Multi-Scale Concepts in the Molecular Theory of Electrolyte Solutions: Mcmillan-Mayer and Quasi-Chemical Theory

Multi-scale concepts in the molecular theory of electrolyte solutions: McMillan-Mayer and Quasi-Chemical Theory

W. Zhang, X. You, and L. R. Pratt

Department of Chemical and Biomolecular Engineering

Tulane University, New Orleans, LA 70118

The role of statistical mechanical theory in understanding electrolyte solutions at a molecular level has shifted substantially over the past few decades. The principal reason is that molecular simulation has steadily improved in precision and scope. By now, simulation calculations bypass dominating obstacles of non-simulation statistical mechanical theory without comment or resolution. Nevertheless, some of those theoretical obstacles reflect our conceptual understanding, and deserve resolution even if alternative brute force computations are widely available. Multi-scale concepts are needed to enable difficult chemical-scale ab initio molecular dynamics (AIMD) calculations to be applied to obtain the thermodynamics of complex electrolyte solution. For example, charges are transferred from charge donor to charge acceptor when one ion approaches another counter-ion. Development of statistical mechanical theory can assist with a better understanding of such chemical effects as charge transfer  between ions.

Here we address one such conceptual point, namely the coarse-graining that eliminates the solvent molecules from direct consideration when treating electrolyte solutions. It is common to consider ions in solutions with the solvent replaced by a uniform dielectric medium. In a primitive case, for example, hard-spherical ions are considered, the solvent is not actively present, and, when the ion hard spheres do not overlap, the interactions between ions is assumed to be q_iq_j/4¹εr where q_i is the formal charge on the i^th ion, (ε/ε₀) is the relative dielectric constant of the solvent, and r is the separation of the ionic charge. Typically called "primitive models," such models are not justified by direct molecular-scale observation of the solvent, i.e., the solvent is not actually a dielectric continuum. The required justification is the target of statistical mechanical theory, and clearly must be more subtle.

Proposed justifications are often intuitive,[1] and sometimes solely empirical as in fitting modeled thermodynamic properties to data.[2] But the theory for elimination of the solvent has been conclusively considered: it is the McMillan- Mayer (MM) theory,[3]^,[4],[5] and it is the pinnacle of the theory of coarse-graining for the statistical mechanics of solutions. "Primitive model" is then synonymous^5,[6] with "McMillan-Mayer model."

The foremost feature of MM theory is that the solvent coordinates are fully integrated out. The statistical mechanical problem that results from MM analysis treats the solute species (ions) only, but with effective interactions that are formally fully specified. Those effective interactions typically are complicated.[7]^,[8] Though it can be argued that no sacrifice of molecular realism is implied by MM theory, cataloging the multi-body potentials implied by a literal MM approach is prohibitively difficult.[9] Use of MM theory to construct a specific primitive model for a system of experimental interest has been limited.^7,8

Figure  SEQ Figure \* ARABIC 1 Evaluation of the excess chemical potential of the interesting ion (red sphere), patterned according to the development of QCT. The blue spheres (TEA⁺) and green spheres (BF₄^-) are other ions in the system, with the solvent in background. Contributions for each step (arrow) are indicated from left to right above the graphic. They are referred to as "packing," "outer shell," and "chemical" contributions, from left to right.

The integrating out does accomplish a coarse-graining, and the coordinates of solvent molecular are fully eliminated. Those coordinates are obvious from the thermodynamic specification of the problem. This is therefore a favored case for theoretical considerations because identification of degrees of freedom to eliminate will never be less arbitrary than that. The effective interactions that result depends on the thermodynamic state of the solvent, of course, and specifically on the chemical potential of the solvent, μ_S. This is consistent with the physical picture of MM analysis that the system under study can be viewed as in osmotic equilibrium with pure solvent at a specific chemical potential. After re-deriving the MM theory, we show how to implement the MM approach without explicit a priori determination of multi-ion potentials of mean force, utilizing Quasi-Chemical Theory (QCT). We will present these derivations and results for the solution of tetra-ethyl ammonium (TEA⁺), tetrafluoroborate (BF₄^-) in propylene carbonate (PC).

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[1] L. D. Landau and E. M. Lifshitz, COURSE IN THEORETICAL PHYSICS, Vol. 5(Pergamon, New York, 1980) ¤ 92.

[2] W. M. Latimer, K. S. Pitzer, and C. M. Slansky, J. Chem. Phys. 7, 108 (1939).

[3] W. G. McMillan Jr and J. E. Mayer, J. Chem. Phys. 13, 276 (1945).

[4] T. L. Hill, STATISTICAL THERMODYNAMICS (Addison-Wesley, Reading, MA USA, 1960) Chap. SS19.1.

[5] H. L. Friedman and W. D. T. Dale, in STATISTICAL MECHANICS PART A: EQUILIBRIUM TECHNIQUES, edited by B. J. Berne (Plenum, New York, 1977) pp. 85-136.

[6] H. L. Friedman, Ann. Rev. Phys. Chem. 32, 1798 (1981).

[7] P. G. Kusalik and G. N. Patey, J. Chem, Phys. 89, 7478 (1988).

[8] C. P. Ursenbach, D. Wei, and G. N. Patey, J. Chem. Phys. 94, 6782 (1991).

[9] S. A. Adelman, Chem. Phys. Letts. 38, 567 (1976).