# (520g) Multi-Scale Concepts in the Molecular Theory of Electrolyte Solutions: Mcmillan-Mayer and Quasi-Chemical Theory

#### AIChE Annual Meeting

#### 2013

#### 2013 AIChE Annual Meeting

#### Engineering Sciences and Fundamentals

#### Thermodynamics At the Nanoscale II

#### Wednesday, November 6, 2013 - 2:30pm to 2:50pm

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 _{i}q_{j}/4¹εr* where

*q*is the formal charge on the i

_{i}^{th}ion,

*(ε/ε*is the relative dielectric constant of

_{0})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 (BF_{4}^{-}) 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 (BF_{4}^{-}) in propylene carbonate (PC).

[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).