Charging Dynamics of Overlapping Double Layers in a Cylindrical Nanopore

Effect of Mixed Morphology (Simple Cubic, Face-Centered Cubic, and Body-Centered Cubic)-Based Electrodes on the Electric Double Layer Capacitance of Supercapacitors

Supercapacitors store energy due to the formation of an electric double layer (EDL) at the interface of the electrodes and electrolyte. The present article deals with the finite element study of equilibrium electric double layer capacitance (EDLC) in the mixed morphology electrodes comprising all three fundamental crystal structures, simple cubic (SC), body-centered cubic (BCC), and face-centered cubic morphologies (FCC). Mesoporous-activated carbon forms the electrode in the supercapacitor with (C2H5)4NBF4/propylene carbonate organic electrolyte. Electrochemical interference is clearly demonstrated in the supercapacitors with the formation of the potential bands, as in the case of interference theory due to the increasing packing factor. The effects of electrode thickness varying from a wide range of 50 nm to 0.04 mm on specific EDLC have been discussed in detail. The interfacial geometry of the unit cell in contact with the electrolyte is the most important parameter determining the properties of the EDL. The critical thickness of the electrodes is 1.71 μm in all the morphologies. Polarization increases the interfacial potential and leads to EDL formation. The Stern layer specific capacitance is 167.6 μF cm-2 in all the morphologies. The maximum capacitance is in the decreasing order of interfacial geometry, as FCC > BCC > SC, dependent on the packing factor. The minimum transmittance in all the morphologies is 98.35%, with the constant figure of merit at higher electrode thickness having applications in the chip interconnects. The transient analysis shows that the interfacial current decreases with increasing polarization in the EDL. The capacitance also decreases with the increase of the scan rate.

A network model to predict ionic transport in porous materials

Understanding the dynamics of electric-double-layer (EDL) charging in porous media is essential for advancements in next-generation energy storage devices. Due to the high computational demands of direct numerical simulations and a lack of interfacial boundary conditions for reduced-order models, the current understanding of EDL charging is limited to simple geometries. Here, we present a network model to predict EDL charging in arbitrary networks of long pores in the Debye-Hückel limit without restrictions on EDL thickness and pore radii. We demonstrate that electrolyte transport is described by Kirchhoff's laws in terms of the electrochemical potential of charge (the valence-weighted average of the ion electrochemical potentials) instead of the electric potential. By employing the equivalent circuit representation suggested by these modified Kirchhoff's laws, our methodology accurately captures the spatial and temporal dependencies of charge density and electric potential, matching results obtained from computationally intensive direct numerical simulations. Our network model provides results up to six orders of magnitude faster, enabling the efficient simulation of a triangular lattice of five thousand pores in 6 min. We employ the framework to study the impact of pore connectivity and polydispersity on electrode charging dynamics for pore networks and discuss how these factors affect the time scale, energy density, and power density of capacitive charging. The scalability and versatility of our methodology make it a rational tool for designing 3D-printed electrodes and for interpreting geometric effects on electrode impedance spectroscopy measurements.

Emerging Frontiers in Multichannel Membrane Capacitive Deionization: Recent Advances and Future Prospects

Capacitive deionization (CDI) has emerged as a promising desalination technology and recently promoted the development of multichannel membrane capacitive deionization (MC-MCDI). In MC-MCDI, the independent control of multiflow channels, including the feed and electrolyte channels, enables the optimization of electrode operation in various modes, such as concentration gradients and reverse voltage discharge, facilitating semicontinuous operation. Moreover, the integration of redox couples into MC-MCDI has led to advancements in redox-mediated desalination. Specifically, the introduction of redox-active species helps enhance the ion removal efficiency and reduce energy consumption during desalination. This systematic approach, combining principles from CDI and electrodialysis, results in more sustainable and efficient desalination. These advancements have contributed to improved desalination performance and practical feasibility, rendering MC-MCDI an increasingly attractive option for addressing water scarcity challenges. Despite the considerable interest in and potential of this process, there is currently no comprehensive review available that covers the operational features and applications of MC-MCDI. Therefore, this Review provides an overview of recent research progress, focusing on the unique cell configuration, vital operation principles, and potential advantages over conventional CDI. Additionally, innovative applications of MC-MCDI are discussed. The Review concludes with insights into future research directions, potential opportunities in industrial desalination technology, and the fundamental and practical challenges for successful implementation.

Electro-osmotic flow in nanoconfinement: Solid-state and protein nanopores

Electro-osmotic flow (EOF) is a phenomenon where fluid motion occurs in porous materials or micro/nano-channels when an external electric field is applied. In the particular example of single-molecule electrophoresis using single nanopores, the role of EOF on the translocation velocity of the analyte molecule through the nanopore is not fully understood. The complexity arises from a combination of effects from hydrodynamics in restricted environments, electrostatics emanating from charge decorations and geometry of the pores. We address this fundamental issue using the Poisson-Nernst-Planck and Navier-Stokes (PNP-NS) equations for cylindrical solid-state nanopores and three representative protein nanopores (α-hemolysin, MspA, and CsgG). We present the velocity profiles inside the nanopores as a function of charge decoration and geometry of the pore and applied electric field. We report several unexpected results: (a) The apparent charges of the protein nanopores are different from their net charge and the surface charge of the whole protein geometry, and the net charge of inner surface is consistent with the apparent charge. (b) The fluid velocity depends non-monotonically on voltage. The three protein nanopores exhibit unique EOF and velocity-voltage relations, which cannot be simply deduced from their net charge. Furthermore, effective point mutations can significantly change both the direction and the magnitude of EOF. The present computational analysis offers an opportunity to further understand the origins of the speed of transport of charged macromolecules in restricted space and to design desirable nanopores for tuning the speed of macromolecules through nanopores.

Tracking Ion Transport in Nanochannels via Transient Single-Particle Imaging

The transport behavior of ions in the nanopores has an important impact on the performance of the electrochemical devices. Although the classical Transmission-Line (TL) model has long been used to describe ion transport in pores, the boundary conditions for the applicability of the TL model remain controversial. Here, we investigated the transport kinetics of different ions, within nanochannels of different lengths, by using transient single-particle imaging with temporal resolution up to microseconds. We found that the ion transport kinetics within short nanochannels may deviate significantly from the TL model. The reason is that the ion transport under nanoconfinement is composed of multi basic stages, and the kinetics differ much under different stage domination. With the shortening of nanochannels, the electrical double layer (EDL) formation would become the "rate-determining step" and dominate the apparent ion kinetics. Our results imply that using the TL model directly and treating the in-pore mobility as an unchanged parameter to estimate the ion transport kinetics in short nanopores/nanochannels may lead to orders of magnitude bias. These findings may advance the understanding of the nanoconfined ion transport and promote the related applications.

Molecular Understanding of Charging Dynamics in Supercapacitors with Porous Electrodes and Ionic Liquids

Porous electrodes and ionic liquids could significantly enhance the energy storage of supercapacitors. However, they may reduce the charging dynamics and power density due to the nanoconfinement of porous electrodes and the high viscosity of ionic liquids. A comprehensive understanding of the charging mechanism in porous supercapacitors with ionic liquids provides a crucial theoretical foundation for their design optimization. Here, we review the progress of molecular simulations of the charging dynamics in supercapacitors consisting of porous electrodes and ionic liquids. We highlight and delve into the breakthroughs in the ion transport and charging mechanism for electrodes with subnanometer pores and realistic porous structures. We also discuss future directions for the charging dynamics of supercapacitors.

Real-time visualisation of ion exchange in molecularly confined spaces where electric double layers overlap

Ion interactions with interfaces and transport in confined spaces, where electric double layers overlap, are essential in many areas, ranging from crevice corrosion to understanding and creating nano-fluidic devices at the sub 10 nm scale. Tracking the spatial and temporal evolution of ion exchange, as well as local surface potentials, in such extreme confinement situations is both experimentally and theoretically challenging. Here, we track in real-time the transport processes of ionic species (LiClO4) confined between a negatively charged mica surface and an electrochemically modulated gold surface using a high-speed in situ sensing Surface Forces Apparatus. With millisecond temporal and sub-micrometer spatial resolution we capture the force and distance equilibration of ions in the confinement of D ≈ 2-3 nm in an overlapping electric double layer (EDL) during ion exchange. Our data indicate that an equilibrated ion concentration front progresses with a velocity of 100-200 μm s-1 into a confined nano-slit. This is in the same order of magnitude and in agreement with continuum estimates from diffusive mass transport calculations. We also compare the ion structuring using high resolution imaging, molecular dynamics simulations, and calculations based on a continuum model for the EDL. With this data we can predict the amount of ion exchange, as well as the force between the two surfaces due to overlapping EDLs, and critically discuss experimental and theoretical limitations and possibilities.

Incorporating ion-specific van der Waals and soft repulsive interactions in the Poisson-Boltzmann theory of electrical double layers

Electrical double layers (EDLs) arise when an electrolyte is in contact with a charged surface, and are encountered in several application areas including batteries, supercapacitors, electrocatalytic reactors, and colloids. Over the last century, the development of Poisson-Boltzmann (PB) models and their modified versions have provided significant physical insight into the structure and dynamics of the EDL. Incorporation of physics such as finite-ion-size effects, dielectric decrement, and ion-ion correlations has made such models increasingly accurate when compared to more computationally expensive approaches such as molecular simulations and classical density functional theory. However, a prominent knowledge gap has been the exclusion of van der Waals (vdW) and soft repulsive interactions in modified PB models. Although short-ranged as compared to electrostatic interactions, we show here that vdW and soft repulsive interactions can play an important role in determining the structure of the EDL via the formation of a Stern layer and in modulating the differential capacitance of an electrode in an electrolyte. To this end, we incorporate ion-ion and wall-ion vdW attraction and soft repulsion via a 12-6 Lennard-Jones (LJ) potential, resulting in a modified PB-LJ approach. The wall-ion LJ interactions were found to have a significant effect on the electrical potential and concentration profiles, especially close to the wall. However, ion-ion LJ interactions do not affect the EDL structure at low bulk ion concentrations (<1 M). We also derive dimensionless numbers to quantify the impact of ion-ion and wall-ion LJ interactions on the EDL. Furthermore, in the pursuit of capturing ion-specific effects, we apply our model by considering various ions such as Na, K+, Mg2+, Cl-, and SO42-. We observe how varying parameters such as the electrolyte concentration and electrode potential affect the structure of the EDL due to the competition between ion-specific LJ and electrostatic interactions. Lastly, we show that the inclusion of vdW and soft repulsion interactions, as well as hydration effects, leads to a better qualitative agreement of the PB models with experimental double-layer differential capacitance data. Overall, the modified PB-LJ approach presented herein will lead to more accurate theoretical descriptions of EDLs in various application areas.

Wafer-Scale Fabrication of Hierarchically Porous Silicon and Silica by Active Nanoparticle-Assisted Chemical Etching and Pseudomorphic Thermal Oxidation

Many biological materials exhibit a multiscale porosity with small, mostly nanoscale pores as well as large, macroscopic capillaries to simultaneously achieve optimized mass transport capabilities and lightweight structures with large inner surfaces. Realizing such a hierarchical porosity in artificial materials necessitates often sophisticated and expensive top-down processing that limits scalability. Here, an approach that combines self-organized porosity based on metal-assisted chemical etching (MACE) with photolithographically induced macroporosity for the synthesis of single-crystalline silicon with a bimodal pore-size distribution is presented, i.e., hexagonally arranged cylindrical macropores with 1 µm diameter separated by walls that are traversed by pores 60 nm across. The MACE process is mainly guided by a metal-catalyzed reduction-oxidation reaction, where silver nanoparticles (AgNPs) serve as the catalyst. In this process, the AgNPs act as self-propelled particles that are constantly removing silicon along their trajectories. High-resolution X-ray imaging and electron tomography reveal a resulting large open porosity and inner surface for potential applications in high-performance energy storage, harvesting and conversion or for on-chip sensorics and actuorics. Finally, the hierarchically porous silicon membranes can be transformed structure-conserving by thermal oxidation into hierarchically porous amorphous silica, a material that could be of particular interest for opto-fluidic and (bio-)photonic applications due to its multiscale artificial vascularization.

Theory and Simulations of Ionic Liquids in Nanoconfinement

Room-temperature ionic liquids (RTILs) have exciting properties such as nonvolatility, large electrochemical windows, and remarkable variety, drawing much interest in energy storage, gating, electrocatalysis, tunable lubrication, and other applications. Confined RTILs appear in various situations, for instance, in pores of nanostructured electrodes of supercapacitors and batteries, as such electrodes increase the contact area with RTILs and enhance the total capacitance and stored energy, between crossed cylinders in surface force balance experiments, between a tip and a sample in atomic force microscopy, and between sliding surfaces in tribology experiments, where RTILs act as lubricants. The properties and functioning of RTILs in confinement, especially nanoconfinement, result in fascinating structural and dynamic phenomena, including layering, overscreening and crowding, nanoscale capillary freezing, quantized and electrotunable friction, and superionic state. This review offers a comprehensive analysis of the fundamental physical phenomena controlling the properties of such systems and the current state-of-the-art theoretical and simulation approaches developed for their description. We discuss these approaches sequentially by increasing atomistic complexity, paying particular attention to new physical phenomena emerging in nanoscale confinement. This review covers theoretical models, most of which are based on mapping the problems on pertinent statistical mechanics models with exact analytical solutions, allowing systematic analysis and new physical insights to develop more easily. We also describe a classical density functional theory, which offers a reliable and computationally inexpensive tool to account for some microscopic details and correlations that simplified models often fail to consider. Molecular simulations play a vital role in studying confined ionic liquids, enabling deep microscopic insights otherwise unavailable to researchers. We describe the basics of various simulation approaches and discuss their challenges and applicability to specific problems, focusing on RTIL structure in cylindrical and slit confinement and how it relates to friction and capacitive and dynamic properties of confined ions.

Effective Modulation of Ion Mobility through Solid-State Single-Digit Nanopores

Many experimental studies have proved that ion dynamics in a single-digit nanopore with dimensions comparable to the Debye length deviate from the bulk values, but we still have critical knowledge gaps in our understanding of ion transport in nanoconfinement. For many energy devices and sensor designs of nanoporous materials, ion mobility is a key parameter for the performance of nanofluidic equipment. However, investigating ion mobility remains an experimental challenge. This study experimentally investigated the monovalent ion dynamics of single-digit nanopores from the perspective of ionic conductance. In this article, we present a theory that is sufficient for a basic understanding of ion transport through a single-digit nanopore, and we subdivided and separately analyzed the contribution of each conductance component. These conclusions will be useful not only in understanding the behavior of ion migration but also in the design of high-performance nanofluidic devices.

Electrostatic Correlation Induced Ion Condensation and Charge Inversion in Multivalent Electrolytes

The study of the electrical double layer lies at the heart of colloidal and interfacial science. The standard mean-field Poisson-Boltzmann (PB) theory is incapable of modeling many phenomena originating from ion correlation. An important example is charge inversion or overcharging of electrical double layers in multivalent electrolyte solutions. Existing theories aiming to include correlations cannot capture the non-monotonic dependence of charge inversion on salt concentration because they have not systematically accounted for the inhomogeneous nature of correlations from surface to the bulk and the excluded volume effect of ions and solvent molecules. In this work, we modify the Gaussian renormalized fluctuation theory by including the excluded volume effect to study ion condensation and charge inversion. A boundary layer approach is developed to accurately model the giant difference in ion correlations between the condensed layer near the surface and the diffuse layer outside. The theory is used to study charge inversion in multivalent electrolytes and their mixtures. We predict a surface charge induced formation of a three-dimensional condensed layer, which is necessary but not sufficient for charge inversion. The value of the effective surface potential is found to depend non-monotonically on the bulk salt concentration. Our results also show a non-monotonic reduction in charge inversion in monovalent and multivalent electrolyte mixtures. Our work is the first to qualitatively reproduce experimental and simulation observations and explains the underlying physics.

Electric-field-induced oscillations in ionic fluids: a unified formulation of modified Poisson-Nernst-Planck models and its relevance to correlation function analysis

We theoretically investigate an electric-field-driven system of charged spheres as a primitive model of concentrated electrolytes under an applied electric field. First, we provide a unified formulation for the stochastic charge and density dynamics of the electric-field-driven primitive model using the stochastic density functional theory (DFT). The stochastic DFT integrates the four frameworks (the equilibrium and dynamic DFTs, the liquid state theory and the field-theoretic approach), which allows us to justify in a unified manner various modifications previously made for the Poisson-Nernst-Planck model. Next, we consider stationary density-density and charge-charge correlation functions of the primitive model with a static electric field. We predict an electric-field-induced synchronization between emergences of density and charge oscillations. We are mainly concerned with the emergence of stripe states formed by segregation bands transverse to the external field, thereby demonstrating the following: (i) the electric-field-induced crossover occurs prior to the conventional Kirkwood crossover without an applied electric field, and (ii) the ion concentration dependence of the decay lengths at the onset of oscillations bears a similarity to the underscreening behavior found by recent simulation and theoretical studies on equilibrium electrolytes. Also, the 2D inverse Fourier transform of the correlation function illustrates the existence of stripe states beyond the electric-field-induced Kirkwood crossover.

Understanding the Electric Double-Layer Structure, Capacitance, and Charging Dynamics

Significant progress has been made in recent years in theoretical modeling of the electric double layer (EDL), a key concept in electrochemistry important for energy storage, electrocatalysis, and multitudes of other technological applications. However, major challenges remain in understanding the microscopic details of the electrochemical interface and charging mechanisms under realistic conditions. This review delves into theoretical methods to describe the equilibrium and dynamic responses of the EDL structure and capacitance for electrochemical systems commonly deployed for capacitive energy storage. Special emphasis is given to recent advances that intend to capture the nonclassical EDL behavior such as oscillatory ion distributions, polarization of nonmetallic electrodes, charge transfer, and various forms of phase transitions in the micropores of electrodes interfacing with an organic electrolyte or ionic liquid. This comprehensive analysis highlights theoretical insights into predictable relationships between materials characteristics and electrochemical performance and offers a perspective on opportunities for further development toward rational design and optimization of electrochemical systems.

On the origins of conductive pulse sensing inside a nanopore

Nanopore sensing is nearly synonymous with resistive pulse sensing due to the characteristic occlusion of ions during pore occupancy, particularly at high salt concentrations. Contrarily, conductive pulses are observed under low salt conditions wherein electroosmotic flow is significant. Most literature reports counterions as the dominant mechanism of conductive events (a molecule-centric theory). However, the counterion theory does not fit well with conductive events occurring via net neutral-charged protein translocation, prompting further investigation into translocation mechanics. Herein, we demonstrate theory and experiments underpinning the translocation mechanism (i.e., electroosmosis or electrophoresis), pulse direction (i.e., conductive or resistive) and shape (e.g., monophasic or biphasic) through fine control of chemical, physical, and electronic parameters. Results from these studies predict strong electroosmosis plays a role in driving DNA events and generating conductive events due to polarization effects (i.e., a pore-centric theory).

Conductive events during nanopore sensing, are seen typically under low salt conditions and widely thought to arise from counterions brought into the pore via analyte. Here, authors show that an imbalance of ionic fluxes lead to conductive events.

Simulating the charging of cylindrical electrolyte-filled pores with the modified Poisson-Nernst-Planck equations

Understanding how electrolyte-filled porous electrodes respond to an applied potential is important to many electrochemical technologies. Here, we consider a model supercapacitor of two blocking cylindrical pores on either side of a cylindrical electrolyte reservoir. A stepwise potential difference $2\Phi$ between the pores drives ionic fluxes in the setup, which we study through the modified Poisson-Nernst-Planck equations, solved with finite elements.We focus our discussion on the dominant timescales with which the pores charge and how these timescales depend on three dimensionless numbers.Next to the dimensionless applied potential $\Phi$, we consider the ratio $R/R_b$ of the pore's resistance $R$ to the bulk reservoir resistance $R_b$ and the ratio $r_{p}/\ld$ of the pore radius $r_p$ to the Debye length $\ld$.We compare our data to theoretical predictions by Aslyamov and Janssen ($\Phi$), Posey and Morozumi ($R/R_b$), and Henrique, Zuk, and Gupta ($r_{p}/\ld$).Through our numerical approach, we delineate the validity of these theories and the assumptions on which they were based.

Relation between Charging Times and Storage Properties of Nanoporous Supercapacitors

An optimal combination of power and energy characteristics is beneficial for the further progress of supercapacitors-based technologies. We develop a nanoscale dynamic electrolyte model, which describes both static capacitance and the time-dependent charging process, including the initial square-root dependency and two subsequent exponential trends. The observed charging time corresponds to one of the relaxation times of the exponential regimes and significantly depends on the pore size. Additionally, we find analytical expressions providing relations of the time scales to the electrode’s parameters, applied potential, and the final state of the confined electrolyte. Our numerical results for the charging regimes agree with published computer simulations, and estimations of the charging times coincide with the experimental values.

Wafer-Scale Electroactive Nanoporous Silicon: Large and Fully Reversible Electrochemo-Mechanical Actuation in Aqueous Electrolytes

Nanoporosity in silicon results in interface-dominated mechanics, fluidics, and photonics that are often superior to the ones of the bulk material. However, their active control, for example, by electronic stimuli, is challenging due to the absence of intrinsic piezoelectricity in the base material. Here, for large-scale nanoporous silicon cantilevers wetted by aqueous electrolytes, electrosorption-induced mechanical stress generation of up to 600 kPa that is reversible and adjustable at will by potential variations of ≈1 V is shown. Laser cantilever bending experiments in combination with in operando voltammetry and step coulombmetry allow this large electro-actuation to be traced to the concerted action of 100 billions of parallel nanopores per square centimeter cross-section and determination of the capacitive charge-stress coupling parameter upon ion adsorption and desorption as well as the intimately related stress actuation dynamics for perchloric and isotonic saline solutions. A comparison with planar silicon surfaces reveals mechanistic insights on the observed electrocapillarity (Hellmann-Feynman interactions) with respect to the importance of oxide formation and wall roughness on the single-nanopore scale. The observation of robust electrochemo-mechanical actuation in a mainstream semiconductor with wafer-scale, self-organized nanoporosity opens up novel opportunities for on-chip integrated stress generation and actuorics at exceptionally low operation voltages.

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.

Transmission Line Circuit and Equation for an Electrolyte-Filled Pore of Finite Length

I discuss the strong link between the transmission line (TL) equation and the TL circuit model for the charging of an electrolyte-filled pore of finite length. In particular, I show how Robin and Neumann boundary conditions to the TL equation, proposed by others on physical grounds, also emerge in the TL circuit subject to a stepwise potential. The pore relaxes with a timescale τ, an expression for which consistently follows from the TL circuit, TL equation, and from the pore's known impedance. An approximation to τ explains the numerically determined relaxation time of the stack-electrode model of Lian et al. [Phys. Rev. Lett. 124, 076001 (2020)PRLTAO0031-900710.1103/PhysRevLett.124.076001].

Electric control of ionic transport in sub-nm nanopores

The ion transport behavior through sub-nm nanopores (length (L) ≈ radius (R)) on a film is different from that in nanochannels (L ≫ R), and even more different from the bulk behavior. The many intriguing phenomena in ionic transport are the key to the design and fabrication of solid-state nanofluidic devices. However, ion transport through sub-nm nanopores is not yet clearly understood. We investigate the ionic transport behavior of sub-nm nanopores from the perspective of conductance via molecular dynamics (MD) and experimental methods. Under the action of surface charge, the average ion concentration inside the nanopore is much higher than the bulk value. It is found that 100 mM is the transition point between the surface-charge-governed and the bulk behavior regimes, which is different from the transition point for nanochannels (10 mM). Moreover, by investigating the access, pores, surface charge, electroosmosis and potential leakage conductance, it is found that the conductive properties of the nanopore at low bulk concentration are determined by the surface charge potential leaks into the reservoir. Specifically, there is a huge increase in cation mobility through a cylindrical nanopore, which implies potential applications for the fast charging of supercapacitors and batteries. Sub-nm nanopores also show a strong selectivity toward Na⁺, and a strong repellence toward Cl⁻. These conclusions presented here will be useful not only in understanding the behavior of ion transport, but also in the design of nanofluidic devices.

The ion transport behavior through sub-nm nanopores (length (L) ≈ radius (R)) on a film is different from that in nanochannels (L ≫ R), and even more different from the bulk behavior.

Ionic Layering and Overcharging in Electrical Double Layers in a Poisson-Boltzmann Model

Electrical double layers (EDLs) play a significant role in a broad range of physical phenomena related to colloidal stability, diffuse-charge dynamics, electrokinetics, and energy storage applications. Recently, it has been suggested that for large ion sizes or multivalent electrolytes, ions can arrange in a layered structure inside the EDLs. However, the widely used Poisson-Boltzmann models for EDLs are unable to capture the details of ion concentration oscillations and the effect of electrolyte valence on such oscillations. Here, by treating a pair of ions as hard spheres below the distance of closest approach and as point charges otherwise, we are able to predict ionic layering without any additional parameters or boundary conditions while still being compatible with the Poisson-Boltzmann framework. Depending on the combination of ion valence, size, and concentration, our model reveals a structured EDL with spatially oscillating ion concentrations. We report the dependence of critical ion concentration, i.e., the ion concentration above which the oscillations are observed, on the counter-ion valence and the ion size. More importantly, our model displays quantitative agreement with the results of computationally intensive models of the EDL. Finally, we analyze the nonequilibrium problem of EDL charging and demonstrate that ionic layering increases the total charge storage capacity and the charging timescale.