Primitive Model Of Electrolyte Solutions Research Articles

Density functional theory of electrolyte solutions in slit-like nanopores I. The RFD/WCA approach extended to non-restricted primitive model

An extended reference fluid density approach/weighted correlation approximation (RFD/WCA) of density functional theory (DFT) is tested for the description of the non-restricted primitive model of electrolyte solutions in slit-like nanopores. The RFD/WCA approach of modeling size-asymmetric ions is validated by reproducing the density and electrostatic potential profiles of planar electrical double layer systems in the presence of mixtures of mono- and multivalent ions. The results from the DFT agree quite well with Monte Carlo simulations and satisfy the contact density sum rules of ionic fluid under a wide range of conditions. These findings suggest that a generic RFD/WCA approach can be formulated within the framework of the restricted and non-restricted primitive model to further investigate the swelling and ion exchange of clay minerals.

Restricted primitive model for electrolyte solutions in slit-like pores with grafted chains: Microscopic structure, thermodynamics of adsorption, and electric properties from a density functional approach

We apply a density functional theory to describe properties of a restricted primitive model of an ionic fluid in slit-like pores. The pore walls are modified by grafted chains. The chains are built of uncharged or charged segments. We study the influence of modification of the pore walls on the structure, adsorption, ion selectivity, and the electric double layer capacitance of ionic fluid under confinement. The brush built of uncharged segments acts as a collection of obstacles in the walls vicinity. Consequently, separation of charges requires higher voltages, in comparison to the models without brushes. At high grafting densities the formation of crowding-type structure is inhibited. The double layer structure becomes more complex in various aspects, if the brushes are built of charged segments. In particular, the evolution of the brush height with the bulk fluid density and with the charge on the walls depends on the length of the blocks of charged spheres as well as on the distribution of charged species along chains. We also investigated how the dependence of the double layer capacitance on the electrostatic potential (or on the charge on the walls) changes with grafting density, the chain length, distribution of charges along the chain, the bulk fluid density, and, finally, with the pore width. The shape of the electric double layer capacitance vs. voltage changes from a camel-like to bell-like shape, if the bulk fluid density changes from low to moderate and high. If the bulk density is appropriately chosen, it is possible to alter the shape of this curve from the double hump to single hump by changing the grafting density. Moreover, in narrow pores one can observe the capacitance curve with even three humps for a certain set of parameters describing brush. This behavior illustrates how strong the influence of brushes on the electric double layer properties can be, particularly for ionic fluids in narrow pores.

Self-diffusion and activity coefficients of ions in charged disordered media

Self-diffusion and single ion activity coefficients of ions of size symmetric electrolytes were studied in the presence of a collection of charged obstacles (called matrix) within a "soft" version of the primitive model of electrolyte solutions. The matrix subsystem possesses a net charge, depending on the concentration and charge of obstacles. The brownian dynamics method was used to calculate the self-diffusion coefficients of mobile species. The replica Ornstein-Zernike theory for the partly quenched systems was used to calculate the individual activity coefficients of mobile ionic species. The results reflect the competition between attractive (obstacle-counterion, co-ion-counterion), and repulsive (obstacle-co-ion) interactions in these model systems. For the simplest possible system of symmetric monovalent ions the latter effect wins: Co-ions are excluded from the area around obstacles, and this slows down their diffusion compared to that of counterions. Introduction of divalent charges into the system changes this result when the concentration of obstacles is low. We compare these results to those obtained for the corresponding fully annealed systems, i.e., where all the species are mobile. In most cases the self-diffusion and activity coefficients of counterions and co-ions in the presence of charged obstacles follow the trends of the fully annealed solution, which are dictated by the composition of the mixture. In few situations, however, the presence of charged obstacles modifies these trends. Our study allows us to clearly identify the effects due to obstacles, and to separate them from those arising from the composition of the solution. In the case of charge and size symmetric systems, the results for the individual activity coefficients fully support the hypothesis of the "electrostatic excluded volume". Thermodynamic and dynamic results are consistent in explaining the behavior of the systems studied.

Density functional theory for encapsidated polyelectrolytes: A comparison with Monte Carlo simulation

Genome packaging inside viral capsids is strongly influenced by the molecular size and the backbone structure of RNA∕DNA chains and their electrostatic affinity with the capsid proteins. Coarse-grained models are able to capture the generic features of non-specific interactions and provide a useful testing ground for theoretical developments. In this work, we use the classical density functional theory (DFT) within the framework of an extended primitive model for electrolyte solutions to investigate the self-organization of flexible and semi-flexible linear polyelectrolytes in spherical capsids that are permeable to small ions but not polymer segments. We compare the DFT predictions with Monte Carlo (MC) simulation for the density distributions of polymer segments and small ions at different backbone flexibilities and several solution conditions. In general, the agreement between DFT and MC is near quantitative except when the simulation results are noticeably influenced by the boundary effects. The numerical efficiency of the DFT calculations makes it promising as a useful tool for quantification of the structural and thermodynamic properties of viral nucleocapsids in vivo and at conditions pertinent to experiments.

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.

Correlation between flexibility of chain-like polyelectrolyte and thermodynamic properties of its solution

Structural and thermodynamic properties of the model solution containing charged oligomers and the equivalent number of counterions were studied by means of the canonical Monte Carlo simulation technique. The oligomers are represented as (flexible) freely jointed chains or as a linear (rigid) array of charged hard spheres. In accordance with the primitive model of electrolyte solutions, the counterions are modeled as charged hard spheres and the solvent as dielectric continuum. Significant differences in the pair distribution functions, obtained for the rigid (rod-like) and flexible model are found but the differences in thermodynamic properties, such as, enthalpy of dilution and excess chemical potential, are less significant. The results are discussed in light of the experimental data an aqueous polyelectrolyte solutions. The simulations suggest that deviations from the fully extended (rod-like) conformation yield slightly stronger binding of counterions. On the other hand, the flexibility of polyions, even when coupled with the ion-size effects, cannot be blamed for qualitative differences between the theoretical results and experimental data for enthalpy of dilution.

Restricted primitive model for electrolyte solutions in contact with solid surface modified by grafted chains: A density functional approach

We have studied the microscopic structure and the thermodynamic and electric properties of the restricted primitive model for electrolyte solutions in contact with a chemically modified uncharged and charged solid surface. The modification of the surface is performed by the tethering of chain particles via a specific single segment at the stage preceeding the adsorption of the restricted primitive model. Some fraction of segments belonging to a chain particle can be charged such that the system models adsorption on an electrode covered by an electrolyte brush. We apply the density functional method which combines previously developed approach for inhomogeneous chain fluids, and the weighted density mean spherical approximation energy route for the description of inhomogeneous ionic fluids. The theory is formulated in the semigrand canonical ensemble. Our principal findings are concerned with the density profiles of the species, the adsorption isotherms of ions in the presence of grafted chain particles, the charge density profile and the potential of zero charge (all as the functions of the grafted density), and other parameters related to the solid surface and its modification.

Electrolyte Exclusion from Charged Adsorbent: Replica Ornstein−Zernike Theory and Simulations

Structural and thermodynamic properties of the restrictive primitive model +1:-1 electrolyte solution adsorbed in a disordered charged media were studied by means of the Grand Canonical Monte Carlo simulation and the replica Ornstein-Zernike theory. Disordered media (adsorbent, matrix) was represented by a distribution of negatively charged hard spheres frozen in a particular equilibrium distribution. The annealed counterions and co-ions were assumed to be distributed within the nanoporous adsorbent in thermodynamic equilibrium with an external reservoir of the same electrolyte. In accordance with the primitive model of electrolyte solutions, the solvent was treated as a dielectric continuum. The simulations were performed for a set of model parameters, varying the net charge of the matrix (i.e., concentrations of matrix ions) and of annealed electrolyte, in addition to the dielectric constant of the invading solution. The concentration of adsorbed electrolyte was found to be lower than the corresponding concentration of the equilibrium bulk solution. This electrolyte "exclusion" depends strongly on the dielectric constant of the invading solution, as also on concentrations of all components. The most important parameter is the net charge of the matrix. Interestingly, the electrolyte rejection decreases with increasing Bjerrum length for the range of parameters studied here. The latter finding can be ascribed to strong inter-ionic correlation in cases where the Bjerumm length is high enough. To a minor extent, the adsorption also depends on the spacial distribution of fixed charges in adsorbent material. The replica Ornstein-Zernike theory was modified to cater for this model and tested against the computer simulations. For the range of parameters explored in this work, the agreement between the two methods is very good. These calculations were also compared with the results of the classical Donnan theory for electrolyte exclusion.

Theoretical aspects and computer simulations of flexible charged oligomers in salt-free solutions

The structural and thermodynamic properties of a model solution containing flexible charged oligomers and an equivalent number of counterions were studied by means of the canonical Monte Carlo simulation and integral equation theory. The oligomers were represented as freely jointed chains of charged hard spheres. In accordance with the primitive model of electrolyte solutions, the counterions were modeled as charged hard spheres and the solvent as a dielectric continuum. Simulations were performed for a set of model parameters, independently varying the chain length and concentration of the oligomers. Structural properties in the form of pair distribution functions were calculated as functions of model parameters. In addition, thermodynamic properties such as the excess energy of solution and the excess chemical potential of counterions were obtained. These properties were correlated with the conformational averages of oligomers as reflected in the end-to-end distances and radii of gyration obtained from the simulations. The relation with the experimental data for heats of dilution and for the activity coefficient is discussed. Finally, theories based on Wertheim's integral equation approach (product reactant Ornstein-Zernike approach) [J. Stat. Phys. 42, 477 (1986)] in the so-called polymer mean spherical and polymer hypernetted chain approximations were tested against the new and existing computer simulations. For the values of parameters examined in this study, the integral equation theory yields semiquantitative agreement with computer simulations.

Simulation of symmetric tricritical behavior in electrolytes

Despite extensive experimental, theoretical, and simulation efforts, a unified description of ionic phase transitions and criticality has not yet emerged. In this work, we investigate the phase behavior of the restricted primitive model of electrolyte solutions on the simple cubic lattice using grand canonical Monte Carlo simulations and finite-size scaling techniques. The phase diagram of the system is distinctly different from its continuum-space analog. We find order-disorder transitions for reduced temperatures T* < or approximately 0.51, where the ordered structures resemble those of the NaCl crystal. The order-disorder transition is continuous for 0.15 < or approximately T* < or approximately 0.51 and becomes first order at lower temperatures. The line of first-order transitions is a line of three-phase coexistence between a disordered and two ordered phases. The line of continuous, second-order transitions meets this line of triple points at a tricritical point at T* approximately equal to 0.1475. We locate the line of continuous transitions, and the line of triple points using finite-size scaling techniques. The tricritical temperature is estimated by extrapolation of the size-dependent tricritical temperatures obtained from a sixth-order Landau expansion of the free energy. Our calculated phase diagram is in qualitative agreement with mean-field theories.

Controlling the Cohesion of Cement Paste

The main source of cohesion in cement paste is the nanoparticles of calcium silicate hydrate (C-S-H), which are formed upon the dissolution of the original tricalcium silicate (C(3)S). The interaction between highly charged C-S-H particles in the presence of divalent calcium counterions is strongly attractive because of ion-ion correlations and a negligible entropic repulsion. Traditional double-layer theory based on the Poisson-Boltzmann equation becomes qualitatively incorrect in these systems. Monte Carlo (MC) simulations in the framework of the primitive model of electrolyte solution is then an alternative, where ion-ion correlations are properly included. In addition to divalent calcium counterions, commercial Portland cement contains a variety of other ions (sodium, potassium, sulfate, etc.). The influence of high concentrations of these ionic additives as well as pH on the stability of the final concrete construction is investigated through MC simulations in a grand canonical ensemble. The results show that calcium ions have a strong physical affinity (in opposition to specific chemical adsorption) to the negatively charged silicate particles of interest (C-S-H, C(3)S). This gives concrete surprisingly robust properties, and the cement cohesion is unaffected by the addition of a large variety of additives provided that the calcium concentration and the C-S-H surface charge are high enough. This general phenomenon is also semiquantitatively reproduced from a simple analytical model. The simulations also predict that the affinity of divalent counterions for a highly and oppositely charged surface sometimes is high enough to cause a "charge reversal" of the apparent surface charge in agreement with electrophoretic measurements on both C(3)S and C-S-H particles.

Phase behavior of the restricted primitive model of ionic fluids with association in slitlike pores. Density-functional approach

We present results of investigations of condensation of restricted primitive model of electrolyte solutions with association between oppositely charged ions confined to slitlike pores. The associative interaction leads to the formation of ionic pairs. It is accounted for by incorporating the first-order thermodynamic perturbation theory into the free energy functional. In order to elucidate the role of association, the phase diagrams are compared with those obtained by us recently [O. Pizio et al., J. Chem. Phys. 121, 11957 (2004)] for the restricted primitive model. The inclusion of the association into the theory leads to lowering the critical temperature for the fluid confined to pores with uncharged and with charged walls. We have observed that the average fraction of bonded ions is high along the coexistence envelope.

Towards the description of the phase behavior of electrolyte solutions in slit-like pores. Density functional approach for the restricted primitive model

We develop a density functional approach for the phase behavior of the restricted primitive model for electrolyte solutions confine d to slit-like pores. The theory permits to evaluate the effects of confinement on the ionic vapor ‐ ionic liquid coexistence envelope. We have shown that due to confinement in pores with uncharged walls the critical temperature of the model decreases compared to the bulk. Also the coexistence envelope of the transition is narrower in comparison to the bulk model. The transition between dense and dilute phase represents capillary evaporation. We have analyzed changes of the density profiles of ions during transition. Possible extensions of this study are discussed.

Critical parameters of the restricted primitive model

The critical parameters for the restricted primitive model of electrolyte solutions were determined from extensive grand canonical Monte Carlo simulations combined with mixed-field finite-size scaling. The fine-lattice discretization method was used for the calculations, with Ewald summation of the long-range coulombic forces. Ising criticality and no pressure mixing were assumed in the finite-size scaling analysis. The critical parameters were obtained as a function of boundary conditions at infinite distance (ε∞), system size L, and lattice discretization parameter ζ. They were found to be sensitive to L for vacuum boundary conditions (ε∞=1), but much less so for “tin-foil” boundary conditions (ε∞=∞). The critical temperature and density decrease with increasing ζ. These calculations are compared to previous estimates of the critical parameters for this model. Extrapolation of our results to the thermodynamic limit in continuous space (L→∞ and ζ→∞) yields Tc*=0.0489±0.0003, ρc*=0.076±0.003.

Application of Double-Layer Theories to the Extensive Crystalline Swelling of Li-Montmorillonite

We have used the Poisson−Boltzmann (PB) theory and DLVO double-layer theory together with the hypernetted chain formalism based on the primitive model for electrolyte solutions to examine recently published results for extensive crystalline swelling of Li-montmorillonite as revealed by d(001) spacings over the pressure range 0.05−0.9 MPa and the concentration range 1−10-4 M LiCl. The analysis based on Langmuir's use of elliptical integrals reveals that the theory satisfactorily predicts surface separation over the range 18−120 A and that the DLVO theory, with a 5.5 A Stern layer, indicates Gouy plane potentials ranging from −58 to −224 mV over the concentration range 0.1−10-4 M LiCl and a constant Gouy plane charge of 0.038 C m-2, which is 30% of the crystal lattice charge. The anisotropic hypernetted chain (HNC) analysis, also based on lattice charge, satisfactorily predicts pressure−distance relationships over the range of concentrations. There is no evidence for the operation of the “secondary hydratio...

Theoretical calculations of the ionic strength dependence of the ionic product of water based on a mean spherical approximation

A modified version of the restricted primitive model for electrolyte solutions based on the mean spherical approximation (MSA) is applied to estimate the ionic strength dependence of the ionic product of water in aqueous solutions containing different salts, which are commonly used as background electrolytes (NaCl, KCl, KNO3, and NaC104). The modification involves the use of permittivity of the solvent as concentration-dependent parameter and a single average effective diameter. This is a way of including effects originated from the solvent which do not exist in the primitive model. In the case of potassium nitrate and sodium perchlorate, a complete methodology to calculate the effective diameter and density dependence of the dielectric constant has been proposed and developed. Fits between calculated and experimental pKw values are possible over wide concentration ranges using a single adjustable parameter, namely, the average hard core diameter of water.

Analytic solution of the restricted primitive model of electrolyte solutions with double sticky interaction between the oppositely charged ions

A model for the description of the association phenomena and chemical reactions in the ionic fluids is considered. It generalizes the sticky electrolyte model and provides simultaneous formation of species with different bond length. The interaction between the oppositely charged ions is characterized by the presence of two narrow and deep attractive square wells inside a hard-core region. It determines the double sticky electrolyte model (DSEM). Application of the simple interpolation scheme by Zhou and Stell and the approximation by Rasaiah and Zhu for the cavity function is investigated for DSEM in order to provide the mass action limiting law. Analytic solution for the DSEM is presented.

Electrochemical potentials in the grand canonical ensemble

The grand canonical ensemble has been used to study electrochemical potentials in confined ionic systems. Grand canonical ensemble Monte Carlo calculations for the primitive model of electrolyte solutions are presented for spherical systems, and the evaluation of single ion activity coefficients by such calculations is discussed. It is also shown that the introduction of finite, confined systems resolves a well-known problem associated with the application of the Kirkwood-Buff theory to ionic systems.

Single-ion activity coefficients and structure of ionic fluids: results for the primitive model of electrolyte solutions

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTSingle-ion activity coefficients and structure of ionic fluids: results for the primitive model of electrolyte solutionsPeter. Sloth and Torben S. SoerensenCite this: J. Phys. Chem. 1990, 94, 5, 2116–2123Publication Date (Print):March 1, 1990Publication History Published online1 May 2002Published inissue 1 March 1990https://doi.org/10.1021/j100368a070RIGHTS & PERMISSIONSArticle Views167Altmetric-Citations51LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (699 KB) Get e-Alerts

On the calculation of single-ion activity coefficients by the Kirkwood-Buff theory

An expression which applies for single-ion activity coefficients is obtained from the theory of Kirkwood and Buff by introduction of a single plausible assumption. When this expression is used in connection with the extended mean spherical approximation for the primitive model of electrolyte solutions, the limiting law of Debye and Hückel is recovered in the limit of low ionic concentration.