McMillan-Mayer approach

For concentrated solutions, there are approaches that are more sophisticated than that of Debye Hiickel. A particularly successful method of describing such solutions is that due to McMillan Mayer (1945) which has subsequently been developed by Ramanathan ... 

Mayer achieved closure in integrals involved in his application of the McMillan-Mayer virial approach to ionic solutions by multiplying this equation by the factor... 

Explain in about 250 words the essential approach of the Mayer theory of ionic solutions and how it differs from the ionic-atmosphere view. The parent of Mayer s theory was the McMillan-Mayer theory of 1950. With what classical equation for imperfect gases might it be likened ... 

The McMillan-Mayer theory is an alternative to the Debye-Htickel theory. It is called the virial coefficient approach and its equations bear some conceptual resemblance to the virial equation of state for gases. The key contribution in... 

In this chapter some aspects of the present state of the concept of ion association in the theory of electrolyte solutions will be reviewed. For simplification our consideration will be restricted to a symmetrical electrolyte. It will be demonstrated that the concept of ion association is useful not only to describe such properties as osmotic and activity coefficients, electroconductivity and dielectric constant of nonaqueous electrolyte solutions, which traditionally are explained using the ion association ideas, but also for the treatment of electrolyte contributions to the intramolecular electron transfer in weakly polar solvents [21, 22] and for the interpretation of specific anomalous properties of electrical double layer in low temperature region [23, 24], The majority of these properties can be described within the McMillan-Mayer or ion approach when the solvent is considered as a dielectric continuum and only ions are treated explicitly. However, the description of dielectric properties also requires the solvent molecules being explicitly taken into account which can be done at the Born-Oppenheimer or ion-molecular approach. This approach also leads to the correct description of different solvation effects. We should also note that effects of ion association require a different treatment of the thermodynamic and electrical properties. For the thermodynamic properties such as the osmotic and activity coefficients or the adsorption coefficient of electrical double layer, the ion pairs give a direct contribution and these properties are described correctly in the framework of AMSA theory. Since the ion pairs have no free electric charges, they give polarization effects only for such electrical properties as electroconductivity, dielectric constant or capacitance of electrical double layer. Hence, to describe the electrical properties, it is more convenient to modify MSA-MAL approach by including the ion pairs as new polar entities. 

In this section we consider the application of the concept of ion association to describe the properties of electrolyte solutions within the ion or McMillan-Mayer level approach. In this approach the effects of solvent molecules are taken into account by introducing the dielectric constant into Coulomb interaction law and by appropriately choosing the short-range part of ion-ion interactions. To simplify, we consider here the restrictive primitive model (RPM)... 

While the McMillan-Mayer theory (Section 4) prescribes the iiabir) as functionals of the Hamiltonian of a BO-level model, little has been learned from this sort of direct approach. The main contributions are an analytical study of charged hard spheres in an uncharged hard-sphere solvent by Stell, " Monte Carlo and molecular dynamics studies of somewhat more realistic models, " " and a study using the mean spherical approximation (Section 7.3). ... 

The crowding approach, which has been based upon McMillan-Mayer solution theory, has anployed Equation 11.7 as a starting point (Davis-Searles et al. 2001 Shimizu and Boon 2004). This is based upon a second virial approximation. FST can even provide the condition upon which this approximation is accurate. Since Equation 11.7 holds under the condition that AN21 is negligibly small, and that this quantity is related to the partial molar volume via Equation 11.5, the proposed condition is... 

Hamiltonian models are classified according to then-level of approximation. The features of Schroedinger (S), Born-Oppenheimer (BO), and McMillan-Mayer (MM) level Hamiltonian models are exemplified in Table I by a solution of NaCl in H2O. The majority of investigations on electrolyte solutions are carried out at the MM level. BO-Level calculations are a precious tool for Monte Carlo and molecular dynamics simulations as well as for integral equation approaches. However, their importance is widely limited to stractural investigations. They, as well as the S-level models, have not yet obtained importance in electrochemical engineering. S-Level quantum-mechanical calculations mainly follow the Car-Parinello ab initio molecular dynamics method. 

A more fundamental approach is to attempt to model electrolyte solutions using statistical mechanical methods, of which there are two kinds of models (reviewed extensively elsewhere ° ) Born-Oppenheimer (BO) level models in which the solvent species as well as the ionic species appear explicitly in the model for the solution and McMillan-Mayer (MM) level models in which the solvent species degrees of freedom are integrated out yielding a continuum solvent approximation. Thus, for a BO level model, in addition to the interionic pair potentials one must specify the ion-solvent and solvent-solvent interactions for all of the ionic and solvent species. In this case, the interionic potentials do not contain the solvent dielectric constant in contrast to the MM-level models. Kusalik and Patey carefully discuss the distinction between these two approaches. 

Another approach to the thermodynamic properties of solutions is to calculate them from the solute-solute distribution functions rather than from the virial coefficients. Approximations to these functions, which correspond to the summation of a certain class of terms in the virial series to all orders in the solute concentration (or density), have already been worked out for simple fluids, and the McMillan-Mayer theory states that the same approximations may be applied to the solute particles in solution provided the solvent-averaged potentials are used to determine the solute distribution functions. Examples of these approximations are the Percus-Yevick (PY) (1958), Hypernetted-Chain (HNC), mean-spherical (MS), and Born-Green-Yvon (BGY) theories. Before discussing them we will review some of the properties of distribution functions and their relationship to the observed thermodynamic variables. 

Our general approach is a proper adaptation and generalization of the gas-type theories of McMillan and Mayer and of Kirkwood and Buff. These were originally developed for simple (monomer) solutions. We use the cluster development of McMillan and Mayer, which itself is an adaptation of the original (Ursell)-Mayer cluster development. We... 

The above expression holds at any solute concentration. The analysis of osmotic pressure data for solutes therefore provides an indication of solute-solute affinity at the concentration of interest (Karunaweera et al. 2012). One can also show that MM theory is obtained from FST in the limiting case of an infinitely dilute solute, where both approaches then provide a series expansion in concentration (McMillan and Mayer 1945 Kirkwood and Buff 1951). A more detailed discussion is provided in the literature (Cabezas and O Connell 1993). 

Kurata and Yamakawa (1958) criticized McMillan and Mayer s theory for lack of experimental support and Rory and Krigbaum s theory for their assumption of Gaussian distribution of polymer segments around the center of the mass. The introduction of the factorization approximation does not help much. As a result, Flory and Krigbaum s theory underestimates the molecular weight dependence of A2. For Kurata and Yamakawa, the excluded volume effect has a non-Gaussian character with respect to the chain configuration. Kurata and Yamakawa s approach, however, basically follows the same line as that of McMiUan and Mayer. 

Clearly a new approach was necessary. As shown by the brilliant work of Mayer and McMillan [1945] and Kirkwood and Buff [1951], interesting results may be obtained by an extension of the powerful techniques used in the theory of imperfect gases (cf. Ch. V). Unfortunately, this appears to be practicable only in the case of very dilute solutions or in polymer solutions in which the diss3unmetry between solvent and solute is such that the molecular structure of the... 

---