Modified Poisson Boltzmann Research Articles

Influence of size of solvent molecules on structural and thermodynamic properties of the electrode/electrolyte interface

The influence of size of solvent molecules on the structural and thermodynamic properties of the interface between the electrode and electrolyte, using the solvent primitive model, was studied by grand canonical Monte Carlo (GCMC) simulations. The computer simulation results are compared with those obtained from the modified Poisson–Boltzmann (MPB) theory. The ionic singlet distribution functions show that the solvent molecules of low diameter favour the counter ion adsorption on the electrode. With increasing diameter of the solvent molecules, the mean electrostatic potential increases, while the integral and differential capacitances decrease. The integral capacitance curves obtained by MPB theory are in qualitative agreement with those obtained by the GCMC simulation although the theoretical results are overestimated.

Influence of the dielectrophoretic force in mixed electrical double layers

The equilibrium properties of a charged plane immersed in an aqueous electrolyte solution are examined using a generalized Poisson–Boltzmann equation that takes into account the finite ion size by modeling the solution as a suspension of polarizable insulating spheres in water. This formalism is applied to a general solution composed of two or more counterion species with different valences, sizes, and effective permittivity values. It is shown that, due to the dependence of the dielectrophoretic force on the ion size and effective permittivity value, the concentration of the smaller counterion strongly increases while that of the larger one decreases in the immediate vicinity of the charged surface. As a result the surface potential value strongly increases as compared to the usual modified Poisson–Boltzmann theory that only includes steric interactions among ions. This effect is particularly important in the case of mixtures of univalent and divalent counterions, being significant even for relatively low surface charge values.

Structure of cylindrical electric double layers: Comparison of density functional and modified Poisson-Boltzmann theories with Monte Carlo simulations

The structure of cylindrical double layers is studied using a modified Poisson Boltzmann theory and the density functional approach. In the model double layer, the electrode is a cylindrical polyion that is infinitely long, impenetrable, and uniformly charged. The polyion is immersed in a sea of equi-sized rigid ions embedded in a dielectric continuum. An in-depth comparison of the theoretically predicted zeta potentials, the mean electrostatic potentials, and the electrode-ion singlet density distributions is made with the corresponding Monte Carlo simulation data. The theories are seen to be consistent in their predictions that include variations in ionic diameters, electrolyte concentrations, and electrode surface charge densities, and are also capable of well reproducing some new and existing Monte Carlo results.

On the interfacial capacitance of an electrolyte at a metallic electrode around zero surface charge

The behaviour of the capacitance of a planar double layer containing a restricted primitive model electrolyte (equi-sized rigid ions moving in a continuum dielectric) at and around zero surface charge is examined for a polarizable electrode with particular emphasis on a metallic surface. Capacitance results are reported for symmetric valency (1:1) salts encompassing a range of concentrations and temperatures covering both electrolyte solution and ionic liquid regimes. Although the modified Poisson–Boltzmann theory is principally employed, at higher concentrations the theoretical calculations have been supplemented by Monte Carlo simulations. Capacitance anomaly, that is, increase of capacitance with temperature at low temperatures, is seen to occur when the electrode is an insulator with a low dielectric constant or when it is unpolarized. No capacitance anomaly is, however, seen for a metallic electrode with an infinite dielectric constant and in this situation the capacitance increases (a) dramatically at low temperatures (strong coupling) at a given concentration, and (b) as concentration increases at a given temperature. These capacitance trends are consistent with earlier works in the presence or absence of surface polarization and, in particular, the results for a conducting electrode at ionic liquid concentrations are consistent with that recently reported by Loth et al. [Phys. Rev. E, 82, 056102 (2010)]. Overall the theoretical predictions are qualitative to semi-quantitative with the simulations.

Unraveling the Combined Effects of Dielectric and Viscosity Profiles on Surface Capacitance, Electro-Osmotic Mobility, and Electric Surface Conductivity

We calculate the electro-osmotic mobility and surface conductivity at a solid-liquid interface from a modified Poisson-Boltzmann equation, including spatial variations of the dielectric function and the viscosity that where extracted previously from molecular dynamics simulations of aqueous interfaces. The low-dielectric region directly at the interface leads to a substantially reduced surface capacitance. At the same time, ions accumulate into a highly condensed interfacial layer, leading to the well-known saturation of the electro-osmotic mobility at large surface charge density regardless of the hydrodynamic boundary conditions. The experimentally well-established apparent excess surface conductivity follows from our model for all hydrodynamic boundary conditions without additional assumptions. Our theory fits multiple published sets of experimental data on hydrophilic and hydrophobic surfaces with striking accuracy, using the nonelectrostatic ion-surface interaction as the only fitting parameter.

Double layer interaction between charged parallel plates using a modified Poisson–Boltzmann equation to include size effects and ion specificity

We present calculations of double layer interaction between charged plates taking into account size effects on colloidal systems. Those are modeled using a modified version of Poisson–Boltzmann equation, including non-electrostatic interactions due to the size and the polarizability of each ion. The van der Waals interactions and size effects can properly describe ion specificity as observed experimentally in many colloidal systems. The differential-algebraic mathematical structure is used to solve the modified Poisson–Boltzmann equation. The DASSL code is recursively applied using a Newton–Raphson method to solve the corresponding boundary-value problem. Density profiles around colloidal particles are calculated for different charged-particles immersed in various electrolytes aqueous solutions. From these profiles, we could calculate the pressure (interaction) between the plates. The influence of charge and size asymmetries of ions on density profiles are studied in different solutions. Results show the adequacy of this modified form of the Poisson–Boltzmann equation and the proposed mathematic procedure to solve the needed equations, describing several properties of colloidal systems and ion size correlation.

Charge reversal of moisturous porous silica colloids by take-up of protons

The net charge of porous Stöber silica colloids is studied using a modified Poisson–Boltzmann theory in a spherical cell, with a focus on the case of water-filled porous silica particles suspended in a non-aqueous solvent. We show that the silica’s usual negative surface charge, due to deprotonisation of the Si–OH group, is counteracted by an excess uptake of protons in the water-filled pores of the particle at low enough pH. A small volume fraction of pores suffices to induce a point of zero charge at pH≈4. Based on the difference in Donnan potential between the porous medium and the solvent a relation can be constructed that describes the location of the point of zero charge analytically. The accuracy of this relation is confirmed by numerical calculations. For Stöber silica in water we find a charge reversal below pH≈3, which is in this case solely a result of the selective uptake of cations in the porous network.

Equilibrium properties of charged spherical colloidal particles suspended in aqueous electrolytes: Finite ion size and effective ion permittivity effects

The equilibrium properties of a charged spherical colloidal particle immersed in an aqueous electrolyte solution are examined using an extension of the Standard Electrokinetic Model that takes into account the finite ion size by modeling the aqueous electrolyte solution as a suspension of polarizable insulating spheres in water. We find that this model greatly amplifies the steric effects predicted by the usual modified Poisson–Boltzmann equation, which only imposes a restriction on the ability of ions to approach one another. This suggests that a solution of the presented model under nonequilibrium conditions could have important consequences in the interpretation of dielectric and electrokinetic data in colloidal suspensions.

Ions near the air/water interface. II: Is the water/air interface acidic or basic? Predictions of a simple model

A simple modified Poisson–Boltzmann model, compatible with zeta potential experiments, is used to calculate the concentrations of ions on the interface. It is shown that, for most pH values and salt concentrations, both H+ and OH− accumulate on the interface, making their interfacial concentrations larger than in the bulk. However, at extreme pH values, the interfacial concentrations of both H+ and OH− have been found to be lower than in bulk, which might explain some contradictory results from literature. The ratio between H+ and OH− concentrations is smaller at the interface than in bulk for most (but not all) pH values and salt concentrations. Recent experimental results, which show that H+ is present on the interface only at very low pH values, are explained on the basis of this model. It is shown that, when the adsorption of Cl− on the interface is sufficiently large compared to that of OH−, the number of H+ ions adsorbed on the interface passes through a maximum with salt concentration, also in agreement with experiment. The concentration at which the maximum occurs is of the order of the adsorption equilibrium constant of Cl− on the interface.

Ions near the air/water interface: I. Compatibility of zeta potential and surface tension experiments

A simple Structure Making/Structure Breaking (SM/SB) Modified Poisson Boltzmann approach is used to describe the distribution of ions in the vicinity of an air/water interface, which is assumed to be a layer possessing a thickness of a few Å. The interface is assumed to be depleted of SM ions, and the adsorption of SB ions on the interface is taken into account via Langmuir adsorption equations. Outside the interface, the ions are assumed to obey a modified Poisson–Boltzmann distribution, which includes the screened image forces. The adsorption constants of SB ions, determined by fitting the existing zeta potential data, are further used to determine the surface excesses of salt ions and the dependence of the surface tension on added salt. The number of sites available for the adsorption of OH− and H+ ions, considered the water molecules of the interface, is (by more than two orders of magnitude) smaller than the available sites in an ice-like layer. This result suggests a (much) lower density of water at the hydrophobic air/water interface than in the bulk. The model can qualitatively explain the dependence of the surface tension upon addition of salt determined experimentally: a large decrease at low concentrations, due to the formation of a double layer, a large increase at intermediate concentrations, where the screened image forces (being less screened than the double layer forces) dominate, and a slower increase at large concentrations, where both double layer and image forces are screened and the ion hydration forces dominate. However, for large surface potentials, the accumulation of ions in the double layer (either as adsorbed on the interface or in the ion clouds in the vicinity of the interface) leads to large surface excesses of ions and consequently to a surface tension that decreases upon addition of salt up to large concentrations, in contrast to experiment. According to the present model, only the zeta potentials experiments that predict low surface potentials and small adsorption of salt anions on the surface are compatible with the surface tension experiments.

Ions at the Water–oil Interface: Interfacial Tension of Electrolyte Solutions

A theory, based on a modified Poisson-Boltzmann equation, is presented that allows us to calculate the excess interfacial tension of an electrolyte-oil interface accurately. The chaotropic (structure-breaking) ions are found to adsorb to the water-oil interface as the result of large polarizability, weak hydration, and hydrophobic and dispersion interactions. However, kosmotropic (structure-making) anions as well as potassium and sodium ions are found to be repelled from the interface. The adsorption of I(-) and ClO(4)(-) is found to be so strong as to lower the interfacial tension of the water-oil interface, in agreement with the experimental data. The agreement between the calculated interfacial tensions and the available experimental data is very good. The theory is used to predict the interfacial tensions of six other potassium salts, for which no experimental data is available at the moment.

Poisson–Boltzmann Description of the Electrical Double Layer Including Ion Size Effects

The electrical double layer is examined using a generalized Poisson-Boltzmann equation that takes into account the finite ion size by modeling the aqueous electrolyte solution as a suspension of polarizable insulating spheres in water. We find that this model greatly amplifies the steric effects predicted by the usual modified Poisson-Boltzmann equation, which imposes only a restriction on the ability of ions to approach one another. This amplification should allow for an interpretation of the experimental results using reasonable effective ionic radii (close to their well-known hydrated values).

‘Blue energy’ from ion adsorption and electrode charging in sea and river water

A huge amount of entropy is produced at places where fresh water and seawater mix, for example at river mouths. This mixing process is a potentially enormous source of sustainable energy, provided it is harnessed properly, for instance by a cyclic charging and discharging process of porous electrodes immersed in salt and fresh water, respectively [D. Brogioli, Phys. Rev. Lett. 103, 058501 (2009)]. Here we employ a modified Poisson–Boltzmann free-energy density functional to calculate the ionic adsorption and desorption onto and from the charged electrodes, from which the electric work of a cycle is deduced. We propose optimal (most efficient) cycles for two given salt baths involving two canonical and two grand-canonical (dis)charging paths, in analogy to the well-known Carnot cycle for heat-to-work conversion from two heat baths involving two isothermal and two adiabatic paths. We also suggest a slightly modified cycle which can be applied in cases that the stream of fresh water is limited.

Application of the singlet reference interaction site model to the primitive model double layer

The Singlet Reference Interaction Site Model (SRISM) is employed to a structural study of the planar electric double layer using the primitive model of the double layer in which the solvent is implicit, being characterized by a dielectric constant, the ions are represented by charged hard spheres, while the electrode is a uniformly charged flat surface. The primitive model double layer has the advantage that simulation results encompassing a wide range of physical states are available for comparison purposes. The predicted results are compared with some recent simulations and an earlier simulation for an extreme case. The SRISM with closures based on the hypernetted chain show the most promise of the closures considered here. Although the SRISM results are not perfect, the accuracy of SRISM with the best closures is often comparable to the modified Poisson–Boltzmann theory, while the density functional theory seems more accurate for the extreme cases. The SRSIM has the advantage that explict water molecules can be included and solutions can be obtained for high densities. Furthermore, since molecular ions can be treated, SRISM seems suitable for the study of ionic liquids.

Simple Description of the Capacitance of the Double Layer of a High Concentration Electrolyte

Recently, the electrochemical properties of an ionic liquid have received increasing attention. The simplest model of an ionic liquid is a fluid of charged hard spheres, all of which have the same diameter (the restricted primitive model or RPM), and the simplest theory of the double layer formed by an electrolyte is the Gouy−Chapman−Stern (GCS) theory. This theory is moderately useful for a low concentration electrolyte but is incorrect for a high concentration electrolyte, such as an ionic liquid. Among other things, the GCS theory predicts that the capacitance has a minimum at small electrode charge. In contrast, the capacitance of the double layer of an ionic liquid often has a maximum at small electrode charge. This is predicted by simulations and the modified Poisson−Boltzmann (MPB) theory. The mean spherical approximation (MSA) seems to agree with the simulations and MPB but is applicable only for small electrode charge. In this note we graft the MSA onto the GCS result and obtain qualitatively reasonable results for the capacitance that may be of use. This result is identical to the GCS theory at small concentrations and to the MSA at small electrode charge. A maximum is found for small electrode charge and high concentration.

Influence of electrode polarization on the capacitance of an electric double layer at and around zero surface charge

The influence of polarization of the electrode on the interfacial capacitance of a restricted primitive model planar double layer at and around zero surface charge is studied using a modified Poisson–Boltzmann theory for 1:1 electrolytes. The polarization, which occurs due to a dielectric discontinuity at the electrode–electrolyte interface, is treated by imagining fictitious image charges within the electrode, which mimic surface polarization charges. Specifically, the cases (i) when the electrode is a metallic conductor with infinite relative permittivity and (ii) when the electrode is a low relative permittivity insulator are investigated. The capacitance around zero surface charge is seen to undergo a gradual transition from having a camel-shaped form with a minimum at low electrolyte concentrations to having a maximum at higher concentrations consistent with the trends observed in earlier works in the absence of surface polarization. However, the transition envelope shifts to the lower concentration side relative to the no polarization situation for case (i), while it shifts to the higher concentration side for case (ii). The overall capacitance behaviour for both the situations indicates that the effect of the image interactions can be substantial, especially at low temperatures.

Electrolytes in a nanometer slab-confinement: Ion-specific structure and solvation forces

We study the liquid structure and solvation forces of dense monovalent electrolytes (LiCl, NaCl, CsCl, and NaI) in a nanometer slab-confinement by explicit-water molecular dynamics (MD) simulations, implicit-water Monte Carlo (MC) simulations, and modified Poisson-Boltzmann (PB) theories. In order to consistently coarse-grain and to account for specific hydration effects in the implicit methods, realistic ion-ion and ion-surface pair potentials have been derived from infinite-dilution MD simulations. The electrolyte structure calculated from MC simulations is in good agreement with the corresponding MD simulations, thereby validating the coarse-graining approach. The agreement improves if a realistic, MD-derived dielectric constant is employed, which partially corrects for (water-mediated) many-body effects. Further analysis of the ionic structure and solvation pressure demonstrates that nonlocal extensions to PB (NPB) perform well for a wide parameter range when compared to MC simulations, whereas all local extensions mostly fail. A Barker-Henderson mapping of the ions onto a charged, asymmetric, and nonadditive binary hard-sphere mixture shows that the strength of structural correlations is strongly related to the magnitude and sign of the salt-specific nonadditivity. Furthermore, a grand canonical NPB analysis shows that the Donnan effect is dominated by steric correlations, whereas solvation forces and overcharging effects are mainly governed by ion-surface interactions. However, steric corrections to solvation forces are strongly repulsive for high concentrations and low surface charges, while overcharging can also be triggered by steric interactions in strongly correlated systems. Generally, we find that ion-surface and ion-ion correlations are strongly coupled and that coarse-grained methods should include both, the latter nonlocally and nonadditive (as given by our specific ionic diameters), when studying electrolytes in highly inhomogeneous situations.

Activity Coefficients of an Electrolyte in a Mixture with a High Density Neutral Component

Activity coefficients for the primitive model electrolyte, containing a high concentration of neutral hard spheres, are calculated for the modified Poisson−Boltzmann (MPB), hypernetted chain (HNC), and mean spherical approximation (MSA) theories. Comparisons are made with Monte Carlo (MC) simulation results for a 1:1 electrolyte having a common ion diameter with the neutral molecule diameter equal or different to that of the ions. Various electrolyte concentrations are treated with the overall packing fraction held fixed at 0.3. A very good agreement with simulation for the MPB theory is found for the charging parameter approach, that via the virial route being less accurate. The MSA activity coefficients via the energy route also give a good representation of the simulation values, while the HNC theory is adequate only for low volume fractions of added neutral particles.

Predicting the surface tension of aqueous 1:1 electrolyte solutions at high salinity

The surface tension of the air/water interface is a phenomenon of particular interest in the water-unsaturated zone of porous media because it influences the contact angle and consequently the capillary water volume. A mechanistic model based on the modified Poisson–Boltzmann equation and the Pitzer theory is described and used to predict, under isothermal and isobaric conditions, the surface tension of 1:1 electrolytes at high salinity. These theories enable the determination of the electrical potential at the air/water interface and the activity coefficient of the ionic species in the bulk pore water, respectively. Hydration free energies of the structure-making and structure-breaking ions that influence the surface tension at high salinity are taken into account. Structure-making ions flee the air/water surface because they can better organize the water dipoles in bulk water than at the interface. Structure-breaking ions are positively adsorbed at the air/water interface because the bulk water can better organize their hydrogen-bonding network without these ions. The resulting surface tension increases and decreases, respectively, compared to the surface tension of pure water. The predictions are in good agreement with the surface tension data of 1:1 electrolytes (NaCl, KCl, HCl, NaNO 3, KNO 3, HNO 3 aqueous solutions) and the optimized parameters depend on the effective electrostatic diameters of cations and on the hydration free energies of the ions at the interface.

Excluded volume effect and orientational ordering near charged surface in solution of ions and Langevin dipoles

The influence of a finite volume of ions and orientational ordering of water Langevin dipoles on the dielectric permittivity profile in the vicinity of charged surface is studied theoretically via a numerical solution of the modified Poisson–Boltzmann equation. It is shown that the dielectric permittivity profile close to the charged surface is mainly determined by two mechanisms; specifically, the depletion of dipoles at the charged surface due to accumulated counterions and the increased orientational ordering of the water dipoles.