Effects of electrostatic correlations on electrokinetic phenomena

Predicting the Electrophoretic Mobility of Charged Particles in an Aqueous Medium

Electrophoresis of charged particles has important applications in biochemical separation processes. The mobility of these particles depends on the surrounding electric double layer (EDL), which is impacted by solvent restructuring because of hydration interactions. Nevertheless, most theoretical estimates ignore such interactions during computation of the electrophoretic mobility. Here, we employ a complementary blend of mean-field analysis and molecular dynamics simulations performed for a peptide-G-quadruplex complex to assess how hydration interactions alter the mobility of a charged particle in an aqueous medium. These interactions are seen to stabilize the EDL, resulting in more significant localized counterion concentrations while strengthening the ensuing electrokinetic flow. The ordering of ions near the particle surface is obtained only upon including hydration interaction, revealing that the hydration water molecules act as a glue for forming a stable EDL, a key finding of this work. Conversely, the observed microstructure of ions near the charged surface as obtained from our theory establishes a bridge link between the micro and continuum model. The presence of larger counter ions enhances the drag on the particle, thus restricting its mobility. The mobility also becomes dependent on size, which may be useful for isolating a wide array of biomolecules. The impact of hydration interactions intensifies with increases in particle size, surface charge density, and bulk ion concentration.

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.

Diffusioosmotic flow reversals due to ion-ion electrostatic correlations

Existing theories of diffusioosmosis have neglected ion-ion electrostatic correlations, which are important in concentrated electrolytes. Here, we develop a mathematical model to numerically compute the diffusioosmotic mobilities of binary symmetric electrolytes across low to high concentrations in a charged parallel-plate channel. We use the modified Poisson equation to model the ion-ion electrostatic correlations and the Bikerman model to account for the finite size of ions. We report two key findings. First, ion-ion electrostatic correlations can cause a unique reversal in the direction of diffusioosmosis. Such a reversal is not captured by existing theories, occurs at ≈ 0.4 M for a monovalent electrolyte, and at a much lower concentration of ≈ 0.003 M for a divalent electrolyte in a channel with the same surface charge. This highlights that diffusioosmosis of a concentrated electrolyte can be qualitatively different from that of a dilute electrolyte, not just in its magnitude but also its direction. Second, we predict a separate diffusioosmotic flow reversal, which is not due to electrostatic correlations but the competition between the underlying chemiosmosis and electroosmosis. This reversal can be achieved by varying the magnitude of the channel surface charge without changing its sign. However, electrostatic correlations can radically change how this flow reversal depends on the channel surface charge and ion diffusivity between a concentrated and a dilute electrolyte. The mathematical model developed here can be used to design diffusioosmosis of dilute and concentrated electrolytes, which is central to applications such as species mixing and separation, enhanced oil recovery, and reverse electrodialysis.

Analytical solutions for viscoelectric effects in electrokinetic nanochannels

Understanding electrokinetic transport in nanochannels and nanopores is essential for emerging biological and electrochemical applications. The viscoelectric effect is an important mechanism implicated in the increase of local viscosity due to the polarization of a solvent under a strong electric field. However, most analyses of the viscoelectric effect have been limited to numerical analyses. In this work, we present a set of analytical solutions applicable to the physical description of viscoelectric effects in nanochannel electrokinetic systems. To achieve such closed-form solutions, we employ the Debye-Hückel approximation of small diffuse charge layer potentials compared to the thermal potential. We analyze critical parameters, including electroosmotic flow profiles, electroosmotic mobility, flow rate, and channel conductance. We compare and benchmark our analytical solutions with published predictions from numerical models. Importantly, we leverage these analytical solutions to identify essential thermophysical and nondimensional parameters that govern the behavior of these systems. We identify scaling parameters and relations among surface charge density, ionic strength, and nanochannel height.

Unraveling the Microscopic Mechanism of Molecular Ion Interaction with Monoclonal Antibodies: Impact on Protein Aggregation

Understanding and predicting protein aggregation represents one of the major challenges in accelerating the pharmaceutical development of protein therapeutics. In addition to maintaining the solution pH, buffers influence both monoclonal antibody (mAb) aggregation in solution and the aggregation mechanisms since the latter depend on the protein charge. Molecular-level insight is necessary to understand the relationship between the buffer-mAb interaction and mAb aggregation. Here, we use all-atom molecular dynamics simulations to investigate the interaction of phosphate (Phos) and citrate (Cit) buffer ions with the Fab and Fc domains of mAb COE3. We demonstrate that Phos and Cit ions feature binding mechanisms, with the protein that are very different from those reported previously for histidine (His). These differences are reflected in distinctive ion-protein binding modes and adsorption/desorption kinetics of the buffer molecules from the mAb surface and result in dissimilar effects of these buffer species on mAb aggregation. While His shows significant affinity toward hydrophobic amino acids on the protein surface, Phos and Cit ions preferentially bind to charged amino acids. We also show that Phos and Cit anions provide bridging contacts between basic amino acids in neighboring proteins. The implications of such contacts and their connection to mAb aggregation in therapeutic formulations are discussed.

Incorporating Electrolyte Correlation Effects into Variational Models of Electrochemical Interfaces

We propose a way for obtaining a classical free energy density functional for electrolytes based on a first-principle many-body partition function. Via a one-loop expansion, we include coulombic correlations beyond the conventional mean-field approximation. To examine electrochemical interfaces, we integrate the electrolyte free energy functional into a hybrid quantum-classical model. This scheme self-consistently couples electronic, ionic, and solvent degrees of freedom and incorporates electrolyte correlation effects. The derived free energy functional causes a correlation-induced enhancement in interfacial counterion density and leads to an overall increase in capacitance. This effect is partially compensated by a reduction of the dielectric permittivity of interfacial water. At larger surface charge densities, ion crowding at the interface stifles these correlation effects. While scientifically intriguing already at planar interfaces, we anticipate these correlation effects to play an essential role for electrolytes in nanoconfinement.

Nature of Overcharging and Charge Inversion in Electrical Double Layers

Understanding overcharging and charge inversion is one of the long-standing challenges in soft matter and biophysics. To study these phenomena, we employ the modified Gaussian renormalized fluctuation theory, which allows for the self-consistent accounting of spatially varying ionic strength as well as the spatial variations in dielectric permittivity and excluded volume effects. The underlying dependence of overcharging on the electrostatic coupling is elucidated by varying the surface charge, counterion valency, and dielectric contrast. Consistent with simulations, three characteristic regimes corresponding to weak, moderate, and strong coupling are identified. Important features like the inversion of zeta potential, crowding, and ionic layering at the surface are successfully captured. For weak coupling, there is no overcharging. In the moderate coupling regime, overcharging increases with the surface charge. Finally, in the strong coupling regime, ionic crowding and saturation in overcharging are observed. Our theory predicts a nonmonotonic dependence of charge inversion on multivalent salt concentration as well as the addition of monovalent salt, in quantitative agreement with experiments.

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.

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.

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.

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.

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.

Structural and electrostatic properties between pH-responsive polyelectrolyte brushes studied by augmented strong stretching theory

In this paper, we study electrostatic and structural properties between pH-responsive polyelectrolyte brushes by using a strong stretching theory accounting for excluded volume interactions, the density of polyelectrolyte chargeable sites and the Born energy difference between the inside and outside of the brush layer.In a free energy framework, we obtain self-consistent field equations to determine electrostatic properties between two pH-responsive polyelectrolyte brushes. We elucidate that in the region between two pH-responsive polyelectrolyte brushes, electrostatic potential at the centerline and osmotic pressure increase not only with excluded volume interaction, but also with density of chargeable sites on a polyelectrolyte molecule. Importantly, we clarify that when two pH-responsive polyelectrolyte brushes approach each other, the brush thickness becomes short and that a large excluded volume interaction and a large density of chargeable sites yield the enhanced contract of polyelectrolyte brushes. In addition, we also demonstrate how the influence of such quantities as pH, the number of Kuhn monomers, the density of charged sites, the lateral separation between adjacent polyelectrolyte brushes, Kuhn length on the electrostatic and structural properties between the two polyelectrolyte brushes is affected by the exclusion volume interaction. Finally, we investigate the influence of Born energy difference on the thickness of polyelectrolyte brushes and the osmotic pressure between two pH-responsive polyelectrolyte brushes.

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.

Thermodynamics of Electrolyte Solutions Near Charged Surfaces: Constant Surface Charge vs. Constant Surface Potential

Electric double layers are ubiquitous in science and engineering and are of current interest, owing to their applications in the stabilization of colloidal suspensions and as supercapacitors. While the structure and properties of electric double layers in electrolyte solutions near a charged surface are well characterized, there are subtleties in calculating thermodynamic properties from the free energy of a system with charged surfaces. These subtleties arise from the difference in the free energy between systems with constant surface charge and constant surface potential. In this work, we present a systematic, pedagogical framework to properly account for the different specifications on charged bodies in electrolyte solutions. Our approach is fully variational---that is, all free energies, boundary conditions, relevant electrostatic equations, and thermodynamic quantities are systematically derived using variational principles of thermodynamics. We illustrate our approach by considering a simple electrolyte solution between two charged surfaces using the Poisson--Boltzmann theory. Our results highlight the importance of using the proper thermodynamic potential and provide a general framework for calculating thermodynamic properties of electrolyte solutions near charged surfaces. Specifically, we present the calculation of the pressure and the surface tension between two charged surfaces for different boundary conditions, including mixed boundary conditions.

Localized X-Ray Photoelectron Impedance Spectroscopy (LoXPIS) for Tapping into Charge-Dynamics of Ionic-Liquid Electrolytes within Energy Storage Devices

Many electrochemical devices are based on the fundamental process of ion migration and accumulation on surfaces. Complex interplay of molecular properties of ions and device dimensions control the entire process...

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.

Comparative Operando XPS and SEM Spatiotemporal Potential Mapping of Ionic Liquid Polarization in a Coplanar Electrochemical Device

The polarization response of a coplanar electrochemical capacitor covered with an ionic liquid as the electrolyte has been examined using a combination of two powerful analytic techniques, X-ray photoelectron spectroscopy (XPS) and scanning electron microcopy (SEM). Spatiotemporal distribution of the ionic liquid surface potential, upon DC or AC (square wave) biasing, has been monitored via chemical element binding energy shifts using XPS and secondary electron intensity variations using SEM. SEM's high spatial resolution and speedy imaging together with application of a data mining algorithm made mapping of the surface potential distribution across the capacitor possible. Interestingly, despite the differences in the detection principles, both techniques yield similar polarization relaxation time constants. The results demonstrate the power of a synergistic combination of the two techniques with complementary capabilities and pave the way to a deeper understanding of liquid/solid interfaces and for performance evaluation and diagnostics of electrochemical devices.

Relaxation dynamics of two interacting electrical double-layers in a 1D Coulomb system

We consider an out-of-equilibrium one-dimensional model for two electrical double-layers. With a combination of exact calculations and Brownian dynamics simulations, we compute the relaxation time (τ) for an electroneutral salt-free suspension, made up of two fixed colloids, withNneutralizing mobile counterions. ForNodd, the two double-layers never decouple, irrespective of their separationL; this is the regime of like-charge attraction, whereτexhibits a diffusive scaling inL²for largeL. On the other hand, for evenN,Lno longer is the relevant length scale for setting the relaxation time; this role is played by the Bjerrum length. This leads to distinctly different dynamics: forNeven, thermal effects are detrimental to relaxation, increasingτ, while they accelerate relaxation forNodd. Finally, we also show that the mean-field theory is recovered for largeNand moreover, that it remains an operational treatment down to relatively small values ofN(N> 3).

A grahame triple-layer model unifies mica monovalent ion exchange, zeta potential, and surface forces

A triple-layer model of the mica/water electrical double layer (EDL) unifies prediction of zeta potential, ion-exchange, and surface-force isotherms. The theory treats cations as partially dehydrated and complexed specifically to the anionic exchange sites of mica. A diffuse layer commencing at the outer Helmholtz plane (OHP) balances the surface charge not neutralized by adsorbed cations in the inner Helmholtz plane (IHP). Ion-binding equilibrium constants are assessed from zeta-potential measurements and used thereafter to predict ion-exchange isotherms and surface forces. Basal-plane mica surface charge is almost completely neutralized by specific binding of cations, including hydronium ions. The charge in the diffuse layer is only a few percent of the mica crystallographic charge density but leads to long-range electrostatic interactions between charged surfaces. The Grahame triple-layer model of the aqueous EDL provides a robust, quantitative, and unified description of the mica/water interface.

Electroosmosis as a probe for electrostatic correlations

We study the role of ionic correlations on the electroosmotic flow in planar double-slit channels, without salt. We propose an analytical theory, based on recent advances in the understanding of correlated systems. We compare the theory with mean-field results and validate it by means of dissipative particle dynamics simulations. Interestingly, for some surface separations, correlated systems exhibit a larger flow than predicted by mean-field. We conclude that the electroosmotic properties of a charged system can be used, in general, to infer and weight the importance of electrostatic correlations therein.

Nanofluidic Charge Transport under Strong Electrostatic Coupling Conditions

The comprehensive depiction of the many-body effects governing nanoconfined electrolytes is an essential step for the conception of nanofluidic devices with optimized performance. By incorporating self-consistently multivalent charges into the Poisson-Boltzmann equation dressed by a background monovalent salt, we investigate the impact of strong-coupling electrostatics on the nanofluidic transport of electrolyte mixtures. We find that the experimentally observed negative streaming currents in anionic nanochannels originate from the collective effect of Cl^- attraction by the interfacially adsorbed multivalent cations and the no-slip layer reducing the hydrodynamic contribution of these cations to the net current. The like-charge current condition emerging from this collective mechanism is shown to be the reversal of the average potential within the no-slip zone. Applying the formalism to surface-coated membrane nanoslits located in the giant dielectric permittivity regime, we reveal a new type of streaming current activated by attractive polarization forces. Under the effect of these forces, multivalent ions added to the KCl solution set a charge separation and generate a counterion current between the neutral slit walls where the pure KCl conductivity vanishes. The adjustability of the current characteristics solely via the valency and amount of the added multivalent ions identifies the underlying process as a promising mechanism for nanofluidic ion separation purposes.

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.

Interfacial Layering in the Electric Double Layer of Ionic Liquids

Ions in ionic liquids and concentrated electrolytes reside in a crowded, strongly interacting environment, leading to the formation of discrete layers of charges at interfaces and spin-glass structure in the bulk. Here, we propose a simple theory that captures the coupling between steric and electrostatic forces in ionic liquids. The theory predicts the formation of discrete layers of charge at charged interfaces. Further from the interface, or at low polarization of the electrode, the model outputs slowly decaying oscillations in the charge density with a wavelength of a single ion diameter, as shown by analysis of the gradient expansion. The gradient expansion suggests a new structure for partial differential equations describing the electrostatic potential at charged interfaces. We find quantitative agreement between the theory and molecular simulations in the differential capacitance and concentration profiles.

Slip Effects on Ionic Current of Viscoelectric Electroviscous Flows through Different Length Nanofluidic Channels

The pressure driven slip flow of an electrolyte solution is studied through different nanofluidic channel lengths at varying salt concentrations. The viscous-thickening due to the electrostatic interactions within the electric double layer and the reverse ionic transport due to the streaming potential are developed. The influence of the Navier slip boundary condition is described under both electroviscous and viscoelectric effects with a surface charge regulation (CR) model while the observed behavior is compared and validated with molecular dynamic (MD) calculations from multiple studies. Results show that electroviscous and viscoelectric effects decrease transport. Earlier studies at the no slip boundary presented an increase of ionic current by increasing salt concentration and decreasing channel length. In contrast, our study found that the ionic current occurred almost independent of both salt concentration and channel length, except for very short channels and very low salt concentrations, when electroviscous and viscoelectric effects were considered. In the case of the constant slip length condition, ionic conduction was enhanced, but velocity slip developing on surfaces showed significant variation based on the salt concentration and channel length. This is due to the natural CR behavior enhancing the surface charge and consequential near surface electrohydrodynamics as a result of increase in salt concentration and/or decrease of channel length. Considering that the electroviscous effect alone creates up to 70% lower velocity slips than Poiseuille flow predictions, while further including the viscoelectric effect, results in an almost no-slip condition at high salt concentrations and/or short channels. As a result, the ionic current of a viscoelectric electroviscous slip flow is found to be equal to 1/3 of an electroviscous slip flow and to decrease with a decrease in the channel length.

On Induced Surface Charge in Solid-State Nanopores

Solid-state nanopores constitute a versatile platform for study of ion transport in nanoconfinement. The electrical double layer (EDL) plays a vital role in such nanoconfinements, but effects of induced surface charge on the EDL in the presence of an external transmembrane electric field are yet to be characterized. Here, the formation of induced charge on the nanopore sidewall surface and its effects, via modulation of the EDL and electroosmotic flow, on the ionic current are elucidated using a novel experimental setup with solid-state truncated-pyramidal nanopores. This study consists of three complementary approaches, i.e., an analytical model for induced surface charge, numerical simulation of induced surface charge, electroosmotic flow, and ionic current, and experimental validation with respect to the ionic current. The induced surface charge is generated by polarization in the dielectric membrane as a response to the applied electric field. This charge generation results in a nonuniform density of surface charge along the nanopore sidewall. It further causes ions in the electrolyte to redistribute, leading to a massive accumulation of single-polarity ions in the EDL and their counterions near the smaller opening of the nanopore. It also alters electrohydrodynamic properties in the nanopore, giving rise to the formation of electroosmotic vortexes in the vicinity of the smaller opening of the nanopore. Finally, the pattern of the electroosmotic flow can significantly influence the transport properties of the nanopore.

Routine, ensemble characterisation of electrophoretic mobility in high and saturated ionic dispersions

With the industrialisation of nanoparticle manufacture, the pervasive incursion of nanoparticles into the environment, the need to characterise nano-scale pharmaceuticals and living systems in replicated in vivo conditions, the continuing development of new theories to describe the electro-kinetic behaviour of nano-particles in representative ionic strengths and numerous other applications, there is an urgent requirement to provide simple and effective experimental tools to validate these models and explore new systems. Micro-electrophoresis implemented with a diffusion barrier, which isolates the dispersed phase from the electrode surface, is demonstrated as enabling such measurements for the first time, preventing the catastrophic outgassing, precipitation and sample degradation observed when the dispersed phase is in close proximity to the electrode surface. Using a measurement of a few minute's duration in a standard laboratory light scattering instrument we reproduce the theoretically predicted phenomena of asymptotic, non-zero electrophoretic mobility with increasing ionic strength, the cationic Hofmeister series dependency, charge inversion and a continuously decreasing variation in mobility with pH as molarity increases. Standard operating procedures are developed and included to encourage further work.

Strong stretching theory for pH-responsive polyelectrolyte brushes in large salt concentrations

In this paper, we develop a theory for describing the thermodynamics, configuration, and electrostatics of strongly-stretched, pH-responsive polyelectrolyte (PE) brushes in the presence of large salt concentrations.

Screening length for finite-size ions in concentrated electrolytes

The classical Debye-Hückel (DH) theory clearly accounts for the origin of screening in electrolyte solutions and works rather well for dilute electrolyte solutions. While the Debye screening length decreases with the ion concentration and is independent of ion size, recent surface-force measurements imply that for concentrated solutions, the screening length exhibits an opposite trend; it increases with ion concentration and depends on the ionic size. The screening length is usually defined by the response of the electrolyte solution to a test charge but can equivalently be derived from the charge-charge correlation function. By going beyond DH theory, we predict the effects of ion size on the charge-charge correlation function. A simple modification of the Coulomb interaction kernel to account for the excluded volume of neighboring ions yields a nonmonotonic dependence of the screening length (correlation length) on the ionic concentration, as well as damped charge oscillations for high concentrations.

Theory of Surface Forces in Multivalent Electrolytes

Aqueous electrolyte solutions containing multivalent ions exhibit various intriguing properties, including attraction between like-charged colloidal particles, which results from strong ion-ion correlations. In contrast, the classical Derjaguin-Landau-Verwey-Overbeek theory of colloidal stability, based on the Poisson-Boltzmann mean-field theory, always predicts a repulsive electrostatic contribution to the disjoining pressure. Here, we formulate a general theory of surface forces, which predicts that the contribution to the disjoining pressure resulting from ion-ion correlations is always attractive and can readily dominate over entropic-induced repulsions for solutions containing multivalent ions, leading to the phenomenon of like-charge attraction. Ion-specific short-range hydration interactions, as well as surface charge regulation, are shown to play an important role at smaller separation distances but do not fundamentally change these trends. The theory is able to predict the experimentally observed strong cohesive forces reported in cement pastes, which result from strong ion-ion correlations involving the divalent calcium ion.

The effects of electrostatic correlations on the ionic current rectification in conical nanopores

Ion-ion electrostatic correlations are recognized to play a significant role in the presence of concentrated multivalent electrolytes. To account for their impact on ionic current rectification phenomenon in conical nanopores, we use the modified continuum Poisson-Nernst-Planck (PNP) equations by Bazant et al. Coupled with the Stokes equations, the effects of the EOF are also included. We thoroughly investigate the dependence of the ionic current rectification ratios as a function of the double layer thickness and the electrostatic correlation length. By considering the electrostatic correlations, the modified PNP model successfully captures the ionic current rectification reversal in nanopores filled with lanthanum chloride LaCl₃ . This finding qualitatively agrees with the experimental observations that cannot be explained by the standard PNP model, suggesting that ion-ion electrostatic correlations are responsible for this reversal behavior. The modified PNP model not only can be used to explain the experiments, but also go beyond to provide a design tool for nanopore applications involving multivalent electrolytes.

Surface forces generated by the action of electric fields across liquid films

We explore the force generation and surface interactions arising when electric fields are applied across fluid films. Using a surface force balance (SFB) we measure directly the force between two electrodes in crossed-cylinder geometry across dielectric and electrolytic fluids. In the case of dielectric films the field between the electrodes exerts a force which can be well explained using classic expressions and with no fitting parameters. However when the electrodes are separated by a film of electrolyte, an alternating electric field induces a force which diverges substantially from the calculated static response of the electrolyte. The magnitude of the force is larger than predicted, and the interaction can switch from attractive to repulsive. Furthermore, the approach to steady state in electrolyte takes place over 102-103 s which is very slow compared to both the charging and viscous timescales of the system. The non-trivial electrolyte response in AC electric fields, measured here directly, is likely to underlie several recent reports of unexpected and bifurcating forces driving colloids in AC fields. Our measurements suggest ways to control colloidal and soft matter using electric fields, as well as providing a direct measure of the length- and time-scales relevant in AC electrochemical and electrokinetic systems.

Revisiting the strong stretching theory for pH-responsive polyelectrolyte brushes: effects of consideration of excluded volume interactions and an expanded form of the mass action law

In this paper, we develop a theory to account for the effect of excluded volume (EV) interactions in the strong stretching theory (SST) based description of pH-responsive polyelectrolyte (PE) brushes. The existing studies have considered the PE brushes to be present in a θ-solvent and hence have neglected the EV interactions; however, such a consideration cannot describe the situations where the pH-responsive brushes are in a "good" solvent. Secondly, we consider a more expanded form of the mass action law, governing the pH-dependent ionization of the PE molecules, in the SST description of the PE brushes. This expanded form of the mass action law considers different values of γa³ (γ is the density of chargeable sites on the PE molecule and a is the PE Kuhn length) and therefore is an improvement over the existing SST models of PE brushes as well as other theories involving pH-responsive PE molecules that always consider γa³ = 1. Our results demonstrate that the EV effects enhance the brush height by inducing additional PE inter-segmental repulsion. Similarly, the consideration of the expanded form of the mass action law would lead to a reduced (enhanced) brush height for γa³ < 1 (γa³ > 1). We also quantify variables such as the monomer density distribution, the distribution of the ends of the PE brush, and the EDL electrostatic potential and explain their differences with respect to those obtained with no EV interactions or γa³ = 1.

Surface charge-dependent hydrodynamic properties of an electroosmotic slip flow

The electroosmosis effects at the interface of an aqueous NaCl solution and a charged silicon surface are studied using a molecular dynamics (MD) method. Considering a plug-like electroosmotic flow, we identified a thin interfacial layer in the immediate vicinity of the charged surface, where the flow velocity experiences almost linear spatial variations. The thickness of this interfacial layer is found to be linearly dependent on the surface charge density, with a negative slope which slightly depends on the surface hydrophobicity while being independent of the salt concentration, electric field strength, and orientation of the surface lattice. It is also found that upon increasing the surface charge density, the effective slip length first increases up to a maximum amount and then follows an almost linear reduction. We found that increasing the salt concentration drastically reduces the surface charge at which the effective slip length reaches its maximum amount. For highly concentrated solutions, therefore, the effective slip length could be assumed to change linearly in the whole range of the surface charge density, with a slope which is proportional to the square root of the electric field strength divided by the depth of the potential well assigned to the surface atoms εwall. Also, in a wide range of the surface charge density, the slip velocity is found to be a constant fraction of the electroosmotic velocity, which could be measured experimentally. Finally, by comparing the electroosmotic velocities calculated from the Stokes equation (considering both the slip and no-slip boundary conditions) with our MD results, we found that the no-slip boundary condition, which is normally used in analytical calculations, leads to a very inaccurate result for the studied system.

Electrical Double Layers: Effects of Asymmetry in Electrolyte Valence on Steric Effects, Dielectric Decrement, and Ion-Ion Correlations

We study the effects of asymmetry in electrolyte valence (i.e., non z: z electrolytes) on mean field theory of the electrical double layer. Specifically, we study the effect of valence asymmetry on finite ion-size effects, the dielectric decrement, and ion-ion correlations. For a model configuration of an electrolyte near a charged surface in equilibrium, we present comprehensive analytical and numerical results for the potential distribution, electrode charge density, capacitance, and dimensionless salt uptake. We emphasize that the asymmetry in electrolyte valence significantly influences the diffuse-charge relations, and prior results reported in the literature are readily extended to non z: z electrolytes. We develop scaling relations and invoke physical arguments to examine the importance of asymmetry in electrolyte valence on the aforementioned effects. We conclude by providing implications of our findings on diffuse-charge dynamics and other electrokinetic phenomena.

Thermodynamics of Ion Separation by Electrosorption

We present a simple, top-down approach for the calculation of minimum energy consumption of electrosorptive ion separation using variational form of the (Gibbs) free energy. We focus and expand on the case of electrostatic capacitive deionization (CDI). The theoretical framework is independent of details of the double-layer charge distribution and is applicable to any thermodynamically consistent model, such as the Gouy-Chapman-Stern and modified Donnan models. We demonstrate that, under certain assumptions, the minimum required electric work energy is indeed equivalent to the free energy of separation. Using the theory, we define the thermodynamic efficiency of CDI. We show that the thermodynamic efficiency of current experimental CDI systems is currently very low, around 1% for most existing systems. We applied this knowledge and constructed and operated a CDI cell to show that judicious selection of the materials, geometry, and process parameters can lead to a 9% thermodynamic efficiency and 4.6 kT per removed ion energy cost. This relatively high thermodynamic efficiency is, to our knowledge, by far the highest thermodynamic efficiency ever demonstrated for traditional CDI. We hypothesize that efficiency can be further improved by further reduction of CDI cell series resistances and optimization of operational parameters.

Diffusiophoresis of charged colloidal particles in the limit of very high salinity

Diffusiophoresis is the migration of a colloidal particle through a viscous fluid, caused by a gradient in concentration of some molecular solute; a long-range physical interaction between the particle and solute molecules is required. In the case of a charged particle and an ionic solute (e.g., table salt, NaCl), previous studies have predicted and experimentally verified the speed for very low salt concentrations at which the salt solution behaves ideally. The current study presents a study of diffusiophoresis at much higher salt concentrations (approaching the solubility limit). At such large salt concentrations, electrostatic interactions are almost completely screened, thus eliminating the long-range interaction required for diffusiophoresis; moreover, the high volume fraction occupied by ions makes the solution highly nonideal. Diffusiophoretic speeds were found to be measurable, albeit much smaller than for the same gradient at low salt concentrations.

On the Impact of Electrostatic Correlations on the Double-Layer Polarization of a Spherical Particle in an Alternating Current Field

At concentrated electrolytes, the ion-ion electrostatic correlation effect is considered an important factor in electrokinetics. In this paper, we compute, in theory and simulation, the dipole moment for a spherical particle (charged, dielectric) under the action of an alternating electric field using the modified continuum Poisson-Nernst-Planck (PNP) model by Bazant et al. [ Double Layer in Ionic Liquids: Overscreening Versus Crowding . Phys. Rev. Lett. 2011 , 106 , 046102 ] We investigate the dependency of the dipole moment in terms of frequency and its variation with such quantities like ζ-potential, electrostatic correlation length, and double-layer thickness. With thin electric double layers, we develop simple models through performing an asymptotic analysis of the modified PNP model. We also present numerical results for an arbitrary Debye screening length and electrostatic correlation length. From the results, we find a complicated impact of electrostatic correlations on the dipole moment. For instance, with increasing the electrostatic correlation length, the dipole moment decreases and reaches a minimum and then it goes up. This is because of initially decreasing of surface conduction and finally increasing due to the impact of ion-ion electrostatic correlations on ion's convection and migration. Also, we show that in contrast to the standard PNP model, the modified PNP model can qualitatively explain the data from the experimental results in multivalent electrolytes.

Anomalies in the equilibrium and nonequilibrium properties of correlated ions in complex molecular environments

Emergent statistical attributes, and therefore the equations of state, of an assembly of interacting charge carriers embedded within a complex molecular environment frequently exhibit a variety of anomalies, particularly in the high-density (equivalently, the concentration) regime, which are not well understood, because they do not fall under the low-concentration phenomenologies of Debye-Hückel-Onsager and Poisson-Nernst-Planck, including their variants. To go beyond, we here use physical concepts and mathematical tools from quantum scattering theory, transport theory with the Stosszahlansatz of Boltzmann, and classical electrodynamics (Lorentz gauge) and obtain analytical expressions both for the average and the frequency-wave vector-dependent longitudinal and transverse current densities, diffusion coefficient, and the charge density, and therefore the analytical expressions for (a) the chemical potential, activity coefficient, and the equivalent conductivity for strong electrolytes and (b) the current-voltage characteristics for ion-transport processes in complex molecular environments. Using a method analogous to the notion of Debye length and thence the electrical double layer, we here identify a pair of characteristic length scales (longitudinal and the transverse), which, being wave vector and frequency dependent, manifestly exhibit nontrivial fluctuations in space-time. As a unifying theme, we advance a quantity (inverse length dimension), g_{scat}^{(a)}, which embodies all dynamical interactions, through various quantum scattering lengths, relevant to molecular species a, and the analytical behavior which helps us to rationalize the properties of strong electrolytes, including anomalies, in all concentration regimes. As an example, the behavior of g_{scat}^{(a)} in the high-concentration regime explains the anomalous increase of the Debye length with concentration, as seen in a recent experiment on electrolyte solutions. We also put forth an extension of the standard diffusion equation, which manifestly incorporates the effects arising from the underlying microscopic collisions among constituent molecular species. Furthermore, we show a nontrivial connection between the current-voltage characteristics of electrolyte solutions and the Landauer's approach to electrical conduction in mesoscopic solids and thereby establish a definite conceptual bridge between the two disjoint subjects. For numerical insight, we present results on the aqueous solution of KCl as an example of strong electrolyte, and the transport (conduction as well as diffusion) of K^{+} ions in water, as an example of ion transport across the voltage-gated channels in biological cells.

From Solvent-Free to Dilute Electrolytes: Essential Components for a Continuum Theory

The increasing number of experimental observations on highly concentrated electrolytes and ionic liquids show qualitative features that are distinct from dilute or moderately concentrated electrolytes, such as self-assembly, multiple-time relaxation, and underscreening, which all impact the emergence of fluid/solid interfaces, and the transport in these systems. Because these phenomena are not captured by existing mean-field models of electrolytes, there is a paramount need for a continuum framework for highly concentrated electrolytes and ionic liquid mixtures. In this work, we present a self-consistent spatiotemporal framework for a ternary composition that comprises ions and solvent employing a free energy that consists of short- and long-range interactions, along with an energy dissipation mechanism obtained by Onsager's relations. We show that the model can describe multiple bulk and interfacial morphologies at steady-state. Thus, the dynamic processes in the emergence of distinct morphologies become equally as important as the interactions that are specified by the free energy. The model equations not only provide insights into transport mechanisms beyond the Stokes-Einstein-Smoluchowski relations but also enable qualitative recovery of three distinct regions in the full range of the nonmonotonic electrical screening length that has been recently observed in experiments in which organic solvent is used to dilute ionic liquids.