Restricted Primitive Model Electrolyte Research Articles

Entropy of electrolytes with off-centre charge cations. Monte Carlo studies

Monte Carlo techniques including Inverse Grand Canonical Monte Carlo simulations are utilized to calculate the entropy of a restricted primitive model electrolyte with the caveat that the cation charge centre is displaced from the geographical centre of the cation sphere towards the surface of the sphere. Calculations have been performed for symmetric valency (1:1) and asymmetric valency (2:1) electrolytes at 298.15 K for a range of electrolyte concentrations. The cation charge centre displacement accompanied by an increase of temperature leads to an increase of entropy. For the 2:1 valency system the entropy as a function of the square root of concentration reveals a broad inflexion region that separates the domains of deformation due to the electrostatic and steric interactions. When the central ion is an anion, and the charge centres of the surrounding cations are oriented towards the anion then the electrostatic interactions are stronger than in the standard restricted primitive model and the entropy decreases, while a central cation deforms the ionic atmosphere surrounding it and the entropy increases. In the low concentration regime good agreement is seen between the results from the Debye-Hückel limiting law and the corresponding simulations.

Comparison of zeta potentials and structure for statistical mechanical theories of a model cylindrical double layer

The zeta potential and the structure of a model cylindrical double layer obtained through the Poisson-Boltzmann theory, the hypernetted chain/mean spherical approximation integral equation theory, the modified Poisson-Boltzmann theory, and the density functional theory are compared with the corresponding Monte Carlo simulation results. The cylindrical double layer consists of an infinitely long cylinder with a uniform surface charge immersed in a restricted primitive model electrolyte (equi-sized, rigid spherical ions in a continuum dielectric). Calculations have been performed at room temperature and in a water-like solvent for 1:1, 2:2, 2:1/1:2 valency electrolytes for different electrolyte concentrations, axial charge parameters (surface charge densities), and ionic diameters. The results for the zeta potential, the mean electrostatic potential, and the electrode-ion singlet distributions predicted by the formal statistical mechanical theories reproduce the simulation data to a high level of accuracy overall for the range of physical parameters studied. The theoretical predictions, except that due to the classical mean field theory, also reveal a remarkable consistency among themselves. The results also (a) explore the relationship between the occurrence of charge reversal and the negative derivative of the mean electrostatic potential,(b) provide insight into the connection between reversed and extremal values of the zeta potential versus surfaced charge density curves and some aspects of electrokinetics, viz., inverted electrophoretic velocities and the possible existence of differently charged colloids with the same mobility, and (c) suggest a plausible identification of the integrated surface charge density at a point near the cylindrical polyion surface with the electrokinetic charge.

An analysis of the fluctuation potential in the modified Poisson-Boltzmann theory for restricted primitive model electrolytes

An approximate analytical solution to the fluctuation potential problem in the modified Poisson-Boltzmann theory of electrolyte solutions in the restricted primitive model is presented. The solution is valid for all inter-ionic distances, including contact values. The fluctuation potential solution is implemented in the theory to describe the structure of the electrolyte in terms of the radial distribution functions, and to calculate some aspects of thermodynamics, viz., configurational reduced energies, and osmotic coefficients. The calculations have been made for symmetric valence 1:1 systems at the physical parameters of ionic diameter $4.25 \times 10^{-10}$ m, relative permittivity 78.5, absolute temperature 298 K, and molar concentrations 0.1038, 0.425, 1.00, and 1.968. Radial distribution functions are compared with the corresponding results from the symmetric Poisson-Boltzmann, and the conventional and modified Poisson-Boltzmann theories. Comparisons have also been done for the contact values of the radial distributions, reduced configurational energies, and osmotic coefficients as functions of electrolyte concentration. Some Monte Carlo simulation data from the literature are also included in the assessment of the thermodynamic predictions. Results show a very good agreement with the Monte Carlo results and some improvement for osmotic coefficients and radial distribution functions contact values relative to these theories. The reduced energy curve shows excellent agreement with Monte Carlo data for molarities up to 1 mol/dm$^{3}$.

The planar electric double layer capacitance for the solvent primitive model electrolyte.

The planar electric double layer capacitance of the solvent primitive model electrolyte is studied using simulation and two versions of the modified Poisson-Boltzmann theory. At small values of the surface charge and varying electrolyte concentration, the capacitance has a behaviour analogous to that of the restricted primitive model electrolyte. As the electrolyte concentration is increased at a fixed total packing fraction, the minimum at zero surface charge changes to a maximum. This qualitative change is predicted by both simulation and the modified Poisson-Boltzmann theories. The transition envelope, separating the change in the capacitance from a minimum to a maximum at zero surface charge, is shifted from the restricted primitive model value to higher ion densities at fixed total packing fractions.

Monte Carlo investigation of structure of an electric double layer formed by a valency asymmetric mixture of charged dimers and charged hard spheres

Canonical and grand canonical Monte Carlo simulation results are reported for the structure of a planar electric double layer containing anisotropic-shaped ions having asymmetric valencies. The model double layer consists of an electrolyte formed by a mixture of charged dimers and charged hard spheres in a dielectric continuum next to a uniformly charged, non-polarisable, planar hard electrode. A dimer is made of two tangentially touching equi-sized hard spheres one of which is positively charged and the other neutral, while the monomer rigid ion of the same size is negatively charged. Results for the electrode-ion, electrode-neutral sphere singlet distributions, and the mean electrostatic potential are obtained for 2:1/1:2 valency asymmetry in the electrolyte solution regime at room temperature, and at a given electrolyte concentration for a series of electrode surface charge densities. Valency asymmetry coupled with anisotropic ion shape leads to a richer double-layer structure than seen previously with dimers in 1:1 symmetric valency situations. Comparisons are also made with the corresponding results for a 2:1/1:2 restricted primitive model electrolyte. The overall asymmetry also leads to a finite, non-zero potential of zero charge indicating a charge separation at the interface.

Structure and thermodynamics of the primitive model electrolyte in a charged matrix: The evaluation of the Madden-Glandt approximation

We compared the results of the Madden-Glandt (MG) integral equation approximation for partly-quenched systems with the commonly accepted formalism of Given and Stell (GS). The system studied was a +1:-1 restricted primitive model (RPM) electrolyte confined in a quenched +1:-1 RPM matrix. A renormalization scheme was proposed for a set of MG replica Ornstein-Zernike equations. Long-ranged direct and total correlation functions, describing the interactions between the annealed electrolyte species within the same replicas and between the annealed and matrix particles, appeared to be the same for MG and GS approach. Both versions of the theory give very similar results for the structure and thermodynamics of an annealed subsystem. Differences between excess internal energy, excess chemical potential, and isothermal compressibility become pronounced only at high concentrations of matrix particles.

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.

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.

Solvent primitive model for electrolyte solutions in disordered porous matrices of charged species. Replica Ornstein-Zernike theory and grand canonical Monte Carlo simulations

We present a theoretical study of a quenched–annealed system in which an annealed component is the restricted primitive model electrolyte in a mixture with an uncharged hard sphere species, i.e. the solvent primitive model (SPM), whereas a disordered quenched medium is modelled as the restricted primitive model (RPM) electrolyte. The annealed mixture is in thermal and chemical equilibrium with an external reservoir containing the same SPM. The system is studied by using the replica Ornstein–Zernike (ROZ) integral equation theory complemented with the hypernetted chain (HNC) approximation and via the grand canonical Monte Carlo simulations. We are primarily interested in collecting computer simulation data and comparing them with theoretical predictions at room temperature (298.15 K). In terms of physical observables, our focus is in the selectivity effects of adsorption of the mixture described by the adsorption isotherms as well as by the composition isotherms. The influence of the ionic matrix density and of the bulk state of the SPM mixture on adsorption and selectivity are examined in detail. Besides, we analyse the dependence of the internal energy and the constant volume heat capacity on the conditions of adsorption. Finally, we explore briefly the effects of quenching conditions of the RPM matrix on the pair distribution functions of fluid ions in the mixture. In general, the theory is in a very good agreement with computer simulations.

Structural and thermodynamic properties of the restricted primitive model electrolyte in a mixture with uncharged hard spheres: a grand canonical Monte Carlo simulation and integral equation study

A restricted primitive model electrolyte in a mixture with uncharged hard spheres was studied at room temperature using grand canonical Monte Carlo computer simulation and Ornstein–Zernike integral equation theory in the hypernetted chain approximation (HNC). The mean spherical approximation results are also presented for a few cases. We obtained the pair distribution functions of species of the system, the dependencies of the total fluid density and the ionic fraction on the chemical potentials, the excess internal energy and the heat capacity at constant volume for a wide range of chemical potentials of the species from the simulations and HNC theory. In the majority of cases, good agreement between the theoretical predictions and simulation data is obtained. The composition of the mixture is determined by the chemical potentials of both species. The pair distribution functions have a Debye-like shape at low densities for various values of the ion fraction. By increasing the chemical potential of the uncharged component, weak trends for structuring of the solution are observed with the formation of ion-hard sphere-ion complexes. At high densities, a tendency for in-phase oscillations of ion–ion functions is observed similar to the pure electrolyte in the restricted primitive model. We analysed the chemical potential–density and the chemical potential–ion fraction projections of the equation of state in detail. Also, the heat capacity at constant volume has been calculated for the first time. The model and the results are useful for the development of the theory of inhomogeneous fluid mixtures.

The Electric Double-Layer Differential Capacitance at and near Zero Surface Charge for a Restricted Primitive Model Electrolyte

The Electric Double-Layer Differential Capacitance at and near Zero Surface Charge for a Restricted Primitive Model Electrolyte

The Electric Double-Layer Differential Capacitance at and near Zero Surface Charge for a Restricted Primitive Model Electrolyte

The behavior of the differential capacitance of a planar electric double layer containing a restricted primitive model electrolyte in the neighborhood of zero surface charge is investigated by theory and simulation. Previous work has demonstrated that at zero surface charge the differential capacitance has a minimum for aqueous electrolytes at room temperature but can have a maximum for molten salts and ionic liquids. The transition envelope separating the two situations is found for a modified Poisson-Boltzmann theory and a Poisson-Boltzmann equation corrected for the exclusion volume term. Good agreement is found between simulation and the modified Poisson-Boltzmann theory in the neighborhood of the envelope at the reduced temperature of 0.8, while the exclusion volume corrected Poisson-Boltzmann theory shows correct qualitative trends.

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.

From restricted towards realistic models of salt solutions: Corrected Debye–Hückel theory and Monte Carlo simulations

The properties of bulk salt solutions over wide concentration ranges are explored by a combination of simple physical theory and Monte Carlo (MC) simulations. The corrected Debye–Hückel (CDH) theory which incorporates ion size effects in a linear response approximation is extended to yield free energy and other thermodynamic properties by integration of the chemical potential over concentration. Charging integration which is usually used to obtain an electrostatic contribution of total free energy of electrolytes is avoided in this new direct approach. MC simulations are performed with a modified Widom particle insertion method, which also provides directly the ionic activity coefficients. The validity of the CDH theory is tested by comparison with the MC simulation data for 1:1, 2:1, 2:2 and 3:1 restricted primitive model (RPM) electrolytes over a wide concentration range and at various ion sizes. Mean ionic activity and osmotic coefficients calculated by the CDH theory in RPM approximation of electrolyte are fitted to experimental data by adjusting only a mean ionic diameter. Good fits up to 1 molal (m) concentration are obtained for a large number of salt solutions. MC simulations data for unrestricted primitive model (UPM) of 1:1 and 2:1 electrolytes are also fitted to the experimental data by varying the cation radius while keeping the anion radius fixed at a crystallographic value. The success of this approach is found to be salt specific. For example good fits up to 2 and 3.5 m concentrations were obtained for LiCl and LiBr, respectively. However in the case of less dissociated salts such as NaCl and KI the experimental data could only be fitted up to one molal concentration. Possibility of extending the applicability range of the CDH theory to concentrations >2 m is explored by including a concentration dependent dielectric constant as measured in experiments. Mean ionic activity coefficients for a number of salts could successfully be fitted up to 3 m concentration by adjusting only a mean ionic diameter. Difficulties encountered in simultaneously fitting the mean ionic activity and osmotic coefficients at salt concentrations > 2 m are discussed.

A modified Poisson–Boltzmann theory and Monte Carlo simulation study of surface polarization effects in the planar diffuse double layer

The equilibrium structural properties of an electric double layer in the presence of a dielectric discontinuity at a plane charged interface are studied through a modified Poisson–Boltzmann equation and some Monte Carlo simulations. The diffuse double layer is modelled by a symmetric valency restricted primitive model electrolyte next to a uniformly charged planar wall whose relative permittivity () is, in general, different from that of the electrolyte () leading to polarization or image forces. The situations when the wall is a conductor (, ), or when the wall is an insulator (1, ) are treated and contrasted with the case , that is, no imaging. The results show that the influence of image forces on the diffuse double layer structure can be substantial, especially at or near zero surface charge density. The theoretical predictions are in nearly quantitative agreement for 1:1 salts or in semi-quantitative or better agreement for 2:2 salts with existing Monte Carlo simulations in the literature and some new simulations presented here.

Some simulation and modified Poisson–Boltzmann theory results for the contact values of an electrolyte near a charged electrode

Abstract Since the development of a contact value theorem by Henderson et al. it has been known that the prediction of the venerable Gouy–Chapman (GC) theory for the contact value of the density profile of an electrolyte near an electrode is incorrect. The more recent semi-empirical work of Fawcett and Henderson and the simulations of Boda and Henderson have shown that the contact value of the charge profile at low electrode charge is also given incorrectly by the GC theory, at least for a binary symmetric restricted primitive model electrolyte. In the present study we examine the contact values of the profiles of the double layer formed by the electrolyte species in a binary symmetric restricted primitive model electrolyte as a function of the electrode charge by means of simulation. It is difficult to extract detailed information from simulation values of the density and charge profiles at large electrode charge because the contact values of these profiles are dominated by the large quadratic term in the electrode charge. However, we find from our simulations for a fairly wide range of concentrations and electrode charges, that the product of the counterion and coion profiles is a sensitive probe with which to study the double layer because it is a direct test of the basis of the GC approximation that the counterion and coion profiles are exponentials of ± zeψ ( x )/ kT and so have a product that is identically one. We find that this product is not one or even a constant as the electrode charge is varied and, thus, the GC contact values do not agree even qualitatively with the simulation results. In contrast, the modified Poisson–Boltzmann theory is quite successful in accounting for the behavior of this product.

Two rods confined by positive plates: effective forces and charge distributionprofiles

The effect of confinement on the interaction force between two negatively charged rods isstudied through Monte Carlo simulations. Confinement is produced by two parallel,charged or uncharged plates. The system is immersed in a 0.1 M 1–1 restricted primitivemodel electrolyte. The effect on the rod–rod effective force by the plate charge distributionis analysed. A strong modification of the rod–rod effective force due to confinement isfound, as compared to the bulk case. In particular, rod–rod attraction was found forplates having a charge equal to that of fully charged bilipid bilayers. In spite ofthe simplicity of the model, these results agree with some DNA–phospholipidexperimental observations. On the other hand, for a model having the plate charges fixedon a grid, very long range, oscillatory rod–rod effective forces were obtained.

Planar Electric Double Layer for a Restricted Primitive Model Electrolyte at Low Temperatures

Monte Carlo simulation and the modified Poisson-Boltzmann theory are used to investigate the planar electric double layer for a restricted primitive model electrolyte at low temperatures. Capacitance as a function of temperature at low surface charge is determined for 1:1, 2:2, 2:1, and 3:1 electrolytes. Negative adsorption can occur for 1:1 electrolytes at low surface charge with low electrolyte concentration. The 1:1 electrolyte diffuse layer potential as a function of surface charge displays a maximum at low densities. At high densities, the diffuse layer potential is negative with a negative slope. The Gouy-Chapman-Stern theory fails in this low-temperature regime, whereas the modified Poisson-Boltzmann theory is fairly successful in this regard.

Singular coexistence-curve diameters: Experiments and simulations

Precise calculations of the coexistence-curve diameters of a hard-core square-well (HCSW) fluid and the restricted primitive model (RPM) electrolyte exhibit marked deviations from rectilinear behavior. The HCSW diameter displays a | t| 1− α singularity that sets in sharply for | t| ≡ | T − T c|/ T c ≲ 10 −3; this compares favorably with extensive data for SF 6, also reflected in C 2H 6, N 2, etc. By contrast, the curvature of the RPM diameter varies slowly over a wide range | t| ≲ 0.1; this behavior mirrors observations for liquid alkali metals, specifically Rb and Cs. Amplitudes for the leading singular terms can be estimated numerically but their values cannot be taken literally.

Yang-Yang anomalies and coexistence diameters: Simulation of asymmetric fluids

A general method for estimating the Yang-Yang ratio, R(mu) , of a model fluid via Monte Carlo simulations is presented on the basis of data for a hard-core square-well (HCSW) fluid and the restricted primitive model (RPM) electrolyte. The isothermal minima of Q(L)(triple bond)<m(2)>2L/<m(4)>L scaling are evaluated at T(c) in an LxLxL box where m=rho-<rho>L is the density fluctuation. The "complete" finite-size scaling theory for the Q(+/-)(min) (T(c);L) incorporates pressure mixing in the scaling fields, thereby allowing for a Yang-Yang anomaly. It yields a dominant term in the asymmetry, Q(+)(min)-Q(-)(min) , varying as L(-beta/nu) with an amplitude proportional to the crucial pressure-mixing coefficient, j(2) . The reliably known critical order-parameter distribution for (d=3) Ising systems then enables one to estimate j(2) , thereby yielding R(mu) , from the Q minima together with information on the nonuniversal amplitudes for the order parameter and the susceptibility. The detailed analysis needed to estimate j(2) for an HCSW fluid and the RPM is presented. Furthermore, the Q-minima below T(c) can also provide the coexistence-curve diameters, rho(diam) (T) (triple bond)1/2 (rho(+) + rho(-)) , very close to T(c) for both models: here rho +/-(T) are the densities of the coexisting liquid and gas phases. The recently developed recursive scaling algorithm for Deltarho(infinity) (T) (triple bond)rho(+)-rho(-) is adapted to investigate the corresponding universal scaling functions. The two extremal forms of these scaling functions are computed with the aid of the exactly soluble decorated lattice-gas model. The critical densities for the RPM and HCSW fluid found via this route are consistent with previous estimates obtained from the data above T(c) ; the magnitudes of the |T- T(c)|(2beta) and |T- T(c)|(1-alpha) corrections to rho(diam)(T) are estimated.