Microfluidics and Nanofluidics

, Volume 4, Issue 6, pp 471–487

Electrohydrodynamics around single ion-permselective glass beads fixed in a microfluidic device

Authors

  • Steffen Ehlert
    • Institut für VerfahrenstechnikOtto-von-Guericke-Universität Magdeburg
  • Dzmitry Hlushkou
    • Institut für VerfahrenstechnikOtto-von-Guericke-Universität Magdeburg
    • Institut für VerfahrenstechnikOtto-von-Guericke-Universität Magdeburg
    • Department of ChemistryPhilipps-Universität Marburg
Research Paper

DOI: 10.1007/s10404-007-0200-5

Cite this article as:
Ehlert, S., Hlushkou, D. & Tallarek, U. Microfluid Nanofluid (2008) 4: 471. doi:10.1007/s10404-007-0200-5

Abstract

This work demonstrates by direct visualization using confocal laser scanning microscopy that the application of electrical fields to a single-fixed, ion-permselective glass bead produces a remarkable complexity in both the coupled mass and charge transport through the bead and the coupled electrokinetics and hydrodynamics in the adjoining bulk electrolyte. The visualization approach enables the acquisition of a wealth of information, forming the basis for a detailed analysis of the underlying effects (e.g., ion-permselectivity, concentration polarization, nonequilibrium electroosmotic slip) and an understanding of electrohydrodynamic phenomena at charge-selective interfaces under more general conditions. The device used for fixing single beads in a microfluidic channel is flexible and allows to investigate the electrohydrodynamics in both transient and stationary regimes under the influence of bead shape, pore size and surface charge density, mobile phase composition, and applied volume forces. This insight is relevant for the design of microfluidic/nanofluidic interconnections and addresses the ionic conductance of discrete nanochannels, as well as nanoporous separation and preconcentration units contained as hybrid configurations, membranes, packed beds, or monoliths in lab-on-a-chip devices.

Keywords

Electrical double layerIon-permselectivityConcentration polarizationElectroosmotic flowNonlinear electroosmosisElectrohydrodynamicsMicrovortex

1 Introduction

Electrokinetic transport of bulk liquid and solutes plays a central role in many analytical, technological, and environmental processes. This includes water removal from industrial slurries and natural porous media, soil remediation, capillary and chip electrochromatography with high-surface-area stationary phase materials (like monoliths or particulate fixed beds), as well as electrophoretic separations, solute focusing strategies, and liquid pumping in microfluidic and lab-on-a-chip devices (Choudhary and Horváth 1997; Cui et al. 2005; Deyl and Svec 2001; Dittmann et al. 1995; Ghosal 2006; Höltzel and Tallarek 2007; Hu et al. 2006; Jung et al. 2006; Kelly and Woolley 2005; Kirby and Hasselbrink 2004; Laser and Santiago 2004; Li 2004; Mulligan et al. 2001; Nischang and Tallarek 2007; Pumera 2005; Raats et al. 2002; Saichek and Reddy 2005; Stachowiak et al. 2004; Stone et al. 2004; Svec 2005; Verpoorte and de Rooij 2003; Virkutyte et al. 2002; Wang et al. 2006; Wong et al. 2004; Yao et al. 2006; Zilberstein et al. 2003).

In the classical description, electroosmotic flow (EOF) along a solid-liquid interface is generated by 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) (Lyklema 1995). The EOF exhibits a linear response to the applied field strength, is laminar and stationary because the Reynolds number (Re) is usually very small (Re << 1) (Stone et al. 2004). Most important, key properties of the primary EDL only depend on the static electrolyte and material characteristics, e.g., the ionic strength or surface charge density, but not on the actual electrohydrodynamics in the system.

This behaviour has been verified extensively for systems with a locally thin EDL that, in particular, remains unaffected by the applied electrical field concerning its charge density and spatial dimension (Stone et al. 2004). Related work includes EOF through open-channel structures (Li 2004) and model porous media like fixed beds of nonporous (solid and dielectric, that is, impermeable and nonconducting) spherical particles (Hlushkou et al. 2005; Kang et al. 2005). In more complex, hierarchically structured materials represented, e.g., by a fixed bed of porous (permeable and conducting) particles, for which a 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. The additional complexity basically stems from the influence of electrical field-induced concentration polarization (CP) on coupled mass and charge transport through the interconnected pore space of the material (Leinweber et al. 2005; Nischang et al. 2006b). CP describes the formation of concentration gradients of charged species (simple ions, solute molecules, or globular and 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 (Probstein 1994).

For example, with a fixed bed of micrometer-sized mesoporous particles (pores with a size between 2 and 50 nm are called mesopores) this scenario refers to species transfer from the macropore space between the particles into the intraparticle mesopores. 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 the 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 solution (Leinweber et al. 2005). 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 (Helfferich 1995), i.e., the tendency of the counterions to leave the mesopores of a particle and that of the co-ions to enter them (cf. Fig. 1a).
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Fig. 1

Schematic view on distribution and transport properties of electrolyte species in a mesoporous, ion-permselective (here, cation-selective) material, e.g., a mesoporous glass bead with a negative surface charge density, and the electrolyte solution adjacent to its anodic interface. Normalized concentration profiles for the mobile counterion and co-ion reflect: a Donnan equilibrium without applied electrical field (Eext = 0); and b, c different intensities of CP induced locally by the normal component of the applied field (En). BGE background electrolyte (electroneutral), CDL convective-diffusion boundary layer (electroneutral), SCR fluid-side space charge region of the secondary EDL containing an excess of mobile counterions. Fel denotes the body force interaction of the mobile SCR with the tangential component of the applied field (Et) leading to nonlinear (induced-charge) electroosmosis. \( J^{ + }_{{\text{int} {\text{ra}}}} \) is the intraparticle counterion flux density due to electrophoretic and electroosmotic velocities (vEP and vEO), and \( J^{ + }_{{{\text{CDL}}}} \) is the flux density through the CDL. Reprinted with permission from Leinweber and Tallarek (2004). Copyright 2004 Americal Chemical Society

Then, as an external electrical field (Eext) 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 (Fig. 1a, b). 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; however, charge transfer through these boundary layers becomes diffusion-limited due to the steep and field-dependent concentration gradients (Fig. 1b).

As illustrated in Fig. 1, the electrokinetics and hydrodynamics at a charge-selective (here cation-selective anodic) interface of a particle is influenced by the applied field strength (Eext). While diffusive flux and, therefore, the overall mass flux of a counterionic species into a mesoporous, cation-selective particle via its anodic hemisphere (cf. Fig. 1b) 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 by electromigration and––due to the prevailing EDL overlap––weak electroosmosis exhibits a much stronger dependence (Leinweber and Tallarek 2004). As a consequence, with increasing field strength a value is approached for which electrokinetic transport within a particle \( (J^{ + }_{{\text{int} {\text{ra}}}} ) \) begins to exceed diffusive charge flux into the particle via its anodic CDL \( (J^{ + }_{{{\text{CDL}}}} ). \) Then, a local deviation from electroneutrality can occur at this interface where counterions enter the charge-selective particle in the direction of the applied field (Fig. 1b, c).

In other words, a secondary, nonequilibrium EDL is generated electrokinetically. It consists of a counterionic mobile space charge region (SCR) in the adjacent macropore space and a co-ionic immobile SCR of unscreened, fixed surface charge inside a particle or the charge-selective spatial domain, in general (Fig. 1c). The nonequilibrium space charges comprising the secondary EDL are induced by the normal component of the applied electrical field (En) and disturb local electroneutrality over a significantly larger distance than characteristic of the primary EDL (Mishchuk and Takhistov 1995). Further, as indicated in Fig. 1c, the interaction of the mobile SCR of the secondary EDL with the tangential field component (Et) results in nonlinear (induced-charge) EOF in the macropore space close to the curved surface of a particle (Ben and Chang 2002; Dukhin 1991; Leinweber and Tallarek 2005; Mishchuk and Dukhin 2002; Mishchuk et al. 2001; Mishchuk and Takhistov 1995). The electrical potential drop in the SCR plays the role of an electrokinetic potential, similar to the classical zeta-potential, but in contrast to the latter, this potential due to nonequilibrium CP depends on the particle size and applied field strength. Relevant conditions are classified as nonlinear or nonequilibrium and underlay a CP-based nonlinear electrokinetics, also referred to as nonequilibrium electrokinetics or electrokinetics of the second kind (Barany 1998; Barany et al. 1998; Ben and Chang 2002; Ben et al. 2004; Dukhin 1991; Leinweber and Tallarek 2005; Mishchuk and Dukhin 2002; Mishchuk et al. 2001; Mishchuk and Takhistov 1995; Nischang et al. 2006a; Wang et al. 2004). The novelty of this electrokinetics is that the classical, primary EDL becomes complemented by a secondary EDL which depends on the applied field strength concerning both its local dimension and charge density.

The CP-based nonequilibrium electrokinetics has a number of implications which, in the past, have been investigated for strong cation-exchange particles (curved interfaces) with respect to the electrochemical macrokinetics of colloids and disperse systems (Mishchuk and Dukhin 2002; Mishchuk and Takhistov 1995), as well as for charge transport through ion-exchange membranes (flat interfaces), particularly in a context of the overlimiting conductance observed in electrodialysis (Belova et al. 2006; Choi et al. 2001a, b; Ibanez et al. 2004; Manzanares et al. 1993; Rubinstein and Zaltzman 1999, 2000; Rubinstein et al. 2002, 2005; Zabolotsky et al. 2002, 2006). For example, with dispersed cation-selective beads 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 (Barany 1998; Ben et al. 2004; Mishchuk and Dukhin 2002). Vice versa, in devices containing fixed particles a nonlinear dependence on the electrical field strength of the local EOF slip velocities at the surface of the beads could be demonstrated (electroosmosis of the second kind) (Ben and Chang 2002; Dukhin 1991; Mishchuk and Takhistov 1995). Since most investigations of electroosmosis of the second kind were carried out in closed electrolysis cells, no net flow through the system was possible (closed devices). For cation-selective membranes a CP-based nonequilibrium electroosmotic slip has been proposed as a mechanism for realizing chaotic convection at the anodic membrane-solution interface (Rubinstein and Zaltzman 2000; Rubinstein et al. 2002). As a consequence, the originally quiescent depleted CP zone is convectively disturbed, and a convective instability tends to destroy the diffusion boundary layer locally. Thus, the (external) diffusion-limitation to the trans-membrane charge transport is removed locally and overlimiting current densities can be realized.

Compared to these topics and fundamental studies, microfluidic and nanofluidic applications of a CP-based nonlinear electrokinetics concerning the pumping and mixing of liquids, or the improvement of performance in, e.g., separation and preconcentration devices until now are rare and the few available have been reported only recently (Chen et al. 2005; Höltzel and Tallarek 2007; Nischang and Tallarek 2007; Wang et al. 2004, 2005). In this line, the present work continues our earlier investigations of coupled mass and charge transport through packed beds of cation-selective particles and silica-based monoliths with a cation-selective skeleton (Leinweber and Tallarek 2004, 2005; Nischang et al. 2006a, b). That work, for example, has resolved the basic dependence of nonlinear EOF velocities and hydrodynamic dispersion in capillary and chip electrochromatography on the variation of applied field and mobile phase ionic strengths. In particular, differences in the pore space morphology of random-close packings of discrete particles and monoliths with a continuous skeleton have been found to criticially affect the local intensity of CP and a macroscopically achievable, CP-based nonlinear EOF in these materials (Nischang et al. 2006a).

In addition, CP effects are increasingly recognized in multifunctional integrated microfluidic devices where discrete nanochannel configurations and nanoporous membranes are currently receiving much attention as modules for on-chip sample preconcentration (Dhopeshwarkar et al. 2005; Foote et al. 2005; Hatch et al. 2006; Kim et al. 2006; Wang et al. 2005). CP in these cases is an intimate companion, either as a prerequisite, or as an undesired side effect, or both. Not only from a most fundamental point of view, but also in considering the practical implications of tunable beneficial and adverse effects, ion-permselective transport and associated CP effects need to be anticipated in a number of microscale analysis systems with applications that involve the passage of ionic currents through microfluidic/nanofluidic interfaces (Chang et al. 2006; Chatterjee et al. 2005; Daiguji et al. 2004; Eijkel and van den Berg 2006; Höltzel and Tallarek 2007; Kuo et al. 2003; Schmuhl et al. 2005, 2006; Siwy 2006; Smeets et al. 2006; Yu et al. 2003). This addresses a variety of geometries (slits, channels, filters, membranes, or random and regular networks of pores), as well as applications, e.g., the gating, sensing, preconcentration, and separation with multifunctional miniaturized devices.

Related to our previous investigations, the present work is motivated by analyzing the electrohydrodynamics around a single ion-permselective particle (fixed in bulk electrolyte) under the influence of superimposed pressure and electrical potential gradients. This specific insight will complement our knowledge on the corresponding phenomena in fixed beds (random-close packings) of particles in which the inherent proximity of the neighbouring beads has been found to strongly affect the local and macroscopic intensity of CP effects with respect to an expected idealized (undisturbed) single-particle picture.

For that purpose, we employed a simple, robust and inexpensive device in this work to fix single, cation-selective glass beads in a microchannel. Thus, the fluid dynamics around a bead is not influenced by the neighbouring particles as in a packed bed. In turn, the device allows to study the origin and consequences of a coupled electrokinetics and hydrodynamics in and around a single bead in dependence of material characteristics (bead shape and diameter, mean intraparticle pore size and porosity, surface charge density) and mobile phase composition (ionic strength, pH, type of electrolyte or buffer), as well as the applied field strength and superimposed pressure gradients.

Visualisation of the electrohydrodynamics around a glass bead, as well as the analysis of intraparticle transport is realized using confocal laser scanning microscopy (CLSM) employing refractive index matching of the liquid electrolyte with respect to a porous glass bead (Tallarek et al. 2003). This approach, which facilitates the microscale flow visualization (Sinton 2004) in optically opaque systems, together with the flexibility of the microfluidic device, provides access over well defined temporal and spatial domains to the distributions of a variety of fluorescent tracers used as indicator for both the transient and stationary electrohydrodynamics in the single-particle system under a given set of conditions.

The manuscript is structured as follows. In Sect. 2 we describe details of the microfluidic device containing single-fixed glass beads and of the CLSM measurements. In Sect. 3 we present the experimental results with increasing complexity, moving from tracer distributions at electrochemical equilibrium (Sect. 3.1) to electrical field-induced CP (Sect. 3.2) observed under the action of both macroscopic pressure and electrical potential gradients and study in more detail the influence of hydraulic flow rate (Sect. 3.2.1) and applied field strength (Sect. 3.2.2). In Sect. 3.3 we visualize the electrohydrodynamics in a closed device, without net flow through the system. Results are then discussed in Sect. 4 with an eye on related phenomena for the membrane geometry and those reported earlier for the single-sphere system under closed-device conditions.

2 Experimental

2.1 Microfluidic device

The microfluidic device has been built for resolving transient and stationary distributions of co-ionic and counterionic fluorescent tracers inside and around a single-fixed glass bead by CLSM, allowing to realize (1) a uniform background electrolyte concentration around the fixed bead; (2) fast and reversible variation of the mobile phase composition; (3) the application of both electrical fields without complex safety precautions and hydraulic flow; (4) efficient dissipation of Joule heat; and (5) a convenient and reversible exchange of individual beads. The device layout is illustrated in Fig. 2.
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Fig. 2

Device for visualisation of the electrohydrodynamics around a single-fixed glass bead, as well as intraparticle transport by CLSM. a Front view onto the device showing the orange PEEK tubing in a PEEK fitting (1) used to fix single beads in the 1.6 mm wide and 0.2 mm deep channel; electrodes (2) for applying voltages up to 1,250 V; and PTFE tubing (3) connecting the channel to a syringe pump. The distance between the platinum electrodes is 20 mm. b Bottom view onto the channel showing a 175 μm-diameter glass bead on top of the orange PEEK tubing. c Enlarged view of a single spherical bead

Makrolon® plates (Bayer MaterialScience AG, Leverkusen, Germany) were used for manufacturing a 46 mm long, 1.6 mm wide and 200 μm deep channel which was tightly closed by a 200 μm thick coverslip adjusted underneath the Makrolon® plate. Spherical porous glass beads (Schuller GmbH, Steinach, Germany) had an intraparticle porosity of 0.7 and a diameter (dp) between 100 and 200 μm. They were used with an intraparticle mean pore size (dintra) of 20 or 122 nm. Porous glass beads of similar size, but irregular shape were purchased from Schott AG (Mainz, Germany), while spherical nonporous (impermeable, nonconducting) glass beads with a mean size of 170 μm came from Polymer Standards Service GmbH (Mainz, Germany).

PEEK tubing (1.6 mm o.d.) was used to fix a single bead inside the channel as in a bench vice (Fig. 2). To ensure that the bead remained fixed the tubing was coated with epoxide glue to form a little nose which could touch and hold the bead like a rubber. To prevent contact between the glue and liquid inside the channel the PEEK tubing was subsequently covered by Parafilm® (American Nation Can, Chicago, IL). This became necessary after preliminary studies had demonstrated poor resistance of the glue against dimethylsulfoxide (DMSO).

Hydraulic flow through the channel and around the fixed bead was generated with a syringe pump (Harvard Apparatus, Holliston, MA) connected to the device by PTFE tubing (Fig. 2). A filter was placed behind the pump to prevent contaminations (lint or dust) from entering the setup and eventually blocking the channel, or adsorbing onto the bead surface. The mobile phase in all experiments was a 90:10 (v/v) mixture of DMSO and aqueous sodium acetate buffer (pH 5.0). The refractive index of this solution was close to that of the glass beads (1.468) in order to minimize the loss of fluorescence light inside a bead caused by aberration. In addition, the coverslip was immersed in anhydrous glycerol. Thus, the refractive index mismatch throughout the device became negligible.

Single positively charged Rhodamine 6G (Fluka, Taufkirchen, Germany) and twice negatively charged BODIPY-disulfonate® (Molecular Probes, Eugene, OR) were employed as fluorescent dyes. These counterionic and co-ionic tracers (with respect to the negative surface charge of the glass beads) were used individually in the buffer at a concentration of 10−5 M. External electrical fields (Eext) were applied by a medium d.c. voltage supply with a maximum of 1,250 V (F.u.G. Elektronik GmbH, Rosenheim, Germany), using 0.5 mm-diameter platinum electrodes which were directly inserted into the channel (Fig. 2). They served as anode towards the inlet and as cathode towards the outlet of the device.

2.2 Confocal laser scanning microscopy

Experiments were run on an Axiovert 100 confocal laser scanning microscope (Carl Zeiss, Jena, Germany) equipped with two continuous noble gas lasers (Argon ion laser: 488 nm, 25 mW output power; Helium-Neon ion laser: 543 nm, 1 mW) and a 40× oil immersion objective (1.3 NA). The geometry of the microfluidic device (Fig. 2) allowed it to be inserted like a conventional microscopy slide into the frame. Glass beads and surrounding solution were analyzed in the section-scanning mode, i.e., in the xy-plane which is perpendicular to the optical axis, but parallel to the axis of the microfluidic channel, generating slices of 325.7 μm × 325.7 μm with voxels of 3.2 μm in z-direction and 0.32 μm × 0.32 μm (1,024 × 1,024 pixels) or 2.54 μm × 2.54 μm (128 × 128 pixels) in xy-directions. The lower resolution was chosen in time series measurements for characterizing the transient dynamics inside and around a glass bead, meaning that a bead was first equilibrated with mobile phase (containing fluorescent tracer) using hydraulic flow through the device, without applied electrical field. Then, the time series was started, an electrical field applied, and images acquired repeatedly. Afterwards, an image with the higher resolution was created (by adding four scans) to characterize stationary tracer distributions.

3 Results

3.1 Electrochemical equilibrium

We begin by analyzing the situation at electrochemical equilibrium between liquid in the cation-selective pore space of a glass bead and surrounding bulk electrolyte solution. As shown in Fig. 3 with a mesoporous and a macroporous glass bead (intraparticle mean pore size, dintra: 20 and 122 nm, respectively) the counterionic fluorescent tracer is enriched by the glass beads with respect to the bulk solution (see also Fig. 1a), except for the larger pore size (122 nm) and highest ionic strength (10−2 M; Fig. 3a), while the co-ionic tracer (Fig. 3b, c) is excluded from the beads meaning that, while it still penetrates the intraparticle pores, its concentration inside the beads is reduced with respect to the bulk solution. In these experiments mobile phase containing counterionic or co-ionic dye is continuously supplied at a volumetric hydraulic flow rate of 20 μl/min. Thus, the liquid flows with a linear velocity of about 1.1 mm/s through the channel and around a glass bead. No electrical field is applied. After the uptake by a particle of the dyes stationary tracer profiles are reached which have an idealized boxcar shape, but different amplitudes. On the basis of the effective intraparticle diffusion coefficient (Dintra), the spherical shape and size (dp) of a glass bead stationary tracer profiles are observed after some characteristic time t = (dp/2)2/Dintra assuming that tracer molecules have to travel a net distance of dp/2 radially inward (from the external surface of the bead to its center) as they are initially flushed around the bead. For the values of Dintra and dp encountered here this equilibration takes several seconds and expectedly needs more time for the counterionic tracer because it adsorbs to the oppositely charged inner surface of a glass bead resulting in a lower Dintra than for the co-ionic tracer with similar size.
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Fig. 3

Stationary fluorescence intensity profiles through spherical glass beads (from the upstream to the downstream pole) and the adjacent bulk electrolyte solution. No electrical field is applied (Eext = 0 kV/m); rate of volumetric hydraulic flow through the channel: 20 μl/min. a Counterionic tracer (single positively charged Rhodamine 6G); bead diameter (dp): 175 μm; intraparticle mean pore size (dintra): 122 nm. b Co-ionic tracer (twice negatively charged BODIPY-disulfonate®); dp = 202 μm; dintra = 20 nm. c Co-ionic tracer; dp = 163 μm (dintra = 122 nm) and 202 μm (dintra = 20 nm); ionic strength: 10−3 M. Fluorescence intensities are normalized with respect to the bulk solution surrounding a bead

These different amplitudes reflect the dependence of pore-scale EDL interaction (or overlap) inside a bead on the mobile phase ionic strength (Fig. 3a, b) and the intraparticle mean pore size (Fig. 3c). It refers to the negative charge at the surface of the small pores of the glass beads (Tallarek et al. 2003) which cannot be sufficiently screened by counterions of the buffer solution (sodium ions) or that, in other words, the thickness of the primary EDL remains comparable with the intraparticle pore size; the counterionic mobile space charge is more or less smeared out over the whole cross-section of a pore by the thermal motion of the ions. Thus, the mesopores and even the macropores of the glass bead under the given conditions are selective for counterions, i.e., they enrich counterionic species and exclude co-ions. At electrochemical equilibrium an electrical phase boundary potential (the so-called Donnan potential) between the particle phase and bulk solution phase balances the tendency of ionic species to level out the chemical potential gradients (Helfferich 1995).

An increase of the ionic strength at a given pore size (Fig. 3a: macroporous glass bead, dintra = 122 nm, counterionic tracer; Fig. 3b: mesoporous glass bead, dintra = 20 nm, co-ionic tracer) or, vice versa, an increase of pore size at a constant ionic strength (Fig. 3c; ionic strength: 10−3 M) reduces the intraparticle pore-scale EDL interaction and cation-selectivity of the glass beads, meaning that the amplitudes of the intraparticle tracer profiles (counterion enrichment; co-ion exclusion) at equilibrium decrease. When the thickness of the primary EDL is much smaller than the intraparticle pore size, the glass bead contains only quasi-electroneutral pore liquid and, as a whole, is charge-nonselective. As seen in Fig. 3a with the largest pore size (122 nm) and highest ionic strength (10−2 M) for counterionic tracer, the glass beads cation-selectivity has become so small that the tracer profile exhibits decreased amplitude with respect to the bulk extraparticle solution. This is caused by the fact that a glass bead naturally contains unpenetrable solid and, consequently, only a reduced void space (about 70% of the bead volume, namely its porosity) with respect to bulk solution surrounding the bead.

3.2 Electrical field-induced concentration polarization

3.2.1 Influence of hydraulic flow rate

The uniform species distribution inside and around a bead at electrochemical equilibrium (as reflected by the tracer profiles in Fig. 3) without applied field breaks down as an electrical field is superimposed (Fig. 4; see also Fig. 1a, b). The schematic in Fig. 4 illustrates the operation of both pressure and electrical potential gradients within the channel containing a single cation-selective glass bead fixed in bulk electrolyte solution. As seen in Fig. 4a through the eyes of the co-ionic tracer (twice negatively charged BODIPY-disulfonate®) and in Fig. 4b via counterionic tracer (single positively charged Rhodamine 6G), CP is already induced by an applied field strength of Eext = 2.5 kV/m. It is characterized by a depleted CP zone along the anodic hemisphere of the cation-selective glass bead and an enriched CP zone along its cathodic hemisphere. Further, as most clearly seen for the counterionic tracer (Fig. 4b), the intraparticle profile has become tilted with respect to Fig. 3a (Eext = 0 kV/m) because of diffusive backflux into the bead from the enriched CP zone.
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Fig. 4

Stationary fluorescence intensity profiles through a spherical glass bead from its upstream anodic to its downstream cathodic pole and the adjacent bulk electrolyte solution for a) co-ionic tracer (twice negatively charged BODIPY-disulfonate®), and b) counterionic tracer (single positively charged Rhodamine 6G). Rate of volumetric hydraulic flow (from left to right): 20 μl/min; Eext = 2.5 kV/m; dp = 175 μm; dintra = 20 nm; ionic strength: 10−3 M. The intraparticle pore space is indicated by the dashed lines. Fluorescence intensities are normalized with respect to the bulk solution surrounding the bead

Both CP zones are affected by the presence of net hydraulic flow through the channel. While the depleted CP zone (around the upstream hemisphere of a bead; Fig. 5a) is compressed against the anodic external surface of the bead, the enriched CP zone (around the downstream hemisphere of a bead) becomes a considerably extended and streamlined tail (Figs. 5a and 6). This is influenced by the flow rate (while Eext = const. = 2.5 kV/m), illustrated in Fig. 5b and c for the depleted CP zone and in Fig. 6 for the enriched CP zone. As demonstrated by Fig. 5b, the electrical field-induced depleted CDL can be analyzed in detail by the visualisation approach employed in this work, e.g., for determination of its local thickness (δCDL) around the anodic hemisphere of the bead (Fig. 5a), or for resolving the dependence of δCDL on hydraulic flow rate at a given field strength (Fig. 5c).
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Fig. 5

Visualization and analysis of CP around a spherical glass bead using the co-ionic fluorescent tracer (twice negatively charged BODIPY-disulfonate®). a Slice through the center of the bead including its anodic and cathodic poles; numbers indicate the thickness of the CDL (δCDL) at selected positions from the anodic pole towards the equator. Rate of volumetric hydraulic flow (from left to right): 10 μl/min; Eext = 2.5 kV/m; dp = 175 μm; dintra = 20 nm; ionic strength: 10−3 M. b Estimation of δCDL from the distributions of co-ionic tracer with and without electrical field; flow rate: 5 μl/min. c Dependence of δCDL at the beads anodic pole (α = 0°) on flow rate; Eext = const. = 2.5 kV/m. The line is drawn as a guide to the eye

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Fig. 6

Influence of volumetric hydraulic flow rate on the appearance of the considerably streamlined enriched CP zone (cf. Fig. 5a where the enriched CP zone is cut off); Eext = const. = 2.5 kV/m. The extracted profiles correspond to the dotted line in the upper image. Glass bead and other conditions as in Fig. 5

While the considerably streamlined enriched CP zone is thinned out at increasing velocity (see images in Fig. 6), it extends significantly downstream of a bead (on the scale of one bead diameter; see profiles in Fig. 6). This dimensional analysis implies that strong interaction occurs between the CP zones of different particles in packed beds (extinguishing effects in dense multiparticle systems), i.e., as individual beads are brought into closer contact (Leinweber and Tallarek 2005; Nischang et al. 2006a; Nischang and Tallarek 2007). In other words, the enriched CP zones feed high ionic strength into the depleted CP zones of downstream particles, which leads to a reduced local and overall intensity of the CP phenomenon in packed beds with respect to a scaling of CP and related effects based on the undisturbed single-particle picture.

In particular, CP-based nonlinear EOF (electroosmosis of the second kind) in packed beds is significantly weaker than expected from a local analysis close to a single-free particle (Mishchuk and Dukhin 2002; Mishchuk and Takhistov 1995). On the other hand, a more dilute packing consisting of, e.g., a single layer of fixed particles with tailored arrangement of individual beads (such that this reduces the adverse, compensating interactions between neighbouring CP zones) may be realized by microfabrication and micromanipulation. The resulting nonlinear EOF is generally attractive in a number of microfluidic applications, e.g., for improved micropumps and electrochromatographic separations in view of overcoming classical, typically not very high, linear EOF velocities (Nischang and Tallarek 2007).

3.2.2 Influence of electrical field strength

Figures 7, 8, 9 illustrate unique phenomena that occur in the depleted CP zone as the electrical field strength is increased at constant volumetric hydraulic flow rate (cf. Fig. 1b, c). Figure 7 demonstrates that stationary enriched and depleted CP zones are formed at Eext = 2.5 kV/m (Fig. 7b) compared to the situation without applied field (Fig. 7a). Particularly, the depleted CP zone is stable and smooth along the anodic hemisphere of the bead (cf. Fig. 4). At higher field strength, however, spatio-temporal fluctuations occur in the depleted CP zone which significantly distort this CDL locally (Fig. 7c, d). These distortions first appear close to the anodic pole of the bead (Fig. 7c; Eext = 5 kV/m), but at still higher field strength (Eext = 7.5 kV/m), chaotic fluctuations exist throughout the whole CDL (Fig. 7d). They effectively destroy the depleted CP zone locally. Indeed, in Fig. 7d the physical boundaries of the bead become discernible again (cf. Fig. 7a; Eext = 0 kV/m) because the CDL exists only partly along the beads anodic hemisphere; it continuously breaks down and tends to be rebuilt in a highly chaotic fashion. While Fig. 7a, b represent stationary distributions, Fig. 7c, d are snapshots of fluctuating patterns which were observed up to the maximum measurement time of about half an hour. In addition, while the depleted CP zone is spatio-temporally destroyed (Fig. 7d) the enriched CP zone becomes more intense, as seen in the profiles. Evidently, the diffusion-limited transport through the depleted CP zone is removed locally, allowing higher current densities through the bead.
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Fig. 7

Visualization and analysis of CP around a spherical glass bead using the co-ionic fluorescent tracer (twice negatively charged BODIPY-disulfonate®). Slices are taken through the center of the bead including its anodic and cathodic poles (center column of images). In addition, the anodic and cathodic bead-solution interfaces are enlarged in the left and right columns, respectively. Rate of volumetric hydraulic flow (from left to right): 20 μl/min; electrical field strength and current as indicated; dp = 200 μm; dintra = 20 nm; ionic strength: 10−3 M. The extracted profiles correspond to the dotted lines in the center images

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Fig. 8

Spatial distribution of the co-ionic fluorescent tracer in the enriched and depleted CP zones around a spherical glass bead and adjacent bulk electrolyte solution in dependence of the applied field strength (as indicated). The extracted profiles correspond to the dotted lines in the images. Glass bead and other conditions as in Fig. 7

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Fig. 9

Temporal distribution of the co-ionic fluorescent tracer in the depleted CP zone opposite to the anodic pole of a spherical glass bead (see white circle). Electrical field strength as indicated. Glass bead and other conditions as in Fig. 7

Thus, as the applied field strength is increased from 0 kV/m (Fig. 7a) a stationary, smooth depleted CP zone first develops (Fig. 7b; Eext = 2.5 kV/m) which, at higher field strength, breaks down locally due to highly chaotic motion within this CDL (Fig. 7d; Eext = 7.5 kV/m). In contrast to the depleted CP zone, the enriched CP zone remains stationary and laminar, but continues to become more intense while the field strength increases, particularly, as the depleted CP zone breaks down. Accordingly, the back-diffusion into the bead from the intensifying enriched CP zone also increases (see profiles in Fig. 7).

Figure 8 compares lateral tracer profiles through the enriched and depleted CP zones adjacent to the fixed glass bead which correspond to the field strengths in Fig. 7. These data demonstrate that the depleted CP zone becomes unstable above a field strength of ca. 2.5 kV/m (Fig. 8a; Eext = 2.5–5 kV/m). For the highest field strength (7.5 kV/m) it can be clearly seen that the spatio-temporal tracer intensity in the CDL (at a lateral distance of about 90 μm; blue profile) increases up to that level observed throughout the extraparticle space when no electrical field is applied (black profile). Thus, the electrical field-induced formation of an extended bulk fluid-side region adjacent to the particle through which coupled mass and charge transport towards the particle becomes diffusion-limited (cf. Fig. 1b) is effectively, yet locally and momentarily counteracted. As a result of these chaotic fluctuations, the originally stationary depleted CP zone (red profile in Fig. 8a; Eext = 2.5 kV/m) is strongly disturbed and the mass transfer limitation reduced or even removed locally. Fig. 8b shows that the enriched CP zone during this scenario remains stationary and laminar, but that part of the upstream depleted CP zone, most probably due to the significant hydraulic flow through the device and around the bead (corresponding to a linear velocity of about 1.1 mm/s), is flushed downstream and thereby enveloping the enriched CP zone, as evidenced by the local minima in the tracer profiles. This is already clearly visible in the center images of Fig. 7b–d, reflected by the dark envelopes around the bead.

Figure 9 complements the foregoing analysis by recording signal intensity over time from a region in the bulk liquid immediately adjacent to the anodic pole of the bead (indicated by the white circle in the image), while the field is switched on and CP develops. For each field strength (up to 12.5 kV/m) the intensity first decreases indicating the formation of a depleted CP zone. However, while for Eext = 2.5 kV/m (red profile) this decrease levels off to approach a constant intensity, which represents the formation of a stationary and stable depleted CP zone (cf. Figs. 7b and 8a), we observe highly chaotic fluctuations for Eext = 12.5 kV/m (Fig. 9; gray profile) after a short period needed to establish CP for the first time. These oscillations occur between the following two extreme cases: (1) no depleted CP zone present (Eext = 0 kV/m; black profile) and (2) stationary, laminar depleted CP zone formed (Eext = 2.5 kV/m; red profile). It demonstrates that the driving forces for building up the depleted CP zone certainly continue to exist, but that a new mechanism, operative at higher field strength and responsible for this chaotic motion in the liquid close to the anodic hemisphere of the bead, effectively counteracts the formation of a stable CDL. When aggravated at increasing field strength, this finally results in a highly chaotic switch between build up and break down of CP (Eext = 12.5 kV/m). At the intermediate field strengths (5 and 7.5 kV/m; green and blue profiles, respectively) intense fluctuations as for Eext = 12.5 kV/m are not discernible, but the actual motion and mixing which takes place in the CDL already at these fields strengths results in a higher intensity with respect to Eext = 2.5 kV/m, implying that CP is effectively not as strong.

3.3 Closed device

For a comparison with the observations made under flow-through conditions (Section 3.2; both pressure and electrical potential gradients are applied, resulting in net hydraulic flow through the device and electrical field-induced CP around the bead), Fig. 10 illustrates the phenomena that occur around a bead under closed-device conditions, i.e., as inlet and outlet are closed. Thus, no net flow through the device can occur. The images in Fig. 10 demonstrate the development of a microvortex flow. As the electrical field with 2.5 kV/m is applied CP first establishes around the bead (t = 2 s); enriched and depleted CP zones are clearly recognizable and early signs of the microvortex become visible by a revolving motion of liquid starting close to the equator of the bead (see also red profile in the graph for t = 2 s). This motion continues (t = 4 s) leading to completely recirculating flow and a clearly visible microvortex pattern (t = 20 s) which was observed up to the maximum measurement time of a few (4–5) min. Yet, images became increasingly blurred with time due to exchange by diffusion of fluorescent tracer between different streamlines. Simultaneously, microvortex flow similar to that in Fig. 10 develops along the other, mirrored side of the beads anodic hemisphere (in the two-dimensional picture). The almost symmetrical events reflect the mirror symmetry of this fixed-bead assembly (with a mirror plane from anode to cathode, through the center of the bead) and result in ideally symmetrical microvortex flow at both free sides of the fixed bead.
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Fig. 10

Generation of electrohydrodynamic nonlinear microvortex flow under closed-device conditions based on the nonequilibrium electroosmotic slip along the anodic hemisphere of a spherical glass bead. The visualization utilizes the counterionic fluorescent tracer (single positively charged Rhodamine 6G). Inlet and outlet of the device (cf. Fig. 2 ) are closed; Eext = 2.5 kV/m; dp = 175 μm; dintra = 20 nm; ionic strength: 10−3 M. The extracted profiles correspond to the dotted line in the image for t = 20 s

We have also fixed irregular-shaped beads (see Sect. 2) in the microfluidic device. It removes this mirror symmetry and, consequently, microvortices on different sides of an irregular bead generally have a different dimension (data not shown here). This depends on the bead orientation with respect to the field direction. As expected, with spherical beads orientation did not visibly affect the symmetry of electrohydrodynamic flow around a bead.

The vortex depicted in Fig. 10 (t = 20 s) basically consists of parts of the depleted and enriched CP zones, which are simply wrapped by liquid motion normal to the equator of the bead (away from its surface) where the CP zones meet. This is clearly seen in the image taken 20 s after switching on the electrical field, as well as in the time-series leading to that situation. Also the extracted profile for t = 20 s (blue) confirms that the higher-intensity and lower-intensity regions of the vortex (with respect to the bulk solution; see black profile for t = 0 s) originate in the enriched and depleted CP zones, respectively. Thus, in contrast to the open device electroosmosis of the second kind along the beads anodic hemisphere in the closed device leads to extended microvortex flow (Mishchuk and Dukhin 2002; Mishchuk and Takhistov 1995). Recirculating motion of liquid associated with the microvortex flow gives the enriched CP zone a considerably streamlined shape (cf. image for t = 6 s; Fig. 10) which is not yet developed at t = 2 s when the microvortex flow is also not fully established.

4 Discussion

The electrokinetic and hydrodynamic phenomena visualized and analyzed in Figs. 4, 5, 6, 7, 8, 9, 10 are now explained and discussed further in view of electrical field-induced CP at the curved surface of a charge-selective glass bead (counterionic conductor). Basic features of coupled mass and charge transport through a curved counterion-selective solid-liquid interface were already stepwise pointed out and explained by Fig. 1, which helps to retrieve the effects of individual phenomena in the experiments.

Prerequisite for the electrical field-induced CP around a glass bead (Fig. 4) is the charge-selectivity of a bead (cf. Figs. 1a and 3) based on the intraparticle surface charge which is insufficiently screened by mobile counterions from the electrolyte solution. While the classical theory of CP assumes ideal-permselectivity, i.e., charge transport in a bead is exclusively achieved by the counterion(s), because co-ions are simply not present in the ion-permselective region, except for fixed surface charges, experimental conditions in this work are nonideal in this respect. They are tunable via EDL overlap (Donnan-exclusion of co-ions) by adjusting the mobile phase ionic strength and/or intraparticle pore size, e.g., as ionic strength (or intraparticle pore size) is decreased the counterion-selectivity of a bead increases (Fig. 3). Consequently, in terms of a simple control experiment (data not shown here), CP and related effects were not observed as a nonporous (solid, dielectric) glass bead (see Sect. 2) had been fixed in the microfluidic channel and similar conditions realized as in Figs. 4, 5, 6, 7, 8, 9, 10.

The most interesting and, until today, least understood phenomena are associated with the transition from Fig. 1b to c, stipulating a complex electrohydrodynamics not explained by the classical theory of CP (Probstein 1994). Only little work exists on their direct and quantitative visualization. This transition (Fig. 1b, c) reflects a violation of the local electroneutrality in the depleted CP zone and adjoining charge-selective domain (Fig. 1c) while the field strength (Eext) continues to increase. As a result, a secondary EDL is induced by the local normal component of the applied field (En) at the charge-selective interface where counterions enter the charge-selective pore space, e.g., that of the glass bead, in the direction of the applied field. For spherical beads investigated in this work it means that the intensity of the field-induced EDL along the anodic hemisphere of the bead becomes a function of the angle with respect to Eext (see definition of α in Fig. 5a). In other words, En is maximal at the anodic pole of the bead (α = 0°), while it is zero at the equator (α = 90°). Vice versa, the tangential field component (Et) is zero at the pole, but maximal at the equator and interacts locally with the fluid-side space charge of mobile counterions (see SCR in Fig. 1c), leading to electroosmosis of the second kind close to the anodic surface of a bead. This induced-charge electroosmosis is characterized by nonlinear local slip velocities because the electrical potential drop in the secondary EDL depends on the magnitude of En, thus, on applied field strength, which is in contrast to the classical zeta-potential characterizing the primary EDL (Ben and Chang 2002; Leinweber and Tallarek 2004; Mishchuk and Dukhin 2002; Mishchuk and Takhistov 1995).

Consequences of this transition (Fig. 1b, c) for the electrohydrodynamics around the glass bead under closed-device conditions (Fig. 10) are the following. Due to the beads curvature CP is weakest along the equator (α = 90°) and strongest at the anodic pole (α = 0°). The nonlinear slip velocity, which scales as sinα(cosα at large fields, has a maximum at α = ¼π. As such, the tangential flow velocity has a negative downstream gradient for ¼π < α < ½π. By continuity, an outward radial flow must then result from the bead. This, in turn, produces a back-pressure gradient due to flow imbalance away from the bead and drives the microvortex (Ben and Chang 2002; Mishchuk and Takhistov 1995). To our knowledge, the sequence of images in Fig. 10 is the first report on the direct visual verification of the early stages of this microvortex flow generation. In particular, the initial ejection of liquid radially outwards, away from the equator of the bead, is clearly visible for t = 2 s. This revolving motion then continues to form a closed microvortex. Consequently, the eye of the vortex (region of zero intensity; see blue profile in Fig. 10) and the high-intensity region (peak in the blue profile) are supplied by the depleted and enriched CP zones, respectively, and exchange between these regions occurs only by diffusion perpendicular to the streamlines.

In contrast to the closed device (Fig. 10), the hydraulic flow through the open device (Figs. 4, 5, 6, 7, 8, 9) provides a dominating hydrodynamics around the bead. In particular, because CP results from the local interplay of diffusion, convection, and electromigration, the hydraulic flow leads to a compressed depleted CP zone along the anodic hemisphere of the bead with a well defined thickness (Figs. 4 and 5), being an inverse function of the velocity (Fig. 5c), as well as a considerably streamlined enriched CP zone (Fig. 6). As a consequence of strong hydraulic flow the depleted CP zone is partly flushed around the bead, towards the cathodic hemisphere (Figs. 7 and 8); large microvortices as in the closed device (Fig. 10) are not generated under these conditions. This complements earlier investigations with closed devices where microvortex flow as in Fig. 10 was always produced (Ben and Chang 2002; Mishchuk and Dukhin 2002; Mishchuk and Takhistov 1995) and means that the superimposed hydraulic flow can suppress the formation of closed whirls which starts with an ejection of liquid radially outwards, away from the equator of the bead, in the closed device (see Fig. 10; t = 2 s). The formation of large microvortices would require that the revolving motion due to electroosmosis of the second kind (Fig. 10; t = 4 s) survives against the countercurrent hydraulic flow.

In addition to the nonlinear microvortex flow in Fig. 10 (Ben and Chang 2002), clear evidence for a CP-based nonequilibrium electroosmotic slip close to the anodic hemisphere of a glass bead, within the depleted CP zone, stems from the chaotic fluctuations observed in this CDL (Figs. 7, 8, 9). To understand this behaviour better, it is helpful to recall associated events at a charge-selective membrane. This parallel is also instructive because the sphere and membrane geometries necessarily share common features concerning CP. In addition, the membrane has been investigated in more detail, particularly in the context of electrodialysis (Belova et al. 2006; Rubinstein and Zaltzman 1999; Rubinstein et al. 2002), while the single-sphere geometry was previously studied only under closed-device conditions (Ben and Chang 2002; Dukhin and Mishchuk 2002; Mishchuk and Takhistov 1995). Thus, from this point of view also the current work, which realizes both macroscopic pressure and electrical potential gradients (see schematic in Fig. 4), provides new insight into the electrohydrodynamics in the single-sphere system under more general conditions relevant to microfluidic applications where a net flow through the system prevails, as in microscale analysis and separation devices (Höltzel and Tallarek 2007), or in which the large microvortices must be destroyed to achieve more efficient mixing of liquid (Wang et al. 2004).

As an electrical field is applied to a highly charge-selective membrane the transport of the electrical current (trans-membrane current) is realized nearly exclusively by the counterions. The transport number of the counterion (or the sum of the transport numbers of all counterionic species) is nearly unity, i.e., the fractional electrical current carried by the counterions within the membrane phase is much larger than the corresponding value within the bulk liquid phases (Helfferich 1995). As a consequence, concentration gradients of ionic species build up at both membrane-solution interfaces; enriched and depleted CP zones form in the bulk, quiescent electrolyte solutions adjacent to cathodic and anodic interfaces of a cation-selective membrane, respectively or, vice versa, at anodic and cathodic interfaces of an anion-selective membrane. For sufficiently low field strengths the stationary voltage-current behaviour follows Ohms law. At increasing field strength, the concentration of ions in the depleted CP zone is reduced towards zero; diffusion-limited transport through this boundary layer approaches a maximum value that, in the classical description of CP, particularly in a context of electrodialysis, is known as limiting current density (Probstein 1994). In this plateau regime the slope is much smaller than in the Ohmic (linear) regime.

The gain of realizing higher current densities than associated with the plateau has stimulated much research on CP. One possibility is to reduce the thickness of the depleted CP zone by stirring (Manzanares et al. 1991). On the other hand, overlimiting current densities through ion-exchange membranes have also been realized with macroscopically quiescent electrolytes by just a further increase in voltages in the plateau regime. The transition to the overlimiting conduction regime is typically accompanied by the appearance of low-frequency excess electrical noise (Rubinstein et al. 2002). A visualisation of the hydrodynamics in this regime has revealed strong fluctuations in the adjoining solution indicating some convective mixing which spontaneously develops in the depleted CP zone (Li et al. 1983).

This chaotic convection has been attributed to gravitational buoyant forces resulting from concentration and temperature gradients in the depleted CP zone. Other arguments, however, suggest that the conditions for creating gravitational instability in the CP zones are not fulfilled (Belova et al. 2006; Rubinstein and Zaltzman 1999). Instead, nonequilibrium electroosmotic slip (electroosmosis of the second kind) was theoretically proposed as a mechanism for the observed chaotic convection; it has been shown that this slip yields an instability of quiescent CP at a homogeneous membrane (Rubinstein and Zaltzman 2000; Rubinstein et al. 2002, 2005). Electroconvection, developing from this instability, results in a destruction of the depleted CP zone causing overlimiting conduction. Not far above the instability threshold, steady-state electroconvective vortices start oscillating in a periodic manner. At higher voltages oscillations soon become chaotic, resulting in the low-frequency excess electrical noise typical of the overlimiting conductance (Rubinstein et al. 2002). Recent experiments by Belova et al. (2006), which systematically excluded the effects of water splitting and gravitational convection, have demonstrated that the main mechanism of overlimiting conductance in narrow membrane cells with low ionic strengths and high applied voltages is electroconvection theoretically described by Rubinstein and Zaltzman (2000) and Rubinstein et al. (2002). In particular, the development of oscillations in the trans-membrane current at increasing overpotential excellently supports the theoretically predicted scenarios assuming that this electroconvection occurs as a nonequilibrium electroosmotic slip (Belova et al. 2006).

This dynamics is influenced by the surface heterogeneity (Belova et al. 2006; Ibanez et al. 2004; Volodina et al. 2005), which facilitates the generation of electroconvection via local tangential electrokinetic forces acting parallel to the “flat” membrane surface (cf. Fig. 1c). For heterogeneous membranes a thresholdless development of the microvortex flow is predicted (Rubinstein and Zaltzman 2000; Rubinstein et al. 2002). On the other hand, it should be noted that even close to a flat homogeneous membrane the SCR (induced by the normal component of the applied field) cannot remain quiescent and uniform. First, the SCR is always nonuniform along the distance normal to the membrane surface and, second, nonuniformities arise spontaneously due to spatio-temporal current fluctuations. Further, the applied electrical force need not necessarily be tangential to the surface. When an electrical force perpendicular to the membrane acts on the induced space charge, an excess pressure develops within the SCR which displaces liquid laterally outwards from the SCR, parallel to the membrane surface. While moving, the displaced liquid experiences inertial resistance from the nonslipping liquid which redirects the moving liquid towards the bulk solution, away from the surface. As a result, a pair of counter-rotating vortices appears causing mixing of the depleted CP zone (Rubinstein and Zaltzman 2000). The presence of electrical nonuniformity on the membrane surface, e.g., due to surface curvature facilitates the development of the nonequilibrium electroosmotic slip (Belova et al. 2006; Mishchuk et al. 2001; Rubinstein and Zaltzman 2000). This, in turn, can significantly shift the onset of overlimiting conductance towards lower voltages (i.e., decrease the plateau length) and, thus, increase electrical current compared to a flat homogeneous membrane (Ibanez et al. 2004). Evidently, the distribution of the SCR near a heterogeneous membrane is nonuniform (Mishchuk and Dukhin 2002; Rubinstein and Zaltzman 2000).

The introduction of surface curvature into the membrane geometry ultimately leads us to the sphere geometry. By returning to the phenomena in Figs. 7, 8, and 9 and the dependence on applied field strength, it appears that the development of chaotic convection in the depleted CP zone for the sphere geometry (in particular, see the intensity profiles in Fig. 9) reflects qualitatively the events for a heterogeneous membrane reported before in the framework of nonequilibrium electroosmotic slip. In agreement with the curved surface of the bead fluctuations start where CP is most intense due to the strongest normal components of the applied field, that is, close to the anodic pole of the cation-selective bead (see anodic hemisphere in Fig. 7c). Already this mixing results in an increase of the average local intensity in the depleted CP zone in immediate proximity to the anodic pole of the bead (green profile in Fig. 9; Eext = 5 kV/m).

After switching on this field strength (5 kV/m; Fig. 9), the intensity decreases faster than for Eext = 2.5 kV/m, but nearly reaches the level (green profile; t ≈ 0.5 s) corresponding to the asymptote for Eext = 2.5 kV/m (red profile). Instead of becoming constant, it soon increases again to level off at a significantly higher value than for Eext = 2.5 kV/m. Further, the noise amplitude in this steady-state (green profile) has increased with respect to the same profile at t < 1 s, as well as the steady-state for Eext = 2.5 kV/m (red profile).

For Eext = 7.5 kV/m (blue profile in Fig. 9) this trend continues, i.e., after switching on the field the intensity first decreases, reaching almost the same minimum at t ≈ 0.5 s as for Eext = 5 kV/m), and earlier increases towards a higher level than for Eext = 5 kV/m. In addition, the oscillations around some appropriate mean value are chaotic (blue profile). With increasing field strength fluctuations in the depleted CP zone turn highly chaotic and finally become so intense (gray profile in Fig. 9; Eext = 12.5 kV/m) that the local intensity adjacent to the anodic pole of the bead covers the complete range spanned by the distributions at Eext = 0 kV/m (maximum intensity; no CP existing) and Eext = 2.5 kV/m (minimum intensity; strong CP, without mixing in the CP zone). Further, the chaotic fluctuations then exist along the whole anodic hemisphere of the bead (Fig. 7d). Evidently, the extracted profile in Fig. 7 for Eext = 7.5 kV/m showing the intense enriched CP zone, together with the destruction of the depleted CP zone (Fig. 7d), reflects overlimiting conduction in the single-sphere geometry.

Compared to the microvortex generation at Eext = 2.5 kV/m in the closed device (Fig. 10), no signs of electroosmosis of the second kind are discernible in the open device at Eext = 2.5 kV/m (Figs. 7b, 8a, and 9). This is caused by the strong hydraulic flow, which dominates the hydrodynamics around a bead and compresses the depleted CP zone against the beads anodic hemisphere (Figs. 4 and 5). In the closed device, by contrast, CP begins to develop in a macroscopically quiescent solution, but finally the thickness of the depleted CP zone is related to the mixing of liquid based on electroosmosis of the second kind (Mishchuk et al. 2001). Due to a compressed CDL in the open device boundary-layer transport with respect to intraparticle electrokinetic transport (cf. Fig. 1) becomes limiting at higher field strength (Leinweber and Tallarek 2004). This is consistent with the observations in Figs. 7, 8, 9, revealing the first signs of electroosmosis of the second kind at a higher field strength (around Eext = 5 kV/m) than for the closed device. In addition, the appearance of the induced space charge (cf. Fig. 1c) is not manifested in the formation of large microvortices as in the closed device (Fig. 10), but results in local recirculating motion on much smaller length scale, beginning close to the beads’ anodic pole where CP is strongest (Figs. 7c and 8a). Thus, the hydrodynamics imposed by the hydraulic flow significantly affects the spatio-temporal pattern of the electrohydrodynamics. Yet, characteristic features of electroosmosis of the second kind in the membrane geometry and the single-sphere system (under closed-device conditions) are essentially recovered.

To conclude, this work demonstrates by direct visualization and the corresponding real-time analysis that the application of electrical fields to a single-fixed, ion-permselective glass bead produces a remarkable complexity concerning the coupled mass and charge transport through the bead itself, as well as the coupled electrokinetics and hydrodynamics in the adjoining bulk electrolyte solution. The employed visualization approach enables the acquisition of a wealth of information which forms the basis for further theoretical refinement of the underlying effects and understanding of the electrohydrodynamic phenomena at charge-selective interfaces under more general conditions. For example, an important result of the present work is the dimensional analysis of CP zones around a single ion-permselective glass bead and instability analysis of the depleted CP zone under the action of both applied pressure and electrical potential gradients. The realized device is flexible and allows to investigate the electrohydrodynamics in both transient and stationary regimes, thereby addressing the influence of particle or surface shape (and surface heterogeneity, in general), multiparticle arrangement, intraparticle pore size and surface charge density, mobile phase composition, and applied volume forces. It is also useful in studying related effects at membranes in dependence of surface heterogeneity.

This fundamental insight into coupled mass and charge transport through charge-selective interfaces and, particularly, into the coupled electrokinetics and hydrodynamics adjacent to these interfaces is further important for a variety of multifunctional microfluidic/nanofluidic devices which have been constructed recently. Specifically, it addresses the ionic conductance of discrete nanochannels, as well as nanoporous separation and preconcentration units contained as hybrid configurations, membranes, packed beds, or monoliths in microscale liquid phase analysis systems (Höltzel and Tallarek 2007). A remarkable complexity arises in these devices as external electrical fields are superimposed on the internal chemical and electrical potential gradients for tailoring molecular transport. Charge transport becomes more complex at this emerging interface between the usually discrete spatial domains of micrometer-sized and nanometer-sized pores or channels (cf. Fig. 1), where microfluidics meets nanofluidics. In this context, an understanding of morphology-related transport in the internal and external electrical potential gradients is critical to the performance of a device concerning the more efficient manipulation and analysis of chemical and biological species (Höltzel and Tallarek 2007).

Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (Bonn, Germany) under grants TA 268/2 and HL 56/1, as well as by the Fonds der Chemischen Industrie (Frankfurt a.M., Germany).

Copyright information

© Springer-Verlag 2007