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Concentration Polarization and Nonequilibrium Electroosmotic Slip in Dense Multiparticle Systems Ivo Nischang,† Udo Reichl,†,‡ Andreas Seidel-Morgenstern,†,‡ and Ulrich Tallarek*,† Institut fu¨r Verfahrenstechnik, Otto-Von-Guericke-UniVersita¨t Magdeburg, UniVersita¨tsplatz 2, 39106 Magdeburg, and Max-Planck-Institut fu¨r Dynamik komplexer technischer Systeme, Sandtorstrasse 1, 39106 Magdeburg, Germany ReceiVed March 9, 2007. In Final Form: June 22, 2007 Electrical field-induced concentration polarization (CP) and CP-based nonequilibrium electroosmotic slip are studied in fixed beds of strong cation-exchange particles using confocal laser scanning microscopy (CLSM) and the macroscopic electroosmotic flow (EOF) dynamics. A key property of the investigated fixed beds is the coexistence of quasielectroneutral macroporous regions between the micrometer-sized particles and the ion-permselective (here, cationselective) intraparticle mesopores with a mean size of 10 nm. The application of an external electrical field to the particles induces depleted and enriched CP zones along their anodic and cathodic interfaces, respectively, by the local interplay of diffusive and electrokinetic transport. The intensity and dimension of the CP zones depend on the applied electrical field strength and the fluid-phase ionic strength. With increasing field strength a limiting current density through a particle is approached, meaning that charge transport locally through a particle becomes controlled by the dynamics in the adjoining extraparticle convective-diffusion boundary layer (depleted CP zone). In this regime a nonequilibrium electrical double layer can be induced electrokinetically in the depleted CP zone and intraparticle pore space, resulting in nonlinear EOF in the interparticle macropore space. The local CP dynamics analyzed by CLSM is successfully correlated with the onset of nonlinearity in the macroscopic EOF dynamics. We further demonstrate that multiparticle effects arising in fixed beds (random close packings) of ion-permselective particles modulate significantly the local pattern of CP and intensity of the nonequilibrium electroosmotic slip with respect to the undisturbed singleparticle picture.
Introduction Electrokinetic transport plays the key role in several analytical, technological, and environmental processes, including water removal from industrial slurries and natural porous media, soil remediation, electrodialysis, electrochromatography, and electrophoretic separations, solute focusing, and liquid pumping in microfluidic and lab-on-a-chip devices.1-27 In the classical * To whom correspondence should be addressed at the Department of Chemistry, Philipps-Universita¨t Marburg, Hans-Meerwein-Strasse, 35032 Marburg, Germany. E-mail:
[email protected]. † Otto-von-Guericke-Universita ¨ t Magdeburg. ‡ Max-Planck-Institut fu ¨ r Dynamik komplexer technischer Systeme. (1) Probstein, R. F. Physicochemical Hydrodynamics; Wiley: New York, 1994. (2) Tsuda, T., Ed. Electric Field Applications in Chromatography, Industrial and Chemical Processes; VCH: Weinheim, Germany, 1995. (3) Sørensen, T. S., Ed. Surface Chemistry and Electrochemistry of Membranes; Marcel Dekker: New York, 1999. (4) Deyl, Z., Svec, F., Eds. Capillary Electrochromatography; Elsevier: Amsterdam, The Netherlands, 2001. (5) Mulligan, C. N.; Yong, R. N.; Gibbs, B. F. Eng. Geol. 2001, 60, 193-207. (6) Delgado, A. V., Ed. Interfacial Electrokinetics and Electrophoresis; Marcel Dekker: New York, 2002. (7) Virkutyte, J.; Sillanpaa, M.; Latostenmaa, P. Sci. Total EnViron. 2002, 289, 97-121. (8) Raats, M. H. M.; van Diemen, A. J. G.; Lave`n, J.; Stein, D. Colloids Surf., A 2002, 210, 231-241. (9) Laser, D. J.; Santiago, J. G. J. Micromech. Microeng. 2004, 14, R35-R64. (10) Wong, P. K.; Wang, T. H.; Deval, J. H.; Ho, C. M. IEEE-ASME Trans. Mechatron. 2004, 9, 366-376. (11) Stone, H. A.; Stroock, A. D.; Ajdari, A. Annu. ReV. Fluid Mech. 2004, 36, 381-411. (12) Kirby, B. J.; Hasselbrink, E. F. Electrophoresis 2004, 25, 187-202. (13) Li, D. Electrokinetics in Microfluidics; Elsevier Academic Press: Burlington, MA, 2004. (14) Svec, F. J. Sep. Sci. 2005, 28, 729-745. (15) Kelly, R. T.; Woolley, A. T. J. Sep. Sci. 2005, 28, 1985-1993. (16) Petsev, D. N.; Lopez, G. P.; Ivory, C. F.; Sibbett, S. S. Lab Chip 2005, 5, 587-597. (17) Cui, H.; Horiuchi, K.; Dutta, P.; Ivory, C. F. Anal. Chem. 2005, 77, 7878-7886. (18) Pumera, M. Talanta 2005, 66, 1048-1062.
description electroosmotic flow (EOF) along a solid-liquid interface is generated by the interaction of the local tangential component of an applied electrical field with the mobile space charge of the primary, quasi-equilibrium electrical double layer (EDL).1,6,13 The EOF exhibits a linear response to the applied field strength and is laminar and stationary because the Reynolds number is usually very small.11 Most important, the key properties of the primary EDL only depend on static electrolyte and material characteristics, e.g., the fluid-phase ionic strength or the surface charge density, but not on the actual electrohydrodynamics in a system. This behavior has been verified extensively for systems with a locally thin EDL which, in particular, remains unaffected by the applied electrical field concerning its charge density and spatial dimension. Related work includes the EOF through openchannel structures1,6,13 and model porous media such as fixed beds of solid, dielectric particles.28,29 Assumptions inherent to hard sphere models are that the particles are essentially impermeable and nonconducting. However, in the engineering and life sciences where fixed beds are employed as solid-phase (19) Squires, T. M.; Quake, S. R. ReV. Mod. Phys. 2005, 77, 977-1026. (20) Saichek, R. E.; Reddy, K. R. Crit. ReV. EnViron. Sci. Technol. 2005, 35, 115-192. (21) Bharadwaj, R.; Santiago, J. G. J. Fluid Mech. 2005, 543, 57-92. (22) Ghosal, S. Annu. ReV. Fluid Mech. 2006, 38, 309-338. (23) Jung, B.; Bharadwaj, R.; Santiago, J. G. Anal. Chem. 2006, 78, 23192327. (24) Yuan, Z.; Garcia, A. L.; Lopez, G. P.; Petsev, D. N. Electrophoresis 2007, 28, 595-610. (25) Nischang, I.; Tallarek, U. Electrophoresis 2007, 28, 611-626. (26) Hlushkou, D.; Khirevich, S.; Apanasovich, V.; Seidel-Morgenstern, A.; Tallarek, U. Anal. Chem. 2007, 79, 113-121. (27) Ho¨ltzel, A.; Tallarek, U. J. Sep. Sci. 2007, 30, 1398-1419. (28) Hlushkou, D.; Seidel-Morgenstern, A.; Tallarek, U. Langmuir 2005, 21, 6097-6112. (29) Kang, Y. J.; Yang, C.; Huang, X. Y. Microfluid. Nanofluid. 2005, 1, 168-176.
10.1021/la700691k CCC: $37.00 © 2007 American Chemical Society Published on Web 08/07/2007
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supports for synthesis, separation, and purification, a large specific surface area is usually required in view of capacity, throughput, and sample complexity. To realize a sufficient surface-to-volume ratio, porous particles are used almost exclusively in these applications.30-32 In these more complex, hierarchically structured materials represented, e.g., by a fixed bed of porous (permeable and conducting) particles, for which the typical pore size inside the particles is about 10 nm and, thus, comparable with the EDL thickness, the classical picture of linear EOF can become substantially modified. This stems from the influence of electricalfield-induced concentration polarization (CP) on coupled mass and charge transport through the material. CP describes the formation of concentration gradients of charged species (simple ions, solute molecules, or colloidal particles) in the bulk electrolyte solution adjacent to an ion-permselective, i.e., charge-selective, interface upon the passage of electrical current normal to that interface.1,3 For a fixed bed of micrometer-sized mesoporous particles (pores with a size between 2 and 50 nm are called mesopores) it refers to species transfer from the macropore space between particles into the intraparticle mesopores.33 In contrast to the quasi-electroneutral interparticle macropore space for which the mean macropore size (micrometer scale) is much larger than the EDL thickness (nanometer scale) at the particles’ external surface, the EDL extends over the whole pore fluid of mesopores inside the particles. Due to this situation, which is also sometimes referred to as EDL overlap, the intraparticle mesopores (and the particles as a whole) are ion-permselective; they enrich counterions and exclude co-ions with respect to the bulk electrolyte. At electrochemical equilibrium, an electrical phase boundary potential, the so-called Donnan potential, balances the tendency of ionic species to level out the chemical potential gradients, i.e., the tendency of the counterions to leave the mesopores of a particle and that of the co-ions to enter them.31 As an external electrical field is superimposed, CP is induced at the anodic and cathodic interfaces of a charge-selective particle due to the local interplay of electromigration, diffusion, and convection. In particular, CP is characterized by the formation of depleted and enriched ion concentration zones in the bulk electrolyte solution adjacent to the anodic and cathodic interfaces of a cation-selective particle, respectively (or, vice versa, at the cathodic and anodic interfaces of an anion-selective particle). In the classical picture, local electroneutrality is preserved in the CP zones, but charge transfer through these boundary layers becomes diffusion-limited due to the steep and field-dependent concentration gradients. While diffusive flux and, therefore, the overall mass flux of counterionic species into a mesoporous, cation-selective particle via its anodic hemisphere depends relatively little on the applied field strength through the generated volumetric EOF and local thickness of the convective-diffusion boundary layer (CDL), the intraparticle counterionic transport through electromigration andsdue to prevailing EDL overlaps weak electroosmosis exhibits a much stronger field strength dependence.34 Consequently, a field strength is approached for which electrokinetic transport within a particle begins to exceed diffusive charge flux into the particle via its anodic CDL. Then, a local deviation from electroneutrality can occur at this interface where counterions enter the charge-selective particle in the (30) Giddings, J. C. Dynamics of Chromatography. Part I: Principles and Theory; Marcel Dekker: New York, 1965. (31) Helfferich, F. Ion Exchange; Dover Publications: New York, 1995. (32) Neue, U. D. HPLC Columns: Theory, Technology, and Practice; WileyVCH: New York, 1997. (33) Leinweber, F. C.; Pfafferodt, M.; Seidel-Morgenstern, A.; Tallarek, U. Anal. Chem. 2005, 77, 5839-5850. (34) Leinweber, F. C.; Tallarek, U. Langmuir 2004, 20, 11647-11648.
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direction of the applied field; a secondary (nonequilibrium) EDL is generated electrokinetically. This consists of a mobile counterionic space charge region in the adjacent macropore space and an immobile co-ionic space charge region of unscreened, fixed surface charge inside a particle (or the charge-selective spatial domain in general). The nonequilibrium space charges comprising the secondary EDL are induced by the normal component of the applied electrical field and disturb local electroneutrality over a significantly longer distance than characteristic of the primary EDL.35 Further, the interaction of the mobile space charge region of the secondary EDL with the tangential field component results in nonlinear EOF in the macropore space close to the curved surface of a particle.34-40 Electrical potential drop in the space charge region plays the role of an electrokinetic potential similar to the classical ζ-potential, but in contrast to the latter this potential (which originates in nonequilibrium CP) depends on the particle size and applied field strength. Relevant conditions are classified as nonlinear or nonequilibrium and underlie a CP-based nonlinear electrokinetics, also referred to as nonequilibrium electrokinetics or electrokinetics of the second kind.34-45 The novelty of this electrokinetics is that the primary EDL is complemented by a secondary EDL which depends on the applied field strength concerning both its local dimension and its charge density. This CP-based nonequilibrium electrokinetics has a number of implications which in the past have been studied in view of the electrochemical macrokinetics of colloids and disperse systems,35,43 the electrohydrodynamics in particulate and monolithic fixed beds,39,40 and the current-voltage characteristics of ion-exchange membranes, particularly in a context of the overlimiting conductance phenomenon observed in electrodialysis.46-57 Figure 1 helps to illustrate similarities and differences for the membrane and sphere geometries (flat and curved, (35) Mishchuk, N. A.; Takhistov, P. V. Colloids Surf., A 1995, 95, 119-131. (36) Dukhin, S. S. AdV. Colloid Interface Sci. 1991, 35, 173-196. (37) Mishchuk, N. A.; Gonzalez-Caballero, F.; Takhistov, P. Colloids Surf., A 2001, 181, 131-144. (38) Ben, Y.; Chang, H. C. J. Fluid Mech. 2002, 461, 229-238. (39) Leinweber, F. C.; Tallarek, U. J. Phys. Chem. B 2005, 109, 2148121485. (40) Nischang, I.; Chen, G.; Tallarek, U. J. Chromatogr., A 2006, 1109, 3250. (41) Barany, S. AdV. Colloid Interface Sci. 1998, 75, 45-78. (42) Barany, S.; Mishchuk, N. A.; Prieve, D. C. J. Colloid Interface Sci. 1998, 207, 240-250. (43) Mishchuk, N. A.; Dukhin, S. S. In Interfacial Electrokinetics and Electrophoresis; Delgado, A. V., Ed.; Marcel Dekker: New York, 2002; pp 241-275. (44) Wang, S. C.; Lai, Y. W.; Ben, Y.; Chang, H. C. Ind. Eng. Chem. Res. 2004, 43, 2902-2911. (45) Ben, Y.; Demekhin, E. A.; Chang, H. C. J. Colloid Interface Sci. 2004, 276, 483-497. (46) Manzanares, J. A.; Murphy, W. D.; Mafe´, S.; Reiss, H. J. Phys. Chem. 1993, 97, 8524-8530. (47) Rubinstein, I.; Zaltzman, B. In Surface Chemistry and Electrochemistry of Membranes; Sørensen, T. S., Ed.; Marcel Dekker: New York, 1999; pp 591621. (48) Rubinstein, I.; Zaltzman, B. Phys. ReV. E 2000, 62, 2238-2251. (49) Choi, J. H.; Lee, H. J.; Moon, S. H. J. Colloid Interface Sci. 2001, 238, 188-195. (50) Choi, J. H.; Kim, S. H.; Moon, S. H. J. Colloid Interface Sci. 2001, 241, 120-126. (51) Zabolotsky, V. I.; Manzanares, J. A.; Nikonenko, V. V.; Lebedev, K. A.; Lovtsov, E. G. Desalination 2002, 147, 387-392. (52) Rubinshtein, I.; Zaltzman, B.; Pretz, J.; Linder, C. Russ. J. Electrochem. 2002, 38, 853-863. (53) Ibanez, R.; Stamatialis, D. F.; Wessling, M. J. Membr. Sci. 2004, 239, 119-128. (54) Rubinstein, I.; Zaltzman, B.; Lerman, I. Phys. ReV. E 2005, 72, 011505. (55) Volodina, E.; Pismenskaya, N.; Nikonenko, V.; Larchet, C.; Pourcelly, G. J. Colloid Interface Sci. 2005, 285, 247-258. (56) Zabolotsky, V. I.; Lebedev, K. A.; Lovtsov, E. G. Russ. J. Electrochem. 2006, 42, 836-846. (57) Belova, E. I.; Lopatkova, G. Y.; Pismenskaya, N. D.; Nikonenko, V. V.; Larchet, C.; Pourcelly, G. J. Phys. Chem. B 2006, 110, 13458-13469.
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Figure 1. Schematic representation of electrical-field-induced CP with respect to the bulk electrolyte concentration (C0) at the interfaces between an ion-permselective, here cation-selective, membrane or a spherical particle and the adjacent fluid phase. The arrows help to illustrate consequences for the (anodically depleted and cathodically enriched) CP zones as the applied field strength (Eext) or mobilephase ionic strength (Imob) is increased. Solid lines reflect ion concentrations in equilibrium CP with local electroneutrality in both CP zones. Dotted lines reflect ion concentrations in nonequilibrium CP with an extended mobile space charge region (SCR) in the depleted CP zone. En and Et denote normal and tangential components of the local electrical field. DBL ) diffusion boundary layer (quiescent solution adjacent to the membrane), CDL ) convective-diffusion boundary layer (flow around the sphere).
respectively) with respect to CP induced in the adjacent bulk solutions by an externally applied electrical field. The CP zones around a sphere each occupy half of its surface and ideally have rotational symmetry with respect to the field direction.35,43 In contrast to the membrane geometry where the adjacent CP zones have no direct contact, although they depend on each other due to the transmembrane electrical current, in a fixed bed (random close packing) of particles CP zones influence each other due to the open, three-dimensional architecture of the chargenonselective interparticle macropore space. In addition, they are considerably streamlined by the net electrokinetic flow through the bed and influenced by the local electrohydrodynamics. Thus, the diffusion boundary layers (enriched and depleted CP zones) at a membrane with adjoining quiescent electrolyte solutions
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become convective-diffusion layers around the particles in a fixed bed.39,40 For cation-selective membranes the CP-based nonequilibrium electroosmotic slip has been proposed as a mechanism for realizing turbulent convection at the anodic membrane-solution interface.48,52 As a consequence, the (originally quiescent) depleted CP zone becomes convectively disturbed, and a convective instability tends to destroy the diffusion boundary layer locally. Thus, the external diffusion limitation to the transmembrane charge transport is removed locally, and overlimiting current densities can be realized. This is influenced by the surface heterogeneity,53,55 which may be tuned to facilitate the generation of electroconvection via local tangential electrokinetic forces acting parallel to the “flat” membrane surface. With dispersed cation-selective beads, on the other hand, a strong nonlinear dependence of the electrophoretic velocities on the applied field strength was observed (electrophoresis of the second kind), especially in electrolytes with low ionic strength.41,43,45 Vice versa, in devices containing fixed particles a nonlinear dependence on the electrical field strength of the local EOF velocities at the surface of the beads could be demonstrated (electroosmosis of the second kind).35,36,38 However, most investigations on electroosmosis of the second kind were carried out in closed electrolysis cells where net flow through the system is impossible. More recently, CP and the evolving nonequilibrium electrokinetics have been investigated in fixed beds of cationselective particles and monoliths with a cation-selective skeleton in view of the tunable nonlinear EOF through a material, as well as the separation and retention behavior of charged analytes in capillary and chip electrochromatography.34,39,40,58,59 Related to our previous studies (with a focus on separation science),40 the present work is motivated by correlating more quantitatively the locally observed CP dynamics with the macroscopically resulting (net) EOF in fixed beds of strong cationexchange particles. The microscopic analysis of CP is realized using confocal laser scanning microscopy (CLSM) employing refractive index matching of the fluid phase with respect to the mesoporous silica-based particles.60 This approach facilitates a microscale flow diagnostics in optically opaque media and provides access over well-defined temporal and spatial domains to transient and stationary distributions of a variety of fluorescent tracers used as indicators for CP under a given set of conditions defined by the material characteristics (e.g., packing density, bead shape and diameter, intraparticle pore size and porosity, surface charge density), mobile-phase composition (ionic strength, pH, type of electrolyte or buffer), and the applied field strength. Results obtained with the dense multiparticle systems (fixed beds of cation-selective particles) are compared with the CP-based nonlinear electrokinetics reported earlier in the literature within the single-free-particle picture.41 Experimental Section Reagents and Materials. Sodium acetate trihydrate (p.a., g99.5%), acetic acid (g99.5%), and DMSO (spectrophotometric grade) were purchased from Sigma-Aldrich Chemie GmbH (Taufkirchen, Germany). Fluorescent tracer molecules BODIPY 493/ 503 (D-3922) and BODIPY 492/515 disulfonate (D-3238) were from Invitrogen GmbH (Karlsruhe, Germany), while Rhodamine 6G (Fluka BioChemika) was from Sigma-Aldrich Chemie GmbH (58) Nischang, I.; Spannmann, K.; Tallarek, U. Anal. Chem. 2006, 78, 36013608. (59) Wang, P.; Chen, Z. L.; Chang, H. C. Sens. Actuators, B 2006, 113, 500509. (60) Tallarek, U.; Rapp, E.; Sann, H.; Reichl, U.; Seidel-Morgenstern, A. Langmuir 2003, 19, 4527-4531.
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Nischang et al. xy-section-scanning mode, generating vertical slices using constant laser and detector settings optimized for each tracer. For the CLSM experiments the respective particulate beds were mounted on a homemade chip device and inserted into fluid-phase reservoirs consisting of poly(ether ether ketone) (PEEK). The geometry of the device allowed it to be used like conventional microscopy slides and to adjust the xy-plane orthogonal to the optical light path, but coaxial with respect to the applied electrical field (Figure 2). A high-voltage power supply was accomplished using a 30 kV dc power generator (FuG Elektronik GmbH, Rosenheim, Germany) and by platinum wire electrodes directly inserted into the fluid-phase-containing vials. In advance of experimental work the microscope was grounded. Data analysis was performed using ImageJ.62 Images were acquired as xy-sections of 28.79 µm × 28.79 µm with a resolution of 256 × 256 data points in a slice of thickness 1 µm. Generally, two consecutive scans were averaged for a better signal-to-noise ratio.
Results and Discussion
(Taufkirchen, Germany). The spherical propanesulfonic acidmodified silica particles (Spherisorb SCX) with a mean diameter (dp) of 5 or 10 µm, an intraparticle mean mesopore size (dintra) of 10 nm, and a surface area of about 220 m2/g were a gift from Waters Co. (Milford, MA). A fluid phase consisting of a 90:10 (v/v) mixture of DMSO and aqueous sodium acetate buffer (pH 5.0) was used in all experiments for refractive index matching to the silica-based materials (porous particles and capillary column).60 An aqueous stock solution of 0.5 M sodium acetate was prepared using doubly distilled water from a Milli-Q Gradient water purification system (Millipore GmbH, Eschborn, Germany). The pH was adjusted to 5.0 by titration with concentrated acetic acid. Appropriate volumes of this stock solution, Milli-Q water, and DMSO were then mixed to yield acetate buffer solutions of the desired ionic strengths in 90:10 (v/v) DMSO/water. The fluid phase contained a 10 µM concentration of either of the aforementioned fluorescent tracer molecules. Microfluidic Device and Sphere Packing. Fused-silica capillaries (75 µm i.d. × 360 µm o.d., Polymicro Technologies, LLC, Phoenix, AZ) were packed by a modified slurry technique using a WellChrom pneumatic pump, K-1900 (Wissenschaftliche Gera¨tebau KNAUER GmbH, Berlin, Germany), as described elsewhere.58,61 After the capillaries were packed, they were flushed overnight with fluid phase (containing the respective fluorescent tracer) at a flow rate of ca. 0.1 µL/min using a syringe pump (Harvard Apparatus, Holliston, MA). For analysis of the CP dynamics, i.e., the analysis of stationary CP dependent on the applied field strength, a packed capillary was electrokinetically conditioned by applying an electrical field of 15 kV/m for several minutes. After the field strength was switched to the desired value, CLSM images were acquired after a suitable time delay. To provide an investigation window for the CLSM experiments, the polyimide coating was removed at the center of a capillary by scraping with a surgical blade. The principal experimental setup is illustrated in Figure 2. All fitting materials were purchased from Upchurch Scientific, Inc. (Oak Harbor, WA). Confocal Laser Scanning Microscopy. Measurements were carried out on an Axiovert 100 confocal laser scanning microscope (Carl Zeiss AG, Jena, Germany) equipped with two continuous gas lasers (argon ion laser, 488 nm, 25 mW maximum output power; helium-neon ion laser, 543 nm, 1 mW) and a 40× oil immersion objective (NA ) 1.3). The CLSM images were acquired in the
Dynamics on the Single-Particle Scale. Figures 3 and 4 show CLSM images of a segment of a fused-silica capillary packed with spherical strong cation-exchange particles (dp ) 10 µm, dintra ) 10 nm) acquired using co-ionic tracer (BODIPY 492/515 disulfonate, Figure 3) or the counterionic tracer (Rhodamine 6G, Figure 4). Under fluid-phase conditions favoring significant EDL overlap inside these mesoporous particles, i.e., as the EDL thickness is comparable to dintra,40 the formation of anodically depleted and cathodically enriched CP zones around a particle in the presence of an applied electrical field (Eext) is readily anticipated. Stationary CP is observed a few hundred milliseconds after application of an electrical field. The time period for this process is governed by the diffusion coefficients of the background electrolyte and fluorescent tracer species on the length scale of a single particle in the packed bed.35 Image contrast without an applied field (Eext ) 0 kV/m) observed for the co-ionic tracer (Figure 3) results from its electrostatic exclusion from the intraparticle pore space and is a well-known phenomenon.31 The application of an electrical field influences the co-ion concentration systematically throughout the whole fixed bed; it displays peculiar features of increase and depletion around the strongly cation-selective particles. From a macroscopic point of view it leads to the induction of “mountains” and “valleys” in electrolyte concentration, reflecting alternating CP zones with increased and reduced ionic strength around the discrete particles in the fixed bed. The axial and lateral profiles (axial profiles are coaxial; lateral profiles are perpendicular to the direction of the applied field and EOF) extracted from the CLSM images in Figure 3 reveal that, while a single particle ideally has rotational symmetry with respect to Eext, the CP zones of the selected particle are significantly distorted by the proximity to neighboring particles in a dense multiparticle system such as the fixed bed. In Figure 4 we observe for the counterionic tracer an image contrast that stems from its intraparticle enrichment (at Eext ) 0 kV/m) and which intensifies with Eext. However, neither distinctive CP zones around the particles nor the slope within a particle characterizing the backward diffusion from the enriched CP zone can be detected in the profiles in Figure 4. The small dimension of the particles (dp ) 10 µm) does not allow the clear visualization of the CP zones against the backdrop of a strong intraparticle enrichment increasing with the current density through a particle with increasing Eext, and backward diffusive flux acts on such a time and length scale that the profiles on average display only a pronounced enrichment. The profiles for
(61) Chen, G.; Pacˇes, M.; Marek, M.; Zhang, Y.; Seidel-Morgenstern, A.; Tallarek, U. Chem. Eng. Technol. 2004, 27, 417-428.
(62) Rasband, W. S. ImageJ; U.S. National Institutes of Health: Bethesda, MD, 1997-2006; http://rsb.info.nih.gov//ij/.
Figure 2. Schematic of the experimental setup used for the CLSM study of CP and EOF in fixed beds of the strong cation-exchange particles. PEEK vials served as inlet and outlet reservoirs, into which stainless steel electrodes were inserted to apply the electrical fields. The polyimide coating was removed at the center of the capillary. The setup allowed three-dimensional visualization of fluorescent tracers and their dynamics in the fixed beds optically matched to the fluid phase.
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Figure 4. Local CP dynamics in and around a single particle in a fixed bed of the strongly cation-selective particles (dp ) 10 µm, dintra ) 10 nm) dependent on the applied electrical field strength (Eext as indicated), as visualized with the counterionic tracer (Rhodamine 6G). The fluid-phase ionic strength was 10 mM. Axial profiles were taken along the dashed lines in the images and normalized with respect to their maximum intensity at Eext ) 0 kV/m.
Figure 3. Local CP dynamics in and around a single particle in a fixed bed of the strongly cation-selective particles (dp ) 10 µm, dintra ) 10 nm) dependent on the applied electrical field strength (Eext as indicated), as visualized with the co-ionic tracer (BODIPY 492/515 disulfonate). The fluid-phase ionic strength was 10 mM. The CLSM images demonstrate the induction of enriched and depleted CP zones throughout the whole packing. Profiles were normalized with respect to their maximum intensity at Eext ) 0 kV/m. Axial and lateral profiles (with respect to the direction of Eext and the resulting macroscopic EOF) were taken along the dashed lines in the images.
both co-ionic and counterionic tracer show a similar dependence on Eext. An upper limit for the maximum co-ion concentration in the enriched CP zone corresponding to a minimum co-ion concentration in the depleted CP zone is approached at Eext ) 40-60 kV/m (Figure 3). The maximum intraparticle concentration of the counterionic tracer is realized in a comparable range of Eext (Figure 4). The relative intensity of the evolving CP is analyzed in more detail in Figure 5 on the basis of the profiles shown in Figure 3 (co-ionic tracer). In view of a classical analysis of CP1,3,6 Figure 5 clearly reflects mutual ion concentration differences at the anodic and cathodic phase boundaries of a particle with respect to the bulk solution. They increase with Eext and can be translated
Figure 5. Dependence of the tracer intensity in the stationary, enriched, and depleted CP zones around a single particle in the fixed bed (dp ) 10 µm, dintra ) 10 nm) on the applied electrical field strength. Data were extracted from the axial profiles in Figure 3.
to an increasing current density through the cation-selective intraparticle pore space of a particle. The transport of ionic species toward a particle (in the direction of Eext) is diffusion-controlled by the depleted CP zone (anodic CDL). At increasing Eext the steepness of the respective concentration gradients increases until a limiting behavior is approached. This should be interpreted as reaching the limiting current density locally through the anodic interface of a particle. The local co-ion concentration in the anodic CDL is reduced toward zero at Eext ) 40-60 kV/m, and this field strength also reflects the maximum ion concentration in the enriched CP zone (Figure 5). In the plateau regime electrokinetic
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Figure 6. Visualization and analysis of multiparticle effects on CP in fixed beds. The fluid-phase ionic strength was 10 mM. The enriched CP zone of the upstream particle directly feeds the electrolyte concentration to the nearest particle downstream. The three graphs compare the intensity profiles obtained with and without an applied field along the three dashed lines (labeled accordingly) in the images. The intensity of local multiparticle effects depends on the distance between the ion-permselective interfaces of the individual particles and their relative orientation. The schematic helps to illustrate the local situation in the fixed bed.
transport through a particle exceeds diffusion-limited transport through the depleted CP zone; thus, in this plateau regime (Eext > 40-60 kV/m) charge transport through a particle is determined by the transport characteristics in the adjoining anodic CDL (depleted CP zone). In other words, with increasing field strength, i.e., while the electrical current through a particle increases and the ionic concentration in the depleted CP zone decreases toward zero (Figure 1), a transition occurs from intraparticle to (extraparticle) boundary-layer-dominated transport behavior on the single-particle scale in the fixed bed. Multiparticle Effects in Fixed Beds. In a random close packing of particles the symmetrical shape with respect to Eext of the anodically depleted and cathodically enriched CP zones observed around a single free particle43 becomes significantly
influenced by neighboring particles as was already indicated in Figure 3. The enriched CP zone of a particle in a fixed bed feeds electrolyte concentration to the depleted CP zones of neighboring downstream particles. This is most pronounced when the particles are located directly behind each other as is shown in more detail in Figure 6 for a cusp region between two particles in the packing (this region is indicated by the white arrow in the image for Eext ) 40 kV/m). The feeding of electrolyte concentration from the enriched CP zone of the upstream particle into the depleted CP zone of the downstream particle can render impossible the local observation of the depleted CP zone at the downstream particle (see the profiles in Figure 6b and the schematic). By contrast, in the upper profiles (Figure 6a) and the lower profiles (Figure 6c) the formation of a depleted CP zone of the downstream
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Figure 7. Ionic strength dependence of the co-ion distribution on a single-particle scale, as reflected by the co-ionic tracer. The fluid phase was 90:10 (v/v) DMSO/aqueous sodium acetate buffer (pH 5.0) at varying effective ionic strengths (from left to right, 10, 20, 30, and 40 mM). The laser and detector settings were optimized for an effective ionic strength of 10 mM and Eext ) 52.6 kV/m. Subsequently, the column was equilibrated with a fluid phase of higher ionic strength until a steady state was achieved. All images were aquired under identical laser and detector settings. Tracer concentrations were normalized with respect to the extraparticle intensity at Eext ) 0 kV/m (as indicated).
particle is detectable upon the application of Eext. At these positions (and illustrated by the schematic) the enriched CP zone of the upstream particle does not extend far enough to interact sufficiently strongly with the depleted CP zone of the downstream particle; along the middle profile (Figure 6b) the two beads actually come closest. The analysis in Figure 6 resolves the mutual interplay between enriched and depleted CP zones in the interparticle pore space of fixed beds and thereby demonstrates the importance of such multiparticle effects for the locally surviving CP. Influence of Fluid-Phase Ionic Strength. The dependence of CP on the ionic strength of the bulk electrolyte was investigated using the co-ionic tracer and is illustrated in Figure 7 for a selected particle in the fixed bed. At the beginning of this experiment the laser and detector settings of the microscope were optimized to include the brightest region of the images, i.e., the enriched CP zone at the lowest investigated ionic strength (10 mM acetate buffer) with Eext ) 52.6 kV/m. The field strength and ionic strengths in this experiment were chosen to avoid Joule heating.
For subsequent experiments, the microscope settings were kept constant and the system was dynamically equilibrated with a higher ionic strength fluid phase until a steady state was achieved. Both with and without an electrical field, we observe a decrease in image contrast with increasing ionic strength. For Eext ) 0 kV/m it results from an increase of the intraparticle co-ion concentration due to the increased screening of surface charge and a corresponding decrease of the particle cation selectivity. With Eext ) 52.6 kV/m the decrease in image contrast at increasing ionic strength corresponds to the attenuation of CP, which is also a consequence of the decreasing cation selectivity of a particle, though even at an effective ionic strength of 40 mM the CP phenomenon is still clearly discernible (Figure 7). The ionic strength dependence of the local co-ion concentration (with and without Eext) is summarized in Figure 8. Data points were extracted from the profiles shown in Figure 7 along the dashed vertical lines. With increasing ionic strength (and Eext ) 0 kV/m), i.e., decreasing EDL overlap inside a particle, the coion exclusion is reduced as is indicated by an increasing
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Figure 8. Ionic strength dependence of the strong cation-exchange particles’ ion permselectivity and of the electrical-field-induced CP, as reflected by the co-ionic tracer. Data were extracted from the axial profiles in Figure 7. The ion permselectivity of the system decreases with increasing ionic strength of the fluid phase, resulting in an increased intraparticle co-ion concentration and attenuated CP.
concentration inside the particle at constant extraparticle concentration; the intraparticle concentration (Figure 8, open circles) approaches a limiting value which is reduced compared to the extraparticle concentration according to the porosity of a particle.60 With increasing ionic strength (and Eext ) 52.6 kV/m) the reduced cation selectivity of a particle clearly results in a decrease of the relative enrichment and depletion of the electrolyte concentration in the CP zones, i.e., in an attenuation of the CP phenomenon in the fixed bed at constant Eext (Figure 8, closed symbols). Macroscopic Electrohydrodynamics. Results from the CLSM data presented so far in view of a local CP dynamics in fixed beds of strongly cation-selective particles dependent on the applied field strength (Figures 3-5) and fluid-phase ionic strength (Figures 7 and 8) are further supported and complemented by the observation of the macroscopic fluid dynamics. In these experiments the uncharged, nonadsorbing fluorescent molecule BODIPY 493/593 was employed as a tracer of the EOF velocity field inside the random close packing. In general, the electroosmotic mobility µeo, which is the ratio of the average EOF velocity ueo and Eext, is determined by measuring ueo as a function of Eext. In previous studies µeo in fixed beds of porous particles was shown to be basically composed of the following contributions: 63 (i) classical EDL behavior at the particles’ external surface, i.e., in the charge-nonselective pore space between the micrometersized particles, leading to a decrease of µeo with increasing ionic strength (this reflects normal electrokinetic behavior insofar that, as the ionic strength increases, the EDL is compressed, which results in a reduced shear plane potential at the solid-liquid interface); (ii) intraparticle EOF, which increases with increasing ionic strength because EDL overlap inside the particles is reduced; (iii) particle porosity. The last contribution results from the fact that a conducting electrolyte inside a particle introduces a normal component to the electrical field at its outer surface. This reduces the field’s tangential component, but because the latter determines the velocity at the particles’ external surface, it is expected to decrease compared to that of a solid particle, more as the porosity increases. In addition, the intraparticle EDL overlap influences (63) Chen, G.; Tallarek, U. Langmuir 2003, 19, 10901-10908.
Figure 9. Electroosmotic mobilities µeo ) ueo/Eext in a capillary (75 µm i.d.) packed with the strong cation-exchange particles (dp ) 10 µm, dintra ) 10 nm). The fluid phase was a 90:10 (v/v) DMSO/ aqueous sodium acetate buffer (pH 5.0). (a) Dependence of µeo on the applied electrical field strength at different effective ionic strengths. (b) Dependence of µeo on the ionic strength at selected values of Eext.
the ion permselectivity of a particle (cf. Figure 8), which, in turn, determines the intensity of CP and a CP-based nonequilibrium EOF at higher field strengths.40 In Figure 9a the average flow velocity (expressed via µeo) of the investigated material (dp ) 10 µm, dintra ) 10 nm) is shown for various effective ionic strengths and applied field strengths up to 120 kV/m. For the case realized in our work, the unique trend of µeo corroborates the operation of a fundamental effect. For example, µeo measured at an effective ionic strength of 10 mM (Figure 9a, solid circles) shows an interesting dependence on Eext with a pronounced slope beginning at approximately 40 kV/m. This indicates the onset of a significantly nonlinear dynamics which can be readily explained by an increasing contribution of the nonequilibrium electroosmotic slip to the overall EOF in the fixed bed. As Eext is increased, ionic concentrations in the depleted CP zone (anodic CDL) of a particle in the bed are decreased toward zero and electrical current through a particle is expected to approach a limiting value. For electrokinetic flow along the conductive, cation-selective surface of a particle this regime (below the limiting current density) corresponds to quasiequilibrium, linear EOF for which the diffusive part of the primary EDL essentially preserves its common structure.1 At higher field
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strength charge transport in the anodic CDL becomes climactic with respect to intraparticle electrokinetic transport, meaning that electrokinetic flux of counterions in the particle begins to exceed their supply through the CDL. In other words, diffusive transport in the CDL has a much weaker dependence on Eext (via the local thickness of the CDL, which is an inverse function of the flow velocity) than counterionic transport in the particle due to EOF and electromigration.34 With an increase in Eext the charge transport through a particle experiences a transition from a regime where it is limited by the intraparticle transport behavior to a regime where it only depends on the transport behavior in the electroneutral part of the CDL.51 The transition toward nonequilibrium CP in the limiting current regime is then accompanied by an induction of regions in the depleted CDL and adjoining intraparticle pore space which carry nonequilibrium space charge of opposite sign, i.e., a mobile counterionic space charge region in the depleted CP zone (in the interparticle macropore space) and an immobile co-ionic space charge region of unscreened and fixed surface charge inside a particle. The nonequilibrium CP induces a fundamental structural change in the EDL as the system moves away from the quasiequilibrium.34-38,43,46-48,52 The mobile space charge region (Figure 1) induced by the normal field component can be regarded as the fluid-side part of a secondary (or nonequilibrium) EDL that interacts locally with the tangential field component to generate nonequilibrium electroosmotic slip along the curved surface of a particle in the fixed bed.35,36,38,39,43 Thus, the secondary EDL is induced by the applied field and interacts with the field to generate nonlinear EOF. With decreasing field strength and increasing ionic strength this nonequilibrium potential (potential drop in the mobile space charge region of the secondary EDL) is expected to turn smoothly into the classical ζ-potential.35,43 This is indicated by Figure 9a. The onset of a significantly nonlinear dynamics in Figure 9a with an effective ionic strength of 10 mM at Eext ≈ 40 kV/m reflects the range where we observed the steepest concentration gradients in the solution adjacent to a cation-selective particle in the fixed bed (Figure 3) which showed a hardly discernible further dependence on Eext (Figure 5). The maximum local intensity of CP is then approached, translating to the limiting current density through a particle. This limiting current density and applied Eext thus mark the transition to the nonlinear EOF behavior dominated by a contribution of the nonequilibrium electroosmotic slip based on the secondary EDL. In agreement with our CLSM studies the observed intensity and onset of nonlinearity depend on the fluid-phase ionic strength and are significantly attenuated and shifted to higher Eext at higher ionic strength (Figure 9a). An increased ionic strength reduces the cation selectivity of a particle and, in turn, attenuates the intensity of the CP phenomenon, as analyzed in Figures 7 and 8. Consequently, higher values of Eext are required to reduce the co-ion concentration in the depleted CP zone toward a value where the secondary EDL is formed. This explains the relatively small nonlinear µeo dynamics for the highest ionic strength in Figure 9a, although CP and its microscopic consequences are still discernible (Figures 7 and 8). In contrast, a factor of approximately 2-3 increase in µeo with respect to the expected classical, linear EOF behavior is observed for a 10 mM effective ionic strength at the highest Eext of 120 kV/m. Figure 9b shows the variation of µeo with the fluid-phase ionic strength. We see a relatively common trend insofar as µeo increases with decreasing ionic strength to approach a plateau or spurious maximum at about 10 mM acetate buffer. While a decrease in µeo with increasing ionic strength above 10 mM represents normal
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behavior when EDL overlap is negligible (i.e., the EDL continues to be compressed, resulting in a reduced shear plane potential),1 the increase in µeo from below that concentration toward a spurious maximum can have several causes.64 The most intriguing conclusion emanating from the data in Figure 9b is that µeo depends significantly on Eext at constant mobile-phase composition, under conditions that can be assumed as isothermal. Further, the increase in µeo with Eext becomes much stronger as the acetate buffer concentration is reduced toward 10 mM as is clearly seen in the different slopes of these curves. This ionic strength dependence (see also Figure 8) demonstrates an increasing contribution of the nonequilibrium electroosmotic slip to ueo, being more pronounced at higher field strength. Finally, it is instructive to compare µeo and its electrical field dependence in the fixed beds of strongly cation-selective particles with data on nonequilibrium electrokinetics (or electrokinetics of the second kind) available from the literature. It should be pointed out, however, that this comparison is limited for the following reasons. First, the theoretical analysis of electroosmosis of the second kind has so far focused on the coupled electrokinetics and hydrodynamics around a single, isolated ion-permselective particle (single-free-particle picture).35,36,38,43 In the present work, however, we investigated the contribution of electroosmosis of the second kind to the overall EOF in fixed beds (random close packings) of ion-permselective particles. Compared to that of the single-free-particle picture, the local and macroscopic electrohydrodynamics around a single particle in a packed bed is influenced by the net flow through the bed and around the particles as well as by the proximity of neighboring particles. Second, we realized a flow regime in which the contribution of classical, linear EOF (electroosmosis of the first kind) to the overall EOF cannot be neglected. Third, electroosmosis of the second kind close to a single free particle was analyzed through locally measured velocities,35 while in this work its contribution to the overall EOF is analyzed through the macroscopic dynamics. Consequently, a quantitative comparison of our data (Figure 9) with the theory of electroosmosis of the second kind as developed for the single-free-particle picture is impossible. A general theory for EOF through a random close packing of ion-permselective particles which takes into account the contribution of nonequilibrium electroosmotic slip to the overall EOF as well as the actual morphology of the material is not available. Electrophoresis of the second kind appears to be a better subject for comparison of our data with the literature on the electrokinetics of the second kind. Electrophoresis reflects the integral characteristic of electroosmotic velocities around a particle, which is better suited for investigating macroscopic net flow through a packed bed of particles and analyzing multiparticle effects with respect to the single-particle picture. Electrophoresis of the second kind has been studied intensively.36,41-43,45 Figure 10 shows electrokinetic mobilities of strongly cation-selective particles as a function of the applied field strength. Electrophoretic mobilities (µep) measured for single spherical particles with diameters from 500 to 1 µm were taken from Barany41 and compared to µeo determined in our work for capillaries packed with 10 and 5 µm sized particles. Our data have been rescaled to the physical properties of a purely aqueous fluid phase used in that study of electrophoresis of the second kind.41 Monitoring µeo at an ionic strength of 10 mM clearly shows that generally higher Eext values are necessary for the smaller particles (dp ) 5 µm) to induce a significantly nonlinear EOF behavior (Figure 10, solid circles). This is expected because nonequilibrium (64) Hidalgo-A Ä lvarez, R.; Martı´n, A.; Ferna´ndez, A.; Bastos, D.; Martı´nez, F.; de las Nieves, F. J. AdV. Colloid Interface Sci. 1996, 67, 1-118.
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This comparison suggests that electroosmosis of the second kind could be intensified by a devised assembly of ionpermselective particles in which the particles are artificially removed from each other (with respect to a random close packing) by a distance on the order of the particle size.40 This aspect concerning an optimum arrangement of ion-permselective particles has previously been addressed by Mishchuk65 in a discussion of electrodialysis intensification based on electroosmosis of the second kind. While this geometrical tuning is practically impossible for conventional fixed beds which represent a random close packing of particles, it bears potential for microfabricated and micropatterned analysis systems where the morphologies of the flow channels and ion-permselective spatial domains may be optimized independently.66
Conclusions Figure 10. Electrokinetic mobility for strong cation-exchange particles as a function of the applied electrical field strength. Values for µeo from this work (fluid-phase ionic strength 10 mM) are compared with µep data from Barany, which are reprinted with permission from ref 41, copyright 1998 Elsevier. The µeo data have been rescaled to the physical properties of a purely aqueous fluid phase used in the study of µep. The numbers in the graph indicate the mean particle size.
electroosmotic slip in first approximation depends linearly on the particle size and squared on the applied field strength.35,43 In other words, the ratio of nonlinear (second kind) to linear (classical or first kind) EOF velocities depends linearly on dp and Eext. The data for µep in Figure 10 demonstrate that the small particles (dp ) 1-10 µm) move with almost the same mobility when Eext is below 2 kV/m. In this regime of Eext and dp electrophoresis follows the classical pattern (linear behavior). As Eext increases, the larger particles begin to move faster. For smaller particles, a higher Eext is required to induce electrophoresis of the second kind.41 For dp ) 10-500 µm and Eext ) 2.5-20 kV/m the electrical field dependence of µep is nearly linear (with almost identical slopes), demonstrating the second-order dependence of electrophoretic velocities on Eext. For larger fields (20-100 kV/ m) the curves taper off to approach saturation. This can be related to the increasing importance of a tangential drift of the mobile space charge region and the fact that the particles move at higher Reynolds numbers. Both factors cause a decrease in electrophoretic velocity.43 Compared to µep measured in dilute particle suspensions, the µeo data obtained with fixed beds (dense multiparticle systems) demonstrate the onset of a noticeably nonlinear behavior only at much higher field strengths (Figure 10). In principle, electrophoresis is an integral characteristic of electroosmosis. However, even when the tortuosity of fixed beds with respect to the single-free-particle picture,63 a purely geometrical effect which reduces the macroscopically measured µeo in dense multiparticle systems relative to µep for a dilute suspension of an electrophoresis experiment, is taken into account, µeo is still significantly smaller than µep for a given field strength and particle size. This can be, in part, due to differences in the cation selectivity of the different particles (those used in this work and by Barany41), but also originates in the multiparticle effects demonstrated here by CLSM. The proximity of particles in a dense multiparticle system (fixed bed) simply means that due to the actual pore space morphology complex extinguishing interactions occur between close or overlapping CP zones (Figure 6). This results in a reduced, locally effective CP and consequently in higher field strengths needed to induce nonequilibrium EOF in a fixed bed of particles with respect to the single-free-particle picture.
In this work we have analyzed the local CP dynamics in fixed beds of strongly cation-selective spherical particles with respect to the macroscopic EOF dynamics, particularly in view of a contribution of CP-based nonequilibrium electroosmotic slip. CP was visualized by CLSM employing refractive index matching of the fluid phase with respect to the solid skeleton of the mesoporous particles (Figures 3-8). The electrical field dependence of the CP pattern (Figure 3) demonstrates that a limiting current density is approached locally through a particle at increasing Eext (Figure 5). The electrical current through a particle in the fixed bed is controlled by the intraparticle transport characteristics at low field strengths, while it becomes controlled by the behavior in the depleted CP zone (anodic CDL) toward the limiting current regime at higher field strengths. The CP pattern in a fixed bed of particles is modulated by the arising multiparticle effects, i.e., the interaction of neighboring CP zones (Figure 6). We were able to directly correlate the induction and development of CP to the evolving nonlinear dynamics accessible through the macroscopically measurable EOF velocities (Figure 9). While the local intensities in the depleted and enriched CP zones approach asymptotic behavior (Figure 5), e.g., the ionic concentration in the depleted CP zone is reduced toward zero at increasing field strength, we observe the onset of a significantly nonlinear contribution to the overall EOF dynamics (Figure 9). In the framework of nonequilibrium CP this suggests that a secondary EDL is electrokinetically induced by the applied field, consisting of a mobile space charge region in the depleted CP zone and an immobile space charge region in the adjacent pore space of a particle. It has been shown that nonequilibrium electroosmosis based on this secondary EDL survives in dense multiparticle systems and is a result of the mutual interplay of a variety of parameters, including pore space morphology and applied field strength, but also factors modulating the counterion selectivity of a particle, e.g., the intraparticle pore size and surface charge density or the fluid-phase ionic strength (Figure 8). Compared to the dimensions of the primary EDL, the sizes of the CP zones around a particle are considerably larger (Figure 3), which results in extinguishing interactions in dense multiparticle systems such as the fixed beds (random close packings). Consequently, electrokinetic mobilities based on µeo in a fixed bed and µep for a dilute suspension of the same (or similar) particles from an electrophoresis experiment cannot be quantitatively compared with each other (Figure 10). The hierarchically structured pore space of fixed beds of particles encountered in this work (charge-nonselective inter(65) Mishchuk, N. A. Desalination 1998, 117, 283-296. (66) Balster, J.; Yildirim, M. H.; Stamatialis, D. F.; Ibanez, R.; Lammertink, R. G. H.; Jordan, V.; Wessling, M. J. Phys. Chem. B 2007, 111, 2152-2165.
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particle macropore space, counterion-selective intraparticle mesopore space) is a good example illustrating the importance of both classical (linear) and nonequilibrium (nonlinear) electroosmosis in the macroscopic EOF dynamics in more complex porous media. Borderline cases are observed as the thickness of the primary EDL becomes much smaller than any pore size (linear EOF behavior)40 and, on the other hand, as the potential drop in the mobile space charge region of the secondary EDL becomes much larger than the classical ζ-potential (nonlinear EOF behavior).43
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Acknowledgment. This work was supported by the Deutsche Forschungsgemeinschaft (Bonn, Germany) under Grants TA 268/ 1-2 and TA 268/2-1, as well as by the Fonds der Chemischen Industrie (Frankfurt a.M., Germany). We are grateful to Dr. Uwe Neue from Waters Co. (Milford, MA) for providing the strong cation-exchange (Spherisorb SCX) particles. We thank Dr. Alexandra Ho¨ltzel for critical reading of the manuscript. LA700691K
