Numerical investigation of the current transition regimes in nanochannels

The concentration polarization phenomena and its effects represent one of the main challenges for the optimal operation of many nanofluidic systems. A numerical investigation of the different electric current transition regimes observed during the concentration polarization phenomena in nanochannels is performed. This included a 2D‐axisymmetric simulation of the nanofluidic system (reservoir‐nanochannel‐reservoir). From these simulations, a novel mechanism is discovered that explains that different current transition regimes. This driving mechanism involves the applied electric field penetration while the convective flow mechanism is found to be negligible. This differs with the classical statement that the mixing process with less depleted areas initiated by an electrokinetic vortex instability starts the overlimiting regime. Additionally, the numerical approach allows us to identify new characteristics of the linear‐limiting transition such as source‐like and saddle‐like points of the electric field streamlines. The three voltage–current regimes (linear, limiting and overlimiting) are explained by observing and quantifying changes in electric field, potential, ion concentration and ion concentration gradients within the system.     

Experimental Verification of the Electroosmotic Mechanism of Overlimiting Conductance Through a Cation Exchange Electrodialysis Membrane

The paper is concerned with convection at an ion-exchange electrodialysis membrane induced by nonequilibrium electroosmosis as a source of overlimiting conductance current through the membrane. Derivation of nonequilibrium electroosmotic slip condition is recapitulated along with the results of linear stability analysis of quiescent electrodiffusion through a flat ion-exchange membrane. Results of numerical calculation pertaining to nonlinear convection, developing from the respective instability, are reported along with those of recent experiments with modified membranes. These latter rule in favor of electroosmotic versus bulk electroconvective origin of overlimiting conductance through ion-exchange membranes.     

Mathematical model for the overlimiting state of an ion-exchange membrane system

A three-layered mathematical model is proposed for describing the overlimiting state in an ion-exchange membrane system. The model’s prominent feature is the allowance for the space-charge region; the water dissociation reaction, which is catalyzed by active ionogenic groups; and the coupled gravitational and electroosmotic convection, which leads to the emergence of dependence of the effective diffusion layer thickness on the electric current density. The model is used for calculating, on the basis of known initial current-voltage curves and dependences of effective transport numbers on the current density, such internal characteristics of the system as the diffusion layer thickness, distribution of concentration of ions, space charge, and electric-field strength at various current densities.     

Ion concentration polarization near microchannel-nanochannel interfaces: effect of pH value

This study investigates the effect of the pH value on the ion concentration polarization phenomenon and the nonlinear current-voltage characteristics of a hybrid soda-lime glass micro/nanochannel for a constant KCl salt concentration of about 1 mM. The experimental results show that the electrical conductance of the nanochannel in the Ohmic regime and the critical threshold voltage of the limiting current are both dependent on the pH value of the salt solution when the electrical double layer thickness is considerable in the nanochannel. Specifically, the nanochannel conductance increases and the critical threshold voltage for the limiting current decreases as the pH value is increased. It also suggests that a higher pH value induces a higher surface charge density on the nanochannel walls, and therefore increases both the ionic conductance and the counter-ion flux within the nanochannel.     

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.     

Induced electrokinetic transport in micro-nanofluidic interconnect devices

Hybrid micro-nanofluidic interconnect devices can be used to control analyte transfer from one microchannel to the other through a nanochannel under rest, injection, and recovery stages of operation by varying the applied potential bias. Using numerical simulations based on coupled transient Poisson-Nernst-Planck and Stokes equations, we examine the electrokinetic transport in a gateable device consisting of two 100 microm long, 1 microm wide negatively charged microchannels connected by a 1 microm long, 10 nm wide positively charged nanochannel under both positive and negative bias potentials. During injection, accumulation of ions is observed at the micro-nano interface region with the positive potential and depletion of ions is observed at the other micro-nano junction region. Net space charge in the depletion region gives rise to nonlinear electrokinetic transport during the recovery stage due to induced pressure, induced electroosmotic flow of the second kind, and complex flow circulations. Ionic currents are computed as a function of time for both positive and negative bias potentials for the three stages. Analytical expressions derived for ion current variation are in agreement with the simulated results. In the presence of multiple accumulation or depletion regions, we show that a hybrid micro-nano device can be designed to function as a logic gate.     

Morphology and microtopology of cation-exchange polymers and the origin of the overlimiting current

In electrodialysis desalination processes, the operating current density is limited by concentration polarization. In contrast to other membrane processes such as ultrafiltration, in electrodialysis, current transport above the limiting current is possible. In this work, the origin of the overlimiting current at cation-exchange polymers is investigated. We show that, under certain experimental conditions, electroconvection is the origin of the overlimiting conductance. The theory concerning electroconvection predicts a shortening of the plateau length of membranes with increased conductive or geometrical heterogeneity. We investigate the influence of these two parameters and show that the creation of line undulations on the membrane surface normal to the flow direction, having distances in the range of approximately 50-200% of the boundary-layer thickness, lead to an earlier onset of the overlimiting current. The plateau length of the undulated membranes is reduced by up to 60% compared to that of a flat membrane. These results verify the existence of electroconvection as a mechanism destabilizing the laminar boundary layer at the liquid-membrane interface and causing ionic transport above the limiting current density.     

Rate of Generation of Ions H<Superscript>+</Superscript> and OH<Superscript>−</Superscript> at the Ion-Exchange Membrane/Dilute Solution Interface as a Function of the Current Density

Partial currents of water dissociation products through cation- and anion-exchange membranes that form thin desalination channels in electrodialyzers are measured. The investigations are performed in a broad interval of flow rates during desalination of dilute sodium chloride solutions at overlimiting currents. A water dissociation theory, which was developed for bipolar membranes, and a mass transfer theory that allows for the space charge formation at overlimiting currents are used to derive an expression, according to which the rate of generation of the H⁺ and OH^? ions is defined by the ratio of the current density to its critical value at which water starts undergoing discernible dissociation.     

Electroviscous effect on the streaming current in a pH-regulated nanochannel

Accurate and rapid estimation of the streaming current in nanochannels is crucial for the development of the nanofluidics based power generation apparatus. In this study, an analytical model is developed for the first time to examine the electroviscous effect on the streaming current/conductance in a pH-regulated nanochannel by considering practical effects of multiple ionic species, surface chemistry reactions, and the Stern layer. Predictions from the model are in good agreement with the experimental results of the streaming conductance in silica nanochannels available in the literature. The electroviscous effect could have a significant reduction of ca. 30% in the streaming conductance at medium pH and low salt concentration.     

A mathematical model describing voltammograms and transport numbers under intensive electrodialysis modes

A three-layer mathematical model of overlimiting state is developed. A reactive layer with a thickness depending on the current density is introduced into the model. A decrease in the thickness of diffusion layer, which donates the counterions, with increasing current density as a result of electroconvection is also taken into account. A boundary-value problem is formulated within the Nernst-Planck and Poisson’s model in the three-layer region with the boundary conditions of constant concentrations in the bulk solution. It is shown that an increase in the reactive layer thickness with increasing current density determines the behavior of effective transport numbers in the overlimiting state of ion-exchange electromembrane system. In the current range under consideration (from 1 to 20 limiting currents), the reactive layer thickness is several tens nanometers and reaches 70 nm at a 100-fold excess over the limiting current. To calculate the voltammograms, the dependence of effective thickness δ_N of diffusion layer on the current density is required. This dependence can be obtained by solving an inverse problem, from the laser interferometry experiments, or calculated by the Navier-Stokes hydrodynamic model. The model enables one to calculate the distribution of electric field strength, potential, concentrations in the diffusion layers and membrane.     

Effect of anion-exchange membrane surface properties on mechanisms of overlimiting mass transfer

Four effects providing overlimiting current transfer in ion-exchange membrane systems are examined. Two of them are related to water splitting: the appearance of additional current 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. Three anion-exchange membranes, which differ in surface morphology and the nature of ion-exchange sites within a surface layer, are examined. The ion transfer across these membranes in NaCl solutions is studied by voltammetry, chronopotentiometry, and pH-metry. By excluding the effects of water splitting and gravitational convection, it is shown that the main mechanism of overlimiting mass transfer in narrow membrane cells at low salt concentrations is electroconvection. The reasons explaining why water splitting suppresses electroconvection are discussed. The scenario of development of potential oscillations with growing current and time is compared with that described theoretically by Rubinstein and Zaltzman.     

Electric Double Layer at the Membrane/Solution Interface in a Three-Layered Membrane System

Transport of ions through a three-layered membrane system at overlimiting currents is simulated mathematically. To allow for the space charge, the Poisson equation is used in both the first diffusion layer and the membrane. Two modes are shown to exist at overlimiting currents: a quasi-equilibrium state of the interface and a Schottky mode.     

Effects of the desalination chamber design on the mass-transfer characteristics of electrodialysis apparatuses at overlimiting current densities

The overlimiting current modes are being increasingly used in electrodialysis (ED) of dilute aqueous solutions. Of great importance is establishing a relationship between the design of ED apparatuses and the character of phenomena observed at overlimiting current densities, primarily, electroconvection and H⁺ and OH^? ion generation during water dissociation at the membrane-solution interface. In this work, we analyze the factors governing the efficiency of dilute solutions using modern theoretical concepts and experimental data obtained in laboratory cells and large-scale electrodialysis apparatuses. We also analyze the relationship between the mechanisms of the overlimiting transfer and the design of the desalinating channel. ED apparatuses of different types are considered, namely, apparatuses with profiled membranes, inert spacers, monolayer of ionite granules, and dipolar fillers of unwoven ionite fibers. The optimum concentration ranges of the desalinated solutions were found, and the operating conditions of membrane stacks, providing maximum overlimiting ion transfer, were determined.     

Effect of modification of ion-exchange membrane MF-4SK on its polarization characteristics

Modifications of perfluorinated membrane MF-4SK are studied by a membrane voltammetry method. Various techniques used for physicochemical modification of the membrane (variation of conditions of its chemical conditioning, insertion of tetrabutylammonium cations) exert predicted action on its structural and electrotransport properties. Correlation is established between variations in the slope of the ohmic segment of a voltammetric curve, the magnitude of the limiting electrodiffusion current, and the potential of the system’s conversion into an overlimiting state and the electrotransport properties of membranes and their structural characteristics that are obtained by independent methods of impedance conductimetry and standard porosimetry. The increase in the limiting current for membranes subjected to a thermal oxidizing treatment is explained by a change in the content and state of water in the membrane bulk. Effects of acceleration of the occurrence of an overlimiting state of the membrane saturated with tetrabutylammonium cations is discovered and possible reasons for this phenomenon are discussed. For membranes saturated with tetrabutylammonium cations, a correlation is established between the limiting current and electroosmotic phenomena.     

Coupled convection of solution near the surface of ion-exchange membranes in intensive current regimes

Mechanisms responsible for the overlimiting ion transfer in membranes systems are discussed. The overlimiting transfer is shown to be due largely to the action of four effects coupled with the concentration polarization of the system. Two of these are connected with the water dissociation near the membrane/solution interface: the emergence of additional charge carriers (ions H⁺ or OH^?) in the depleted solution layer and the exaltation of transfer of salt counterions. The latter effect is connected with the perturbation of electric field caused by the water dissociation products. The other two effects are two versions of coupled convection, which leads to partial destruction of the depleted diffusion layer. These include gravitational convection and electroconvection. The former is caused by the emergence of the solution’s density gradient. The latter develops via a mechanism of electroosmotic slip. In this work, methods of voltammetry and chronopotentiometry and pH measurements are used to study the transfer of ions through homogeneous membranes Nafion-117 and AMX as a function of the concentration of sodium chloride solutions in the underlimiting and overlimiting current regimes. In a 0.1 M NaCl solution, gravitational convection makes a considerable contribution to the transfer of salt ions near the membrane surface in intensive current regimes. The influence of this effect on the electrochemical behavior of membrane systems weakens with the solution dilution and with increasing relative transfer of the H⁺ and OH^? ions that are generated at the membrane/solution interface. In conditions where gravitational convection is suppressed and the water dissociation near the membrane/solution interface is not great, the major contribution to the overlimiting growth of current is made by electroconvection. Topics for discussion in the paper include the mutual influence of effects on one another, in particular, the effect the rate of generation of the H⁺ and OH^? ions exerts on the gravitational convection and electroconvection and the reasons for the different behavior of cation-and anion-exchange membranes in intensive current regimes.     

Understanding flow enhancement in graphene‐coated nanochannels

In this work, we investigate pressure‐driven water flows in graphene‐coated copper nanochannels through molecular dynamics simulations. It is found that the flow rate in bare copper nanochannel can be significantly enhanced by a factor of 45 when the nanochannel is coated with monolayer graphene. The enhancement factor for the flow rate reaches about 90 when the nanochannel is modified with 3 or more graphene layers. The dipole relaxation time and the hydrogen bond lifetime of interfacial water molecules show that the graphene coating promotes the mobility of water molecules at the interface. The distribution of the potential of mean force and the free energy barriers also confirm that graphene coating reduces the flow resistance and 3 layers of graphene can fully screen the surface effects. The results in this work provide important information for the design of graphene‐based nanofluidic systems for flow enhancement.     

Structural change of ion-exchange membrane surfaces under high electric fields and its effects on membrane properties

Structural change of an ion-exchange membrane under a high electric field was investigated by comparing water dissociation and the FTIR spectra between the virgin membrane and that used at an overlimiting current density. From a series of water dissociation experiments at overlimiting current densities, it was observed that water dissociation in an anion-exchange membrane used at an overlimiting current density was higher than that in a virgin membrane at the same current density. The FTIR study revealed that the tertiary amine groups are formed from the quaternary ammonium groups on the anion-exchange membrane surface where ion depletion occurs under the influence of the applied strong electric field. The occurrence of increased water dissociation is considered to be caused by the protonation and deprotonation of the tertiary amine groups in the anion-exchange membrane. On the other hand, there was no structural change for the cation-exchange membrane under the electric field investigated in this study, which is coincident with the results of water dissociation experiments for the CMX membrane. In addition, we found that membrane resistance, permselectivity, and plateau length of the current-voltage curve were affected by the converted tertiary amine groups depending on the solution pH.     

Decoupling of the nernst-planck and poisson equations. Application to a membrane system at overlimiting currents

This paper deals with one-dimensional stationary Nernst-Planck and Poisson (NPP) equations describing ion electrodiffusion in multicomponent solution/electrode or ion-conductive membrane systems. A general method for resolving ordinary and singularly perturbed problems with these equations is developed. This method is based on the decoupling of NPP equations that results in deduction of an equation containing only the terms with different powers of the electrical field and its derivatives. Then, the solution of this equation, analytical in several cases or numerical, is substituted into the Nernst-Planck equations for calculating the concentration profile for each ion present in the system. Different ionic species are grouped in valency classes that allows one to reduce the dimension of the original set of equations and leads to a relatively easy treatment of multi-ion systems. When applying the method developed, the main attention is paid to ion transfer at limiting and overlimiting currents, where a significant deviation from local electroneutrality occurs. The boundary conditions and different approximations are examined: the local electroneutrality (LEN) assumption and the original assumption of quasi-uniform distribution of the space charge density (QCD). The relations between the ion fluxes at limiting and overlimiting currents are discussed. In particular, attention is paid to the "exaltation" of counterion transfer toward an ion-exchange membrane by co-ion flux leaking through the membrane or generated at the membrane/solution interface. The structure of the multi-ion concentration field in a depleted diffusion boundary layer (DBL) near an ion-exchange membrane at overlimiting currents is analyzed. The presence of salt ions and hydrogen and hydroxyl ions generated in the course of the water "splitting" reaction is considered. The thickness of the DBL and its different zones, as functions of applied current density, are found by fitting experimental current-voltage curves.     

Effect of an ion-exchange filler on the deionized water quality in electrodialysis

Desalination channels, containing an inert separator and a monolayer of ionites AV-17 and KU-2 taken in various volume ratios, are studied while maintaining concentrations of all solution components invariant. It is shown that the composition of the ion-exchange filler of the desalination channel affects the rate of transport of salt ions through relevant membranes, pH, and specific resistance of desalinated solution. The behavior of membrane systems in an overlimiting state is explained by using notions about different mechanisms governing the so-called overlimiting current through anion-exchange and cation-exchange membranes     

Transport and separation of charged macromolecules under nonlinear electromigration in nanochannels

In this work, we theoretically investigate the implications of nonlinear electrophoretic effects on the transport and size-based separation of charged macromolecules in nanoscale confinements. By employing a regular perturbation analysis, we address certain nontrivial features of interconnection among wall-induced transverse migrative fluxes, electrophoretic and electroosmotic transport, confinement-induced hindered diffusive effects, and hydrodynamic interactions in detail. We demonstrate that there occurs an optimal regime of influence of the nonlinear electrophoretic effects, within which high values of separation resolution may be achieved. This size-based optimal regime, however, can be effectively exploited only for nanochannel flows, as attributed to the strong electric double layer interactions prevalent within the same.