Hypernetted Chain/mean Spherical Approximation Research Articles

Topological impact of nanopore electrodes on the structure of the electrical double layer and the differential capacitance

The electrical double layer for three different topologies of nanopore electrodes is studied, i.e., the interior and exterior electrical double layers of planar, cylindrical and spherical nanopores immersed into a point-ions electrolyte, and not connected to a power source, are analytically attained through the linearized Poisson-Boltzmann equation. Thus, analytical formulas for the mean electrostatic potential, electrolyte's reduced concentration, and electrical field profiles, are exhibited. Their corresponding analytical expressions for their differential capacitances are presented. All the nanopores are treated as permeable, so the electrolyte outside and inside the electrodes are at the same chemical potential. Analogous analytical formulas for solid nano-electrodes are obtained as a corollary of those for nanopores. In particular, their analytical expressions for the differential capacitance here derived are shown to be consistent with the capacitive compactness proposed in the past by one of us. Numerical results of all of the above functions are analyzed as a function of the nanopores geometrical parameters and the electrolyte's temperature and molar concentration. It is found that the spherical topology, at lower temperatures, has the higher differential capacitance. It is demonstrated that for the three nanopore topologies here considered their capacitances reduce to that of a single planar electrode, in the limit of infinitely wide nanopores. The electrical double layer and mean electrostatic potential of the three topologies are in qualitatively agreement with those from the non-linearized Poisson-Boltzmann, hypernetted chain/mean-spherical approximation (HNC/MSA) equations and computer simulations results presented in the past, within the low mean electrostatic potential assumption.

Model charged cylindrical nanopore in a colloidal dispersion: charge reversal, overcharging and double overcharging

Using the hypernetted-chain/mean spherical approximation (HNC/MSA) integral equations we study the electrical double layer inside and outside a model charged cylindrical vesicle (nanopore) immersed into a primitive model macroions solution, so that the macroions are only present outside the nanopore, i.e., the vesicle wall is impermeable only to the external macroions. We calculate the ionic and local linear charge density profiles inside and outside the vesicle, and find that the correlation between the inside and outside ionic distributions causes the phenomena of overcharging (also referred to as surface charge amplification) and/or charge reversal. This is the first time overcharging is predicted in an electrical double layer of cylindrical geometry. We also report the new phenomenon of double overcharging. The present results can be of consequence for relevant systems in physical-chemistry, energy storage and biology, e.g., nanofilters, capacitors and cell membranes.

Electrokinetic properties of a restricted primitive model electrolyte in slit-like nanopores: Effects of enhanced ionic excluded volume

The effects associated with the migration of charged particles in solution, driven by an external pressure gradient and/or electric field, are known as electrokinetic phenomena. Among these are the electroosmotic volume flux, capillary conductance, streaming potential, and the apparent electroviscosity. In this work we study the electrokinetic properties of an electrolyte solution confined in a slit-like nanopore. The pore is made up of two charged parallel plates and the fluid is considered through the restricted primitive model electrolyte. The hypernetted chain/mean spherical approximation (HNC/MSA) and Poisson–Boltzmann theory are used to obtain the equilibrium ionic concentration profiles inside the nanopore, thus, the electrokinetic coefficients are calculated by means hydrodynamics equations. The considered monovalent electrolytes are characterized by different ionic diameters and they are at fixed molar concentration. It is exhibited that the electrokinetic coefficients are strongly sensitive to the ionic size, especially for small pore widths. Important variations of electrokinetic coefficients are produced for large ionic diameters associated with their enhanced excluded volume. The quantitative differences between HNC/MSA and Poisson–Boltzmann theories are discussed throughout the paper.

Overcharging and charge reversal in the electrical double layer around the point of zero charge

The ionic adsorption around a weakly charged spherical colloid, immersed in size-asymmetric 1:1 and 2:2 salts, is studied. We use the primitive model (PM) of an electrolyte to perform Monte Carlo simulations as well as theoretical calculations by means of the hypernetted chain/mean spherical approximation (HNC/MSA) and the unequal-radius modified Gouy-Chapman (URMGC) integral equations. Structural quantities such as the radial distribution functions, the integrated charge, and the mean electrostatic potential are reported. Our Monte Carlo "experiments" evidence that near the point of zero charge, the smallest ionic species is preferentially adsorbed onto the macroparticle, independently of the sign of the charge carried by this tiniest electrolytic component, giving rise to the appearance of the phenomena of charge reversal (CR) and overcharging (OC). Accordingly, colloidal CR, due to an excessive attachment of counterions, is observed when the macroion is slightly charged and the coions are larger than the counterions. In the opposite situation, i.e., if the counterions are larger than the coions, the central macroion acquires additional like-charge (coions) and hence becomes "overcharged," a feature theoretically predicted in the past [F. Jiménez-Angeles and M. Lozada-Cassou, J. Phys. Chem. B 108, 7286 (2004)]. In other words, here we present the first simulation data on OC in the PM electrical double layer, showing that close to the point of zero charge, this novel effect surges as a consequence of the ionic size asymmetry. We also find that the HNC/MSA theory captures well the CR and OC phenomena exhibited by the computer experiments, especially as the macroion's charge increases. On the contrary, even if URMGC also displays CR and OC, its predictions do not compare favorably with the Monte Carlo data, evidencing that the inclusion of hard-core correlations in Monte Carlo and HNC/MSA enhances and extends those effects. We explain our findings in terms of the energy-entropy balance. In the field of electrophoresis, it has been generally agreed that the charge of a colloid in motion is partially decreased by counterion adsorption. Depending on the location of the macroion's slipping surface, the OC results of this paper could imply an increase in the expected electrophoretic mobility. These observations aware about the interpretation of electrokinetic measurements using the standard Poisson-Boltzmann approximation beyond its validity region.

Simulational and theoretical study of the spherical electrical double layer for a size-asymmetric electrolyte: The case of big coions

Monte Carlo simulations of a spherical macroion, surrounded by a size-asymmetric electrolyte in the primitive model, were performed. We considered 1:1 and 2:2 salts with a size ratio of 2 (i.e., with coions twice the size of counterions), for several surface charge densities of the macrosphere. The radial distribution functions, electrostatic potential in all the space and at the Helmholtz surfaces, and integrated charge are reported. We compare these simulational data with original results obtained from the Ornstein-Zernike integral equation, supplemented by the hypernetted chain-hypernetted chain (HNC-HNC) and hypernetted chain-mean spherical approximation (HNC-MSA) closures, and with the corresponding calculations using the modified Gouy-Chapman and unequal-radius modified Gouy-Chapman theories. The HNC-HNC and HNC-MSA integral equations formalisms show good concordance with Monte Carlo "experiments," whereas the notable limitations of point-ion approaches are evidenced. Most importantly, the simulations confirm our previous theoretical predictions of the nondominance of the counterions in the size-asymmetric spherical electrical double layer [J. Chem. Phys. 123, 034703 (2005)], the appearance of anomalous curvatures at the outer Helmholtz plane, and the enhancement of the charge reversal and screening at high colloidal surface charge densities due to the ionic size asymmetry.

Electrokinetic properties of monovalent electrolytes confined in charged nanopores: Effect of geometry and ionic short-range correlations

The electrokinetic properties (such as capillary conductance, electroviscosity, and the streaming potential) are obtained for a restricted primitive model electrolyte confined in a slitlike nanopore made up of two infinite parallel plates and in a cylindrical cavity of infinite extension. The hypernetted chain/mean spherical approximation (HNC/MSA) is used to obtain the equilibrium ionic concentration profiles inside the pores, which in turn are used to calculate the electrokinetic properties via linear hydrodynamic equations. Our results are compared with those obtained via the classical Poisson–Boltzmann (PB) theory. Important quantitative and qualitative effects, attributed to geometry and to the proper consideration of short-range correlations by HNC/MSA, are discussed.

Charge reversal in real colloids: Experiments, theory and simulations

In the last years, the inclusion of ionic short-range correlations in the study of colloidal stability has led to significant disagreements with the predictions obtained from classical treatments. An example of these discrepancies is the occurrence of charge reversal of charged particles. In order to shed light on this issue, the charge reversal of latex particles in the presence of asymmetric electrolytes has been investigated through Monte Carlo (MC) simulations. In particular, experimental results concerning electrophoretic mobility reversals and the Hyper-Netted-Chain/Mean-Spherical-Approximation (HNC/MSA) predictions have been compared with simulations in which two alternative methods for evaluating energies have been applied. A realistic hydrated ion size is used in the HNC/MSA calculations and simulations. In this way, the existence of a reversal in the electrophoretic mobility due to ion size correlations and without requiring specific counterion adsorption is probed. Moreover, the simulations appears as a useful tool for explaining those results in which the HNC/MSA does not reproduce the experimental trends.

Van der Waals-Like Isotherms in a Confined Electrolyte by Spherical and Cylindrical Nanopores

Electrolytes confined by spherical, cylindrical, and slit-like charged nanopores are studied. Results for ionic distribution profiles, pressures of the confined fluid, and absorption isotherms are obtained through the hypernetted chain/mean spherical approximation (HNC/MSA) integral equations theory. In spherical and cylindrical geometries, an inward, non-monotonic behavior of the pressure is found as confinement increases, implying a negative compressibility. The pressure vs volume isotherms resemble liquid-vapor van der Waals-like phase transition diagrams. This effect is correlated with a charge separation inside a spherical pore previously reported (Phys. Rev. Lett., 79, 3656, 1997). Here, the mechanism of charge separation and negative compressibility are explored in detail. When compared with the slit-like pore pressure, important qualitative differences are found.

Electric Double Layers with Electrolyte Mixtures: Integral Equations Theories and Simulations

A study of a planar electric double layer (EDL) in the presence of mixtures of electrolyte is presented. In particular, results from the Hyper-Netted-Chain/Mean-Spherical-Approximation (HNC/MSA) theory are compared with Monte Carlo (MC) simulations. In this way, the charge inversion induced by mixtures of multivalent and monovalent counterions is probed. Since overcharging phenomena in nature emerge under such conditions, the role of ion-ion correlations in the EDL appears as a crucial point in this kind of study. Unlike previous related works, a realistic hydrated ion size is used in the HNC/MSA calculations and simulations. In this way, a qualitative agreement between the results obtained from the theory and MC simulations is found. However, some discrepancies arise when the charge inversion is expected to be more noticeable, namely at high surface charges and/or elevated concentrations of multivalent electrolytes. Such differences are explained in terms of an overestimation of the charge inversion by the integral equation (IE) formalism.

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

The hypernetted chain/mean spherical approximation (HNC/MSA) integral equation for a totally asymmetric primitive model electrolyte around a spherical macroparticle is obtained and solved numerically in the case of size-asymmetric systems. The ensuing radial distribution functions show a very good agreement when compared to our Monte Carlo and molecular-dynamics simulations for spherical geometry and with respect to previous anisotropic reference HNC calculations in the planar limit. We report an analysis of the potential versus charge relationship, radial distribution functions, mean electrostatic potential, and cumulative reduced charge for representative examples of 1:1 and 2:2 salts with a size-asymmetry ratio of 2. Our results are collated with those of the modified Gouy-Chapman (MGC) and unequal radius modified Gouy-Chapman (URMGC) theories and with those of HNC/MSA in the restricted primitive model (RPM) to assess the importance of size-asymmetry effects. One of the most striking characteristics found is that, contrary to the general belief, away from the point of zero charge the properties of an asymmetric electrical double layer (EDL) are not those corresponding to a symmetric electrolyte with the size and charge of the counterion, i.e., counterions do not always dominate. This behavior suggests the existence of a new phenomenology in the EDL that genuinely belongs to a more realistic size-asymmetric model where steric correlations are taken into account consistently. Such novel features cannot be described by traditional mean-field theories such as MGC, URMGC, or even by enhanced formalisms, such as HNC/MSA, if they are based on the RPM.

Primitive models and electrophoresis: an experimental study

Several theories (developed in the context of the so-called primitive models) argue that the classical Poisson–Boltzmann (PB) approach could fail to describe the electrical double layer in the case of high surface charge densities and/or ionic strengths. Under such conditions, the concentration of counterions in the vicinity of the surface would be larger than expected. What is more, the particle charge could be overcompensated (which is known as overcharging). This would lead to a considerable reduction in the magnitude of the ζ-potential (as compared with PB predictions) and, even, a reversal in its sign. Although these predictions are quite remarkable, these new approaches have been applied just scarcely in practice. Obviously, electrophoresis offers an excellent possibility of testing these theories. Accordingly, we have measured the electrophoretic mobility for latex particles with moderate and large surface charge densities at high ionic strengths. A commercial setup specifically designed for measuring extremely low mobilities has been used. The results have been interpreted with the help of the so-called hyper-netted-chain/mean-spherical approximation (HNC/MSA), a theory that includes ion size effects. In this way, overcharging has been experimentally investigated. Although the HNC/MSA is not a completely successful approach, our analysis suggests that the role of ion size correlations is important, particularly in electrolytes with a counterion valency of 2 or greater.

Looking into overcharging in model colloids through electrophoresis: Asymmetric electrolytes

Some theories claim that the Poisson–Boltzmann approach could fail to describe the electric double layer of colloids under certain conditions as a result of neglecting ion size correlations. For instance, if the surface charge density and/or the electrolyte concentration are high enough, the counterion local density in the vicinity of charged surface could become so large that the particle charge would be overcompensated. This phenomenon is theoretically known as overcharging and, sometimes, should involve a ζ-potential reversal. Accordingly, this work looks into overcharging through electrophoresis experiments. The electrophoretic mobility has been measured for latex particles with moderate and large surface charge density in solutions of asymmetric electrolytes z:1 (symmetric electrolytes have been studied in a previous work). In order to find out the relevance of ion size correlations, results are analyzed within the so-called hypernetted-chain/mean-spherical approximation (HNC/MSA) as well as a Poisson–Boltzmann approach. In the case of divalent counterions (z=2), the HNC/MSA seems to justify why a mobility reversal is hardly observed. For z=3, our results suggest that ion correlations could play an important role and be mostly (or partially) responsible for mobility reversal in certain cases.

Electrophoretic mobility of model colloids and overcharging: theory and experiment

Several theories claim that ion-ion correlations play an important role in the electric double layer of colloids. One of the most outstanding predictions is overcharging, which would take place at high electrolyte concentrations and surface charge densities. The counterion concentration next to the surface can become so large that the particle charge is overcompensated and the overcharging occurs. Sometimes this would also involve a reversal of the ζ potential, but this phenomenon has been observed rarely through mobility measurements. This study explores the matter further. The electrophoretic mobility is measured for latex particles with moderate and extremely large surface charge densities at high ionic strengths (up to 2 M) in solutions of symmetric electrolytes. The results are analysed within the so-called hypernetted chain/mean-spherical approximation (HNC/MSA) and a Poisson-Boltzmann approach. In this way, the relevance of ion-ion correlations in practice and the occurrence of overcharging are probed experimentally.

Overcharging of DNA in the Presence of Salt: Theory and Simulation

A study of a model rodlike polyelectrolyte molecule immersed in a monovalent or divalent electrolyte is presented. Results for the local concentration profile, mean electrostatic potential, charge distribution function, and ζ-potential are obtained from hypernetted-chain/mean spherical approximation (HNC/MSA) theory and compared with molecular dynamics (MD) simulations. As a particular case, the parameters of the polyelectrolyte molecule are mapped to those of a DNA molecule. Both HNC/MSA and MD predict the occurrence of overcharging, which is not present in the Poisson−Boltzmann theory. Further an excellent qualitative, and in some cases quantitative, agreement between HNC/MSA and MD is found. Oscillations observed in the mean electrostatic potential, local concentration profiles, and the curvature of the ζ-potential are discussed in terms of the observed overcharging effect. Particularly interesting results are a very nonmonotonic behavior of the ζ-potential, as a function of the rod charge density, and...

Effect of pore geometry on a confined hard sphere fluid

The structure of a hard sphere fluid confined by model slit and cylindrical pores is investigated. Results from grand canonical Monte Carlo (GCMC) simulations and from the hypernetted chain/mean spherical approximation (HNC/MSA) equation are reported. GCMC results are compared with those from the HNC/MSA equation, and agreement is good. The effect of confinement on liquids at the same chemical potentials is that the absorption of the hard sphere fluid into the pores decreases with increasing confinement, i.e., when going from planar to cylindrical geometry or by narrowing the pores. The adsorption on the pore walls has, in general, the opposite behaviour. For high bulk concentrations and certain values of cylindrical pore diameter the concentration profile is higher at the centre of the pore than next to the pore wall. A very strong, but continuous, transition occurs in the concentration profile, as a function of the cylinder's diameter. These results could be of some interest in catalysis studies.

Theory of gas–gas phase transition in rare-gas binary mixtures

The lines of critical points of rare-gas binary mixtures are evaluated within the framework of a statistical mechanics theory of thermodynamic stability, recently developed by Chen and Forstmann. The phase instabilities are related to the local compositional and density order in the fluid through a density functional expansion of the grand canonical potential. The structural inputs are determined by using the thermodynamically self-consistent integral equation scheme known as hypernetted chain/mean spherical approximation (HMSA). A good agreement is obtained with neutron scattering data of He–Xe and Ne–Xe mixtures near the phase transition. Although the accuracy in reproducing the experimental critical point lines is only semi-quantitative, the qualitative behaviors are properly described. For the first time, a detailed microscopic interpretation of the so called behavior of the ‘‘second type’’ of the critical line is given.

Ion Distribution and Force between Two Charged Plates: A Variational Solution of the Three Point Extension Hypernetted Chain Equations

Abstract We have employed a variational method for the solution of the three point extension hypernetted chain/mean spherical approximation (HNC/MSA) equations for the distribution of ions between two charged plates of infinite extent, a system of great interest in colloid physics. We have previously shown, for the case of a single charged plate, that this method of solution has several features which make it convenient for use in applications. The calculated electrolyte structure within this model slit pore can be used to find the net pressure between the two plates. Comparisons are made throughout the paper with the classical Verwey-Overbeek theory of interacting electrical double layers. Additionally we will compare our theoretical results with experimental clay swelling pressure data. Finally, some results for hard sphere fluids confined to an uncharged pore are presented.

Monte Carlo simulation for a symmetrical electrolyte next to a charged spherical colloid particle

Results from Monte Carlo (MC) simulations for a restricted primitive model symmetrical electrolyte next to an isolated spherical macroion are reported. Calculations were made for various 1:1 and 2:2 electrolyte concentrations and macroion’s radii and charges. The MC results are compared to those from the hypernetted chain/mean spherical approximation (HNC/MSA) integral equation and the fifth version of the modified Poisson–Boltzmann (MPB5) and Poisson–Boltzmann (PB) differential equations. An overall good agreement of the HNC/MSA and MPB5 results with the MC data is found. On the other hand, the widely used PB theory is found to have important quantitative and qualitative disagreements with the MC results.

The structure of electrolytes in cylindrical pores

The hypernetted chain/mean-spherical approximation (HNC/MSA) equations are derived for ion distributions within and surrounding a charged cylindrical pore for the restricted primitive model of electrolytes. The equilibrium density profiles, mean electrostatic potential profiles, as well as other electrical double-layer structural parameters are presented and compared with the results of the Poisson–Boltzmann (PB) equation and with grand Canonical Monte Carlo data. Quantitative and qualitative differences between the HNC/MSA and the PB theories are found which are especially large for small pore diameters. The effect of geometry is investigated through a comparison with HNC/MSA results for electrolyte confined within a slit pore formed by two parallel plates of infinite extension.

Thermodynamics and structure of the primitive model near its gas–liquid transition

The hypernetted-chain/mean spherical approximation (HNC/MSA) is employed to calculate the thermodynamic properties of the restricted primitive model (symmetric binary mixture of charged hard spheres) in the low density–high coupling region, where the gas–liquid phase transition is located. The spinodal curve is calculated from the virial and the compressibility equations. The HNC/MSA total correlation function does not show long-range behavior as the spinodal curve is approached. This is due to the hybrid nature of this approximation. However, a consistent HNC/MSA total correlation function is defined which does show the expected long-range behavior. This long-range behavior is found to be related to the phenomenon of ionic clustering, in accordance with previous studies.