Solvent primitive model

The simplest way to treat the solvent molecules of an electrolyte explicitly is to represent them as hard spheres, whereas the electrostatic contribution of the solvent is expressed implicitly by a uniform dielectric medium in which charged hard-sphere ions interact. A schematic representation is shown in Figure 2(a) for the case of an idealized situation in which the cations, anions, and solvent have the same diameters. This is the solvent primitive model (SPM), first named by Davis and coworkers [15,16] but appearing earlier in other studies [17]. As shown in Figure 2(b), the interaction potential of a pair of particles (ions or solvent molecule), i and j, in the SPM are ... 

FIG. 2 (a) Solvent primitive model, with charged hard spheres representing the ions, neutral hard... 

Kristof, T., Boda, D., Szalai, I., and Henderson, D. A Gibbs ensemble Monte Carlo study of phase coexistence in the solvent primitive model. J. Chem. Phys., 2000,113, p. 7488-91. 

Figure 10. Osmotic coefficient as a function of the reduced density of monomer units pp = pm.03, where prn is the number density of monomer units. Solvent primitive model (continuous lines) McMillan-Mayer model results (broken lines). From top to bottom a = 0.125,0.5, and 1.0 (each bead is charged).

Various attempts have been made in going beyond the primitive models towards civilized models. In the most simple of such extensions, called the solvent-primitive model [152], the solvent is still treated as a dielectric continuum. In addition, individual solvent molecules are treated as hard spheres. This takes into account the molecular natme of the solvent in hut the crudest fashion. Replacing the description of the solvent from dielectric continuum by molecular point dipoles at the center of soft spheres representing solvent molecules (the Stockmayer model) leads to the ion-dipole models. Various extensions of these models incorporate higher-order electrostatic moments (up to octupole moments [153]) and molecular polarizability for the description of the solvent bound charge density. 

Lamperski, S., and A. Zydor. 2007. Monte Carlo study of the electrode solvent primitive model electrolyte interface. Electrochimica Acta 52, no. 7 (February 1) 2429-2436. doi 10.1016. electacta.2006.08.045. 

In principle, simulation teclmiques can be used, and Monte Carlo simulations of the primitive model of electrolyte solutions have appeared since the 1960s. Results for the osmotic coefficients are given for comparison in table A2.4.4 together with results from the MSA, PY and HNC approaches. The primitive model is clearly deficient for values of r. close to the closest distance of approach of the ions. Many years ago, Gurney [H] noted that when two ions are close enough together for their solvation sheaths to overlap, some solvent molecules become freed from ionic attraction and are effectively returned to the bulk [12]. 

The popular and well-studied primitive model is a degenerate case of the SPM with = 0, shown schematically in Figure (c). The restricted primitive model (RPM) refers to the case when the ions are of equal diameter. This model can realistically represent the packing of a molten salt in which no solvent is present. For an aqueous electrolyte, the primitive model does not treat the solvent molecules exphcitly and the number density of the electrolyte is umealistically low. For modeling nano-surface interactions, short-range interactions are important and the primitive model is expected not to give adequate account of confinement effects. For its simphcity, however, many theories [18-22] and simulation studies [23-25] have been made based on the primitive model for the bulk electrolyte. Ap-phcations to electrolyte interfaces have also been widely reported [26-30]. 

Figure 2. Solvent-averaged potential for charged hard-sphere ions in a dipolar hard-sphere solvent. MC approximation by Patey and Valleau (16) and LHNC approximation by Levesque, Weis, and Patey (11). Also shown are the primitive model functions for solvent dielectric constants 9.6 and 6.

Issue is taken here, not with the mathematical treatment of the Debye-Hiickel model but rather with the underlying assumptions on which it is based. Friedman (58) has been concerned with extending the primitive model of electrolytes, and recently Wu and Friedman (159) have shown that not only are there theoretical objections to the Debye-Hiickel theory, but present experimental evidence also points to shortcomings in the theory. Thus, Wu and Friedman emphasize that since the dielectric constant and relative temperature coefficient of the dielectric constant differ by only 0.4 and 0.8% respectively for D O and H20, the thermodynamic results based on the Debye-Hiickel theory should be similar for salt solutions in these two solvents. Experimentally, the excess entropies in D >0 are far greater than in ordinary water and indeed are approximately linearly proportional to the aquamolality of the salts. In this connection, see also Ref. 129. 

For molten salts one sets so = 1. For electrolyte solutions solvent-averaged potential [37]. Then, in real fluids, eo in Eq. (11) depends on the ion density [167]. Usually, one sets so = s, where e is the dielectric constant of the solvent. A further assumption inherent in all primitive models is in = , where is the dielectric constant inside the ionic spheres. This deficit can be compensated by a cavity term that, for electrolyte solutions with e > in, is repulsive. At zero ion density this cavity term decays as r-4 [17, 168]. At... 

The simplest model of an electrolyte is charged hard spheres (the ions) in a continuum dielectric whose dielectric constant is s (the solvent). This is called the primitive model of an electrolyte. This model has been studied using the MSA and HNC approximations. The PY approximation is not successful for this system. 

To move beyond the primitive model, we must include a molecular model of the solvent. A simple model of the solvent is the dipolar hard sphere model, Eq. (16). A mixture of dipolar and charged hard spheres has been called the civilized model of an electrolyte. This is, perhaps, an overstatement as dipolar hard spheres are only partially satisfactory as a model of most solvents, especially water still it is an improvement. 

An alternative to integral equation theories of the nonprimitive inhomogeneous electric double layer is a mean electrostatic field analysis of an ion-solvent dipole mixture against a charged wall [83-90]. Although this approach has been successful with the primitive model and avoids the difficult problem with the bridge function, it is still in the early stages of development with the nonprimitive electric double layer model. 

One way to test and compare these various statistical approaches is by computer simulation. Molecular dynamics (MD) simulations are based on the classical equations of motion to be solved for a limited number of molecules. From such simulations information about equilibrium properties as well as the dynamics of the system are obtained. In order to test theories based on primitive models for the solvent, Monte Carlo simulations are more appropriate. In Monte... 

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