Electrokinetics in nanochannels: part I. Electric double layer overlap and channel-to-well equilibrium

In this paper a new model is described for calculating the electric potential field in a long, thin nanochannel with overlapped electric double layers. Electrolyte concentration in the nanochannel is predicted self-consistently via equilibrium between ionic solution in the wells and within the nanochannel. Differently than published models that require detailed iterative numerical solutions of coupled differential equations, the framework presented here is self-consistent and predictions are obtained solving a simple one-dimensional integral. The derivation clearly shows that the electric potential field depends on three new parameters: the ratio of ion density in the channel to ion density in the wells; the ratio of free-charge density to bulk ion density within the channel; and a modified Debye-Hückel thickness, which is the relevant scale for shielding of surface net charge. For completeness, three wall-surface boundary conditions are analyzed: specified zeta-potential; specified surface net charge density; and charge regulation. Predictions of experimentally observable quantities based on the model proposed here, such as depth-averaged electroosmotic flow and net ionic current, are significantly different than results from previous overlapped electric double layer models. In this first paper of a series of two, predictions are presented where channel depth is varied at constant well concentration. Results show that under conditions of electric double layer overlap, electroosmosis contributes only a small fraction of the net ionic current, and that most of the measurable current is due to ionic conduction in conditions of increased counterion density in the nanochannel. In the second of this two-paper series, predictions are presented where well-concentration is varied and the channel depth is held constant, and the model described here is employed to study the dependence of ion mobility on ionic strength, and compare predictions to measurements of ionic current as a function of channel depth and ion density.     

Electrohydrodynamics of charge separation in droplet-based ion sources with time-varying electrical and mechanical actuation

Charge transport and separation in mechanically-driven, droplet-based ion sources are investigated using computational analysis and supporting experiments. A first-principles model of electrohydrodynamics (EHD) and charge migration is formulated and implemented using FLUENT CFD software for jet/droplet formation. For validation, classical experiments of electrospraying from a thin capillary are simulated, specifically, the transient EHD cone-jet formation of a fluid with finite electrical conductivity, and the Taylor cone formation in a perfectly electrically-conducting fluid. The model is also used to investigate the microscopic physics of droplet charging in mechanically-driven droplet-based ion sources, such as array of micromachined ultrasonic electrospray (AMUSE). Here, AMUSE is subject to DC and AC electric fields of varying amplitude and phase, with respect to a time-varying mechanical force driving the droplet formation. For the DC-charging case, a linear relationship is demonstrated between the charge carried by each droplet and an applied electric field magnitude, in agreement with previously reported experiments. For the AC-charging case, a judiciously-chosen phase-shift in the time-varying mechanical (driving ejection) and electrical (driving charge transport) signals allows for a significantly increased amount of charge, of desired polarity, to be pumped into a droplet upon ejection. Complementary experimental measurements of electrospray electrical current and charge-per-droplet, produced by the AMUSE ion source, are performed and support theoretical predictions for both DC- and AC-charging cases. The theoretical model and simulation tools provide a versatile and general analytical framework for fundamental investigations of coupled electrohydrodynamics and charge transport. The model also allows for the exploration of different configurations and operating modes to optimize charge separation in atmospheric pressure electrohydrodynamic ion sources under static and dynamic electrical and mechanical fields.     

Evidence in support of levitation effect as the reason for size dependence of ionic conductivity in water: a molecular dynamics simulation

We report extensive molecular dynamics simulations of (i) model ions in water at high concentrations as a function of the size and charge of the ion as well as (ii) realistic simulation of Cl- and Br- ions at low concentrations in water at room temperature. We also analyze existing experimental data in light of the results obtained here. The halide ion simulations have been carried out using the interaction potentials of Koneshan et al. (J. Phys. Chem. B 1998, 102, 4193). We compute structural and dynamical properties of ions in water and explore their variation with size and charge of the ion. We find that ions of certain intermediate sizes exhibit a maximum in self-diffusivity in agreement with previous experimental measurements and computer simulations. We analyze molecular dynamics trajectories in light of the previous understanding of the levitation effect (LE) and the recent suggestion that ionic conductivity has its origin in LE (J. Phys. Chem. B 2005, 109, 8120). We report the distribution of void and neck radii that exist amidst water. Our analysis suggests that the ion with maximum self-diffusivity is characterized by a lower activation energy and a single-exponential decay of F(s)(k,t). The behavior of these and other related quantities of the ion with maximum self-diffusivity are characteristic of the anomalous regime of the LE. The simulation results of Br- and Cl- ions in water also yield results in agreement with the predictions of LE. A plot of experimental conductivity data in the literature for alkali ions in water by Kay and Evans (J. Phys. Chem. 1966, 70, 2325) also yields a lower activation energy for the ion with maximum conductivity in excellent agreement with the LE. To the best of our knowledge, none of the existing theories predict a lower activation energy for the ion with maximum conductivity.     

Ion current rectification at nanopores in glass membranes

The origin of ion current rectification observed at conical-shaped nanopores in glass membranes immersed in KCl solutions has been investigated using finite-element simulations. The ion concentrations and fluxes (due to diffusion, migration, and electroosmotic convection) were determined by the simultaneous solution of the Nernst-Planck, Poisson, and Navier-Stokes equations for the two-ion (K+ and Cl-) system. Fixed surface charge on both the internal and external glass surfaces that define the pore structure was included to account for electric fields and nonuniform ion conductivity within the nanopores and electric fields in the external solution near the pore mouth. We demonstrate that previous observations of ion current rectification in conical-shaped glass nanopores are a consequence of the voltage-dependent solution conductivity in the vicinity of the pore mouth, both inside and outside of the pore. The simulations also demonstrate that current rectification is maximized at intermediate bulk ion concentrations, a combination of (i) the electrical screening of surface charge at high concentrations and (ii) a fixed number of charge-carrying ions in the pore at lower concentration, which are physical conditions where the voltage dependence of the conductivity disappears. In addition, we have quantitatively shown that electroosmotic flow gives rise to a significant but small contribution to current rectification.     

Polarization and interactions of colloidal particles in ac electric fields

Micrometer-sized polystyrene particles form two-dimensional crystals in alternating current (ac) electric fields. The induced dipole-dipole interaction is the dominant force that drives this assembly. We report measurements of forces between colloidal particles in ac electric fields using optical tweezers and find good agreement with the point dipole model. The magnitude of the pair interaction forces depends strongly on the bulk solution conductivity and decreases as the ionic strength increases. The forces also decrease with increasing field frequency. The salt and frequency dependences are consistent with double layer polarization with a characteristic relaxation frequency omega(CD) approximately a(2)/D, where a is the particle radius and D is the ion diffusivity. This enables us to reinterpret the order-disorder transition reported for micrometer-sized polystyrene particles [Lumsdon et al., Langmuir 20, 2108 (2004)], including the dependence on particle size, frequency, and ionic strength. These results provide a rational framework for identifying assembly conditions of colloidal particles in ac fields over a wide range of parameters.     

Quantification of pH gradients and implications in insulator-based dielectrophoresis of biomolecules

Direct current (DC) insulator-based dielectrophoretic (iDEP) microdevices have the potential to replace traditional alternating current dielectrophoretic devices for many cellular and biomolecular separation applications. The use of large DC fields suggest that electrode reactions and ion transport mechanisms can become important and impact ion distributions in the nanoliters of fluid in iDEP microchannels. This work tracked natural pH gradient formation in a 100?μm wide, 1?cm-long microchannel under applicable iDEP protein manipulation conditions. Using fluorescence microscopy with the pH-sensitive dye FITC Isomer I and the pH-insensitive dye TRITC as a reference, pH was observed to drop drastically in the microchannels within 1?min in a 3000?V/cm electric field; pH drops were observed in the range of 6-10 min within a 100?V/cm electric field and varied based on the buffer conductivity. To address concerns of dye transport impacting intensity data, electrokinetic mobilities of FITC were carefully examined and found to be (i) toward the anode and (ii) 1 to 2 orders of magnitude smaller than H? transport which is responsible for pH drops from the anode toward the cathode. COMSOL simulations of ion transport showed qualitative agreement with experimental results. The results indicate that pH changes are severe enough and rapid enough to influence the net charge of a protein or cause aggregation during iDEP experiments. The results also elucidate reasonable time periods over which the phosphate buffering capacity can counter increases in H? and OH? for unperturbed iDEP manipulations.     

Modeling and simulation of chemo-electro-mechanical behavior of pH-electric-sensitive hydrogel

A chemo-electro-mechanical multi-field model, termed the multi-effect-coupling pH-electric-stimuli (MECpHe) model, has been developed to simulate the response behavior of smart hydrogels subject to pH and electric voltage coupled stimuli when the hydrogels are immersed in a pH buffer solution subject to an externally applied electric field. The MECpHe model developed considers multiphysics effects and formulates the fixed charge density with the coupled buffer solution pH and electric voltage effects, expressed by a set of nonlinear partial differential governing equations. The model can be used to predict the hydrogel displacement and the distributive profiles of the concentrations of diffusive ionic species and the electric potential and the fixed charge density in both the hydrogels and surrounding solution. After validation of the model by comparison of current numerical results with experiment data extracted from the literature, one-dimensional steady-state simulations were carried out for equilibrium of the smart hydrogels subject to pH and electric coupled stimuli. The effects of several important physical conditions, including the externally applied electric voltage, on the distributions of the concentrations of diffusive ionic species, the electric potential, the fixed charge density, and the displacement of the hydrogel strip were studied in detail. The effects of the ionic strength on the bending deformation of the hydrogels under the solution pH and electric voltage coupled stimuli are also discussed.     

Effects of ampholyte concentration on protein behavior in on-chip isoelectric focusing

The effects of mobility corrections on carrier ampholytes are studied at various ampholyte concentrations to understand protein behavior during IEF. IEF simulations are conducted in the presence of 25 biprotic carrier ampholytes within a pH range of 6-9 after applying the Onsager-Debye-Hückel correction to the carrier ampholytes. Two model proteins with ten charge states but without ionic strength corrections are allowed to focus under an electric field of 300 V/cm in a 1 cm long channel. The IEF simulation results show that higher ionic strengths (50 - 100 mM) cause significant changes in the transient movement as well as the final focused profiles of both ampholytes and proteins. The time required for a single, well-defined peak to form increases with ionic strength when Onsager corrections are applied to the carrier ampholytes. For a particular ampholyte concentration, the space-averaged conductivity does not change during the final focusing stage, but the magnitude of space averaged conductivity is different for different ampholyte concentration. The simulation results also reveal that at steady-state ionic strength profiles remain flat throughout the channel except at the locations of proteins where a significant change in ampholyte concentration is obtained.     

Characterization of the surface charge property and porosity of track-etched polymer membranes

As an important property of porous membranes, the surface charge property determines many ionic behaviors of nanopores, such as ionic conductance and selectivity. Based on the dependence of electric double layers on bulk concentrations, ionic conductance through nanopores at high and low concentrations is governed by the bulk conductance and surface charge density, respectively. Here, through the investigation of ionic conductance inside track-etched single polyethylene terephthalate (PET) nanopores under various concentrations, the surface charge density of PET membranes is extracted as ∼−0.021 C/m² at pH 10 over measurements with 40 PET nanopores. Simulations show that surface roughness can cause underestimation in surface charge density due to the inhibited electroosmotic flow. Then, the averaged pore size and porosity of track-etched multipore PET membranes are characterized by the developed ionic conductance method. Through coupled theoretical predictions in ionic conductance under high and low concentrations, the averaged pore size and porosity of porous membranes can be obtained simultaneously. Our method provides a simple and precise way to characterize the pore size and porosity of multipore membranes, especially for those with sub-100 nm pores and low porosities.     

The water-amorphous silica interface: analysis of the Stern layer and surface conduction

To explain why dynamical properties of an aqueous electrolyte near a charged surface seem to be governed by a surface charge less than the actual one, the canonical Stern model supposes an interfacial layer of ions and immobile fluid. However, large ion mobilities within the Stern layer are needed to reconcile the Stern model with surface conduction measurements. Modeling the aqueous electrolyte-amorphous silica interface at typical charge densities, a prototypical double layer system, the flow velocity does not vanish until right at the surface. The Stern model is a good effective model away from the surface, but cannot be taken literally near the surface. Indeed, simulations show no ion mobility where water is immobile, nor is such mobility necessary since the surface conductivity in the simulations is comparable to experimental values.     

Diffusiophoresis in a suspension of charge-regulating colloidal spheres

An analytical study of diffusiophoresis in a homogeneous suspension of identical spherical charge-regulating particles with an arbitrary thickness of the electric double layers in a solution of a symmetrically charged electrolyte with a uniform prescribed concentration gradient is presented. The charge regulation due to association/dissociation reactions of ionogenic functional groups on the particle surface is approximated by a linearized regulation model, which specifies a linear relationship between the surface charge density and the surface potential. The effects of particle-particle electrohydrodynamic interactions are taken into account by employing a unit cell model, and the overlap of the double layers of adjacent particles is allowed. The electrokinetic equations that govern the electric potential profile, the ionic concentration distributions, and the fluid flow field in the electrolyte solution surrounding the particle in a unit cell are linearized assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved with the equilibrium surface charge density (or zeta potential) of the particle as the small perturbation parameter. Closed-form formulas for the diffusiophoretic velocity of the charge-regulating sphere correct to the second order of its surface charge density or zeta potential are derived. Our results indicate that the charge regulation effect on the diffusiophoretic mobility is quite sensitive to the boundary condition for the electric potential specified at the outer surface of the unit cell. For the limiting cases of a very dilute suspension and a very thin or very thick electric double layer, the particle velocity is independent of the charge regulation parameter.     

Electro-osmosis at inhomogeneous charged surfaces: hydrodynamic versus electric friction

Electrokinetic methods are efficient in probing the electrostatic surface properties of charged systems. However, anomalies observed in experiments indicate that the classical electrokinetic theory should be reconsidered. Using Green's function methods and hydrodynamic simulations, we investigate electro-osmosis driven by electric-field-induced ion motion near a charged planar substrate with smooth or rough boundary. First, a reformulation of electro-osmotic theory for planar charged surfaces employing Green's functions shows that the Helmholtz-Smoluchowski (HS) relation between electrostatic potential and solvent velocity is exact for smooth surfaces, even in the presence of ion correlations. Deviations from HS theory are caused by combined hydrodynamic and electric surface friction, as our hydrodynamic simulations of ions at smooth and corrugated charged surfaces in lateral electric fields demonstrate. Within the simulations, hydrodynamic interactions are treated in the continuum limit and the presence of a no-slip boundary condition at the surface is taken into account. While electrofriction is relevant in highly charged system and/or for multivalent ions, hydrodynamic friction is dominant in systems with moderate surface charge density and/or low ionic valency. We also derive the effective electrokinetic surface charge from the electro-osmotic solvent profiles, which is substantially reduced when compared with the bare value and shows qualitative agreement with the experimental tendency.     

Electrical properties of electrochemically doped organic semiconductors using light-emitting electrochemical cells

We present a study of the electrical properties of electrochemically doped conjugated polymers using polymeric light-emitting electrochemical cells (PLECs) and interpreting the results according to a phenomenological model (PM) which assumes that, above the device turn-on voltage, the bulk transport properties of the doped organic semiconductor are responsible for the main contribution to the whole device conductivity. To confirm the predictions of this model, the dependence of the conductivity of PLECs with different parameters is evaluated and compared with the behavior expected for a doped semiconducting polymeric material. The organic semiconductor doping level, the blend concentration of organic semiconducting molecules, the device thickness, the charge carrier mobility, and the temperature are the parameters varied to perform this analysis. We observed that the device conductivity is independent of the active layer thickness, weakly dependent on the temperature, but strongly dependent on the semiconductor doping level, on the semiconductor fraction in the blend, and on the intrinsic charge carrier mobility. These results were well described by the variable range hopping (VRH) model, which has been widely employed to describe the charge transport in doped semiconducting polymeric materials, confirming the prediction of the phenomenological model. The current analysis demonstrates that PLECs are a suitable system for studying, in situ, the electrochemical doping of semiconducting polymers, permitting the evaluation of material properties as, for instance, the density of electronic charge carriers (and, consequently, the ionic charge carrier concentration) necessary to achieve the maximum electrochemical doping level of the organic semiconductor.     

Charge stabilization in nonpolar solvents

While the important role of electrostatic interactions in aqueous colloidal suspensions is widely known and reasonably well-understood, their relevance to nonpolar suspensions remains mysterious. We measure the interaction potentials of colloidal particles in a nonpolar solvent with reverse micelles. We find surprisingly strong electrostatic interactions characterized by surface potentials, |ezeta|, from 2.0 to 4.4 k(B)T and screening lengths, kappa(-1), from 0.2 to 1.4 microm. Interactions depend on the concentration of reverse micelles and the degree of confinement. Furthermore, when the particles are weakly confined, the values of |ezeta| and kappa extracted from interaction measurements are consistent with bulk measurements of conductivity and electrophoretic mobility. A simple thermodynamic model, relating the structure of the micelles to the equilibrium ionic strength, is in good agreement with both conductivity and interaction measurements. Since dissociated ions are solubilized by reverse micelles, the entropic incentive to charge a particle surface is qualitatively changed from aqueous systems, and surface entropy plays an important role.     

Charge reversal in real colloids: Experiments, theory and simulations

In the last years, the inclusion of ionic short-range correlations in the study of colloidal stability has led to significant disagreements with the predictions obtained from classical treatments. An example of these discrepancies is the occurrence of charge reversal of charged particles. In order to shed light on this issue, the charge reversal of latex particles in the presence of asymmetric electrolytes has been investigated through Monte Carlo (MC) simulations. In particular, experimental results concerning electrophoretic mobility reversals and the Hyper-Netted-Chain/Mean-Spherical-Approximation (HNC/MSA) predictions have been compared with simulations in which two alternative methods for evaluating energies have been applied. A realistic hydrated ion size is used in the HNC/MSA calculations and simulations. In this way, the existence of a reversal in the electrophoretic mobility due to ion size correlations and without requiring specific counterion adsorption is probed. Moreover, the simulations appears as a useful tool for explaining those results in which the HNC/MSA does not reproduce the experimental trends.     

Electrophoresis of soft particles with charged rigid core coated with pH-regulated polyelectrolyte layer

We present a theoretical study on the electrophoresis of a soft particle with a dielectric charged rigid core grafted with a charge-regulated polyelectrolyte layer. The polyelectrolyte layer possesses either an acidic or a basic functional group and the charge dissociation depends on the local pH and ionic concentration of the electrolyte. The dielectric rigid core is considered to possess a uniform volumetric charge density. The electric potential distribution is determined by computing the Poisson-Boltzmann equation outside the core coupled with a Poisson equation inside the impermeable core along with suitable matching conditions at the core-shell interface. The computed electric field is used to determine the mobility of the particle through an existing analytic expression based on the Debye-Huckel approximation. Our results are found to be in good agreement with the existing solutions for the limiting cases. The influence of the core charge density, ionic concentration, and pH of the electrolyte on the particle mobility is studied for different choice of hydrodynamic penetration length of the polyelectrolyte and dissociation constant of the functional group. The critical value of the pH required to achieve zero mobility is estimated. We find that in a monovalent electrolyte solution, the soft particle with a net negative (positive) charge can have positive (negative) mobility.     

Physicochemical study of the ‘anatase/electrolyte solution’ interface in suspension and closely packed particles of the solid

The electric double layer (EDL) developed at the interface of anatase in contact with aqueous electrolyte solutions was investigated at 25 °C. Potentiometric titrations (PT), measurements of the electrophoretic mobility (EM) in suspensions, and streaming potential (SP) measurements were taken. The surface charge over a wide pH range (ca. 3–10) and the point of zero charge (pzc = 6.3 ± 0.1) of anatase was easily determined by means of the suspension titration curve and the blank one, obtained at a single ionic strength value. Streaming potential measurements were conducted in anatase particles appropriately packed to form plugs. Two different plugs were prepared differing in the degree of particles' packing and, consequently, in the respective porosities. It was found that surface conductivity is lower at higher packing (lower porosity), because of the reduction of the total surface area in contact with the electrolyte. Moreover, it was found that the surface conductivity of the anatase samples increased at pH values away from pzc, while the mobility of the counter ions behind the shear plane decreased. This trend was attributed to the increase of the absolute surface charge. This increase caused an increase in the amount of the counter ions and, therefore, in the conductivity due to these ions. On the other hand, stronger electrostatic interactions between the surface of the solid and the counter ions reduced their mobility. The packing density of the anatase particles in the respective plugs, affected the values of ζ‐potential calculated from SP measurements when the effect of surface conductivity was neglected. Copyright © 2010 John Wiley & Sons, Ltd.     

The Electrophoretic Mobility and Electric Conductivity of a Concentrated Suspension of Colloidal Spheres with Arbitrary Double-Layer Thickness

The electrophoresis in a monodisperse suspension of dielectric spheres with an arbitrary thickness of the electric double layers is analytically studied. The effects of particle interactions are taken into account by employing a unit cell model, and the overlap of the double layers of adjacent particles is allowed. The electrokinetic equations, which govern the ionic concentration distributions, the electric potential profile, and the fluid flow field in the electrolyte solution surrounding the charged sphere in a unit cell, are linearized assuming that the system is only slightly distorted from equilibrium. Using a perturbation method, these linearized equations are solved with the surface charge density (or zeta potential) of the particle as the small perturbation parameter. Analytical expressions for the electrophoretic mobility of the colloidal sphere in closed form correct to O(zeta) are obtained. Based on the solution of the electrokinetic equations in a cell, a closed-form formula for the electric conductivity of the suspension up to O(zeta(2)) is derived from the average electric current density. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made for both the electrophoretic mobility and the electric conductivity. Copyright 2001 Academic Press.     

Transport properties of the ionic liquid 1-ethyl-3-methylimidazolium chloride from equilibrium molecular dynamics simulation. The effect of temperature

We present here equilibrium molecular dynamics simulation results for self-diffusion coefficients, shear viscosity, and electrical conductivity in a model ionic liquid (1-ethyl-3-methylimidazolium chloride) at different temperatures. The Green-Kubo relations were employed to evaluate the transport coefficients. When compared with available experimental data, the model underestimates the conductivity and self-diffusion, whereas the viscosity is overpredicted, showing only a semiquantitative agreement with experimental data. These discrepancies are explained on the basis of the rigidity and lack of polarizability of the model. Despite this, the experimental trends with temperature are remarkably well reproduced, with a good agreement on the activation energies when available. No significant deviations from the Nernst-Einstein relation can be assessed on the basis of the statistical uncertainty of the simulations, although the comparison between the electric current and the velocity autocorrelation functions suggests some degree of cross-correlation among ions in a short time scale. The simulations reproduce remarkably well the slope of the Walden plots obtained from experimental data of 1-ethyl-3-methylimidazolium chloride, confirming that temperature does not alter appreciably the extent of ion pairing.