Fluctuation Theory of Solutions

The advantages of the direct correlation function approach include the ability to treat systems close to critical points, together with a general simplification of the expressions such as those in Section 1.2.3. The disadvantages are that the DCF or DCFI cannot be calculated directly from simulation, only through the defining expression in Equation 1.40, and their molecular interpretation is less clear compared to the KBIs. 

The previous background material covered many aspects of thermodynamics, statistical thermodynamics, and solution thermodynamics. At this point, we have all we need to derive the main expressions provided by FST. There are many derivations of the principal expressions in the literature, including a matrix approach that is general for any number of components, all of which involve a series of thermodynamic manipulations (Kirkwood and Buff 1951 Hall 1971 O Connell 1971b Valdeavella, Perkyns, and Pettitt 1994 Ben-Naim 2006 Kang and Smith 2008 Nichols, Moore, and Wheeler 2009). We will not use that type of approach here as our primary concern is binary and ternary solutions for which a more transparent and, in our opinion, simpler approach is available. 

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Effect of a third component on the interactions in a binary mixture determined from the fluctuation theory of solutions... 

The KB theory of solution [15] (often called fluctuation theory of solution) employed the grand canonical ensemble to relate macroscopic properties, such as the derivatives of the chemical potentials with respect to concentrations, the isothermal compressibility and the partial molar volumes to microscopic properties in the form of spatial integrals involving the radial distribution function. 

Equations 13 and 14 can be used to calculate the solubilities of mixed gases if the solubilities of the pure constituent gases in the same solvent and the values of 22> 33> and K23 are known. Whereas the values of 22 and kss can be determined from the solubilities of the individual gases, an expression for K23 will be obtained below using the fluctuation theory of solution. 

Another approach is to employ rigorous statistical thermodynamic theories. In this paper, the Kirkwood-Buff (KB) theory of solutions (fluctuation theory of solutions) is employed to analyze the thermodynamics of multicomponent mixtures, with the emphasis on quaternary mixtures. This theory connects the macroscopic properties of re-component solutions, such as the isothermal compressibility, the concentration deriva-... 

The present paper is devoted to the extension of the theory developed by the authors for the solubility of proteins to the solubility of gases. Because this theory is based on the Kirkwood-Buff fluctuation theory of solutions, the next section summarizes the expressions which are involved. This is followed by a summary of the derivation of an equation for the solubility of proteins and finally its extension to the solubility of gases. 

The application of the Kirkwood-Buff fluctuation theory of solutions to the activity coefficients in ternary and multicomponent solutions... 

The authors of the present paper have shown previously [21-31] that the fluctuation theory of solution can provide a new approach to the solubility of gases, drugs, protein, etc., in binary and multicomponent aqueous mixed solvents. 

The present paper deals with the application of the fluctuation theory of solutions to the solubility of poorly soluble drugs in aqueous mixed solvents. The fluctuation theory of ternary solutions is first used to derive an expression for the activity coefficient of a solute at infinite dilution in an ideal mixed solvent and, further, to obtain an equation for the solubility of a poorly soluble solid in an ideal mixed solvent. Finally, this equation is adapted to the solubility of poorly soluble drugs in aqueous mixed solvents by treating the molar volume of the mixed solvent as nonideal and including one adjustable parameter in its expression. The obtained expression was applied to 32 experimental data sets and the results were compared with the three parameter equations available in the literature. 

In this paper, the fluctuation theory of solutions was applied to the solubility of drugs in aqueous mixed solvents. A rigorous expression for the activity coefficient of a solute at infinite dilution in a real mixed solvent was used to derive an equation for the sol-... 

The results obtained previously by Ruckenstein and Shulgin [Int. J. Pharm. 258 (2003a) 193 Int. J. Pharm. 260 (2003b) 283] via the fluctuation theory of solutions regarding the solubility of drugs in binary aqueous mixed solvents were extended in the present paper to multicomponent aqueous solvents. The multicomponent mixed solvent was considered to behave as an ideal solution and the solubility of the drug was assumed small enough to satisfy the infinite dilution approximation. 

Recently (Ruckenstein and Shulgin, 2003c), a method was suggested to calculate the activity coefficient of a poorly soluble solid in an ideal multicomponent solvent in terms of its activity coefficients at infinite dilution in some subsystems of the multicomponent solvent. The method, based on the fluctuation theory of solutions (Kirkwood and Buff, 1951), provided the following expression for the activity coefficient of a poorly soluble solid solute in an ideal multicomponent solvent ... 

As for infinite dilution, the main difficulty in predicting the solid solute solubility in a mixed solvent for a dilute solution is provided by the calculation of the activity coefficient of the solute in a ternary mixture. To obtain an expression for the activity coefficient of a low concentration solute in a ternary mixture, the fluctuation theory of solution will be combined with the assumption that the system is dilute with respect to the solute. 

Eq. (13) will be used for the drug solubility when its saturated solution in a binary solvent can be considered dilute. First, expressions for the two partial derivatives in Eq. (13) will be derived on the basis of the fluctuation theory of solutions (Kirkwood and Buff, 1951). 

In contrast to previous papers (Ruckenstein and Shulgin, 2003a-d), the solubility of the drug in a binary solvent is considered to be finite, and the infinite dilution approximation is replaced by a more realistic one, the dilute solution approximation. An expression for the activity coefficient of a solute at low concentrations in a binary solvent was derived by combining the fluctuation theory of solutions (Kirkwood and Buff, 1951) with the dilute approximation. This procedure allowed one to relate the activity coefficient of a solute forming a dilute solution in a binary solvent to the solvent properties and some parameters characterizing the nonidealities of the various pairs of the ternary mixture. 

Abstract Fluctuation Theory of Solutions or Fluctuation Solution Theory (FST) combines aspects of statistical mechanics and solution thermodynamics, with an emphasis on the grand canonical ensemble of the former. To understand the most common applications of FST one needs to relate fluctuations observed for a grand canonical system, on which FST is based, to properties of an isothermal-isobaric system, which is the most common type of system studied experimentally. Alternatively, one can invert the whole process to provide experimental information concerning particle number (density) fluctuations, or the local composition, from the available thermodynamic data. In this chapter, we provide the basic background material required to formulate and apply FST to a variety of applications. The major aims of this section are (i) to provide a brief introduction or recap of the relevant thermodynamics and statistical thermodynamics behind the formulation and primary uses of the Fluctuation Theory of Solutions (ii) to establish a consistent notation which helps to emphasize the similarities between apparently different applications of FST and (iii) to provide the working expressions for some of the potential applications of FST. 

The Fluctuation Theory of Solutions—also known as Fluctuation Solution Theory, Kirkwood-Buff Theory, or simply Fluctuation Theory— provides an elegant approach relating solution thermodynamics to the underlying molecular distributions or particle number fluctuations. Here, we provide the background material required to develop the basic theory. More details can be found in standard texts on thermodynamics and statistical mechanics (Hill 1956 Munster 1970). Indeed, the experienced reader may skip this chapter completely, or jump to Section 1.2. A list of standard symbols is also provided in the Prolegomenon to aid the reader, and we have attempted to use the same set of symbols and notations in all subsequent chapters. Throughout this work we refer to a collection of species (1, 2, 3,...) in a systan of interest. We consider this to represent a primary solvent (1), a solute of interest (2), and a series of additional cosolutes or cosolvents (3,4,...) which may also be present in the solution. However, other notations such as A/B or u/v is also used in the various chapters. All summations appearing here refer to the set of thermodynamically independent components (n in the mixture unless stated otherwise. Derivatives of the chemical potentials with respect to composition form a central component of the theory. The primary derivative of interest here is defined as... 

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