Analysis of electrolyte transport through charged nanopores

Electrophoresis of ions and electrolyte conductivity: From bulk to nanochannels

When electrolyte solutions are confined in micro- and nanochannels their conductivity is significantly different from those in a bulk phase. Here we revisit the theory of this phenomenon by focusing attention on the reduction in the ion mobility with the concentration of salt and a consequent impact to the conductivity of a monovalent solution, from bulk to confined in a narrow slit. We first give a systematic treatment of electrophoresis of ions and obtain equations for their zeta potentials and mobilities. The latter are then used to obtain a simple expression for a bulk conductivity, which is valid in a concentration range up to a few molars and more accurate than prior analytic theories. By extending the formalism to the electrolyte solution in the charged channel the equations describing the conductivity in different modes are presented. They can be regarded as a generalization of prior work on the channel conductivity to a more realistic case of a nonzero reduction of the electrophoretic mobility of ions with salt concentration. Our analysis provides a framework for interpreting measurements on the conductivity of electrolyte solutions in the bulk and in narrow channels.

Spiers Memorial Lecture: Towards understanding of iontronic systems: electroosmotic flow of monovalent and divalent electrolyte through charged cylindrical nanopores

In many practical applications, ions are the primary charge carrier and must move through either semipermeable membranes or through pores, which mimic ion channels in biological systems. In analogy to electronic devices, the "iontronic" ones use electric fields to induce the charge motion. However, unlike the electrons that move through a conductor, motion of ions is usually associated with simultaneous solvent flow. A study of electroosmotic flow through narrow pores is an outstanding challenge that lies at the interface of non-equilibrium statistical mechanics and fluid dynamics. In this paper, we will review recent works that use dissipative particle dynamics simulations to tackle this difficult problem. We will also present a classical density functional theory (DFT) based on the hypernetted-chain approximation (HNC), which allows us to calculate the velocity of electroosmotic flows inside nanopores containing 1 : 1 or 2 : 1 electrolyte solution. The theoretical results will be compared with simulations. In simulations, the electrostatic interactions are treated using the recently introduced pseudo-1D Ewald summation method. The zeta potentials calculated from the location of the shear plane of a pure solvent are found to agree reasonably well with the Smoluchowski equation. However, the quantitative structure of the fluid velocity profiles deviates significantly from the predictions of the Smoluchowski equation in the case of charged pores with 2 : 1 electrolyte. For low to moderate surface charge densities, the DFT allows us to accurately calculate the electrostatic potential profiles and the zeta potentials inside the nanopores. For pores with 1 : 1 electrolyte, the agreement between theory and simulation is particularly good for large ions, for which steric effects dominate over the ionic electrostatic correlations. The electroosmotic flow is found to depend very strongly on the ionic radii. In the case of pores containing 2 : 1 electrolyte, we observe a reentrant transition in which the electroosmotic flow first reverses and then returns to normal as the surface change density of the pore is increased.

Ion Steric Effect Induces Giant Enhancement of Thermoelectric Conversion in Electrolyte-Filled Nanochannels

Ionic thermoelectricity in nanochannels has received increasing attention because of its advantages, such as high Seebeck coefficient and low cost. However, most studies have focused on dilute simple electrolytes that neglect the effects of finite ion sizes and short-range electrostatic correlation. Here, we reveal a new thermoelectric mechanism arising from the coupling of the ion steric effect due to finite ion sizes and ion thermodiffusion in electric double layers, using both theoretical and numerical methods. We show that this mechanism can significantly enhance the thermoelectric response in nanoconfined electrolytes depending on the properties of electrolytes and nanochannels. Compared to the previously known mechanisms, the new mechanism can increase the Seebeck coefficient by 100% or even 1 order of magnitude enhancement under optimal conditions. Moreover, we demonstrate that the short-range electrostatic correlation can help preserve the Seebeck coefficient enhancement in a weaker confinement or in more concentrated electrolytes.

Fluids and Electrolytes under Confinement in Single-Digit Nanopores

Confined fluids and electrolyte solutions in nanopores exhibit rich and surprising physics and chemistry that impact the mass transport and energy efficiency in many important natural systems and industrial applications. Existing theories often fail to predict the exotic effects observed in the narrowest of such pores, called single-digit nanopores (SDNs), which have diameters or conduit widths of less than 10 nm, and have only recently become accessible for experimental measurements. What SDNs reveal has been surprising, including a rapidly increasing number of examples such as extraordinarily fast water transport, distorted fluid-phase boundaries, strong ion-correlation and quantum effects, and dielectric anomalies that are not observed in larger pores. Exploiting these effects presents myriad opportunities in both basic and applied research that stand to impact a host of new technologies at the water-energy nexus, from new membranes for precise separations and water purification to new gas permeable materials for water electrolyzers and energy-storage devices. SDNs also present unique opportunities to achieve ultrasensitive and selective chemical sensing at the single-ion and single-molecule limit. In this review article, we summarize the progress on nanofluidics of SDNs, with a focus on the confinement effects that arise in these extremely narrow nanopores. The recent development of precision model systems, transformative experimental tools, and multiscale theories that have played enabling roles in advancing this frontier are reviewed. We also identify new knowledge gaps in our understanding of nanofluidic transport and provide an outlook for the future challenges and opportunities at this rapidly advancing frontier.

Ion and Water Transport in Ion-Exchange Membranes for Power Generation Systems: Guidelines for Modeling

Artificial ion-exchange and other charged membranes, such as biomembranes, are self-organizing nanomaterials built from macromolecules. The interactions of fragments of macromolecules results in phase separation and the formation of ion-conducting channels. The properties conditioned by the structure of charged membranes determine their application in separation processes (water treatment, electrolyte concentration, food industry and others), energy (reverse electrodialysis, fuel cells and others), and chlore-alkali production and others. The purpose of this review is to provide guidelines for modeling the transport of ions and water in charged membranes, as well as to describe the latest advances in this field with a focus on power generation systems. We briefly describe the main structural elements of charged membranes which determine their ion and water transport characteristics. The main governing equations and the most commonly used theories and assumptions are presented and analyzed. The known models are classified and then described based on the information about the equations and the assumptions they are based on. Most attention is paid to the models which have the greatest impact and are most frequently used in the literature. Among them, we focus on recent models developed for proton-exchange membranes used in fuel cells and for membranes applied in reverse electrodialysis.

Electrical Conductance of Charged Nanopores

A nanopore's response to an electrical potential drop is characterized by its electrical conductance, . For the last two decades, it has been thought that at low electrolyte concentrations, , the conductance is concentration-independent such that . It has been recently demonstrated that surface charge regulation changes the dependency to , whereby the slope typically takes the values α = 1/3 or 1/2. However, experiments have observed slopes of 2/3 and 1 suggesting that additional mechanisms, such as convection and slip-lengths, appear. Here, we elucidate the interplay between three mechanisms: surface charge regulation, convection, and slip lengths. We show that the inclusion of convection does not change the slope, and when the effects of hydrodynamic slip are included, the slope is doubled. We show that when all effects are accounted for, α can take any value between 0 and 1 where the exact value of the slope depends on the material properties. This result is of utmost importance in designing any electro-kinetically driven nanofluidic system characterized by its conductance.

Thermodynamics of Charge Regulation during Ion Transport through Silica Nanochannels

Ion-surface interactions can alter the properties of nanopores and dictate nanofluidic transport in engineered and biological systems central to the water-energy nexus. The ion adsorption process, known as "charge regulation", is ion-specific and is dependent on the extent of confinement when the electric double layers (EDLs) between two charged surfaces overlap. A fundamental understanding of the mechanisms behind charge regulation remains lacking. Herein, we study the thermodynamics of charge regulation reactions in 20 nm SiO₂ channels via conductance measurements at various concentrations and temperatures. The effective activation energies (E_a) for ion conductance at low concentrations (strong EDL overlap) are ∼2-fold higher than at high concentrations (no EDL overlap) for the electrolytes studied here: LiCl, NaCl, KCl, and CsCl. We find that E_a values measured at high concentrations result from the temperature dependence of viscosity and its influence on ion mobility, whereas E_a values measured at low concentrations result from the combined effects of ion mobility and the enthalpy of cation adsorption to the charged surface. Notably, the E_a for surface reactions increases from 7.03 kJ mol^-1 for NaCl to 16.72 ± 0.48 kJ mol^-1 for KCl, corresponding to a difference in surface charge of -8.2 to -0.8 mC m^-2, respectively. We construct a charge regulation model to rationalize the cation-specific charge regulation behavior based on an adsorption equilibrium. Our findings show that temperature- and concentration-dependent conductance measurements can help indirectly probe the ion-surface interactions that govern transport and colloidal interactions at the nanoscale─representing a critical step forward in our understanding of charge regulation and adsorption phenomena under nanoconfinement.

Colloidal Suspensions Displaying Anomalous Phoretic Behavior: Field and Mobility Reversal

We show that colloidal suspensions that acquire a surface charge by capturing ions from the surrounding solution display unexpected and remarkable phoretic behavior. Depending on suspension volume fraction, a critical zeta potential ζ exists where the effective electrophoretic mobility diverges, becoming virtually infinite. Beyond such critical value, a ζ-range is identified where mobility reversal occurs, i.e., the effective mobility becomes negative. This counterintuitive behavior is due to the salt gradient engendered by phoretic drift of this kind of particles, which capture and release ions (salt), respectively, at the start and the end of the phoretic path. This salt gradient deeply influences the electric field in the bulk electrolyte where the particles migrate: it can make the field vanish, hence the mobility divergence, or even entail inversion of the field, which is reflected in the mobility reversal. These findings should spur new concepts in a variety of traditional and emerging technologies involving, for example, the separation or targeting of colloids as well as in applications where the creation or manipulation of chemical gradients or electric fields in solution is critical.

Electrochemical Methods for Water Purification, Ion Separations, and Energy Conversion

Agricultural development, extensive industrialization, and rapid growth of the global population have inadvertently been accompanied by environmental pollution. Water pollution is exacerbated by the decreasing ability of traditional treatment methods to comply with tightening environmental standards. This review provides a comprehensive description of the principles and applications of electrochemical methods for water purification, ion separations, and energy conversion. Electrochemical methods have attractive features such as compact size, chemical selectivity, broad applicability, and reduced generation of secondary waste. Perhaps the greatest advantage of electrochemical methods, however, is that they remove contaminants directly from the water, while other technologies extract the water from the contaminants, which enables efficient removal of trace pollutants. The review begins with an overview of conventional electrochemical methods, which drive chemical or physical transformations via Faradaic reactions at electrodes, and proceeds to a detailed examination of the two primary mechanisms by which contaminants are separated in nondestructive electrochemical processes, namely electrokinetics and electrosorption. In these sections, special attention is given to emerging methods, such as shock electrodialysis and Faradaic electrosorption. Given the importance of generating clean, renewable energy, which may sometimes be combined with water purification, the review also discusses inverse methods of electrochemical energy conversion based on reverse electrosorption, electrowetting, and electrokinetic phenomena. The review concludes with a discussion of technology comparisons, remaining challenges, and potential innovations for the field such as process intensification and technoeconomic optimization.

Simultaneous thermoosmotic and thermoelectric responses in nanoconfined electrolyte solutions: Effects of nanopore structures and membrane properties

HYPOTHESIS

Nanofluidic systems provide an emerging and efficient platform for thermoelectric conversion and fluid pumping with low-grade heat energy. As a basis of their performance enhancement, the effects of the structures and properties of the nanofluidic systems on the thermoelectric response (TER) and the thermoosmotic response (TOR) are yet to be explored.

METHODS

The simultaneous TER and TOR of electrolyte solutions in nanofluidic membrane pores on which an axial temperature gradient is exerted are investigated numerically and semi-analytically. A semi-analytical model is developed with the consideration of finite membrane thermal conductivity and the reservoir/entrance effect.

FINDINGS

The increase in the access resistance due to the nanopore-reservoir interfaces accounts for the decrease of short circuit current at the low concentration regime. The decrease in the thermal conductivity ratio can enhance the TER and TOR. The maximum power density occurring at the nanopore radius twice the Debye length ranges from several to dozens of mW K^-2 m^-2 and is an order of magnitude higher than typical thermo-supercapacitors. The surface charge polarity can heavily affect the sign and magnitude of the short-circuit current, the Seebeck coefficient and the open-circuit thermoosmotic coefficient, but has less effect on the short-circuit thermoosmotic coefficient. Furthermore, the membrane thickness makes different impacts on TER and TOR for zero and finite membrane thermal conductivity.

Ionic current magnetic fields in 3D finite-length nanopores and nanoslits

Deoxyribonucleic acid (DNA) encodes all genetic information, and in genetic disorders, DNA sequencing is used as an effective diagnosis. Nanopore/slit is one of the recent and successful tools for DNA sequencing. Passage of DNA along the pores creates non-uniform ionic currents which creates non-uniform electric and magnetic fields, accordingly. Sensing the electric field is usually used for sequencing application. We suggest to use the magnetic field induced by pressure-driven ionic currents as a secondary signal. We systematically compared the induced magnetic field of nanopores and nanoslits with equal cross-sectional area. The 3D magnetic field is numerically obtained by solving the Poisson-Nernst-Planck, Ampere, and Navier-Stokes equations. As expected, the maximum value of the maximum magnetic flux occurs near the wall and inside the channel, and increasing the pressure gradient along the pore/slit increases the flowrate and magnetic field, consequently. At a given pressure difference across the pore/slit, nanopores are better than nanoslits in sensing the magnetic flux. For example, by applying 2 MPa across the pore/slit, the maximum magnetic flux density for nanopore, nanoslit A R = 1 and nanoslit A R = 5 are 1.10 pT, 1.08 pT and 0.45 pT, accordingly. Also, at a given flowrate across the pore/slit, nanoslits are the better choice. It should be noted the external magnetic fields as small as pico-Tesla are detectable and measurable in voltage/pressure driven electrokinetic flow slits.

Mathematical Description of the Increase in Selectivity of an Anion-Exchange Membrane Due to Its Modification with a Perfluorosulfonated Ionomer

In this paper, we simulate the changes in the structure and transport properties of an anion-exchange membrane (CJMA-7, Hefei Chemjoy Polymer Materials Co. Ltd., China) caused by its modification with a perfluorosulfonated ionomer (PFSI). The modification was made in several stages and included keeping the membrane at a low temperature, applying a PFSI solution on its surface, and, subsequently, drying it at an elevated temperature. We applied the known microheterogeneous model with some new amendments to simulate each stage of the membrane modification. It has been shown that the PFSI film formed on the membrane-substrate does not affect significantly its properties due to the small thickness of the film (≈4 µm) and similar properties of the film and substrate. The main effect is caused by the fact that PFSI material “clogs” the macropores of the CJMA-7 membrane, thereby, blocking the transport of coions through the membrane. In this case, the membrane microporous gel phase, which exhibits a high selectivity to counterions, remains the primary pathway for both counterions and coions. Due to the above modification of the CJMA-7 membrane, the coion (Na⁺) transport number in the membrane equilibrated with 1 M NaCl solution decreased from 0.11 to 0.03. Thus, the modified membrane became comparable in its transport characteristics with more expensive IEMs available on the market.

Nonlinear electrohydrodynamic ion transport in graphene nanopores

Mechanosensitivity is one of the essential functionalities of biological ion channels. Synthesizing an artificial nanofluidic system to mimic such sensations will not only improve our understanding of these fluidic systems but also inspire applications. In contrast to the electrohydrodynamic ion transport in long nanoslits and nanotubes, coupling hydrodynamical and ion transport at the single-atom thickness remains challenging. Here, we report the pressure-modulated ion conduction in graphene nanopores featuring nonlinear electrohydrodynamic coupling. Increase of ionic conductance, ranging from a few percent to 204.5% induced by the pressure—an effect that was not predicted by the classical linear coupling of molecular streaming to voltage-driven ion transport—was observed experimentally. Computational and theoretical studies reveal that the pressure sensitivity of graphene nanopores arises from the transport of capacitively accumulated ions near the graphene surface. Our findings may help understand the electrohydrodynamic ion transport in nanopores and offer a new ion transport controlling methodology.

Charging dynamics of electrical double layers inside a cylindrical pore: predicting the effects of arbitrary pore size

Porous electrodes are found in energy storage devices such as supercapacitors and pseudocapacitors. However, the effect of electrode-pore-size distribution on their energy storage properties remains unclear. Here, we develop a model for the charging of electrical double layers inside a cylindrical pore for arbitrary pore size. We assume small applied potentials and perform a regular perturbation analysis to predict the evolution of electrical potential and ion concentrations in both the radial and axial directions. We validate our perturbation model with direct numerical simulations of the Poisson-Nernst-Planck equations, and obtain quantitative agreement between the two approaches for small and moderate potentials. Our analysis yields two main characteristic features of arbitrary pore size: (i) a monotonic decrease of the charging timescale with an increase in relative pore size (pore size relative to Debye length); (ii) large potential changes for overlapping double layers in a thin transition region, which we approximate mathematically by a jump discontinuity. We quantify the contributions of electromigration and charge diffusion fluxes, which provide mechanistic insights into the dependence of charging timescale and capacitance on pore size. We develop a modified transmission circuit model that captures the effect of arbitrary pore size and demonstrate that a time-dependent transition-region resistor needs to be included in the circuit. We also derive phenomenological expressions for average effective capacitance and charging timescale as a function of pore-size distribution. We show that the capacitance and charging timescale increase with smaller average pore sizes and with smaller polydispersity, resulting in a gain of energy density at a constant power density. Overall, our results advance the mechanistic understanding of electrical-double-layer charging.

Osmotic Pressure and Diffusion of Ions in Charged Nanopores

The transport of ions and water in nanopores is of interest for a number of natural and technological processes. Due to their practically identical long straight cylindrical pores, nanoporous track-etched membranes are suitable materials for investigation of its mechanisms. This communication reports on simultaneous measurements of osmotic pressure and salt diffusion with a 24 nm pore track-etched membrane. Due to the use of dilute electrolyte solutions (1-4 mM KCl and LiCl), this pore size was commensurate with the Debye screening length. Advanced interpretation of experimental results using a full version of the space-charge model has revealed that osmotic pressure and salt diffusion can be quantitatively correlated with electrostatic interactions of ions with charged nanopore walls. The surface-charge density is shown to increase with electrolyte concentration in agreement with the mechanism of deprotonation of weakly acidic surface groups. Moreover, a lack of significant surface-charge dependence on the kind of cation (K+ or Li+) demonstrates that binding of salt counterions does not play a major role in this system.

Enhanced Salt Removal Performance Using Graphene-Modified Sodium Vanadium Fluorophosphate in Flow Electrode Capacitive Deionization

Designing electrode materials with excellent comprehensive properties was of top priority in promoting development of flow electrode capacitive deionization (FCDI). To date, most FCDI studies involved the application and modification of carbon-based materials, which suffered the contradiction between rheological behavior and electrochemical performance. In this study, a Na⁺ superionic conductor (NASICON) sodium vanadium fluorophosphate@reduced graphene oxide (NVOPF@rGO) was synthesized and applied as a flow electrode in FCDI. Benefiting from the confinement effect of the three-dimensional (3D) reduced graphene oxide (rGO) network, thin and uniform NVOPF nanosheets formed and provided abundant active sites for adsorbing Na⁺. Moreover, the interconnected rGO network formed a 3D conductive network for Na⁺ and electron transport. Compared with an activated carbon (AC)-AC system (AC was used as an anode and a cathode), a NVOPF@rGO-AC system (NVOPF@rGO was used as a cathode and AC was used as an anode) exhibited preferable dispersibility and stability of electrode dispersion, lower internal resistance, higher desalination rate, and lower energy consumption. Besides, the average salt adsorption rate (ASAR) reached 5.32 μg·cm^-2·min^-1 by adjusting the concentration of the electrode (4.73 wt %), the flow rate of the electrode (25 mL·min^-1), and the operation voltage (1.6 V). This study demonstrated the potential of faradic flow electrodes for promoting the development and application of FCDI.

Electroneutrality breakdown in nanopore arrays

The electrostatic screening of charge in one-dimensional confinement leads to long-range breakdown in electroneutrality within a nanopore. Through a series of continuum simulations, we demonstrate the principles of electroneutrality breakdown for electrolytes in one-dimensional confinement. We show how interacting pores in a membrane can counteract the phenomenon of electroneutrality breakdown, eventually returning to electroneutrality. Emphasis is placed on applying simplifying formulas to reduce the multidimensional partial differential equations into a single ordinary differential equation for the electrostatic potential. Dielectric mismatch between the electrolyte and membrane, pore aspect ratio, and confinement dimensionality are studied independently, outlining the relevance of electroneutrality breakdown in nanoporous membranes for selective ion transport and separations.

Conditions for electroneutrality breakdown in nanopores

It has recently been suggested that a breakdown of electroneutrality occurs in highly confined nanopores that are encompassed by a dielectric material. This work elucidates the conditions for this breakdown. We show that the breakdown within the pore results from the response of the electric field within the dielectric. Namely, we show that this response is highly sensitive to the boundary condition at the dielectric edge. The standard Neumann boundary condition of no-flux predicts that the breakdown does not occur. However, a Dirichlet boundary condition for a zero-potential predicts a breakdown. Within this latter scenario, the breakdown exhibits a dependence on the thickness of the dielectric material. Specifically, infinite thickness dielectrics do not exhibit a breakdown, while dielectrics of finite thickness do exhibit a breakdown. Numerical simulations confirm theoretical predictions. The breakdown outcomes are discussed with regard to single pore systems and multiple pore systems.

Regional and functional division of functional elements of solid-state nanochannels for enhanced sensitivity and specificity of biosensing in complex matrices

Solid-state nanochannels (SSNs) provide a promising approach for biosensing due to the confinement of molecules inside, their great mechanical strength and diversified surface chemical properties; however, until now, their sensitivity and specificity have not satisfied the practical requirements of sensing applications, especially in complex matrices, i.e., media of diverse constitutions. Here, we report a protocol to achieve explicit regional and functional division of functional elements at the outer surface (FE_OS) and inner wall (FE_IW) of SSNs, which offers a nanochannel-based sensing platform with enhanced specificity and sensitivity. The protocol starts with the fabrication and characterization of the distribution of FE_OS and FE_IW. Then, the evaluation of the contributions of FE_OS and FE_IW to ionic gating is described; the FE_IW mainly regulate ionic gating, and the FE_OS can produce a synergistic effect. Finally, hydrophobic or highly charged FE_OS are applied to ward off interference molecules, non-target molecules that may affect the ionic signal of nanochannels, which decreases false signals and helps to achieve the highly specific ionic output in complex matrices. Compared with other methods currently available, this method will contribute to the fundamental understanding of substance transport in SSNs and provide high specificity and sensitivity in SSN-based analyses. The procedure takes 3-6 d to complete.

Nanopore-Based Power Generation from Salinity Gradient: Why It Is Not Viable

In recent years, the development of nanopore-based membranes has revitalized the prospect of harvesting salinity gradient (blue) energy. In this study, we systematically analyze the energetic performance of nanopore-based power generation (NPG) at various process scales, beginning with a single nanopore, followed by a multipore membrane coupon, and ending with a full-scale system. We confirm the high power densities attainable by a single nanopore and demonstrate that, at the coupon scale and above, concentration polarization severely hinders the power density of NPG, revealing the common, yet significant, error in linearly extrapolating single-pore performance to multipore membranes. Through our consideration of concentration polarization, we also importantly show that the development of materials with exceptional nanopore properties provides limited enhancement of practical process performance. For a full-scale NPG membrane module, we find an inherent tradeoff between power density and thermodynamic energy efficiency, whereby achieving a high power density sacrifices the energy efficiency. Furthermore, we derive a simple expression for the theoretical maximum energy efficiency of NPG, showing it is solely related to the membrane selectivity (i.e., S²/2). Through this relation, it is apparent that the energy efficiency of NPG is limited to only 50% (for a completely selective membrane, i.e., S = 1), reinforcing our optimistic full-scale simulations which result in a (practical) maximum energy efficiency of 42%. Finally, we assess the net extractable energy of a full-scale NPG system which mixes river water and seawater by including the energy losses from pretreatment and pumping, revealing that the NPG process-both in its current state of development and in the case of highly optimistic performance with minimized external energy losses-is not viable for power generation.

Electric-field-based Poisson-Boltzmann theory: Treating mobile charge as polarization

Mobile charge in an electrolytic solution can in principle be represented as the divergence of ionic polarization. After adding explicit solvent polarization a finite volume of an electrolyte can then be treated as a composite nonuniform dielectric body. Writing the electrostatic interactions as an integral over electric-field energy density we show that the Poisson-Boltzmann functional in this formulation is convex and can be used to derive the equilibrium equations for electric potential and ion concentration by a variational procedure developed by Ericksen for dielectric continua [J. L. Ericksen, Arch. Rational Mech. Anal. 183, 299 (2007)AVRMAW0003-952710.1007/s00205-006-0042-4]. The Maxwell field equations are enforced by extending the set of variational parameters by a vector potential representing the dielectric displacement which is fully transverse in a dielectric system without embedded external charge. The electric-field energy density in this representation is a function of the vector potential and the sum of ionic and solvent polarization making the mutual screening explicit. Transverse polarization is accounted for by construction, lifting the restriction to longitudinal polarization inherent in the electrostatic potential based formulation of Poisson-Boltzmann mean field theory.

Ion transport in nanopores with highly overlapping electric double layers

Investigation of ion transport through nanopores with highly overlapping electric double layers is extremely challenging. This can be attributed to the non-linear Poisson-Boltzmann equation that governs the behavior of the electrical potential distribution as well as other characteristics of ion transport. In this work, we leverage the approach of Schnitzer and Yariv [Phys. Rev. E 87, 054301 (2013)] to reduce the complexity of the governing equation. An asymptotic solution is derived, which shows remarkable correspondence to simulations of the non-approximated equations. This new solution is leveraged to address a number of highly debated issues. We derive the equivalent of the Gouy-Chapman equation for systems with highly overlapping electric double layers. This new relationship between the surface charge density and the surface potential is then utilized to determine the power-law scaling of nanopore conductances as a function of the bulk concentrations. We derive the coefficients of transport for the case of overlapping electric double layers and compare it to the renowned uniform potential model. We show that the uniform potential model is only an approximation for the exact solution for small surface charges. The findings of this work can be leveraged to uncover additional hidden attributes of ion transport through nanopores.

Theory of shock electrodialysis I: Water dissociation and electrosmotic vortices

Shock electrodialysis (shock ED), an emerging electrokinetic process for water purification, leverages the new physics of deionization shock waves in porous media. In previous work, a simple leaky membrane model with surface conduction can explain the propagation of deionization shocks in a shock ED system, but it cannot quantitatively predict the deionization and conductance (which determines the energy consumption), and it cannot explain the selective removal of ions in experiments. This two-part series of work establishes a more comprehensive model for shock ED, which applies to multicomponent electrolytes and any electrical double layer thickness, captures the phenomena of electroosmosis, diffusioosmosis, and water dissociation, and incorporates more realistic boundary conditions. In this paper, we will present the model details and show that hydronium transport and electroosmotic vortices (at the inlet and outlet) play important roles in determining the deionization and conductance in shock ED. We also find that the results are quantitatively consistent with experimental data in the literature. Finally, the model is used to investigate design strategies for scale up and optimization.

Insights into the estimation of capacitance for carbon-based supercapacitors

Carbon-based materials are broadly used as the active component of electric double layer capacitors (EDLCs) in energy storage systems with a high power density. Most of the reported computational studies have investigated the electrochemical properties under equilibrium conditions, limiting the direct and practical use of the results to design electrochemical energy systems. In the present study, for the first time, the experimental data from more than 300 published papers have been extracted and then analyzed through an optimized support vector machine (SVM) by a grey wolf optimization (GWO) algorithm to obtain a correlation between carbon-based structural features and EDLC performance. Several structural features, including calculated pore size, specific surface area, N-doping level, I D/I G ratio, and applied potential window were selected as the input variables to determine their impact on the respective capacitances. Sensitivity analysis, which has only been performed in this study for approximating the EDLC capacitance, indicated that the specific surface area of the carbon-based supercapacitors is of the greatest effect on the corresponding capacitance. The proposed SVM-GWO, with an R 2 value of 0.92, showed more accuracy than all the other proposed machine learning (ML) models employed for this purpose.

Breakdown of electroneutrality in nanopores

Ion transport in extremely narrow nanochannels has gained increasing interest in recent years due to unique physical properties at the nanoscale and the technological advances that allow us to study them. It is tempting to approach this confined regime with the theoretical tools and knowledge developed for membranes and microfluidic devices, and naively apply continuum models, such as the Poisson-Nernst-Planck and Navier-Stokes equations. However, it turns out that some of the most basic principles we take for granted in larger systems, such as the complete screening of surface charge by counter-ions, can break down under extreme confinement. We show that in a truly one-dimensional system of ions interacting with three-dimensional electrostatic interactions, the screening length is exponentially large, and can easily exceed the macroscopic length of a nanotube. Without screening, electroneutrality breaks down within the nanotube, with fundamental consequences for ion transport and electrokinetic phenomena. In this work, we build a general theoretical framework for electroneutrality breakdown in nanopores, focusing on the most interesting case of a one-dimensional nanotube, and show how it provides an elegant interpretation for the peculiar scaling observed in experimental measurements of ionic conductance in carbon nanotubes.

Novel ionic separation mechanisms in electrically driven membrane processes

Electromembrane processes including electrodialysis (ED) and related processes are usually limited by diffusion transport of ions from a bulk solution to ion exchange membranes. The diffusion limited current (DLC) occurs when the concentration at membrane surfaces vanishes and approaches zero. Increasing the applied potential difference above this point has no substantial effect on ion transport and causes operational problems such as low current efficiency, high energy consumption, and mineral scaling. However, it is evident from numerous studies that operating at overlimiting current (OLC) is possible and allows one to enhance the mass transfer of an electromembrane process. While OLC is sometimes possible by electrochemical means, such as water splitting or current induced membrane discharge, it has been found that exotic ion transport mechanisms, such as ion concentration polarization in micro/nanofluidic system, deionization shock waves, and ionic bridges, can provide novel electrokinetic means of achieving OLC. In this paper, these novel ionic separation mechanisms and their role in enhanced current transfer are reviewed in the context of emerging electromembrane processes, such as shock ED and electrodeionization (EDI).

Ion Transport in Electrically Imperfect Nanopores

Ionic transport through a charged nanopore at low ion concentration is governed by the surface conductance. Several experiments have reported various power-law relations between the surface conductance and ion concentration, i.e., G_surf ∝ c₀^α. However, the physical origin of the varying exponent, α, is not yet clearly understood. By performing extensive coarse-grained Molecular Dynamics simulations for various pore diameters, lengths, and surface charge densities, we observe varying power-law exponents even with a constant surface charge and show that α depends on how electrically "perfect" the nanopore is. Specifically, when the net charge of the solution in the pore is insufficient to ensure electroneutrality, the pore is electrically "imperfect" and such nanopores can exhibit varying α depending on the degree of "imperfectness". We present an ionic conductance theory for electrically "imperfect" nanopores that not only explains the various power-law relationships but also describes most of the experimental data available in the literature.

Modeling the Effective Conductance Drop Due to a Particle in a Solid State Nanopore Towards Optimized Design

An understanding of the current change in a solid state nanopore due to particle movement or capture is crucial for improvement of nanopore based sensing technologies. For lower aspect ratio pores, which are gaining importance due to their high sensitivity, there is interplay between access and pore resistance and the existing theories for computation of access resistance cannot explain most of the experimental observations. Hence, there is a need to develop a comprehensive model for calculating the effective conductance drop in presence of particles in a solid state nanopore. In this paper, we develop analytical models to calculate both the access and pore resistance in presence of particle at different positions during translocation and also when captured by receptors in functionalized nanopores. A wide range of pore geometry and molar strength has been investigated. Taking into consideration the positional uncertainty during particle translocation, the effective resistance sensitivity has been found to agree very well with the experimental observations in low aspect ratio pore. Additionally, we observe that in functionalized nanopores, a pore of higher diameter results in around 50% increase in sensitivity compared to a pore with half its diameter, which indicates the scope of design optimization in such systems.

Towards Electrochemical Water Desalination Techniques: A Review on Capacitive Deionization, Membrane Capacitive Deionization and Flow Capacitive Deionization

Electrochemical water desalination has been a major research area since the 1960s with the development of capacitive deionization technique. For the latter, its modus operandi lies in temporary salt ion adsorption when a simple potential difference (1.0-1.4 V) of about 1.2 V is supplied to the system to temporarily create an electric field that drives the ions to their different polarized poles and subsequently desorb these solvated ions when potential is switched off. Capacitive deionization targets/extracts the solutes instead of the solvent and thus consumes less energy and is highly effective for brackish water. This paper reviews Capacitive Deionization (mechanism of operation, sustainability, optimization processes, and shortcomings) with extension to its counterparts (Membrane Capacitive Deionization and Flow Capacitive Deionization).

Glass/Au Composite Membranes with Gold Nanoparticles Synthesized inside Pores for Selective Ion Transport

Nanocomposite membranes have been actively developed in the last decade. The involvement of nanostructures can improve the permeability, selectivity, and anti-fouling properties of a membrane for improved filtration processes. In this work, we propose a novel type of ion-selective Glass/Au composite membrane based on porous glass (PG), which combines the advantages of porous media and promising selective properties. The latter are achieved by depositing gold nanoparticles into the membrane pores by the laser-induced liquid phase chemical deposition technique. Inside the pores, gold nanoparticles with an average diameter 25 nm were formed, which was confirmed by optical and microscopic studies. To study the transport and selective properties of the PG/Au composite membrane, the potentiometric method was applied. The uniform potential model was used to determine the surface charge from the experimental data. It was found that the formation of gold nanoparticles inside membrane pores leads to an increase in the surface charge from −2.75 mC/m² to −5.42 mC/m². The methods proposed in this work allow the creation of a whole family of composite materials based on porous glasses. In this case, conceptually, the synthesis of these materials will differ only in the selection of initial precursors.

Asymmetric double-layer charging in a cylindrical nanopore under closed confinement

This article presents a physical-mathematical treatment and numerical simulations of electric double layer charging in a closed, finite, and cylindrical nanopore of circular cross section, embedded in a polymeric host with charged walls and sealed at both ends by metal electrodes under an external voltage bias. Modified Poisson-Nernst-Planck equations were used to account for finite ion sizes, subject to an electroneutrality condition. The time evolution of the formation and relaxation of the double layers was explored. Moreover, equilibrium ion distributions and differential capacitance curves were investigated as functions of the pore surface charge density, electrolyte concentration, ion sizes, and pore size. Asymmetric properties of the differential capacitance curves reveal that the structure of the double layer near each electrode is controlled by the charge concentration along the pore surface and by charge asymmetry in the electrolyte. These results carry implications for accurately simulating cylindrical capacitors and electroactuators.

Coupled water, charge and salt transport in heterogeneous nano-fluidic systems

We theoretically study the electrokinetic transport properties of nano-fluidic devices under the influence of a pressure, voltage or salinity gradient. On a microscopic level the behaviour of the device is quantified by the Onsager matrix L, a generalised conductivity matrix relating the local driving forces and the induced volume, charge and salt flux. Extending L from a local to a global linear-response relation is trivial for homogeneous electrokinetic systems, but in this manuscript we derive a generalised conductivity matrix G from L that applies also to heterogeneous electrokinetic systems. This extension is especially important in the case of an imposed salinity gradient, which gives necessarily rise to heterogeneous devices. Within this formalism we can also incorporate a heterogeneous surface charge due to, for instance, a charge regulating boundary condition, which we show to have a significant impact on the resulting fluxes. The predictions of the Poisson-Nernst-Planck-Stokes theory show good agreement with exact solutions of the governing equations determined using the finite element method under a wide variety of parameters. Having established the validity of the theory, it provides an accessible method to analyse electrokinetic systems in general without the need of extensive numerical methods. As an example, we analyse a reverse electrodialysis "blue energy" system, and analyse how the many parameters that characterise such a system affect the generated electrical power and efficiency.

Carbon Black Flow Electrode Enhanced Electrochemical Desalination Using Single-Cycle Operation

Flow-electrode electrochemical desalination (FEED) processes (e.g., flow-electrode capacitive deionization), which use flowable carbon particles as the electrodes, have attracted increasing attention, holding the promise for continuous desalination and high desalting efficiency. While it is generally believed that carbon particles with abundant microporous and large specific capacitances (e.g., activated carbon, AC) should be ideal candidates for FEED electrodes, we provide evidence to the contrary, showing that highly conductive electrodes with low specific surface area can outperform microporous AC-based electrodes. This study revealed that FEED using solely high surface area AC particles (∼2000 m² g^-1, specific capacitance of ∼44 F g^-1, average salt adsorption rate of ∼0.15 μmol cm^-2 min^-1) was vastly outperformed by electrodes based solely on low-surface area carbon black (CB, ∼70 m² g^-1, ∼0.5 F g^-1, ∼0.75 μmol cm^-2 min^-1). Electrochemical impedance spectroscopy results suggest that the electrode formed by CB particles led to more effective electronic charge percolation, likely contributing to the improved desalination performance. In addition, we propose and demonstrate a novel operation mode, termed single cycle (SC), which greatly simplified the FEED cell configuration and enabled simultaneous charging and discharging. Using SC mode with CB flow electrodes delivered an increased average salt removal rate relative to the more traditional short-circuited closed cycle (SCC) mode, achieving up to 1.13 μmol cm^-2 min^-1. Further investigations demonstrate that up to 50% of energy input would be avoided when using CB flow electrodes operated under SC mode as compared to that of AC flow electrodes operated under SCC mode. In summary, the FEED process presented in this study provided an innovative and promising approach toward high-efficient and low-cost brackish water desalination.

Electrochemical impedance of electrodiffusion in charged medium under dc bias

An immobile charged species provides a charged medium for transport of charge carriers that is exploited in many applications, such as permselective membranes, doped semiconductors, biological ion channels, as well as porous media and microchannels with surface charges. In this paper, we theoretically study the electrochemical impedance of electrodiffusion in a charged medium by employing the Nernst-Planck equation and the electroneutrality condition with a background charge density. The impedance response is obtained under different dc bias conditions extending above the diffusion-limiting bias. We find a transition in the impedance behavior around the diffusion-limiting bias and present an analytical approximation for a weakly charged medium under an overlimiting bias.

Entrance effects in concentration-gradient-driven flow through an ultrathin porous membrane

Transport of liquid mixtures through porous membranes is central to processes such as desalination, chemical separations, and energy harvesting, with ultrathin membranes made from novel 2D nanomaterials showing exceptional promise. Here, we derive, for the first time, general equations for the solution and solute fluxes through a circular pore in an ultrathin planar membrane induced by a solute concentration gradient. We show that the equations accurately capture the fluid fluxes measured in finite-element numerical simulations for weak solute-membrane interactions. We also derive scaling laws for these fluxes as a function of the pore size and the strength and range of solute-membrane interactions. These scaling relationships differ markedly from those for concentration-gradient-driven flow through a long cylindrical pore or for flow induced by a pressure gradient or an electric field through a pore in an ultrathin membrane. These results have broad implications for transport of liquid mixtures through membranes with thickness on the order of the characteristic pore size.

Non-Negligible Roles of Pore Size Distribution on Electroosmotic Flow in Nanoporous Materials

Electroosmotic flow in nanoporous materials is of fundamental importance for the design and development of filtration membranes and electrochemical devices such as supercapacitors and batteries. Recent experiments suggest that ion transport in a porous network is substantially different from that in individual nanochannels due to the pore size distribution and pore connectivity. Herein, we report a theoretical framework for ion transport in nanoporous materials by combing the classical density functional theory to describe the electrical double layer (EDL) structure, the Navier-Stokes equation for the fluid flow, and the effective medium approximation to bridge the gap between individual nanopores and the network connectivity. We find that ion conductivity in nanoporous materials is extremely sensitive to the pore size distribution when the average size of micropores is comparable to the EDL thickness. The theoretical predictions provide an explanation of the giant gap between the conductivity of a single pore and that of a porous network and highlight the mechanism of ion transport through nanoporous materials important for numerous practical applications.

Modelling nanofiltration of electrolyte solutions

This review critically examines current models for nanofiltration (NF) of electrolyte solutions. We start from linear irreversible thermodynamics, we derive a basic equation set for ion transfer in terms of gradients of ion electrochemical potentials and transmembrane volume flux. These equations are extended to the case of significant differences of thermodynamic forces across the membrane (continuous version of irreversible thermodynamics) and solved in quadratures for single salts and trace ions added to single salts in the case of macroscopically-homogeneous membranes. These solutions reduce to (quasi)analytical expressions in the popular Spiegler-Kedem approximation (composition-independent phenomenological coefficients), which we extend to the case of trace ions. This enables us to identify membrane properties (e.g. ion permeances, ion reflection coefficients, electrokinetic charge density) that control its performance in NF of multi-ion solutions. Further, we specify the phenomenological coefficients of irreversible thermodynamics in terms of ion partitioning, hindrance and diffusion coefficients for the model of straight cylindrical capillaries. The corresponding expressions enable assessment of the applicability of the popular nanopore model of NF. This model (based on the use of macroscopic approaches at nanoscale) leads to a number of trends that have never been observed experimentally. We also show that the use of the Born formula (frequently employed for the description of dielectric exclusion) hardly leads to meaningful values of solvent dielectric constant in membrane pores because this formula disregards the very solvent structure whose changes are supposed to bring about the reduction of dielectric permittivity in nanopores. We conclude that the effect should better be quantified in terms of ion excess solvation energies in the membrane phase. As an alternative to the nanopore description of NF, we review recent work on the development of an advanced engineering model for NF of multi-ion solutions in terms of a solution-diffusion-electromigration mechanism. This model (taking into account spontaneously arising transmembrane electric fields) captures several trends observed experimentally, and the use of trace ions can provide model parameters (ion permeances in the membrane) from experiment. We also consider a recent model (ultrathin barrier layers with deviations from local electroneutrality) that may reproduce observed feed-salt concentration dependences of membrane performance in terms of concentration-independent properties like excess ion solvation energies. Due to its complexity, practical modelling of nanofiltration will probably be performed with advanced engineering models for the foreseeable future. Although mechanistic studies are vital for understanding transport and developing membranes, future simulations in this area will likely need to depart from typical continuum models to provide physical insight. For enhancing the quality of modelling input, it is essential to improve the control of concentration polarization in membrane test cells.