(520g) 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).