The electrical double layer for a fully asymmetric electrolyte around a spherical colloid: An integral equation study

Effect of Dielectric Constant on the Zeta Potential of Spherical Electric Double Layers

Zeta potential refers to the electrokinetic potential present in colloidal systems, exerting significant influence on the diverse properties of nano-drug delivery systems. The impact of the dielectric constant on the zeta potential and charge inversion of highly charged colloidal particles immersed in a variety of solvents spanning from polar, such as water, to nonpolar solvents and in the presence of multivalent salts was investigated through primitive Monte Carlo (MC) model simulations. Zeta potential, ξ, is decreased with the decreasing dielectric constant of the solvent and upon further increase in the salinity and the valency of the salt. At elevated levels of salt, the colloidal particles become overcharged in all solvents. As a result, their apparent charge becomes opposite in sign to the stoichiometric charge. This reversal of charge intensifies until reaching a saturation point with further increase in salinity.

Theoretical description of the electrical double layer for a mixture of n ionic species with arbitrary size and charge asymmetries. I. Spherical geometry

In this work, we propose a theoretical finite element description of the ionic profiles of a general mixture of n species of spherical charged particles dissolved in an implicit solvent, with arbitrary size and charge asymmetries, neutralizing a spherical macroion. This approach aims to close the gap between the nano- and the micro-scales in macroion solutions, taking into account the ion correlations and ionic excluded volume effects consistently. When these last two features are neglected, the classical non-linear Poisson-Boltzmann theory for n ionic species-with different ionic closest approach distances to the colloidal surface-is recovered as a limit case. As a proof of concept, we study the electrical double layer of an electroneutral mixture of oppositely charged colloids and small microions, with an asymmetry 1:333 in size and 1:10 in valence, in salt-free and added salt environments. Our theoretical approach displays a good agreement regarding the ionic profiles, the integrated charge, and the mean electrostatic potential obtained from molecular dynamics simulations with explicit-sized microions. Although the non-linear Poisson-Boltzmann colloid-colloid and colloid-microion profiles differ notably from those obtained via molecular dynamics simulations with explicit small-sized ions, the associated mean electrostatic potential agrees well with the corresponding explicit microion simulations.

Structural Forces in Ionic Liquids: The Role of Ionic Size Asymmetry

Ionic liquids (ILs) are charged fluids composed of anions and cations of different size and shape. The ordering of charge and density in ILs confined between charged interfaces underlies numerous applications of IL electrolytes. Here, we analyze the screening behavior and the resulting structural forces of a representative IL confined between two charge-varied plates. Using both molecular dynamics simulations and a continuum theory, we contrast the screening features of a more-realistic asymmetric system and a less-realistic symmetric one. The ionic size asymmetry plays a nontrivial role in charge screening, affecting both the ionic density profiles and the disjoining pressure distance dependence. Ionic systems with size asymmetry are stronger coupled systems, and this manifests itself both in their response to the electrode polarization and spontaneous structure formation at the interface. Analytical expressions for decay lengths of the disjoining pressure are obtained in agreement with the pressure profiles computed from molecular dynamics simulations.

Differential capacitance of ionic liquids according to lattice-gas mean-field model with nearest-neighbor interactions

The Bragg-Williams free energy is used to incorporate nearest-neighbor interactions into the lattice gas model of a solvent-free ionic liquid near a planar electrode. We calculate the differential capacitance from solutions of the mean-field consistency relation, arriving at an explicit expression in the limit of a weakly charged electrode. The two additional material parameters that appear in the theory-the degree of nonideality and the resistance to concentration changes of each ion type-give rise to different regimes that we identify and discuss. As the nonideality parameter, which becomes more positive for stronger nearest-neighbor attraction between like-charged ions, increases and the electrode is weakly charged, the differential capacitance is predicted to transition through a divergence and subsequently adopt negative values just before the ionic liquid becomes structurally unstable. This is associated with the spontaneous charging of an electrode at vanishing potential. The physical origin of the divergence and the negative sign of the differential capacitance is a nonmonotonic relationship between the surface potential and surface charge density, which reflects the formation of layered domains alternatingly enriched in counterions and coions near the electrode. The decay length of this layered domain pattern, which can be many times larger than the ion size, is reminiscent of the recently introduced concept of "underscreening."

The non-dominance of counterions in charge-asymmetric electrolytes: non-monotonic precedence of electrostatic screening and local inversion of the electric field by multivalent coions

The asymptotic convergence of the thermodynamic and structural properties of unequally-sized charge-symmetric ions in strong electric fields was postulated more than thirty years ago by Valleau and Torrie as the dominance of counterions via the non-linear Poisson-Boltzmann theory [Valleau and Torrie, J. Chem. Phys., 1982, 76, 4623]. According to this mean field prescription, the properties of the electrical double layer near a highly charged electrode immersed in a size-asymmetric binary electrolyte converge to those of a size-symmetric electrolyte if the properties of counterions are the same in both instances. On the other hand, some of the present authors have shown that, in fact, counterions do not dominate the electrical properties of a spherical macroion in the presence of unequally-sized ions, symmetric in valence, if ion correlations and ionic excluded volume effects are taken into account consistently. These ingredients are neglected in the classical Poisson-Boltzmann picture. In the present work, we show the occurrence of the non-dominance of counterions in the opposite scenario, that is, when ions are equally-sized but asymmetric in valence. This is performed in the presence of highly charged colloidal surfaces of spherical and planar geometries for different ionic volume fractions. In addition to the phenomenon of non-dominance of counterions, our simulations and theoretical data also exhibit a non-monotonic order or precedence in the mean electrostatic potential, or electrostatic screening, at the Helmholtz plane of a charged colloid. This interesting behaviour is analyzed as a function of the coion's valence, the ionic volume fraction, and the charge and size of the colloidal particle. All these phenomena are explained in terms of the decay of the electric field near the colloidal surface, and by the appearance of a local inversion of both the electric field and the integrated surface charge density of the colloidal particle in the presence of monovalent counterions and multivalent coions.

Structure of fully asymmetric mixed electrolytes around a charged nanoparticle: a density functional and simulation investigation

A systematic study on the structure of mixed electrolytes with arbitrary size and charge asymmetry around a charged nanoparticle is carried out using density functional theory and Monte Carlo simulation.

Structure of spherical electric double layers with fully asymmetric electrolytes: a systematic study by Monte Carlo simulations and density functional theory

A systematic investigation of the spherical electric double layers with the electrolytes having size as well as charge asymmetry is carried out using density functional theory and Monte Carlo simulations. The system is considered within the primitive model, where the macroion is a structureless hard spherical colloid, the small ions as charged hard spheres of different size, and the solvent is represented as a dielectric continuum. The present theory approximates the hard sphere part of the one particle correlation function using a weighted density approach whereas a perturbation expansion around the uniform fluid is applied to evaluate the ionic contribution. The theory is in quantitative agreement with Monte Carlo simulation for the density and the mean electrostatic potential profiles over a wide range of electrolyte concentrations, surface charge densities, valence of small ions, and macroion sizes. The theory provides distinctive evidence of charge and size correlations within the electrode-electrolyte interface in spherical geometry.

Temperature effects on energy production by salinity exchange

In recent years, the capacitance of the interface between charged electrodes and ionic solutions (the electric double layer) has been investigated as a source of clean energy. Charge is placed on the electrodes either by means of ion-exchange membranes or of an external power source. In the latter method, net energy is produced by simple solution exchange in open circuit, due to the associated decrease in the capacitance of the electric double layer. In this work, we consider the change in capacitance associated with temperature variations: the former decreases when temperature is raised, and, hence, a cycle is possible in which some charge is put on the electrode at a certain potential and returned at a higher one. We demonstrate experimentally that it is thus viable to obtain energy from electric double layers if these are successively contacted with water at different temperatures. In addition, we show theoretically and experimentally that temperature and salinity variations can be conveniently combined to maximize the electrode potential increase. The resulting available energy is also estimated.

Ion permeation inside microgel particles induced by specific interactions: from charge inversion to overcharging

In this work we have performed a theoretical study of a system formed by ionic microgels in the presence of monovalent salt with the help of Ornstein-Zernike integral equations within the hypernetted-chain (HNC) approximation. We focus in particular on analysing the role that the short-range specific interactions between the polymer fibres of the microgel and the incoming ions have on the equilibrium ion distribution inside and outside the microgel. For this purpose, a theoretical model based on the equilibrium partitioning effect is developed to determine the interaction between the microgel particle and a single ion. The results indicate that when counterions are specifically attracted to the polymer fibres of the microgel, an enhanced counterion accumulation occurs that induces the charge inversion of the microgel and a strong increase of the microgel net charge (or overcharging). In the case of coions, the specific attraction is also able to provoke the coion adsorption even though they are electrostatically repelled, and so increasing the microgel charge (true overcharging). Moreover, we show that ion adsorption onto the microgel particle is very different in swollen and shrunken states due to the competition between specific attraction and steric repulsion. In particular, ion adsorption occurs preferentially in the internal core of the particle for swollen states, whereas it is mainly concentrated in the external shell for de-swollen configurations. Finally, we observe the existence of a critical salt concentration, where the net charge of the microgels vanishes; above this inversion point the net charge of the microgels increases again, thus leading to reentrant stability of microgel suspensions.

Ionic asymmetry and solvent excluded volume effects on spherical electric double layers: a density functional approach

In this article, we present a classical density functional theory for electrical double layers of spherical macroions that extends the capabilities of conventional approaches by accounting for electrostatic ion correlations, size asymmetry, and excluded volume effects. The approach is based on a recent approximation introduced by Hansen-Goos and Roth for the hard sphere excess free energy of inhomogeneous fluids [J. Chem. Phys. 124, 154506 (2006); Hansen-Goos and Roth, J. Phys.: Condens. Matter 18, 8413 (2006)]. It accounts for the proper and efficient description of the effects of ionic asymmetry and solvent excluded volume, especially at high ion concentrations and size asymmetry ratios including those observed in experimental studies. Additionally, we utilize a leading functional Taylor expansion approximation of the ion density profiles. In addition, we use the mean spherical approximation for multi-component charged hard sphere fluids to account for the electrostatic ion correlation effects. These approximations are implemented in our theoretical formulation into a suitable decomposition of the excess free energy which plays a key role in capturing the complex interplay between charge correlations and excluded volume effects. We perform Monte Carlo simulations in various scenarios to validate the proposed approach, obtaining a good compromise between accuracy and computational cost. We use the proposed computational approach to study the effects of ion size, ion size asymmetry, and solvent excluded volume on the ion profiles, integrated charge, mean electrostatic potential, and ionic coordination number around spherical macroions in various electrolyte mixtures. Our results show that both solvent hard sphere diameter and density play a dominant role in the distribution of ions around spherical macroions, mainly for experimental water molarity and size values where the counterion distribution is characterized by a tight binding to the macroion, similar to that predicted by the Stern model.

Ionic density distributions near the charged colloids: spherical electric double layers

We have studied the structure of the spherical electric double layers on charged colloids by a density functional perturbation theory, which is based both on the modified fundamental-measure theory for the hard spheres and on the one-particle direct correlation functional (DCF) for the electronic residual contribution. The contribution of one-particle DCF has been approximated as the functional integration of the second-order correlation function of the ionic fluids in a bulk phase. The calculated result is in very good agreement with the computer simulations for the ionic density distributions and the zeta potentials over a wide range of macroion sizes and electrolyte concentrations, and compares with the results of Yu et al. [J. Chem. Phys. 120, 7223 (2004)] and modified Poisson-Boltzmann approximation [L. B. Bhuiyan and C. W. Outhwaite, Condens. Matter Phys. 8, 287 (2005)]. The present theory is able to provide interesting insights about the charge inversion phenomena occurring at the interface.

Integral Equation Theory of Biomolecules and Electrolytes

The so-called three-dimensional version (3D-RISM) can be used to describe the interactions of solvent components (here we treat water and ions) with a chemical or biomolecular solute of arbitrary size and shape. Here we give an overview of the current status of such models, describing some aspects of “pure” electrolytes (water plus simple ions) and of ionophores, proteins and nucleic acids in the presence of water and salts. Here we focus primarily on interactions with water and dissolved salts; as a practical matter, the discussion is mostly limited to monovalent ions, since studies of divalent ions present many difficult problems that have not yet been addressed. This is not a comprehensive review, but covers a few recent examples that illustrate current issues.

Macroion solutions in the cell model studied by field theory and Monte Carlo simulations

Aqueous solutions of charged spherical macroions with variable dielectric permittivity and their associated counterions are examined within the cell model using a field theory and Monte Carlo simulations. The field theory is based on separation of fields into short- and long-wavelength terms, which are subjected to different statistical-mechanical treatments. The simulations were performed by using a new, accurate, and fast algorithm for numerical evaluation of the electrostatic polarization interaction. The field theory provides counterion distributions outside a macroion in good agreement with the simulation results over the full range from weak to strong electrostatic coupling. A low-dielectric macroion leads to a displacement of the counterions away from the macroion.

Coulomb interactions in charged fluids

The use of Ewald summation schemes for calculating long-range Coulomb interactions, originally applied to ionic crystalline solids, is a very common practice in molecular simulations of charged fluids at present. Such a choice imposes an artificial periodicity which is generally absent in the liquid state. In this paper we propose a simple analytical O(N(2)) method which is based on Gauss's law for computing exactly the Coulomb interaction between charged particles in a simulation box, when it is averaged over all possible orientations of a surrounding infinite lattice. This method mitigates the periodicity typical of crystalline systems and it is suitable for numerical studies of ionic liquids, charged molecular fluids, and colloidal systems with Monte Carlo and molecular dynamics simulations.

Electrostatics at the nanoscale

Electrostatic forces are amongst the most versatile interactions to mediate the assembly of nanostructured materials. Depending on experimental conditions, these forces can be long- or short-ranged, can be either attractive or repulsive, and their directionality can be controlled by the shapes of the charged nano-objects. This Review is intended to serve as a primer for experimentalists curious about the fundamentals of nanoscale electrostatics and for theorists wishing to learn about recent experimental advances in the field. Accordingly, the first portion introduces the theoretical models of electrostatic double layers and derives electrostatic interaction potentials applicable to particles of different sizes and/or shapes and under different experimental conditions. This discussion is followed by the review of the key experimental systems in which electrostatic interactions are operative. Examples include electroactive and "switchable" nanoparticles, mixtures of charged nanoparticles, nanoparticle chains, sheets, coatings, crystals, and crystals-within-crystals. Applications of these and other structures in chemical sensing and amplification are also illustrated.

Structure of spherical electric double layers containing mixed electrolytes: a systematic study by Monte Carlo simulations and density functional theory

The structure of spherical electric double layers in the presence of mixed electrolytes is studied using Monte Carlo simulation and density functional theory within the restricted primitive model. The macroion is modeled as an impenetrable charged hard sphere carrying a uniform surface charge density, surrounded by the small ions represented as charged hard spheres, and the solvent is taken as a dielectric continuum. The density functional theory uses a partially perturbative scheme, where the hard-sphere contribution to the one-particle correlation function is evaluated using weighted density approximation and the ionic interactions are calculated using a second-order functional Taylor expansion with respect to a bulk electrolyte. The Monte Carlo simulations have been performed in canonical ensemble. The system is studied at varying ionic concentrations, at different concentration ratios of mono- and multivalent counterions of mixed electrolytes, at different diameters of hard spheres, at different macroion radius, and at varying polyion surface charge densities. The theoretical predictions in terms of the density profiles and the mean electrostatic potential profiles are found to be in good agreement with the simulation results. This model study shows clear manipulations of ionic size and charge correlations in dictating a number of interesting phenomena relating to width of the diffuse layer and charge inversion under different parametric conditions.

The behavior of ions near a charged wall-dependence on ion size, concentration, and surface charge

A renormalization of the 3D-RISM-HNC integral equation is used to study the solvent and ion distributions at neutral and negatively charged planar atomistic surfaces. The charge density of the surfaces ranged from 0.0 to 0.4116 C/m(2), and the modeled electrolyte solutions consist of the salts NaCl, KCl, and CsCl at concentrations of 0.1, 0.25, and 1.0 M in SPC/E water. The results are qualitatively compared to the results from other integral equation methods and simulations for similar models. We find that the 3D-IEs predict an electric multilayer screening behavior in the solvent and ion distributions in contrast to the double layer anticipated from Poisson-Boltzmann theory. It is observed that the cation size has a significant effect on the distributions near the surface up to three solvation layers beyond which the behavior is the same among the different cations. The response of the distributions to the charged surface is described as an increase in ion and solvent density near the wall. The higher concentration solutions screen the electrostatic source more strongly at the wall as expected. The importance of ion-solvent and ion-ion correlations near the surface is shown through three-body correlation functions which are obtainable from the 3D-IEs in this study.

Molecular solvent model of spherical electric double layers: a systematic study by Monte Carlo simulations and density functional theory

The structure of spherical electric double layers is studied using Monte Carlo simulation and density functional theory by considering solvent as the third component. In this molecular solvent model (MSM), ions and solvent molecules are considered as charged and neutral hard spheres, respectively, having equal diameter. The macroion is considered as an isolated hard sphere having uniform surface charge density surrounded by the electrolyte and the solvent. The theory is partially perturbative as the hard-sphere contribution to the one particle correlation function is evaluated using suitably averaged weighted density, and the ionic part is obtained through a second-order functional Taylor expansion around the bulk electrolyte. The Monte Carlo simulations have been performed in a canonical ensemble. The system is studied at varying concentrations of electrolytes, and the solvent molecules, at different valences of the electrolyte, at different macroion radii, and at varying surface charge densities. The theory is found to be in good agreement with the simulation results over a wide range of parametric conditions. The excluded volume effects due to the molecular nature of the solvent are shown to have much richer features in diffuse layer phenomena like layering and charge inversion.

Additional considerations about the role of ion size in charge reversal

The effect of the ion size on the charge reversal process is studied via canonical Monte Carlo simulation. To this end, a primitive model of electrolyte is used to analyze the electric double layer formed by an asymmetric electrolyte in the presence of a charged planar wall. Different values of ion diameters and surface charge densities are used so as to determine the conditions at which the charge reversal first occurs. For each case, the apparent surface charge density is calculated as a function of the distance from the charged wall for the different electrolyte concentrations in order to establish the minimal salt concentration required for the charge reversal. We will refer to this electrolyte concentration as the reversal concentration and will show how it depends on the surface charge density and on the ion size. From the apparent surface charge density profiles, the distance from the wall at which the charge reversal arises as well as its intensity can be also inferred.