Local Fluctuations in Solution: Theory and Applications.

Liquids are complicated. Liquid mixtures are even more complicated.1 While our knowledge of gases and solids is quite extensive, and in many cases can be considered essentially complete, a deep understanding of the properties of solutions remains elusive. The major problems involve our simplistic understanding of how molecules interact, especially at high densities, and how these interactions are affected by changes in the conditions – usually temperature, pressure and composition. Even in the absence of strong intermolecular interactions, our ability to rationalize the packing of molecules in solution remains rather rudimentary. One possible solution to this problem involves the use of molecular simulation.2 Computer simulations have now advanced to a stage where they have provided a wealth of information using reasonably accurate models of liquid solutions. The ability to investigate the properties of solution mixtures is clearly aided by computer simulation and research in this direction will continually evolve. However, consistent comparison with experiment is still required in order to validate many of the results. Furthermore, our ability to relate the properties of solution mixtures to the underlying interactions between the molecules is still in its infancy. For instance, when a particular model (or force field) provides an incorrect description of a given thermodynamic property, for example the isothermal compressibility or the enthalpy of mixing, it is often extremely difficult to trace this to a specific incorrectly modeled interaction between molecules. Conversely, it is also difficult to predict the thermodynamic consequences arising from variations in the intermolecular interactions. Both issues limit the effectiveness of computer driven research. Another, more subtle, problem arises when attempting to analyze experimental data concerning solution mixtures. Experimental data is typically obtained under isothermal isobaric conditions where the system is closed to matter exchange. This is clearly the most convenient and relevant set of conditions for most experiments and real systems. However, it is well known that the thermodynamic constraints placed on a system affect the behavior of the system.3 The statistical thermodynamics of large systems ensures that ensemble averages between systems under the same average conditions are equal.4 In fact, many theories of solutions have traditionally been developed starting from the grand canonical ensemble where the required manipulations are often much easier to perform. However, this equivalence does not hold for fluctuating quantities. Fluctuations depend on the ensemble. We will see here that fluctuations characteristic of open systems can be used to help understand solution behavior in closed systems. In this type of approach the fluctuations then represent the fluctuations associated with a relatively small local volume of solution. We will argue that these local fluctuations should be much easier to understand and probe compared to fluctuations of the system as a whole. One then simply needs to relate these local fluctuations to thermodynamic properties of the solution under the appropriate thermodynamic constraints. This is the role that the Fluctuation Theory (FT) of solutions seeks to play. Fluctuations play an important role in the properties of systems. This was well established by the work of Einstein and others.5 However, this work has primarily focused on the fluctuations in bulk systems under different thermodynamic constraints. A major step forward was provided by Kirkwood and Buff when they related particle number fluctuations and integrals over molecular distribution functions to the properties of closed systems - specifically the changes in chemical potentials, partial molar volumes and isothermal compressibility - and to changes in the osmotic pressure for semi-open systems.6 This approach is generally known as the Kirkwood-Buff (KB) theory of solutions. However, the theory lay relatively unused for over 25 years. Presumably, this initial lack of interest was because the theory appeared to be unsuited for the study of salt solutions,7,8 and/or the theory required radial distribution functions, which are generally unknown, as input data. More recently, KB theory has developed into a powerful tool for probing the microstructure of solution mixtures.9–12 This can be attributed to two main factors. First, the development of the KB inversion procedure by Ben-Naim,13 whereby the experimental data can be used to obtain the integrals over the rdfs, the so called KB integrals (KBIs). Second, the ability of modern computer simulations to provide realistic rdfs as input for the theory has improved significantly. These two factors have prompted the development of approaches to investigate the local microstructure in solutions, i.e. deviations in the local composition from the bulk distribution, which has in turn led to a rigorous description of preferential solvation.10 Fluctuation Theory has now been widely used to understand the basic properties of solutions,9 to understand the effects of additives on the solubility of solutes (from small hydrocarbons to proteins) and biomolecular equilibria,14–20 to investigate the local composition of solutions in the context of preferential solvation,21 to study the effects of additives on the surface tension of liquids,22 to study critical behavior,23–26 to interpret computer simulation data,19,27–29 to develop models for many of the above effects,30 and to provide expressions for the volume, enthalpy, compressibility and heat capacity corresponding to simple association equilibria.31,32 The general formulation and application of KB theory has been outlined in detail by several workers.9–12 KB theory is restricted to the use of particle number fluctuations. It was recognized quite early that one could extend this type of approach to include particle-energy and energy-energy fluctuations.33 Further work by Debenedetti has also highlighted the additional insight this provides for computer simulations.34–36 However, a direct link to experimental data was absent until very recently when Ploetz and Smith provided a framework for the analysis of additional solution properties - specifically the excess partial molar enthalpies, the thermal expansion coefficient and the constant pressure heat capacity - all as a function of solution composition.37 Furthermore, all the fluctuating properties can also be obtained directly from available experimental data. The approach has since been extended to treat molecular association and conformational equilibria.32 Here, we outline a practical general theory of solution fluctuations, representing local regions within a solution, which relates the local fluctuations to thermodynamic properties of bulk closed isothermal isobaric systems. We argue that this type of approach leads to a much simpler, though certainly not trivial, description of the behavior of solution mixtures. Furthermore, this appears to be the most appropriate and convenient approach for analyzing a variety of computer simulation data.