Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions

Electrorotation of metallic microspheres

Electrorotation (ER) experiments of gold-coated micrometer-sized spheres suspended in an electrolyte are presented for three different ionic conductivities over the frequency range of 100 Hz to 40 MHz. The direction of rotation was observed to be counter-field (opposite to the rotating field vector) with a single rotation peak. The maximum in rotation occurs for a frequency on the order of the reciprocal RC time constant for charging the double layer at the gold surface. Dielectrophoresis (DEP) experiments showed that the gold-coated particles undergo negative DEP at low frequencies and positive DEP at high frequencies with the same relaxation frequency as electrorotation. No induced charge electrophoresis was noticeable in ER or DEP experiments.

Rotation of a microvalve near conductive electrodes via induced-charge electrophoresis

The development of a high-speed microactuator in water is difficult because of hydrodynamic resistance and the lack of the knowledge of complex electrostatic problems combined with flow fields and ion dynamics. Previously, to overcome these problems, we proposed rotary microvalves in water using hydrodynamic force due to induced-charge electrophoresis (ICEP). In this study, by using an elliptical conductive carbon element fabricated by the pyrolysis of a photoresist film coated with gold, we experimentally demonstrate that microvalves can rotate near conductive electrodes. Namely, by numerically analyzing video data, we show the time evolution of the rotation angle, the flow field, and the center position of the microvalve. Further, we compare them with the theoretical results and find that they are in good agreement qualitatively. In the future, by using ICEP valves as a latch device, we can significantly improve the size and processing speed of a fluidic integrated circuit.

Double layer in ionic liquids: overscreening versus crowding

We develop a simple Landau-Ginzburg-type continuum theory of solvent-free ionic liquids and use it to predict the structure of the electrical double layer. The model captures overscreening from short-range correlations, dominant at small voltages, and steric constraints of finite ion sizes, which prevail at large voltages. Increasing the voltage gradually suppresses overscreening in favor of the crowding of counterions in a condensed inner layer near the electrode. This prediction, the ion profiles, and the capacitance-voltage dependence are consistent with recent computer simulations and experiments on room-temperature ionic liquids, using a correlation length of order the ion size.

Utility of continuum diffusion models for analyzing mobile-ion immittance data: electrode polarization, bulk, and generation-recombination effects

Consequences of the well-known Poisson-Nernst-Planck (PNP) continuum equations of charge motion in liquids or solids for ordinary or anomalous diffusion are investigated for an electrochemical cell with completely blocking electrodes. Previous work is summarized and much of it is shown to be independent of earlier published results and incomplete, with little comparison made between ordinary and anomalous diffusion. Such comparison is provided here and also includes variation of the mobility ratio of the mobilities of positive and negative charges from equality to charge of only one sign mobile. New generation-recombination effects are demonstrated for a range of mobility ratios, with particular attention given to those present for the case of charge of only one sign mobile. No previous analyses of experimental data with PNP models using complex-least-squares fitting have been published. Here such a model is found to fit frequency response data well for a hydrogel and to lead to estimates of physically meaningful parameters such as the diffusion constant and ionic concentration. PNP analysis of a synthetic data set derived from experimental results for liquid electrolytes refutes claims made in the original publication dealing with it, but verifies and extends an interesting analysis equation proposed there. PNP fitting of data for solids, including ones showing colossal low-frequency-limiting dielectric constants, suggests that they may often be well described as arising from simple diffuse-charge double-layer effects, and that continuum microscopic models such as the PNP, in series with a conducting Debye response model, may be sufficient for fitting well an appreciable amount of data involving ion hopping and trapping behavior.

Intensive current transfer in membrane systems: modelling, mechanisms and application in electrodialysis

Usually in electrochemical systems, the direct current densities not exceeding the limiting current density are applied. However, the recent practice of electrodialysis evidences the interest of other current modes where either the imposed direct current is over the limiting one or a non-constant asymmetrical (such as pulsed) current is used. The paper is devoted to make the mechanisms of mass transfer under these current regimes more clear. The theoretical background for mathematical modelling of mass transfer at overlimiting currents is described. Four effects providing overlimiting current conductance are examined. Two of them are related to water splitting: the appearance of additional charge carriers (H(+) and OH(-) ions) and exaltation effect. Two others are due to coupled convection partially destroying the diffusion boundary layer: gravitational convection and electroconvection. These effects result from formation of concentration gradients (known as concentration polarization) caused by the current flowing under conditions where ionic transport numbers are different in the membrane and solution. Similar effects take place not only in electrodialysis membrane systems, but in electrode ones, in electrophoresis and electrokinetic micro- and nanofluidic devices such as micropumps. The relation of these effects to the properties of the membrane surface (the chemical nature of the fixed groups, the degree of heterogeneity and hydrophobicity, and the geometrical shape of the surface) is analyzed. The interaction between the coupled effects is studied, and the conditions under which one or another effect becomes dominant are discussed. The application of intensive current modes in electrodialysis, the state-of-the-art and perspectives, are considered. It is shown that the intensive current modes are compatible with new trends in water treatment oriented towards Zero Liquid Discharge (ZLD) technologies. The main idea of these hybrid schemes including pressure- and electro-driven processes as well as conventional methods is to provide the precipitation of hardness salts before the membrane modules and that of well dissolved salts after.

Suppression of electro-osmotic flow by surface roughness

We show that nanoscale surface roughness, which commonly occurs on microfabricated metal electrodes, can significantly suppress electro-osmotic flows when excess surface conductivity is appreciable. We demonstrate the physical mechanism for electro-osmotic flow suppression due to surface curvature, compute the effects of varying surface conductivity and roughness amplitudes on the slip velocities of a model system, and identify scalings for flow suppression in different regimes of surface conduction. We suggest that roughness may be one factor that contributes to large discrepancies observed between classical electrokinetic theory and modern microfluidic experiments.

An automated, high-throughput experimental system for induced charge electrokinetics

Recent experiments in induced charge electrokinetics (ICEK) have shown that the standard theory generally overpredicts experimentally observed velocities. Such discrepancies reduce the efficacy of practical ICEK devices, and highlight our incomplete understanding of electrokinetic phenomena. Here, we present an automated experimental system that allows for the rapid collection of ICEK data under a variety of conditions ( approximately 1000 per day) to help develop and constrain new theories. We demonstrate this system by studying the ICEK slip flows over electrodes that have been controllably "contaminated" with a dielectric layer, either SiO(2) or an alkanethiol self-assembled monolayer, of known thickness. We also develop a theory that accounts for the effects of the dielectric coatings surface chemistry that yields quantitative agreement with experiments over nearly a thousand distinct conditions in the SiO(2) system and present an additional three thousand experiments of flows over alkanethiol monolayers. Our experimental system allows the direct interrogation of the physico-chemical effects that influence ICEK flows and for the optimization of these flows in lab-on-a-chip systems.

Extended space charge in concentration polarization

This paper is concerned with ionic currents from an electrolyte solution into a charge-selective solid, such as, an electrode, an ion-exchange membrane or an array of nano-channels in a micro-fluidic system, and the related viscous fluid flows on the length scales varying from nanometers to millimeters. All systems of this kind have characteristic voltage-current curves with segments in which current nearly saturates at some plateau values due to concentration polarization--formation of solute concentration gradients under the passage of a DC current. A number of seemingly different phenomena occurring in that range, such as anomalous rectification in cathodic copper deposition from a copper sulfate solution, super-fast vortexes near an ion-exchange granule, overlimiting conductance in electrodialysis and the recently observed non-equilibrium electroosmotic instability, result from the formation of an additional extended space charge layer next to that of a classical electrical double layer at the solid/liquid interface. In this paper we review the peculiar features of the non-equilibrium electric double layer and extended space charge and the possibility of their direct probing by harmonic voltage/current perturbations through a linear and non-linear system's response, by the methods of electrical impedance spectroscopy and via the anomalous rectification effect. On the relevant microscopic scales the ionic transport in the direction normal to the interface is dominated by drift-diffusion; hence, the extended space charge related viscous flows remain beyond the scope of this paper.

Finite ion-size effects dominate the interaction between charged colloidal particles at an oil-water interface

The electrostatic interaction of charged spherical colloids trapped at an interface between a nonpolar medium and water is analyzed. Complementary experiments provide consistent values for the dipole-dipole interaction potential over a wide range of interparticle distances. After accounting for the contribution from the compact inner double layer arising from the finite size of the counterions, we demonstrate quantitative agreement between experiments and nonlinear Poisson-Boltzmann theory. We find that the inner layer contribution dominates the electrostatic interaction in the far field for particles pinned at the interface. This result is fundamentally different from screened electrostatic interactions in the bulk and could contribute to the further understanding of the structure of the compact counterion layer in highly charged systems.

Fast three dimensional ac electro-osmotic pumps with nonphotolithographic electrode patterning

Three dimensional (3D) stepped electrodes dramatically improve the flow rate and frequency range of ac electro-osmotic pumps, compared to planar electrodes. However, the fabrication of 3D stepped electrodes for ac electro-osmosis (ACEO) pumps usually involves several processing steps. This paper demonstrates results from ACEO pumps produced by a faster and less expensive method to fabricate the 3D electrodes-extending the previous work to disposable devices. The method is based on shadowed evaporation of metal on an insulating substrate that can be injection molded. Flow velocities through the 3D ACEO pump are similar to those seen in the previous work.

Strongly nonlinear dynamics of electrolytes in large ac voltages

We study the response of a model microelectrochemical cell to a large ac voltage of frequency comparable to the inverse cell relaxation time. To bring out the basic physics, we consider the simplest possible model of a symmetric binary electrolyte confined between parallel-plate blocking electrodes, ignoring any transverse instability or fluid flow. We analyze the resulting one-dimensional problem by matched asymptotic expansions in the limit of thin double layers and extend previous work into the strongly nonlinear regime, which is characterized by two features--significant salt depletion in the electrolyte near the electrodes and, at very large voltage, the breakdown of the quasiequilibrium structure of the double layers. The former leads to the prediction of "ac capacitive desalination" since there is a time-averaged transfer of salt from the bulk to the double layers, via oscillating diffusion layers. The latter is associated with transient diffusion limitation, which drives the formation and collapse of space-charge layers, even in the absence of any net Faradaic current through the cell. We also predict that steric effects of finite ion sizes (going beyond dilute-solution theory) act to suppress the strongly nonlinear regime in the limit of concentrated electrolytes, ionic liquids, and molten salts. Beyond the model problem, our reduced equations for thin double layers, based on uniformly valid matched asymptotic expansions, provide a useful mathematical framework to describe additional nonlinear responses to large ac voltages, such as Faradaic reactions, electro-osmotic instabilities, and induced-charge electrokinetic phenomena.

Effective zero-thickness model for a conductive membrane driven by an electric field

The behavior of a conductive membrane in a static (dc) electric field is investigated theoretically. An effective zero-thickness model is constructed based on a Robin-type boundary condition for the electric potential at the membrane, originally developed for electrochemical systems. Within such a framework, corrections to the elastic moduli of the membrane are obtained, which arise from charge accumulation in the Debye layers due to capacitive effects and electric currents through the membrane and can lead to an undulation instability of the membrane. The fluid flow surrounding the membrane is also calculated, which clarifies issues regarding these flows sharing many similarities with flows produced by induced charge electro-osmosis (ICEO). Nonequilibrium steady states of the membrane and of the fluid can be effectively described by this method. It is both simpler, due to the zero thickness approximation which is widely used in the literature on fluid membranes, and more general than previous approaches. The predictions of this model are compared to recent experiments on supported membranes in an electric field.

Nonlinear dynamics of capacitive charging and desalination by porous electrodes

The rapid and efficient exchange of ions between porous electrodes and aqueous solutions is important in many applications, such as electrical energy storage by supercapacitors, water desalination and purification by capacitive deionization, and capacitive extraction of renewable energy from a salinity difference. Here, we present a unified mean-field theory for capacitive charging and desalination by ideally polarizable porous electrodes (without Faradaic reactions or specific adsorption of ions) valid in the limit of thin double layers (compared to typical pore dimensions). We illustrate the theory for the case of a dilute, symmetric, binary electrolyte using the Gouy-Chapman-Stern (GCS) model of the double layer, for which simple formulae are available for salt adsorption and capacitive charging of the diffuse part of the double layer. We solve the full GCS mean-field theory numerically for realistic parameters in capacitive deionization, and we derive reduced models for two limiting regimes with different time scales: (i) in the "supercapacitor regime" of small voltages and/or early times, the porous electrode acts like a transmission line, governed by a linear diffusion equation for the electrostatic potential, scaled to the RC time of a single pore, and (ii) in the "desalination regime" of large voltages and long times, the porous electrode slowly absorbs counterions, governed by coupled, nonlinear diffusion equations for the pore-averaged potential and salt concentration.

Induced charge electro-osmosis over controllably contaminated electrodes

Recent studies in nonlinear electrokinetics reveal the standard theory to generally overpredict measured velocities, sometimes dramatically. Contamination of the driving surface provides a natural mechanism for electrokinetic suppression. We measure induced charge electro-osmosis over gold electrodes "contaminated" with silica layers of controlled thickness for nearly a thousand distinct conditions, in a system that enables direct comparisons between theoretical predictions and experimental measurements. Both the magnitude and frequency dependence of the measured slip velocity are captured quantitatively over the entire range of experiments by accounting for the physical capacitance and surface chemistry of the dielectric layer. More generally, the quantitative characterization enabled by our apparatus will prove invaluable for the rational design and prediction of electrokinetic systems.

Diffuse-charge effects on the transient response of electrochemical cells

We present theoretical models for the time-dependent voltage of an electrochemical cell in response to a current step, including effects of diffuse charge (or "space charge") near the electrodes on Faradaic reaction kinetics. The full model is based on the classical Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions to describe electron-transfer reactions across the Stern layer at the electrode surface. In practical situations, diffuse charge is confined to thin diffuse layers (DLs), which poses numerical difficulties for the full model but allows simplification by asymptotic analysis. For a thin quasi-equilibrium DL, we derive effective boundary conditions on the quasi-neutral bulk electrolyte at the diffusion time scale, valid up to the transition time, where the bulk concentration vanishes due to diffusion limitation. We integrate the thin-DL problem analytically to obtain a set of algebraic equations, whose (numerical) solution compares favorably to the full model. In the Gouy-Chapman and Helmholtz limits, where the Stern layer is thin or thick compared to the DL, respectively, we derive simple analytical formulas for the cell voltage versus time. The full model also describes the fast initial capacitive charging of the DLs and superlimiting currents beyond the transition time, where the DL expands to a transient non-equilibrium structure. We extend the well-known Sand equation for the transition time to include all values of the superlimiting current beyond the diffusion-limiting current.

Effect of the combined action of Faradaic currents and mobility differences in ac electro-osmosis

In this work, we extend previous analyses of ac electro-osmosis to account for the combined action of two experimentally relevant effects: (i) Faradaic currents from electrochemical reactions at the electrodes and (ii) differences in ion mobilities of the electrolyte. In previous works, the ac electro-osmotic motion has been analyzed theoretically under the assumption that only forces in the diffuse (Debye) layer are relevant. Here, we first show that if the ion mobilities of a 1-1 aqueous solution are different, the charged zone expands from the Debye layer to include the diffusion layer. We later include the Faradaic currents and, as an attempt to explore both factors simultaneously, we perform a thin-layer, low-frequency, linear analysis of the system. Finally, the model is applied to the case of an electrolyte actuated by a traveling-wave signal. A steady liquid motion in opposite direction to the applied signal is predicted for some ranges of the parameters. This could serve as a partial explanation for the observed flow reversal in some experiments.