as well as the positively or negatively charged tracer; zp, zn, and ztr denote the corresponding charge numbers. (t and n are tangential and normal unit vectors.) b Settings. T ) 298 K; ωcfix ) -1.65 mol/m3; cp,in ) 25.62 mol/m3; ctr,in ) 10-2 mol/m3; Dm,i ) 7.0 × 10-10 m2/s; kp ) 0.33; kn ) 0.16; ktr,p ) 0.11; ktr,n ) 0.20; fluid phase, η ) 1.87 × 10-3 kg/(m s); F ) 1.09 × 103 kg/m3; Fv ) 0.1 × 10-9 m3/s.

the applied pressure and electrical potential gradients. Parameter settings for the two subdomains (Figure 3, particle and external fluid domain, Ωp and Ωf) are given in Table 1. Assuming ideal solution behavior, the activity coefficients βi in eq 1 approach unity and the flux density for each mobile ionic species reduces to

Jtotal,i ) - Di(∇ci + zici(F/RT)∇φ) + ciu

(2)

The electrophoretic mobility was replaced according to the Nernst-Einstein relation16

µep,i ) DiF/RT

(3)

which relates local space charge densities by spatial variations in electrical potential to preserve electroneutrality. 0 is the permittivity of vacuum and r is the relative permittivity of the electrolyte solution. In the present case, eq 4 was modified for the particle domain (Ωp) by including the concentration of negative charges fixed at the inner surface of the particle (cfix; ω ) -1). Electrical potential drop over the model space was adjusted to the experimental field strength with appropriate conditions at the domain boundaries (Table 1, δΩin and δΩout). The velocity field of the incompressible Newtonian electrolyte solution surrounding the particle is governed by the NavierStokes equation

F(∂u/∂t + (u‚∇)u) ) - ∇p + η∇2u + f where F, R, and T are the Faraday constant, universal gas constant, and absolute temperature. In the whole model space, the continuity condition for electrical current (∇i ) 0) is assumed to hold. Concentrations of ionic species at the model space inlet (Figure 3, δΩin) were defined to facilitate the adjustment of model parameters toward experimental conditions. For example, charge number and concentration of the tracer can be varied according to experimental settings, while the electroneutrality of the whole entering electrolyte solution remains preserved via slightly modified concentrations of the buffer components. The corresponding boundary conditions are listed in Table 1. At the symmetry axis (δΩsym) and capillary wall (δΩwall), the normal flux density is zero and continuity of mass flux through the particle surface (δΩp) is preserved. Within the particle (Ωp), only an axial velocity is assumed.49 This EOF component (ueo,exp) was obtained as described in the Experimental Section using the neutral, nonadsorbing tracer, which yielded an effective intraparticle electroosmotic mobility µeo ) 2.5 × 10-9 m2/(V s). This rather small value basically is a result of the significant EDL overlap in the mesopores of the ion-permselective glass beads.19,49 The mass transport of ions inherently involves the transfer of charges so the three flux equations for ionic species (two for the buffer components and one for co-ionic or counterionic tracer) were subjected to the Poisson equation

-∇‚(∇φ) ) (F/0r)(

∑z c + ωc i i

fix)

(4)

i

5844


(5)

where F and η are the density and dynamic viscosity of the fluid, u represents the divergence-free velocity field (∇‚u ) 0), p is hydrostatic pressure, and f denotes the body force (either electrical or gravitational) acting on the bulk fluid. The numerical solution of the set of coupled balance equations was performed with Femlab 2.3 (Comsol, Go¨ttingen, Germany). Our developed numerical code implementing the model configuration (geometry, differential equations, and boundary conditions) is provided as Supporting Information. To reduce the complexity of the underlying calculations, careful simplifications were applied to the description of the velocity field. First, the contribution of EOF to the overall convective transport in the macropore space (external fluid domain, Ωf) was assumed as negligible because, under the present conditions, the highest extraparticle EOF velocity (∼0.11 mm/s) remained a decade below the average velocity due to the hydraulic flow. The neglect of EOF in the external fluid domain allows us to omit the electrical body force in eq 5 without introducing significant error in the model. Consequently, the velocity field in the external fluid domain becomes independent of applied field strength, a characteristic that also prevails in the experiments with strong hydraulic flow through the microfluidic device. Thus, flow velocities (uax, urad) in the external fluid domain (Ωf) were obtained a priori by solution of the momentum balance equation, i.e., prior to the solution of the coupled Poisson-Nernst-Planck problem. Further, a fully established parabolic velocity profile was assumed at the model

Figure 4. Experimental and simulated steady-state axial centerline concentration profiles through an ion-permselective particle and the adjoining CP zones for counterionic tracer (positively charged Rhodamine 6G, enriched by the particle) and the co-ionic tracer (negatively charged Bodipydisulfonate, excluded from the particle). Experimental profiles are indicated in the corresponding CLSM images. The electrical field (E ) 16 kV/m) is applied from left to right.

space inlet (Table 1, inlet boundary settings, δΩin), and the noslip velocity boundary condition was applied to the capillary and particle surfaces (δΩwall, δΩp). The complete numerical implementation was realized in cylindrical coordinates because of lateral rotational symmetry in the model, which allowed us to reduce the three-dimensional sphere to a two-dimensional semicircle (Figure 3). Background buffer components (sodium and acetate ions) were assumed as nonadsorbing to the particle surface, and the whole model space was considered as isothermal at 298 K. RESULTS AND DISCUSSION Steady-State Properties. The combination of quantitative CLSM with the numerical simulations not only enables a detailed analysis of dynamic concentration distributions but also provides quantitative data on the spatial variation of electrical and chemical potential gradients, as well as the flux densities due to diffusion, electromigration, and convection for each ionic species. The ionic strength of the electrolyte (buffer) solution was 25.6 × 10-3 mol/L to introduce an intermediate EDL overlap in the mesopores of a glass bead characterized by an electrokinetic radius (i.e., ratio of mesopore radius to Debye screening length) of rmeso/δEDL ) 5.1. Due to resulting ion-permselective (charge-selective) transport through the glass beads, the steady-state distribution of ion concentrations in the whole material is affected when an electrical field is superimposed externally on internal potential gradients. The distributions of (enriched) counterionic and (excluded) coionic tracer shown in Figure 4 differ considerably from those at electrochemical equilibrium without superimposed electrical potential gradient.51 CP in the external fluid domain leads to a zone with reduced ion concentrations at the anodic hemisphere of the negatively charged particle (depleted CP zone) where more counterions than co-ions enter the cation-selective intraparticle pore space per unit time in the direction of the applied field. At the opposite, cathodic hemisphere, the electrolyte concentration in the adjoining bulk solution increases due to the flux of counterions from the intraparticle pore space into the external fluid domain where they are neutralized by co-ions moving toward

this interface (enriched CP zone). Simulations indicate that with the employed conditions electroneutrality is preserved locally in the complete material over the investigated spatiotemporal scales including the CP zones around the particle. Higher applied electrical field strengths or ion permselectivity of the particles (realized by reduced ionic strength, for example), or both, may lead to a fundamental structural change in the EDL of the depleted CP zone accompanied by the induction of (mobile) fluid-side and (immobile) particle-side space charges.59 However, in the present work, conditions were adjusted to remain in quasi-equilibrium characterized by a locally quasi-electroneutral convective-diffusion boundary layer at the anodic hemisphere of the particle and a quasi-equilibrium EDL. We also observe in Figure 4 variations between experimental and simulated axial concentration profiles, which become most pronounced for the enriched CP zone. Actually, its spatial dimension is overestimated by the simulations; the decline toward background electrolyte concentration is too slow compared with the experimental data. The origin for this artifact in simulations is the velocity field in the external fluid domain Ωf (cf. Figure 3). Because of the no-slip velocity boundary condition at the particle’s external surface and the local geometry of the model space, a region of stagnant fluid exists close to the centerline downstream of the particle. In this region, there are no flow components either in axial or radial direction (Figure 5a). As a consequence, the results of the simulations in this region contrast with the flow behavior typical for fixed multiparticle systems. In a random-close sphere packing, the actual flow path is affected by many beads and becomes tortuous. The interparticle hydrodynamics then does not reveal extended zones of stagnant mobile phase (as in the external fluid domain of the model space), except in cusp regions close to the contact points between particles.14 Lateral dispersion thus is effective closer to the particle’s external surface, and the CP zones show a faster relaxation toward the bulk concentration level, which is confirmed by the CLSM data in Figure 4. Analytical Chemistry, Vol. 77, No. 18, September 15, 2005

5845

Figure 5. (a) Distribution of the axial and radial flow velocities in the external fluid domain (Ωf) obtained by solution of the Navier-Stokes equation with no-slip velocity boundary condition at the capillary wall (δΩwall) and particle surface (δΩp). (b) Electrical potential gradients in axial and radial direction were calculated using the coupled Poisson and Nernst-Planck equations. For parameters and settings see Table 1.

The CP phenomenon not only causes a more inhomogeneous steady-state distribution of the ionic species compared to electrochemical equilibrium without applied field, but also leads to locally varying electrical field strengths (Figure 5b). Electrical potential gradients in an axial direction reveal a strong deviation from the linear behavior observed in a single, straight capillary filled only with electrolyte solution. This contrasts with the widespread opinion that a homogeneous, linear decay of electrical potential (with negligible radial field components) can also be expected in random porous media.52 In fact, the local axial field strength increases considerably within the particle and the equatorial region of the extraparticle fluid domain (Figure 5b) but becomes almost zero at the poles of the particle. Maximum values are more than doubled compared to the nominal field strength. In addition, radial field components have developed (absent in the open-tubular geometry) that account for about two-thirds of the nominal axial field strength. An explanation for the electrical field inhomogeneity (Figure 5b) can be found in the continuity requirement for electrical current, implying that a constant ion flux density must be obtained through any cross-sectional area of the model space. Therefore, an increased ion flux occurs in the bypass between particle and capillary wall because of decreased flux through the particle. It was mentioned earlier that convective transport in the external fluid domain remains unaffected by the applied electrical fields due to the dominating and constant hydraulic flow component. This means that in the bypass region continuity of ion flux is satisfied by increased electromigration flux (Jel,ax) because concentration gradients are absent (Figure 6a). In this context, the increased axial and radial field components arise along the curved interface of the particle (Figure 5b). As a consequence of this continuity requirement also, transport in the CP zones along anodic and cathodic hemispheres of the particle is influenced. Caused by concentration gradients in the depleted and enriched CP zones (cf. Figure 4), diffusive flux is the dominating contribution within these zones to the transport of charged species (Figure 6b); i.e., mass transfer into the ion-permselective particle is 5846


diffusion-limited by both CP zones, which may be considered as somehow streamlined extraparticle fluid-side diffusion boundary layers. In addition, the axial electrical field strength is strongly reduced locally (Figure 5b) in order to satisfy continuity of electrical current in the model space. Dynamic Elution Profiles. The influence of external electrical fields on the transient distribution of co- and counterionic tracer was investigated experimentally by injecting short pulses of dyed electrolyte solution into the microfluidic device. Quantitative fluorescence signals were recorded for a region covering a single particle, as well as the surrounding solution (due to the developing CP phenomenon). Concentrations of eluting tracer were acquired at the same axial location in the material, once at the particle center and, for comparison, in the external fluid domain. These profiles are shown in Figure 7. More quantitative information was derived by analyzing the distributions with an exponentially modified Gauss function

f(t) ) y0 +

x [( )

h0σt τ

π 1 σt exp 2 2 t

2

-

]{ [ ( )]}

t - t0 τ

1 σt x2 t t - t0 (6) τ

1 - erf

where y0, h0, and t0 denote the baseline offset, signal height, and time at signal maximum and σt and τ are the standard deviation of the Gauss term and time constant of the first-order decay function.60 This data analysis reveals a substantial improvement of the particle-based statistical figures of merit with respect to elution efficiency for both co- and counterionic tracers, while elution data in the extraparticle fluid domain remain almost unaffected by the superimposed electrical fields (see Table 2, signal variances σ2p and σ2f ). As evidenced by the profiles in Figure 7, not only the peak variance is improved by the application of electrical fields but also the time delay between the centers of (60) Marco, V. B. D.; Bombi, G. G. J. Chromatogr., A 2001, 931, 1-30.

Figure 6. (a) Simulated absolute molar flux densities due to axial electromigration (Jel,ax) for co- and counterionic buffer components. Note that the direction of migration is opposite for both kinds of ions. (b) Molar flux densities due to axial diffusion (Jdiff,ax) for both buffer components. Data were obtained by solution of the coupled Poisson-Nernst-Planck problem using the solution of the Navier-Stokes equation for the velocity field in the external fluid domain (Ωf). For parameters and settings see Table 1. Table 2. Statistical Figures of Merit for Elution Profiles (Figure 7)a

E (kV/m)

Peintra (neutral tracer)

Peintra

0 6.7 13.4 33.5

0 14.8 29.5 73.8

0 88.5 177.1 442.7

counterionic tracer σp2 σf2 (s2) (s2) 123.9 15.1 10.3 10.6

12.1 12.5 11.6 10.9

∆t0 (s)

τ (s)

Peintra

σp2 (s2)

12.1 3.3 1.3 0.3

3.1 1.0 1.1 1.0

0 25.7 51.4 128.4

66.9 39.9 17.8 13.9

co-ionic tracer σf2 (s2) 10.4 9.9 9.6 11.4

∆t0 (s)

τ (s)

8.3 5.9 3.4 1.4

2.2 1.4 0.8 0.6

a Analyte-specific effective intraparticle Peclet numbers (Pe intra,i ) uintra,effdp/Dintra,i) were calculated by accounting for electroosmotic and electrophoretic contributions, if applicable, to the intraparticle effective velocity (uintra,eff) experienced by the neutral, co-ionic, or counterionic analyte. These values were obtained using the experimentally determined electroosmotic and electrophoretic mobilities (µeo and µep,i).

gravity of particle-based concentration distributions and their fluidside pendant (∆t0) is minimized until both signals almost coincide. Further, the asymmetry of the elution profiles is strongly reduced as shown by the decrease in τ. This behavior indicates vanishing intraparticle mass-transfer resistance due to enhanced transport by forced electroconvection and electromigration. In fact, mass transfer is increased well beyond the diffusion-limited regime as reflected by the intraparticle Peclet numbers (Peintra) in Table 2. Overall electrokinetic improvement is much stronger for the counterionic analyte than for co-ionic and neutral analytes. To understand better the dynamic elution behavior of the counterionic analyte spatially resolved centerline, concentration profiles were extracted from the quantitative CLSM data in Figure 7. These profiles represent the axial distribution of tracer inside the particle and external fluid during passage of the tracer pulse and, thus, reflect local concentrations during the tracer uptake by and release from the particle. This allows us to analyze the influence of combined intraparticle electrophoresis and electroos-

mosis on the elution dynamics. Experimental CLSM data were complemented by a time-dependent solution to the set of physical balance equations for pulse injections of the co- or counterionic tracer (Figures 8 and 9). For this purpose, the transient tracer concentrations at the model space inlet (Figure 3, δΩin) were realized by a Gauss function

[

cin ) cin,0 exp -

]

(t - t0)2 w

(7)

where t0 is the time at which the signal maximum reaches the confocal detection volume. This value and the width (w) of the Gaussian were derived from elution curves in the external fluid domain (Figure 7) to represent best the experimental conditions. The experimental and simulated data reveal differences in the elution characteristics of counter- and co-ionic analytes due to the opposite direction of their electrophoretic motion. Under the Analytical Chemistry, Vol. 77, No. 18, September 15, 2005

5847

Figure 7. Tracer elution (following pulse injection) monitored at the particle center for purely diffusive and electrokinetically enhanced intraparticle transport. (a) Positively charged, enriched counterionic tracer. (b) Twice negatively charged co-ionic tracer, partly excluded from the mesopore space. The solid line represents the respective profile in the surrounding solution, at the same axial position. All signal intensities were normalized with respect to maximum intensity in the external fluid domain.

present conditions, the counterionic analyte migrates in the elution direction because pressure and electrical potential gradients are parallel (Figure 8). Its effective intraparticle electrophoretic mobility calculated with the Nernst-Einstein relation (eq 3) using the experimentally determined intraparticle diffusion coefficient is 3.0 × 10-9 m2/(V s). This value slightly exceeds the electroosmotic mobility, and after a superposition of both contributions, the transport enhancement becomes twice as high for the counterionic as compared to the neutral analyte. Thus, overall electrokinetic mobility for the counterionic analyte increases to ∼5.5 × 10-9 m2/(V s), which accelerates intraparticle transport far beyond the diffusion-limited regime and significantly improves the statistical figures of merit for elution profiles, especially signal variance and peak asymmetry, over purely diffusive behavior (Table 2). The consequences of this drastically reduced intraparticle mass-transfer resistance deserve a closer examination, particularly with respect to the observed signal asymmetry. For purely diffusive intraparticle transport, axial concentration profiles reveal 5848


radial symmetry, but the uptake of counterionic tracer by the particle is much faster than its release (Figure 8a) and the whole elution signal becomes strongly asymmetric (see Figure 7a and Table 2). The origin for a relatively slow diffusive release compared with the uptake of tracer can be found in much weaker concentration gradients between intraparticle pore space and external fluid domain during the release. When the tracer pulse reaches the particle, extraparticle fluid-side tracer concentration quickly increases to the maximum value. Steep concentration gradients arise between bulk and intraparticle fluid phase, but the particle-based tracer concentration increases only slowly due to the actual intraparticle mass-transfer resistance. Then, after passage of the center of gravity of the pulse, extraparticle fluidside tracer concentration quickly falls to zero, but the resulting concentration gradient between the intraparticle and the bulk solution is now much smaller than during tracer uptake. Consequently, the net driving force for particle emptying is weaker and diffusive release takes longer. We therefore observe asymmetric overall elution signals. This signal asymmetry is strongly reduced by electrokinetically enhanced intraparticle transport. As indicated by Peclet numbers far beyond unity (Table 2), electrokinetic mobility due to convection and migration dominates intraparticle transport of the counterionic tracer. Thus, its uptake by and release from a particle during the passage of a tracer pulse become independent of the actual concentration gradients. Since electrokinetic transport is independent of the external fluid-side and intraparticle concentrations, tracer uptake and release can be assumed as occurring on very similar time scales. As a consequence, the asymmetry of elution profiles for the counterionic tracer and the signal variances are substantially reduced by the intraparticle electrokinetic transport (cf. Figure 7a and Table 2), particularly as the directions of both the intraparticle EOF and electromigration coincide with the elution direction. This situation changes for the co-ionic tracer because it experiences electrophoretic motion in the direction opposite to the EOF. Consequently, an improvement of the elution characteristics depends on the difference between the intraparticle electroosmotic and electrophoretic mobilities. The electrophoretic mobility of the twice negatively charged co-ionic tracer is calculated as 5.4 × 10-9 m2/(V s) and exceeds the intraparticle electroosmotic mobility by a factor of more than two. Further, as the direction of motion originating in both electrokinetic phenomena is diametrical, the high electrophoretic mobility causes the co-ionic tracer to move opposite to the elution direction within a particle. As reflected by the respective Peclet numbers for coionic tracer, the overall enhancement of intraparticle transport then is less substantial than for the counterionic tracer (Table 2). Yet another implication arises from the fact that with applied electrical fields the co-ionic tracer in the intraparticle pore space moves opposite to the elution direction, which is still determined by hydraulic flow in the interparticle pore space. Under the present conditions, uptake of co-ionic tracer by the spherical particle occurs from its downstream hemisphere. This is demonstrated in both the experiments and simulations, as shown by the axial concentration profiles in Figure 9. In turn, tracer release occurs via the upstream hemisphere. This particle-scale uptake and

Figure 8. Axial centerline concentration profiles of counterionic analyte through a single particle at different times after pulse injection, as indicated. (a) Purely diffusive intraparticle transport (experimental data, no external field applied). (b) Electrokinetically enhanced mass transfer (experimental data, E ) 13.4 kV/m). (c) Numerical simulation of concentration profiles for E ) 13.0 kV/m. Experimental data were extracted from elution signals in Figure 7. The dashed lines represent the physical boundaries of the particle.

release dynamics is disadvantageous for any improvement in elution efficiency because co-ionic tracer molecules that have been released already can be subjected to repeated uptake by the same particle. An additional, important aspect is related to the occurrence of CP. The co-ionic tracer molecules are enriched close to the cathodic downstream hemisphere of the particle (cf. Figure 4). Due to the small flow components and low electrical field strengths in this region, co-ionic tracer may get trapped in the enriched CP zone (Figure 9). This trapping provides continuous backflux into the particle, even if it has been passed already by the main tracer pulse. Asymmetric elution signals result, and signal asymmetry for the elution of co-ionic tracer indeed cannot be significantly reduced, even at elevated field strengths (Table 2). As a consequence of CP and the actual interplay of intraparticle electrophoretic and electroosmotic mobilities, the electrokinetic improvement of statistical figures of merit for pulse injections of the co-ionic tracer remains less pronounced when compared to the counterionic tracer. CONCLUSIONS The superimposition of external electrical fields on the internal electrical potential gradients in hierarchically structured (bi)porous media with discrete ion-permselective regions (sphere packings

or monoliths, Figure 1) influences substantially local concentration distributions of the background ionic species (buffer components), as well as the elution dynamics of charged (co- and counterionic) analytes. Electrical and chemical potential gradients develop due to charge-selective transport through the mesoporous compartment (intraparticle or intraskeleton pore space, Figures 4-6). To satisfy the continuity of coupled mass and charge transport in the whole material, increased electromigration flux results in the external fluid domain close to the equator of a particle (bypass in the model, Figure 6a), while diffusion dominates in the vicinity to its poles (Figure 6b). Thus, both the distribution and transport of ionic species depend on location within a hierarchically structured material and vary considerably from those in a relatively simple open-tubular geometry. In particular, the elution characteristics of injected tracer pulses relevant to electrochromatographic separations (Figures 7-9) grounds on these transport properties and concentration distributions (Figures 4-6). The observed CP phenomenon affects the elution process in the external fluid domain by trapping tracer molecules in the enriched CP zone (Figure 4) where convection is reduced and diffusion dominates the overall mass flux (Figures 5a and 6b). In the charge-selective domain, relative sign and magnitude of electroosmosis and electrophoresis determine Analytical Chemistry, Vol. 77, No. 18, September 15, 2005

5849

Figure 9. Axial centerline concentration profiles of co-ionic analyte through a single particle at different times after pulse injection, as indicated. (a) Purely diffusive intraparticle transport (experimental data, no external field applied). (b) Electrokinetically enhanced mass transfer (experimental data, E ) 13.4 kV/m). (c) Numerical simulation of concentration profiles for E ) 13.0 kV/m. Experimental data were extracted from elution signals in Figure 7. The dashed lines represent the physical boundaries of the particle.

transport of charged analytes (Figures 8 and 9). For a counterionic analyte, both contributions point in the same direction, which, in the present work, coincides with the elution direction. Already the use of gentle electrical field strengths results in a substantial enhancement of intraparticle mass transfer beyond the diffusionlimited regime (Table 2). It appears that more systematic studies are needed to resolve and understand relative contributions of the documented electrokinetic phenomena (electroosmosis, electrophoresis, and CP) to the macroscopic transport behavior in electrochromatographic separations and electrical field-assisted processes, in general, depending on the most relevant system parameters such as mobile-phase ionic strength, applied field strength, and ion permselectivity, which interactively influence effective migration, retention, and zone dispersion of charged analytes. Based on those results, the devised application of electrical fields may also improve the performance of selected applications in downstream process(61) Hlushkou, D.; Kandhai, D.; Tallarek, U. Int. J. Numer. Methods Fluids 2004, 46, 507-532. (62) Hlushkou, D.; Seidel-Morgenstern, A.; Tallarek, U. Langmuir 2005, 21, 6097-6112.

5850 Analytical Chemistry, Vol. 77, No. 18, September 15, 2005

ing and preparative chromatography. This insight further will certainly lead to a better appreciation and hopefully reproducibility of capillary electrochromatography. These studies may be complemented by more advanced numerical simulations in random sphere packings or monoliths to address with a sufficiently highresolution salient features of the (three-dimensional) electrohydrodynamics over relevant time and length scales.61,62 ACKNOWLEDGMENT This work was supported by the Deutsche Forschungsgemeinschaft (Bonn, Germany) under Grants SE 586/7-2 and TA 268/ 1-1 and by Fonds der Chemischen Industrie (Frankfurt a.M., Germany). We thank Dr. Dzmitry Hlushkou (Otto-von-GuerickeUniversita¨t Magdeburg, Germany) for helpful discussions. 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 April 11, 2005. Accepted July 3, 2005. AC050609O

Electrokinetic Effects on the Transport of Charged Analytes in

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