Fluid Of Charged Hard Spheres Research Articles

Binding Debye-Hückel theory for associative electrolyte solutions.

This study presents a new equation of state (EOS) for charged hard sphere fluids that incorporates ion-ion association. The EOS is developed using the Debye-Hückel (DH) theory, reference cavity approximation, and Wertheim's theory. Predictive accuracy is evaluated by comparing the model's predictions with Monte Carlo simulations for various charged hard-sphere fluids. The assessment focuses on mean ionic activity coefficient, individual ionic activity coefficient, and osmotic coefficients. The results demonstrate good agreement between the model and simulations, indicating its success for different electrolyte systems. Incorporating ion-ion association improves accuracy compared to the DH theory. The importance of the cavity function and ion-dipole interactions is emphasized in accurately representing structural properties. Overall, the developed EOS shows promising predictive capabilities for charged hard sphere fluids, providing validation and highlighting the significance of ion-ion association in thermodynamic predictions of electrolyte solutions.

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

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

Solidification of the charged hard-sphere fluid

Monte Carlo simulations of a hard-sphere model with positive and negative charges are carried out to determine the range of temperatures and densities in which the fluid and solid phases are thermodynamically stable. This is a simple model of an ionic liquid. The limiting densities of fluid and solid phases for a given temperature are determined from the discontinuity of the compressibility factor against density dependence at a constant volume. The dependence of density on the potential energy and the heat capacity confirms these predictions. The temperature–density curves showing the limits of fluid and solid phases increase with the increasing density at low temperatures and are nearly temperature independent at high temperatures. The energy drop between the fluid and solid phases depends on temperature. This energy is positive at low temperatures and negative at high temperatures. This unusual behaviour observed at low temperatures appears to be due to the formation of a tetragonal primitive quasi-structure before solid phase. This quasi-structure is stabilised by the electrostatic interactions. The solid phase also forms a tetragonal primitive lattice but with a different c/a ratio, which is a result of the high-density hard core interactions. The translational order parameter of quasi- and crystal structures amounts to 0.565 and 0.576, respectively. Our results are in general in agreement with earlier studies, but some differences were found.

A Short History of Phase Transitions in Ionic Fluids

AbstractThe paper reviews the development of theory and experiments concerning the nature of the critical point in ionic fluids. Because of the long‐range nature of the Coulomb interactions the possibility of mean‐field critical behaviour was discussed as a possibility. Although some experiments supported mean‐field criticality, simulations on the model fluid of charged hard spheres and later experiments on ionic solutions have shown that phase transition of ionic systems belong to the Ising universality class like phase transitions in non‐ionic fluids. Experiments concerning the crossover from Ising to mean field‐behaviour are discussed as well as systematic differences between the phase behaviour predicted for the model fluid of charged hard spheres and that observed in ionic solutions (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Thermodynamics and diffusion in size-symmetric and asymmetric dense electrolytes

MD simulation results for model size-symmetric and asymmetric electrolytes at high densities and temperatures (well outside the liquid-gas coexistence region) are generated and analyzed focusing on thermodynamic and diffusion properties. An extension of the mean spherical approximation for electrolytes originally derived for charged hard sphere fluids is adapted to these systems by exploiting the separation of short range and Coulomb interaction contributions intrinsic to these theoretical models and is found to perform well for predicting equation of state quantities. The diffusion coefficients of these electrolytes can also be reasonably well predicted using entropy scaling ideas suitably adapted to charged systems and mixtures. Thus, this approach may provide an avenue for studying dense electrolytes or complex molecular systems containing charged species at high pressures and temperatures.

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.

Density Functional Theory Approach for Charged Hard Sphere Fluids Confined in Spherical Micro-Cavity

Within the framework of the density functional theory for classical fluids, the equilibrium density profiles of charged hard sphere fluid confined in micro-cavity are studied by means of the modified fundamental measure theory. The dimension of micro-cavity, the charge of hard sphere and the applied electric field are found to have significant effects on the density profiles. In particular, it is shown that Coulomb interaction, excluded volume interaction and applied electric field play the central role in controlling the aggregated structure of the system.

Liquid–liquid phase transition in the ionic solutions of tetra-n-butylammonium chloride in o-xylene and ethylbenzene: Phase diagrams and corresponding state analysis

The phase diagrams of the liquid–liquid phase transition of solutions of tetra-n-butyl ammonium chloride (N 4444Cl) in o-xylene and ethylbenzene are reported and compared with the phase diagrams of solutions of other ionic liquids (IL) with halide anions (tri-n-hexyl-n-tetradecyl phosphonium (P 666 14 +)) chloride and bromide and 1-n-alkyl-3-methyl imidazolium (R nmim +) chlorides) in non-polar solvents such as alkanes, arenes and CCl 4 and also with solutions of various 1-n-alkyl-3-methyl imidazolium tetrafluoroborates (R nmim + BF 4) in alcohols and water. All phase diagrams have an upper critical solution point (UCSP) and are consistent with the presumption of Ising criticality. For the corresponding state analysis different concentration variables (mole fraction, volume fraction, and density) are applied. Using the variables of the model fluid of charged hard spheres in a dielectric continuum, termed restricted primitive model (RPM) a systematic overview of the critical properties of the mixtures is obtained. The critical temperatures of the solutions in the non-polar solvents are in the region predicted by the RPM but somewhat smaller. They are very similar, and do not correlate with the dielectric constant of the solvents but show a weak correlation with the boiling temperatures of the solvents, indicating the influence of the cohesion of the solvents on the phase separation driven by Coulomb interactions. The critical concentrations, width of the phase diagrams and the slope of the diameter are noticeably different for the solutions in protic and aprotic solvents but show no further systematic trends.

Liquid-liquid phase transition in solutions of ionic liquids with halide anions: Criticality and corresponding states

Abstract Measurements of the liquid-liquid phase diagrams of solutions of the ionic liquids (ILs) 1-dodecyl-3-methylimidazolium chloride (C12mimCl) in arenes (benzene, toluene, o-xylene, tetraline) and 1-tetradecyl-3-methylimidazolium chloride (C14mimCl) in CCl4 are reported and compared with those of solutions of trihexyl-tetradecyl-phosphonium halides (P666 14Cl, P666 14Br) in hydrocarbons and 1-alkyl-3-methylimidazolium tetrafluoroborates (CnmimBF4) in alcohols and water. The phase diagrams of solutions of tetrapentyl-ammonium bromide (N5555Br) in water and KI in SO2 are also discussed. Except for the KI/SO2 system, which features a lower critical solution point (LCSP), all systems have an upper critical solution point (UCSP) and show corresponding-states behavior. The experimental data are compared with results from simulations and theory concerning the model fluid of charged hard spheres in a dielectric continuum, termed restricted primitive model (RPM). The analysis in terms of of RPM variables shows agreement with the location of the critical point (CP) of the model with noticeable systematic deviations. However, for protic solvents, the CP becomes an LCSP, while in aprotic solvents the CP is a UCSP as expected for Coulomb systems. This indicates that in aprotic solvents, the phase transition is essentially determined by the Coulomb interactions, while in the solutions in protic solvents with hydrogen bonds, both Coulomb and solvophobic interactions are important.

Phase coexistence in a polydisperse charged hard-sphere fluid: Polymer mean spherical approximation

We have reconsidered the phase behavior of a polydisperse mixture of charged hard spheres (CHSs) introducing the concept of minimal size neutral clusters. We thus take into account ionic association effects observed in charged systems close to the phase boundary where the properties of the system are dominated by the presence of neutral clusters while the amount of free ions or charged clusters is negligible. With this concept we clearly pass beyond the simple level of the mean spherical approximation (MSA) that we have presented in our recent study of a polydisperse mixture of CHS [Yu. V. Kalyuzhnyi, G. Kahl, and P. T. Cummings, J. Chem. Phys. 120, 10133 (2004)]. Restricting ourselves to a 1:1 and possibly size-asymmetric model we treat the resulting polydisperse mixture of neutral, polar dimers within the framework of the polymer MSA, i.e., a concept that--similar as the MSA--readily can be generalized from the case of a mixture with a finite number of components to the polydisperse case: again, the model belongs to the class of truncatable free-energy models so that we can map the formally infinitely many coexistence equations onto a finite set of coupled, nonlinear equations in the generalized moments of the distribution function that characterizes the system. This allows us to determine the full phase diagram (in terms of binodals as well as cloud and shadow curves), we can study fractionation effects on the level of the distribution functions of the coexisting daughter phases, and we propose estimates on how the location of the critical point might vary in a polydisperse mixture with an increasing size asymmetry and polydispersity.

Phase coexistence in polydisperse charged hard-sphere fluids: mean spherical approximation.

Taking advantage of the availability of the analytic solution of the mean spherical approximation for a mixture of charged hard spheres with an arbitrary number of components we show that the polydisperse fluid mixture of charged hard spheres belongs to the class of truncatable free energy models, i.e., to those systems where the thermodynamic properties can be represented by a finite number of (generalized) moments of the distribution function that characterizes the mixture. Thus, the formally infinitely many equations that determine the parameters of the two coexisting phases can be mapped onto a system of coupled nonlinear equations in these moments. We present the formalism and demonstrate the power of this approach for two systems; we calculate the full phase diagram in terms of cloud and shadow curves as well as binodals and discuss the distribution functions of the coexisting daughter phases and their charge distributions.

A phenomenological description of the anomalous behavior of the electrical double layer at low temperatures

A simple description that accounts for some recent simulations of the electric double layer of a charged hard sphere fluid that show anomalously large negative adsorption of the ions at a surface and an unusual positive temperature derivative for the capacitance at low temperatures is discussed. The treatment is phenomenological since it does not result from any controlled approximation but it is reasonably well founded. The mean spherical approximation expressions for the density profiles of this system, which do not exhibit anomalous behavior, are modified in light of an exact contact value condition, the Ornstein–Zernike theory of the critical point, and some plausible reasoning. The agreement with the simulation results is fairly good. At the moment, there is no theory that accounts for these recent simulation results.

Scattering functions for multicomponent mixtures of charged hard spheres, including the polydisperse limit: Analytic expressions in the mean spherical approximation

We present a closed analytical formula for the scattering intensity from charged hard sphere fluids with any arbitrary number of components. Our result is an extension to ionic systems of Vrij’s analogous expression for uncharged hard sphere mixtures. Use is made of Baxter’s factor correlation functions within the mean spherical approximation (MSA). The polydisperse case of an infinite number of species with a continuous distribution of hard sphere diameters and charges is also considered. As an important by-product of our investigation, we present some properties of a particular kind of matrices (sum of the identity matrix with a dyadic matrix) appearing in the solution of the MSA integral equations for both uncharged and charged hard sphere mixtures. This analysis provides a general framework to deal with a wide class of MSA solutions having dyadic structure and allows an easy extension of our formula for the scattering intensity to different potential models. Finally, the relevance of our results for the interpretation of small angle neutron scattering experimental data is briefly discussed.

Monte Carlo simulation and integral-equation studies of a fluid of charged hard spheres near the critical region.

Monte Carlo simulation and integral-equation studies of a fluid of charged hard spheres near the critical region.

Free energy model for inhomogeneous fluid mixtures: Yukawa-charged hard spheres, general interactions, and plasmas

A free energy model for the inhomogeneous hard-sphere fluid mixture was derived recently [Phys. Rev. Lett. 63, 980 (1989)], which is based on the fundamental geometric measures of the particles. Along with an updated assessment of its accuracy, this model is first generalized for charged hard-sphere fluid mixtures, in which every particle carries a central Yukawa charge, and it is then extended to general fluid mixtures in external fields. The Yukawa-charged hard-sphere mixture provides a quite general reference system for many interesting physical systems including plasmas, molten salts, and colloidal dispersions, the screening parameter enabling to interpolate between the long range Coulomb forces and the short range hard cores. A special renormalization property of the Yukawa potential provides the means to derive the exact Onsager-type lower bound for the potential energy of the mixture, and its related asymptotic strong-coupling limit of the liquid pair correlation functions. These results are obtained analytically for the general homogeneous mixture with Yukawa interactions. They enable to extend the fundamental measure free energy model to inhomogeneous charged Yukawa mixtures, with the charge contributions given by a truncated second order expansion from the uniform (bulk) fluid limit. The resulting free energy model, which interpolates between the ideal-gas and ‘‘ideal-liquid’’ limits, then leads to a self-consistent method for calculating the density profiles for general fluid mixtures in external fields. This method is equivalent to an ansatz of ‘‘universality of the bridge functional.’’ The ‘‘bridge functional’’ consists of all the terms beyond the second order, in the expansion of the excess free energy functional around a reference uniform fluid. The self-consistency is imposed by applying the general method in the special case when the external potential is generated by a ‘‘test particle’’ at the origin of coordinates. In this limit, our general method for nonuniform fluids corresponds to an established and successful theory for the bulk uniform fluid pair structure, namely the thermodynamically consistent modified-hypernetted-chain theory, with the bridge functions now generated by an explicit and demonstratively accurate, ‘‘universal,’’ hard-sphere bridge functional. As a stringent test for the general model, the strongly coupled one-component plasma, in the bulk and near a hard wall, is considered in some detail.

Phase stability of dense multicomponent charged and uncharged hard-sphere fluid mixtures

We report calculations of the phase diagram and spinodal decomposition line of a charged hard-sphere fluid mixture constituted of two different cations, one anion and one neutral particle species. In the high-density regime the model is suitable to mimic molten silicate and metal halides mixtures, as those forming a magmatic fluid; in the low-density regime it can conveniently describe the properties of charged colloidal suspensions or micellar solutions. In this work the attention is focused mainly on the high-density regime of the mixture, and phase stability conditions are determined through the knowledge of the Gibbs free energy of mixing, Gmix at constant pressure, as calculated in the mean spherical approximation. It turns out that the solubility of the neutral component in the ionic mixture depends substantially on the potential-to-kinetic-energy ratio, that is, on the so-called coupling strength or plasma parameter, and increases when this decreases. This result is rationalized by considering the different terms contributing to Gmix and, specifically, the confi-gurational Coulombic internal energy term and entropic term. The fine balance between these two quantities is detailed and the onset of phase segregation discussed on these same bases. Such a representation also allows us to interpret why mixtures like those envisaged here are characterized by a phase diagram with an upper consolution point and why the critical temperature turns out to be an increasing function of the pressure. The implications of this last result seem interesting since, e.g., molten salt-silicate mixtures and, therefore, presumably fused rocks or magmas, could segregate not only under cooling, but also at constant temperature as the result of pressure variations.

A mean spherical approximation study of phase stability in charged colloidal solutions

The phase stability of charged hard sphere fluid mixtures modelling colloidal solutions is studied through the Mean Spherical Approximation (MSA). The isothermal compressibility and the Gibbs free energy of mixing are calculated, and from them, the liquid-vapor and the liquid-liquid decomposition spinodal line are determined, respectively. The MSA results favorably compare with those available from other theories as far as the liquid-vapor instabiity is concerned. For liquid-liquid decomposition the theory allows one to construct the full temperature concentration spinodal which shows an upper consolution point in temperature. Discussion and interpretation of this and related features of the phase stability line are reported

Charged hard spheres in a uniform neutralizing background: the role of thermodynamic self consistence

Calculations of the thermodynamic properties and pair distribution function of a one-component classical fluid of charged hard spheres in a uniform neutralizing background are reported and compared with Monte Carlo results of Hansen and Weis. Thermodynamic self consistence between the virial pressure and the fluctuations formula for the isothermal compressibility is enforced in the calculations by various alternative approaches. The role of thermodynamic self consistence is crucial to obtain a satisfactory quantitative description of this model fluid, in view of its applications in the theory of liquid metals and of dispersions of charged colloidal particles.

Thermodynamics and structure of liquid metals from the charged hard-sphere reference fluid

Perturbative variational calculations of thermodynamic and structural properties of liquid metals, based on the use of ab initio and highly reliable nonlocal pseudopotentials for the electron-ion interactions and of the fluid of charged hard spheres as a reference system, have been reported recently for the liquid alkali metals from Na to Cs near the freezing point. We extend in this work the above-mentioned calculations in two directions. Firstly, we discuss the predicted temperature dependence of the liquid structure factor for the same alkali metals over a limited temperature range above the freezing point. Secondly, we examine the usefulness of the approach for metals with relatively strong electron-ion interactions, namely Li and several polyvalent metals (Mg, Cd, Al, In, Tl and Pb). The charged hard-sphere reference system leads to lower values of the Helmholtz free energy and to slightly improved values of the excess entropy for all the liquid metals that we evaluate, even though polyvalent ones overall appear to be relatively close to fluids of neutral hard spheres. For the liquid alkali metals at elevated temperatures, the calculated structure factors are of similar quality as in our previous work, that is, they show a systematic shift in the positions of peaks and valleys to slightly larger wave numbers and peak heights that are somewhat underestimated with increasing temperature. However, for liquid polyvalent metals, our approach yields quite good agreement with experiment for the positions of maxima and minima in the liquid structure factor, while it tends to overemphasize somewhat these structures.

Thermodynamics and structure of liquid alkali metals from the charged-hard-sphere reference fluid.

The evaluation of thermodynamic properties of liquid alkali metals is reexamined in the approach based on the Gibbs-Bogoliubov inequality and using the fluid of charged hard spheres in the mean-spherical approximation as reference system, with a view to achieving consistency with the liquid structure factor. The perturbative variational calculation of the Helmholtz free energy is based on an ab initio and highly reliable nonlocal pseudopotential. Only limited improvement is found in the calculated thermodynamic functions, even when full advantage is taken of the two variational parameters inherent in this approach. The role of thermodynamic self-consistency between the equations of state of the reference fluid derived from the routes of the internal energy and of the virial theorem is then discussed, using previous results by Ho?ye and Stell [J. Chem. Phys. 67, 439 (1977)]. An approximate evaluation of the corresponding contribution to the free energy of liquid alkali metals yields appreciable improvement in both the thermodynamic functions and the liquid structure factor. It thus appears that an accurate treatment of thermodynamic self-consistency in the charged-hard-sphere system may help to resolve some of the difficulties that are commonly met in the evaluation of thermodynamic and structural properties of liquid metals.