Mixture Of Charged Hard Spheres Research Articles

Calculation of liquidus in eutectic alkali halide mixtures using thermodynamic perturbation theory

A statistical-thermodynamic model that makes it possible to describe with good accuracy phase equilibria between melt and crystal in binary alkali halide mixtures with common cations and anions is proposed. This model is based on the thermodynamic perturbation theory, which takes into account the charge-dipole contribution to the interionic interaction of binary molten salts using a multicomponent mixture of charged hard spheres as a reference system. Specific expressions for chemical potentials, taking into account the charge-induced dipole interaction in the molten phase, are presented within the developed approach. Considering binary eutectic mixtures NaF–NaCl, CsF–CsCl, NaF–CsF, and NaCl–CsCl as an example, the liquidus curves obtained in two series of calculations are shown: with the equation of state and without it. The discrepancies between the calculated and experimental data on the position of eutectic equilibrium for most of the considered mixtures do not exceed about 10 percent on the temperature scale and 5 percent on the composition when using only the tabulated values of ionic radii and polarizabilities without any adjustment.

On the stability of (MX6)3− complexes in halide melts of three-valent metals

ABSTRACTThe effects of volume change at dissociation equilibrium of possible anion complexes of the (MX6)3− type in halide melts of trivalent metals MX3 within the statistical thermodynamics based on the MSA approximation were analysed. With the help of a simplified Akdeniz–Tosi model of a mixture of charged hard spheres of different diameters and valencies, we obtained the full system of equilibrium equations including the mass action law (MAL) and the equation of state (EoS). It was shown that the simplest approximation of the complex diameter as a tripled diameter of simple ions leads to a significant overestimation of the effects of volume changes at dissociation. It was found that the complex dissociation should be accompanied by a significant increase in density in a narrow temperature interval. It can be associated with the specific manifestation of electrolytic dissociation in the case of the trivalent metal halides’ auto-complexation.

Electrochemical Properties of the Double Layer of an Ionic Liquid Using a Dimer Model Electrolyte and Density Functional Theory

Most theoretical studies of the properties of an electrolyte, including an ionic liquid, treat the electrolyte as a mixture of charged hard spheres in a solvent modeled as a dielectric continuum. However, ionic liquids generally consist of nonspherical ions that are not dissolved in a solvent. A simple extension of the primitivel model of electrolytes is to represent an ionic liquid as a mixture of charged hard spheres (negative monovalent ions in our case) and nonspherical ions consisting of a dimer of two touching hard spheres, one of which is charged (monovalent and positive in our case) and the other is neutral. This simple model has been used previously by Federov et al. and by ourselves. Here, we use the classical density functional theory to study the interfacial properties of the model ionic liquid over a range of electrode charges and two electrolyte concentrations. For simplicity, all of the spheres have the same diameter. In contrast to the simulations of Federov and Kornyshev, we find that a plot of the differential capacitance of the dimer electrolyte versus the surface potential typically exhibits only a single hump. Also, differing from the studies of Lamperski et al. for a spherical electrolyte, which showed a decline of the maximal differential capacitance as the ionic concentration decreases, the maximum of the differential capacitance of the dimer electrolyte increases slightly with decreasing ionic concentration. Our theoretical results show other unexpected effects of the geometry of ionic species on the electrochemical properties of the electric double layer of an ionic liquid.

Ionic and Wigner Glasses, Superionic Conductors, and Spinodal Electrostatic Gels: Dynamically Arrested Phases of the Primitive Model

Based on the recently proposed self-consistent generalized Langevin equation theory of dynamic arrest, in this letter we show that the ergodic-nonergodic phase diagram of a classical mixture of charged hard spheres (the so-called "primitive model" of ionic solutions and molten salts) includes arrested phases corresponding to nonconducting ionic glasses, partially arrested states that represent solid electrolytes (or "superionic" conductors), low-density colloidal Wigner glasses, and low-density electrostatic gels associated with arrested spinodal decomposition.

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.

Self-consistent effective interactions in charged colloidal suspensions

We use an integral equation scheme to obtain self-consistently the effective interaction between colloids in salt-free charged colloidal suspensions. The colloid–counterion direct correlation function (DCF) is obtained for the fixed colloid–colloid pair structure by solving the corresponding hypernetted-chain equation (HNC). This DCF is then used to formulate an effective colloid–colloid pair potential for which the one-component reference hypernetted-chain equation is solved. Both processes are iterated until self-consistency is achieved. Counterion–counterion correlations are considered linear and uncoupled from the rest of the correlations. The method is based on a similar treatment utilized in liquid metals [Phys. Rev. B 61, 11400 (2000)] and provides equivalent results to those obtained using the standard multicomponent HNC equation for mixtures of charged hard spheres. The theory proves rather accurate when compared with molecular dynamic simulations of charged hard and soft spheres for colloidal charges of up to 300. We study in detail the existence of net attractions between colloids in certain cases (especially in the presence of divalent and trivalent counterions) and how this attraction may lead to phase instability. The problem of the lack of solution of the integral equation for more realistic cases (larger charges) is also discussed.

Peristaltic transport in a slip flow

The Enskog-Landau kinetic equation is considered to describe non-equilibrium processes of a mixture of charged hard spheres. This equation has been obtained in our previous papers by means of the non-equilibrium statistical operator method. The normal solution of this kinetic equation found in the first approximation using the standard Chapman-Enskog method is given. On the basis of the found solution the flows and transport coefficients have been calculated. All transport coefficients for multicomponent mixture of spherical Coulomb particles are presented analytically for the first time. Numerical calculations of thermal conductivity and thermal diffusion coefficient are performed for some specific mixtures of noble gases of high density. We compare the calculations with those ones for point-like neutral and charged particles.

STATIC CORRELATIONS IN THE ORDERED COLLOIDAL SYSTEMS

The macroion-macroion correlation function and structure factor are studied within the Ornstein-Zernike (OZ) integral equation closed by hypernetted-chain approximation using the method of critical modes. Developing the results which have recently been received in \cite{1,2,3} for a case of a model system consisting of a mixture of charged hard spheres, the role of detalization in describing the compound characters is investigated. In particular, it is shown that adding the uncharged particles to the model of the solution of macroparticles and counterions significantly affects the structural parameter of ordering extracted within the critical modes method providing close theoretical and relating experimental 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.

Solution of the polymer mean spherical approximation for the totally flexible sticky two-point electrolyte model

The general solution of the polymer MSA (mean spherical approximation) for an arbitrary mixture of charged hard spheres with multiple bonding obtained recently by Blum, Holovko and Protsykevytch [J. Stat. Phys,] is applied to the case of totally flexible linear polymer chains. The approximation is an extension of the associative MSA proposed by M.Holovko and Yu.Kalyuzhnyi, [Mol.Phys., 73, 1145(1991)] to study the effects of association in ionic systems. The model is defined by adding charges to the multicomponent version of the totally-flexible sticky two-point model of associating monomers introduced by Wertheim.

A solution of the multiple-binding mean spherical approximation for ionic mixtures

The mean spherical approximation (MSA) for an arbitrary mixture of charged hard spheres with saturating bonds is solved in the Wertheim formalism. Any number of bonds is allowed. It is shown that the general solution is given in terms of a screening MSA-like parameterΓT, a cross-interaction parameterηβ that will depend on the binding association equations, the set of binding association fractions, and an additional algebraic equation. The equation forΓT is given for the general case. The equation forηβ, however, depends strongly on the particular closure that is used to compute the contact pair correlation function. The full solution requires, as in the dimer case recently solved by Blum and Bernard, solvingm+2 equations and additionally the inversion of a matrix of size [(ν−1)m] for a system withm components and ν bonds. We recall that when ν=1, only dimers are allowed; for ν=2, only linear chains are formed: and when ν≥3, branching of the polymers occurs. It can be shown that the excess entropy for the polymer case is as before,ΔSMSA=(ΓT)3/3π + sticky terms, where the sticky terms depend on the model and will be given in future work.

Binding mean spherical approximation for pairing ions: An exponential approximation and thermodynamics

The thermodynamics of mean spherical approximation for an arbitrary mixture of charged hard spheres with sticky-point attraction in the Wertheim formalism is derived for the most general case. It is shown that the electrostatic excess thermodynamics can be still computed from relatively simple formulas which depend on the screening parameter ΓB. We introduce also a new exponential approximation, which unlike the simpler mean spherical-like approximation discussed in our previous work, has the correct low density behavior. The restricted case of a binary mixture with a single association constant is discussed explicitly.

Phase instabilities in charged hard-sphere mixtures. II. Binary mixtures of salts

Charged hard-sphere mixtures consisting of two positively charged species and one negatively charged species (common anion mixtures) are used to represent binary mixtures of salts. Phase separation in the mixture is studied using the Gibbs free energy expression for common anion mixtures derived in paper I of this series. A detailed description of the dependence of the resulting phase diagrams on molecular size and charge of the species, and on the osmotic pressure of the mixture is presented. Binary mixtures of salts containing equal-sized ions exhibit type III phase behavior whereas binary mixtures of salts containing ions of unequal size exhibit either type II or type IV phase behavior. The type of phase behavior observed in binary mixtures of salts is characterized as a function of the critical pressures and critical volumes of the pure salts. Our results suggest that phase separation in mixtures of charged hard spheres is influenced by a competition between mixing effects (entropy), which encourage miscibility and ion-pairing effects (enthalpy), which encourage phase separation. Potential applications of the model to experimental systems are discussed.

The general solution of the binding mean spherical approximation for pairing ions

The mean spherical approximation (MSA) for an arbitrary mixture of charged hard spheres with saturating bonds is solved in the Wertheim formalism. Only pairs are allowed. It is shown that the general solution is given in terms of two scaling parameters and the set of binding fractions. One of the scales is the screening parameter, and the other is a coupling parameter analogous to that of the simple MSA, but that now is found solving a cubic equation. Therefore the full solution requires solvingm+2 nonlinear algebraic equations for a system withm components. A brief discussion of the thermodynamics is given.

Structural relaxations in a Simple Model Molten SaltM

ABSTRACTThe structural relaxations of a dense, binary mixture of charged hard spheres are studied using the Mode Coupling Theory (MCT). Qualitative differences to non-ionic systems are shown to result from the long-range Coulomb interaction and charge ordering in dense molten salts. The presented non-equilibrium results are determined by the equilibrium structure, which is input using the well studied Mean Spherical Approximation.

Mean spherical approximation for an arbitrary mixture of ions with surface adhesion in a neutralizing background

The mean spherical approximation for a multicomponent mixture of charged hard spheres with sticky interactions in a uniform neutralizing background is solved analytically. When the sticky parameters are of the form Λ ij ∼ w i w j ± v i v j , then the correlation functions and the excess thermodynamics properties are explicitly expressed through a single parameter Γ, which can be obtained by solving an algebraic equation. In the one-component case, an explicit expression for the parameter Γ is obtained.

Theory of precipitation of protein mixtures by nonionic polymer

AbstractA theory is developed to predict the solubility of protein mixtures in solutions containing nonionic polymer. Effective protein‐protein interactions due to polymer are taken to be volume‐exclusion potentials derived using statistical mechanics. Statistical‐mechanical perturbation theory is used to calculate chemical potentials. The effects of protein size, mole fraction and polymer concentration on solubility are explored. The theory is extended to include electrostatic interactions. The excess chemical potential of the proteins due to the charges on all species is calculated using the mean spherical approximation for a mixture of charged hard spheres. The theory predicts: the larger protein is preferentially precipitated over the smaller one; the more concentrated protein is more likely to precipitate; and increasing the charge of a particular protein reduces its ability to precipitate.

HNCR — a program to calculate the structure and thermodynamics of binary mixtures of charged hard spheres

HNCR is a program to determine the structure and thermodynamics of the primitive model of electrolytes, i.e. binary mixtures of charged hard spheres with arbitrary sizes and charges. This is achieved by solving the hypernetted chain equation by means of a hybrid Newton-Raphson procedure. The algorithm used is both fast and stable, and it is particularly useful for calculations involving a wide range of temperatures or concentrations for a given system.

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

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