Nonadditive Diameters Research Articles

The restricted primitive model of ionic fluids with nonadditive diameters

The restricted primitive model with nonadditive hard-sphere diameters is shown to have interesting and peculiar clustering properties. We report accurate calculations of the cluster concentrations. Implementing efficient and ad hoc Monte Carlo algorithms we determine the effect of nonadditivity on both the clustering and the gas-liquid binodal. For negative nonadditivity, tending to the extreme case of completely overlapping unlike ions, the prevailing clusters are made of an even number of particles having zero total charge. For positive nonadditivity, the frustrated tendency to segregation of like particles and the reduced space available to the ions favors percolating clusters at high densities.

On the critical non-additivity driving segregation of asymmetric binary hard sphere fluids

A previously proposed version of thermodynamic perturbation theory, appropriate for singular pair interactions between particles, is applied to binary mixtures of hard spheres with non-additive diameters. The critical non-additivity ΔC required to drive fluid–fluid phase separation is determined as a function of the ratio ξ ≤ 1 of the diameters of the two species. ΔC(ξ) is found to decrease with ξ and to go through a minimum for ξ ≃ 0.015 before increasing sharply as ξ → 0, irrespective of the total packing-fraction η of the mixture. These results are the basis of an estimate of the range of size ratios for which a binary mixture of additive hard spheres exhibit a fluid–fluid miscibility gap. This range is conjectured to be 0.01 ≲ ξ ≲ 0.1.

Fundamental measure density functional theory for nonadditive hard-core mixtures: The one-dimensional case

We treat a one-dimensional binary mixture of hard-core particles that possess nonadditive diameters. For this model, a density functional theory is constructed following similar principles as an earlier extension of Rosenfeld's fundamental measure theory to three-dimensional nonadditive hard-sphere mixtures. The theory applies to arbitrary positive and moderate negative nonadditivity and reduces to Percus' exact functional in the additive case. Bulk direct correlation functions are obtained as functional derivatives of the excess free energy functional. Results for the partial pair correlation functions in bulk, as calculated via the Ornstein-Zernike route and using the direct correlation functions as input, show very good agreement with results from our Monte Carlo computer simulations of the mixture.

Mixtures of charged colloid and neutral polymer: Influence of electrostatic interactions on demixing and interfacial tension

The equilibrium phase behavior of a binary mixture of charged colloids and neutral, nonadsorbing polymers is studied within free-volume theory. A model mixture of charged hard-sphere macroions and ideal, coarse-grained, effective-sphere polymers is mapped first onto a binary hard-sphere mixture with nonadditive diameters and then onto an effective Asakura-Oosawa model [S. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255 (1954)]. The effective model is defined by a single dimensionless parameter-the ratio of the polymer diameter to the effective colloid diameter. For high salt-to-counterion concentration ratios, a free-volume approximation for the free energy is used to compute the fluid phase diagram, which describes demixing into colloid-rich (liquid) and colloid-poor (vapor) phases. Increasing the range of electrostatic interactions shifts the demixing binodal toward higher polymer concentration, stabilizing the mixture. The enhanced stability is attributed to a weakening of polymer depletion-induced attraction between electrostatically repelling macroions. Comparison with predictions of density-functional theory reveals a corresponding increase in the liquid-vapor interfacial tension. The predicted trends in phase stability are consistent with observed behavior of protein-polysaccharide mixtures in food colloids.

Electrostatic energies in an ionic micellar solution in the mean spherical approximation

We present a method for calculating electrostatic energies of an ionic micellar solution under the conditions of quasi-spherical micelles and point-like counterions. It is based on a multi-component mean spherical approximation (MMSA) theory with non-additive diameters. The partial structure factors can be evaluated analytically according to a method due to Khan and Ronis, and used to calculate the electrostatic energies. As an example, a micellar solution formed by Cesium Dodecyl Sulfate in water has been studied with small angle neutron scattering. The micellar size and effective charge are extracted from analysis of small angle neutron scattering data, and used to evaluate the internal energy and Gibbs free energy of the micellar solution.

Ion correlations and counter-ion condensation in ionic micellar solutions

We present a general method for obtaining various ion correlations in ionic micellar solutions. This is based on combined small-angle x-ray and neutron scattering intensity measurements and their analyses using Khan - Ronis theory. The K - R theory of ionic solutions is a multicomponent mean-spherical approximation with non-additive diameters within the primitive model. We first extract an accurate macro-ion - macro-ion structure factor from analyses of SANS and SAXS data, including an accurate estimate of the renormalized micellar charge, , and then use the K - R theory to compute all the other partial structure factors. We demonstrate the validity and practicality of the method using micellar solutions made of caesium alkyl sulphate of various chain lengths. The charge renormalization phenomenon is then shown to be the consequence of condensation of counter-ions into the outer hydrophilic layers of micelles. The renormalized charge of a micelle is obtained for different types of monovalent counter-ion. In particular, we are able to determine the macro-ion - counter-ion pair correlation function which gives the distribution of counter-ions near the surface of a micelle.

Structure, melting, and order tendencies ofA13B13Lennard-Jones heteroclusters with additive and nonadditive collision diameters

Using constant-energy molecular-dynamics simulations, we studied the lowest-energy structures, melting behavior, and ordering tendencies of ${\mathit{A}}_{13}$${\mathit{B}}_{13}$ Lennard-Jones (LJ) clusters with additive and nonadditive collision diameters. We found that nonadditivity can significantly affect the properties of the LJ clusters. In particular, small negative nonadditivities give rise to nonicosahedral configurations.

Molecular dynamics and mean spherical approximation results for symmetric nonadditive hard core Yukawa mixtures

The thermodynamic properties of some symmetric hard core Yukawa mixtures with positively and negatively nonadditive diameters are studied by molecular dynamics (MD) simulation, and the results compared with the predictions of the mean spherical approximation (MSA). MSA results for the compressibility factors are obtained by taking as reference the semiempirical equation of state proposed by Gazzillo and Pastore for symmetric nonadditive hard sphere fluids. Our calculations show that the ‘‘exact’’ computer simulation data are, in general, closely approximated by the MSA energy route to the thermodynamic properties. For the largest positive nonadditivity considered in this paper, some discrepancies appear between the compressibility factors obtained by MD and the MSA, but it is shown that the origin of the discrepancies lies in the hard core part of the compressibility factor rather than in the MSA theory of liquids.

Inability of the hypernetted chain integral equation to exhibit a spinodal line

The hypernetted chain (HNC) integral equation is solved for a variety of systems including the simple fluid of hard spheres with attractive Yukawa tail, the symmetrical and highly asymmetrical mixtures of charged particles, and the mixtures of hard spheres with positively nonadditive diameters. All these systems exhibit a liquid–gas or fluid–fluid phase separation. We are concerned with the behavior of the HNC solution near the two-phase region in the phase diagram. In all cases, the HNC equation does not have solutions inside a certain region whose boundary line is not a spinodal line. As the boundary is approached, the isothermal compressibility deduced from the compressibility route does not diverge, but tends to finite values. The isotherms and isochores terminate at the boundary line in square root branch points. This behavior is directly correlated to the existence of multiple HNC solutions. Physical and nonphysical solutions coincide on the boundary line. This universal HNC behavior is compared with that of the mean spherical approximation and Percus–Yevick integral equations.

Fluid–fluid phase separation of nonadditive hard-sphere mixtures as predicted by integral-equation theories

We investigate the existence of a fluid–fluid phase separation in binary mixtures of equal-size hard spheres with positively nonadditive diameters [i.e., d11=d22≡d, d12=(1+Δ)d with Δ>0]. An integral-equation approach is used to evaluate both thermodynamics and structure of many symmetric (equal to equimolar) mixtures (with Δ=0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 1) and some asymmetric cases. We present the results obtained via the Percus–Yevick, the Martynov–Sarkisov, and the Ballone–Pastore–Galli–Gazzillo closures; the thermodynamic consistency of these approximations is discussed and some possible ways to get further improvements are proposed too. The integral-equation results are then compared with the available ‘‘exact’’ simulation data, a first-order perturbation approach, and a scaled particle theory. Our study predicts that there exists a demixing for each considered value of the nonadditivity parameter Δ.

The nonadditive hard-sphere mixture as a reference system in equation-of-state calculations

A new reference system is presented for equation-of-state calculations for highly asymmetric mixtures: a mixture of hard spheres with nonadditive diameters. This system allows the introduction of an adjustable binary parameter in the reference term of the EOS while preserving the Boublik-Mansoori mixed-hard-sphere form which is advantageous for mixtures whose components differ greatly in size. The new reference term is tested against a one-fluid reference term of the Carnahan-Starling form; in each case the EOS uses one adjustable parameter in the reference term and one in the perturbation term. For binary systems, the new method offers little improvement over the one-fluid theory. However, when binary parameters are used to predict multicomponent vapor-liquid equilibria, the new method may have some advantages.

Equation of state for symmetric non-additive hard-sphere fluids: An approximate analytic expression and new Monte Carlo results

We present a simple extension of the Carnahan-Starling equation of state to symmetric mixtures of hard spheres with positively or negatively non-additive diameters. New Monte Carlo results are generated and show an excellent quality of our equation of state over all the fluid densities and non-additivities of physical interest.

Direct correlation functions for negatively non-additive hard spheres in the PY approximation

We investigate, within the Percus-Yevick (PY) approximation, the functional form of the direct correlation functions cij (r) for symmetric binary mixtures of hard spheres with negatively non-additive diameters Rij (i.e. two-component systems with R 11=R 22=≡R and R 12<R, at equimolar concentration). A partial exact solution is obtained when−R/2ˇ-α≡R 12-R<0 and, by introducing appropriate, simple, polynomial approximations to two functions X 12(r) and Y 11(r) contributing, respectively, to c 12(r) and c 11(r), the problem of solving the Ornstein-Zernike (OZ) integral equation is reduced to an algebraic one, i.e. to solving a system of a few (usually non-linear) algebraic equations in some unknown parameters. A very good agreement is found between our approximate solutions and the exact PY ones. Finally, a numerical Carnahan-Starling-type equation of state is presented and a successful comparison with Monte Carlo simulation data of Adams and McDonald is made.

Symmetric mixtures of hard spheres with positively nonadditive diameters: An approximate analytic solution of the Percus–Yevick integral equation

We study the properties of symmetric binary mixtures of hard spheres with positive nonadditive diameters Rij, namely two-component systems with R11=R22=R and R12&amp;gt;R, at equimolar concentration. The functional form of the direct correlation functions cij(r) is investigated, within the Percus–Yevick approximation, by using Hiroike and Fukui’s version of the Ornstein–Zernike integral equation for multicomponent fluids. It is shown that, by introducing simple polynomial expressions for the cross term C12(r)=rc12(r), the problem of finding an approximate analytic solution of the abovementioned integral equations can be reduced to an algebraic one, i.e., to solving a closed set of a few nonlinear algebraic equations for some unknown parameters. Results corresponding to three different approximations are presented for the radial distribution functions at contact, the virial pressure and the so called bulk modulus. Comparison is made with our exact numerical solutions of the Percus–Yevick integral equation and a very good agreement is found. Finally, calculations based on a simple first-order perturbation method, which gives a slight extension of analytic expressions derived from the Barker–Henderson perturbation theory, are reported and discussed.

Ionic pairing in binary liquids of charged hard spheres with nonadditive diameters

We examine types of short range order that arise in binary liquids from a combination of Coulombic interactions and nonadditivity of excluded volumes, the initial motivation being observations of complex formation by hydrated ions in concentrated aqueous solutions. The model is a fluid of charged hard spheres with contact distances σ+−≠ 1/2 (σ+++σ−−), its structural functions being evaluated in the mean spherical approximation and in the hypernetted chain approximation. Cation–anion pairing is clearly seen in the calculated structural functions for negative deviations from additivity (σ+−&amp;lt;σ++=σ−−), though the absence of true chemical bonding in the model does not allow long-lived complexes. Positive deviations from additivity (σ+−&amp;gt;σ++=σ−−) favor long-wavelength concentration fluctuations and demixing in a neutral mixture: these are suppressed by Coulombic interactions in favor of microscopic intermixing of the two species in the local liquid structure, up to like-ion pairing. Comparison is made with diffraction from concentrated aqueous solutions of cadmium sulphate and other instances of possible applicability of the model are pointed out.

Percus-Yevick results for a binary mixture of hard spheres with non-additive diameters

We have carried out extensive calculations, within the Percus-Yevick approximation, for a binary mixture of hard spheres with negative departures from additivity in their hard sphere diameters. The results presented in this work are restricted to the case of equal size spheres at equimolar composition. We have calculated the virial equation of state and show that the more negative the non-additive parameter, the better the agreement between our results and the Monte Carlo results of Adams and McDonald. Moreover, we find no evidence of a phase transition for this system. Our calculations of the radial distribution functions gij(r) show clearly the important changes brought about by non-additivity, in particular, to g 12(r). We have also calculated the number-concentration partial structure factors, which highlights the ordering effects taking place in the system as a results of non-additivity. Our results indicate that, if size difference between the hard spheres is included, non-additive hard spheres with...

Nonadditive hard discs, a model for partially localized adsorption

It is shown how one can approximate partially localized adsorbed layers of molecules on surfaces by a mixture of hard discs with negatively nonadditive diameters. The thermodynamic properties for this reference state are found with a particular version of scaled particle theory. The contribution of the attractive forces is obtained from a perturbation expansion where we keep the two leading terms. The radial-distribution function of the reference state for this contribution is obtained from a virial expansion. The calculations are carried out for a square-well potential. We calculate numerically the following results: The degree of localization, i.e., the concentration of particles in the localized state, the equation of state, i.e., spreading pressure as a function of coverage, the isotherms, and the isosteric heat of adsorption. These quantities are calculated for different values of the adsorbate-adsorbent lattice mismatch, the height of the barrier to lateral diffusion on the surface, the strength of the attractive forces, and the temperature and pressure. The attractive forces favor out-of-registry adsorption against localized adsorption. The localization effects lead to a noticeable decrease in the critical temperature for two-dimensional condensation even in highly mobile adsorbed layers.

Transformation of the Ornstein-Zernike Relation for Fluid Mixtures

where r= lrl, r' = lr'l, and Pr is the number density of particles of species r. Equation (1) is called the Ornstein-Zernike (OZ) relation for the fluid mixture. Baxter presented two transformations of the OZ relation for a single-component fluid_n·' 1 Both of these are suitable for use with approximate theories where the direct correlation functions can be assumed to be of finite range. The first transformation vvas generali~ed to the case of fluid mixtures by Hiroike and Fukui. 31 The generalized transformation was applied to a mixture of hard spheres by Hiroike,'1 and recently to a binary mixture of hard spheres with non-additive diameters by Perry and Silbert. 51 The second transformation was generalized to the case of fluid mixtures by Baxter himsel£. 61 The purpose of the present paper is to derive Baxter's generalized trsnsformation without the assumption that the direct correlation functions are of finite range. The derivation itself is much simpler than Baxter's, in which usc was made of the Wiener-Hopf technique.

Approximate solution of the Percus-Yevick equation for a binary mixture of hard spheres with non-additive diameters

We present an approximate solution of the Percus-Yevick integral equation for a binary mixture of hard spheres with non-additive diameters. Defining Rij the distance of closest approach between particles of species i and j by R 12 = ½(R 11 + R 22) + α, we obtain a closed set of equations for the direct correlation functions cij (r) when 0 < α ⩽ min [½(R 22 - R 11), ½R 11]. Our expressions for cii (r), and for c 12(r) in the range 0 < r ⩽ ½[R 22 - R 11] - α, agree with those previously obtained by Lebowitz and Zomick.

Scaled particle theory for nonadditive hard spheres: Solutions for general positive nonadditivity

A complete scheme for a scaled particle theory for hard spheres with nonadditive diameters is developed. As macroscopic representation of the nonadditive hard spheres a cavity surrounded by a family of concentric selectively permeable membranes is used. The resulting equations are solved explicitly for a binary mixture with positive nonadditivity. Independent of the scheme used, scaled particle theory predicts a demixing phase transition at high density and sufficient nonadditivity. The critical value for the nonadditivity parameter is found to be ${\ensuremath{\Delta}}^{c}=0.215$. Agreement with the few Monte Carlo results available is good.