Inversion of the Kirkwood-Buff theory

The Kirkwood-Buff theory of solutions was originally formulated to obtain thermodynamic quantities from molecular distribution functions. This formulation is useful whenever distribution functions are available either from analytical calculations or from computer simulations. The inversion procedure of the same theory reverses the role of the thermodynamic and molecular quantities, i.e., it allows the evaluation of integrals over the pair correlation functions from thermodynamic quantities. These integrals Gy, referred to as the Kirkwood-Buff integrals (KBIs), were found useful in the study of mixtures on 

Having information on the Gy, one can compute the thermodynamic quantities. However, the original KB theory could have been used only in rare cases where Gy could be obtained from theoretical work. In principle, having an approximate theory for computing the various pair correlation functions gij(R), it is possible to evaluate the integrals Gy and then compute the thermodynamic quantities through the KB theory. Comparison between the thermodynamic quantities thus obtained, and the corresponding experimental data, could serve as a test of the theory that provides the pair correlation functions. 

In this form, the thermodynamic quantities are used as input to compute the molecular quantities Gy. Since it is relatively easier to measure the required thermodynamic quantities, the inversion procedure provides a new and powerful tool to investigate the characteristics of the local environments of each species in a multicomponent system. 

It should be noted that there are some difficulties in obtaining accurate values of the KBI from the available thermodynamic data (Kato 1984 Zaitsev et al. 1985, 1989). Matteoli and Lepori (1984) have made an extensive comparison between the values of Gy calculated by different authors (e.g., Ben-Naim 1977 Donkersloot 1979a, b Patil 1981) and found large discrepancies between the reported results. Another method of obtaining the KBI is the small-angle x-ray or neutron scattering intensities from mixtures see, for example, Nishikawa (1986), Nishikawa et al. (1989), Hayashi et al. (1990), Misawa and Yoshida (2000), Almasy et al. (2002), and Dixit et al. (2002). 

In the following we shall discuss only the mathematical aspects of the inversion procedure and not delve into the problem of the accuracies of the results. 

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Ben-Naim, A. Inversion of the Kirkwood—Buff theory of solutions apphcation to the water—ethanol system. J. Chem. Phys. 1977, 67, 4884-4890. 

A more dramatic turning point for the Kirkwood-BufF theory occurred in 1978 after the publication of the inversion of the Kirkwood-BufF theory (Ben-Naim 1978). Symbolically, the inversion theory may be written as... 

Recall that rj is a measurable quantity through the inversion of the Kirkwood-Buff theory. Since t] > 0, the entire quantity kT(pA +pB)2/t] is always positive. Therefore, the sign of the derivative on the lhs of (8.25) is the same as the sign of GBs — GAs. 

We now extend the result obtained above for two-component system of A and B. We first write the general result from the inversion of the Kirkwood-Buff theory (section 4.4). 

Ben-Naim A (1977) Inversion of the Kirkwood-Buff theory of solutions appUcation to the water-ethanol system. J Chem Phys 67 4884- 4890... 

All the G(j( 3 quantities defined above may be computed from the inversion of the Kirkwood-Buff theory (Ben-Naim 2006). The relevant relations are provided in Section 1.2.4 in Chapter 1. 

Local Properties Calculated from the Inversion OF THE Kirkwood-Buff Theory... 

In this section we present the results of a recent paper (Ben-Naim and Santos 2009) where we directly recalculated the KBIs for two component mixtures of particles interacting via square-well potential. The theoretical background is lengthy and will not be presented here. Instead, we show a sample of results for mixtures of square-well particles. It is shown that the results are in quantitative agreement with those obtained from the inversion of the Kirkwood-Buff theory of solution. We also... 

There are basically two main developments in the molecular theory of solutions in the sense of route —IV one based on the inversion of the Kirkwood-Buff (KB) theory the second is the introduction of a new measure to study solvation properties. Both of these use measurable macroscopic, or global quantities to probe into the microscopic, or the local properties of the system. The types of properties probed by these tools are local densities, local composition, local change of order, or structure (of water and aqueous solutions) and many more. These form the core of properties discussed in this book. Both use exact and rigorous tools of statistical mechanics to define and to calculate local properties that are not directly accessible to measurements, from measurable macroscopic quantities. 

Thus, the main scope of this book is to cover the two topics the Kirkwood-Buff theory and its inversion and solvation theory. These theories were designed and developed for mixtures and solutions. I shall also describe briefly the two important theories the integral equation approach and the scaled particle theory. These were primarily developed for studying pure simple liquids, and later were also generalized and applied for mixtures. 

The book is organized into eight chapters and some appendices. The first three include more or less standard material on molecular distribution functions and their relation to thermodynamic quantities. Chapter 4 is devoted to the Kirkwood-Buff theory of solutions and its inversion which I consider as... 

The Kirkwood-Buff (KB) theory is the most important theory of solutions. This chapter is therefore central to the entire book. We devote this chapter to derive the main results of this theory. We start with some general historical comments. Then we derive the main results, almost exactly as Kirkwood and Buff did, only more slowly and in more detail, adding occasionally a comment of clarification that was missing in the original publication. We first derive the results for any multicomponent system, and thereafter specialize to the case of two-components system. In section 4, we present the inversion of the KB theory, which has turned a potentially useful theory into an actually useful, general and powerful tool for investigating solutions on a molecular level. Three-component systems and some comments on the application of the KB theory to electrolyte solutions are discussed in the last sections. 

There are essentially three significant quantities that can be derived from the inversion of the KB theory. The first is a measure of the extent of deviation from symmetrical ideal (SI) solution behavior, A b. defined below in the next section. It also provides a necessary and sufficient condition for SI solution. The second is a measure of the extent of preferential solvation (PS) around each molecule. In a binary system of A and B, there are only two independent PS quantities these measure the preference of, say, molecule A to be solvated by either A or B molecules. Deviations from SI solution behavior can be expressed in terms of either the sum or difference of these PS quantities. Finally, the Kirkwood-Buff integrals (KBIs) may be obtained from the inversion of the KB theory. These provide information on the affinities between any two species for instance, PaGaa measures the excess of the average number of A particles around A relative to the average number of A particles in the same region chosen at a random location in the mixture. All these quantities can be obtained from the KB integrals. 

The preferential solvation parameters calculated from the KBIs G a. Gab nd Gbb and Equations 3.2 and 3.3, with the aid of Equations 3.5 and 3.6 for the correlation volumes, represent the total values. These are the formally correctly derived values from the inverse Kirkwood-Buff theory applied to thermodynamic data. 

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