Hard sphere ionic solutions

Another model which combined a model for the solvent with a jellium-type model for the metal electrons was given by Badiali et a/.83 The metal electrons were supposed to be in the potential of a jellium background, plus a repulsive pseudopotential averaged over the jellium profile. The solvent was modeled as a collection of equal-sized hard spheres, charged and dipolar. In this model, the distance of closest approach of ions and molecules to the metal surface at z = 0 is fixed in terms of the molecular and ionic radii. The effect of the metal on the solution is thus that of an infinitely smooth, infinitely high barrier, as well as charged surface. The solution species are also under the influence of the electronic tail of the metal, represented by an exponential profile. 

Until very recently there was no information about how an MM pair potential should look, based upon calculations from the deeper BO level. In the simplest BO level model for an ionic solution the solvent molecules are represented as hard spheres with centered point dipoles and the ions as hard spheres with centered charges. Now there are two sets of calculations, (16,17) by very different approximation methods, for this model where all of the spheres are 3A in diameter, where the dipole moments are near 1 Debye, and where the ions are singly charged. The temperature is 25° and the solvent concentration is about 50M, corresponding to a liquid state. The dielectric constant of the model solvent is believed to be near 9 6. 

One of the further refinements which seems desirable is to modify Eq. (9) so that it has wiggles (damped oscillatory behavior). Wiggles are expected in any realistic MM-level pair potential as a consequence of the molecular structure of the solvent (2,3,10,11,21,22) they would be found even for two hard sphere solute particles in a hard-sphere liquid or for two H2I80 solute molecules in ordinary liquid HpO, and are found in simulation studies of solutions based on BO-level models. In ionic solutions in a polar solvent another source of wiggles, evidenced in Fig. 2, may be associated with an oscillatory nonlocal dielectric function e(r). ( 36) These various studies may be used to guide the introduction of wiggles into Eq. (9) in a realistic way. 

The interfacial solution layer contains h3 ated ions and dipoles of water molecules. According to the hard sphere model or the mean sphere approximation of aqueous solution, the plane of the center of mass of the excess ionic charge, o,(x), is given at the distance x. from the jellium metal edge in Eqn. 5-31 ... 

More realistic approaches should, of course, comprise solvent models that give rise to electrostatic interactions. Shelley and Patey [273] used grand canonical MC simulations to investigate the demixing transition in model ionic solutions where the solvent is explicitly included. Charged hard-sphere ions in neutral, dipolar, and quadrupolar hard-sphere solvents were consi-... 

The perturbation theories [2, 3] go a step beyond corresponding states the properties (e.g., Ac) of some substance with potential U are related to those for a simpler reference substance with potential Uq by a perturbation expansion (Ac = Aq + A + Aj + ). The properties of the simple reference fluid can be obtained from experimental data (or from simulation data for model fluids such as hard spheres) or corresponding states correlations, while the perturbation corrections are calculated from the statistical mechanical expressions, which involve only reference fluid properties and the perturbing potential. Cluster expansions involve a series in molecular clusters and are closely related to the perturbation theories they have proved particularly useful for moderately dense gases, dilute solutions, hydrogen-bonded liquids, and ionic solutions. 

The solution for hs(r) formally coincides with the Wertheim solution [25] for dimerizing hard spheres with a calculated from Eq. (5). The solution for hci(r) was also obtained [6, 7] for the RPM and for the asymmetrical ionic fluids [8] using the Wertheim-Baxter factorization technique. 

In conclusion, the MSA provides an excellent description of the properties of electrolyte solutions up to quite high concentrations. In dilute solutions, the most important feature of these systems is the influence of ion-ion interactions, which account for almost all of the departure from ideality. In this concentration region, the MSA theory does not differ significantly from the Debye-Hiickel model. As the ionic strength increases beyond 0.1 M, the finite size of all of the ions must be considered. This is done in the MSA on the basis of the hard-sphere contribution. Further improvement in the model comes from considering the presence of ion pairing and by using the actual dielectric permittivity of the solution rather than that of the pure solvent. 

One of the simplest realistic molecular models of an ionic solution is the restricted primitive model. This consists of an equimolar mixture of oppositely charged but equal-sized hard spheres in a dielectric continuum. 

The LHNC and QHNC approximations have been solved for three-component mixtures consisting of charged hard spheres in dipolar hard-sphere solvents. Only 1 1 electrolytes with ions of equal diameter have been considered. Thus denoting the positive and negative ionic species with the obvious subscripts, 4- and —, one has qj = —q =q, d =d, and the system is characterized by the reduced variables p%=p+d, P p =fip /d, and q = pq /d +. In addition, the solvent-ion diameter ratio, d /dmust be specified. For some values of these parameters, convergence problems are encountered in the QHNC theory, and these difficulties are discussed in Ref. 143. However, for many ionic solutions numerical results can be obtained. 

Y. Hyon, B. S. Eisenberg, and C. Liu. A mathematical model for the hard sphere repulsion in ionic solution. Commun. Math. Sci., 9 459-475,2011. 

Ions can exist in solution as well as in a crystalline, or solid form. Ideal ionic crystals are composed of atoms which may be represented as hard spheres, of varying size and opposing charge. Ionic crystals are often brittle because electrons are taken up by anions, and they often have high melting points. 

We have evaluated the simplest of these second-order results for 1-1 ionic solutions and a Lennard-Jones liquid [where < (1 2) is treated as a hard-sphere reference term 4>o(l 2) with state-dependent R plus an attractive Lennard-Jones w(l 2) in the manner prescribed by Andersen er As shown in the Figs. 1-5 and Tables 1 and 2, the results are encouraging. 

We conclude this chapter with a brief description of some tests that may be applied to judge the accuracy or self-consistency of a particular approximation method. Ideally, one could compare the approximation method results with the exact results for the measurable properties of the same model. Since such exact solutions are seldom available except for very simple models (e.g., hard spheres in one dimension), one must use Monte Carlo or molecular dynamics calculations as the standard. These too represent approximations, however, and are more expensive and less reliable for ionic solution models than for nonionic systems. 

It is possible to take into account the short range ion-ion interaction effect on the volumetric properties of electrolytes by resorting to integral equation theories, as the mean spherical approximation (MSA). The MSA model renders an analytical solution (Blum, 1975) for the umestricted primitive model of electrolytes (ions of different sizes immersed in a continuous solvent). Thus, the excess volume can be described in terms of an electrostatic contribution given by the MSA expression (Corti, 1997) and a hard sphere contribution obtained form the excess pressure of a hard sphere mixture (Mansoori et al, 1971). The only parameters of the model are the ionic diameters and numerical densities. 

This investigation has clearly shown that the dissociation-recombination processes of ionic species in low polar media can be treated successfully as a problem of diffusional motion of hard spheres in a continuous medium. The structural properties of the ionic aggregate and the macroscopic properties of the solvent-medium are sufficient to evaluate quantitatively the dynamic properties of ionic phenomena. Eventual discrepancies between experimental data and theoretical values are therefore indicative for specific solute-solvent interactions. 

Calculations of departures from ideality in ionic solutions using the MSA have been published in the past by a number of authors. Effective ionic radii have been determined for the calculation of osmotic coefficients for concentrated salts [13], in solutions up to 1 mol/L [14] and for the computation of activity coefficients in ionic mixtures [15]. In these studies, for a given salt, a unique hard sphere diameter was determined for the whole concentration range. Also, thermodynamic data were fitted with the use of one linearly density-dependent parameter (a hard core size o C)., or dielectric parameter e C)), up to 2 mol/L, by least-squares refinement [16]-[18], or quite recently with a non-linearly varying cation size [19] in very concentrated electrolytes. 

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