Linking Solution Microstructure and Solvation Thermodynamics of Mixed-Solvent Systems: Formal Results, Critical Observations, and Modeling Pitfalls

Abstract

This review provides a critical assessment of the current state of affairs regarding the solvation thermodynamics involving mixed-solvent systems. It focuses specifically on (i) its rigorous molecular-based foundations, (ii) the underlying connections between the microstructural behavior of the mixed-solvent environment and its thermodynamic responses, (iii) the microstructural characterization of the behavior of the mixed-solvent environment around the dilute solute via unique fundamental structure-making/-breaking functions and the universal preferential solvation function, (iv) the discussion of potential drawbacks associated with the molecular simulation-based determination of thermodynamic preferential interaction parameters, and (v) the forensic examination of frequent modeling pitfalls behind the interpretation of preferential solvation from experimental data of Gibbs free energy of solute transfer.

1. Introduction

An effective approach to gaining an understanding of solution nonidealities involves the formal interfacing between the microstructural evolution of a system as described by statistical mechanics, and its macroscopic manifestation rationalized by chemical thermodynamics. The sources of success behind this strategy can usually be attributed to the following elements: (a) the inhomogeneous distribution of species in a one-component homogeneous (i.e., one phase diffusively stable) system, i.e., its microstructure, results from the interspecies interactions, so that the non-zero correlation functions reflect the strength of the interactions and translate into their deviations from the uniform (i.e., ideal gas) distribution counterpart [1]; (b) the microstructure of a homogeneous mixture stems from the differences (e.g., interspecies interaction asymmetries) between the unlike pairs giving rise to contrasting pair correlation functions [2,3]; (c) the Kirkwood–Buff fluctuation formalism that connects rigorously the interspecies correlation function integrals, characterizing the microstructure, with the isothermal–isobaric composition dependence of the volumetric and chemical partial molar properties of the system [4]; and (d) the fundamental solution thermodynamics that provides the formal description of the excess properties, i.e., the deviations from a precisely defined ideal solution reference, in terms of the residual properties, i.e., their departure from the ideal gas model [5,6,7].

Naturally, these elements have been the foundational support for the drawing of rigorous microscopic-to-macroscopic interpretations of solvation processes based on the determination of the isobaric–isothermal solute-induced perturbation of the solvent environment, resulting from the solute–solvent intermolecular interaction asymmetry, and the ensuing perturbation propagation across the system [3,8,9]. In fact, this local microstructural distortion, formally described by the related direct correlation function, propagates across the system up to a distance given by the correlation length of the environment as described by the corresponding indirect correlation function [8,10], and allows its macroscopic interpretation in terms of the thermodynamics of solution non-idealities [2,9,11,12,13,14,15].

Unsurprisingly, a significantly large effort in solution chemistry has focused on the development of experimental techniques for the microstructural characterization of fluid mixtures to interpret, and gain an understanding of, the thermophysical properties of real solutions in terms of the structure of the first few solvation shells centered at the species of interest. For that purpose, these techniques have comprised scattering approaches, such as X-ray [16,17,18,19,20,21,22,23] and neutron diffraction [21,24,25,26,27], assisted by empirical microstructural refinement [28,29,30,31,32,33,34], molecular simulation [35,36,37,38,39,40,41], and complemented with infrared [42,43], Raman [44,45,46,47,48], NMR [49,50], and femtosecond pump-probe spectroscopies [51,52]. The effort has frequently led to the unfortunate depiction of the solvation behavior of a species as either structure-making or structure-breaking [53,54,55,56,57], later accompanied by the fancier kosmotrope (order-maker) and chaotrope (disorder-maker) monikers [58], which have contributed further to the ambiguity and confusion when they were used as interchangeable alternative notation [58,59,60,61,62]. In fact, the recurrent factor leading to the structure-making/-breaking confusion has been, until recently, the lack of an explicit and unambiguous formal relation between the microstructural impact of the solute on the surrounding solvent environment, also known as solute-induced local density perturbation, and the macroscopic (thermodynamic) response that can be experimentally measured [63,64,65].

The need for a fundamental understanding of solvation phenomena becomes even more pressing when dealing with species in binary mixed-solvent environments, where the presence of a second solvent, also known as cosolvent or entrainer, modifies the solubility of a solute and the diffusive stability of the solvating environment, through the combination of the contrasting solvating power of the individual solvents, and resulting in significant deviations from solution ideality [66,67,68,69,70,71]. This need has been typically addressed via a variety of binding models [72,73,74,75], supplemented with molecular simulations and integral equation approximations [76,77,78,79,80,81,82], as well as fluctuation-based modeling [83,84,85,86].

On the one hand, the identification of so-called thermodynamic preferential interaction parameters to describe how the chemical potential of the $i$-solute species in solution, ${\mu }_i\left(T,P,{\chi }_i,{\chi }_{\alpha }\right)$, could be modified by changing the concentration ${\chi }_{\alpha }$ of the $\alpha$-species in an $\alpha =\left(j,k\right)$ mixed-solvent system pioneered a formal thermodynamic description for the interpretation of the experimental evidence [87,88,89]. These parameters have been explicitly defined as isothermal $\chi$-composition derivatives under specific thermodynamic constraints $\left({\mathcal{Q}}_{\alpha },{\mathcal{Q}}_{\beta }\right)$, including $\left({\mu }_{\alpha },P\right)$ as ${\Gamma }_{{\mu }_{\alpha }}^{({\chi }_i\to 0)}\left(x_j\right)\equiv {\left(\partial {\chi }_k/\partial {\chi }_i\right)}_{TP{\mu }_{\alpha }}$ and $\left({\mu }_j,{\mu }_k\right)$ as ${\Gamma }_{{\mu }_j{\mu }_k}^{({\chi }_i\to 0)}\left(x_j\right)\equiv {\left(\partial {\chi }_k/\partial {\chi }_i\right)}_{T{\mu }_j{\mu }_k}$, where $\chi =\left(\rho ,m,x\right)$ denotes the species concentration as either molar, molal, or mole fraction scales, while ${\mu }_{\alpha }$ identifies the chemical potential of the $\alpha$-species at the prevailing state conditions and composition [90,91,92].

On the other hand, under this thermodynamic approach, we typically encounter a variety of thermodynamic preferential interaction parameters leading to a lack of uniqueness in the resulting microscopic markers for a preferential solvation ($\mathcal{P}\mathcal{S}$) event. In fact, depending on the chosen thermodynamic constraints $\left({\mathcal{Q}}_{\alpha },{\mathcal{Q}}_{\beta }\right)$, the composition derivatives within an open (grand canonical) ${\Gamma }_{{\mu }_i{\mu }_k}^{({\chi }_i)}\left(x_j\right)$, semi-open (semi-grand), ${\Gamma }_{{\mu }_{\alpha }}^{({\chi }_i)}\left(x_j\right)$, and a closed (isothermal–isobaric) ${\Gamma }_{i\alpha }^{({\chi }_i)}\left(x_j\right)$ systems translate formally into different functional microstructural descriptions of the system [93]. In other words, how could we assert anything about the $\mathcal{P}\mathcal{S}$ behavior of a mixed-solvent system when we have neither a precise microstructural manifestation of the differential solute affinity of the solvent environment relative to that of the cosolvent around the solute, nor a unique macroscopic preferential interaction parameter counterpart?

This mystifying scenario was originally underscored by Ben-Naim when first asking “… how do we measure the $\mathcal{P}\mathcal{S}$ of a given solute in a given mixture?” and then,“… what are the molecular reasons that cause a solute to prefer one component over the other and, hence, alter the composition in its local environment?” [83]. Adding to the confusion was the need for an identification of the formal microstructure-to-macroscopic (i.e., statistical mechanics to thermodynamics) property relationships required to connect a precisely defined molecular-based signature of $\mathcal{P}\mathcal{S}$ with the underlying solute–solvent, solute–cosolvent, and solvent–cosolvent intermolecular interaction asymmetries leading to the thermodynamic solution non-idealities [93].

The preferential solvation of solute species, momentarily portrayed loosely as the differential affinity of the solute in the mixed-solvent environment (see Section 2.3 for a formal definition), embodies the differential solute-induced perturbations between the solvent and the cosolvent environments. Therefore, it implies a microstructural signature linked explicitly to the experimentally available macroscopic counterparts [91,93,94,95,96], that leads to the rigorous interpretation of the preferential solvation phenomena in terms of the fundamental structure-making/-breaking functions ${\mathcal{S}}_{\alpha \beta }\left(T,P,x_{\alpha }\right)$ for any solute in mixed solvents [65,93]. Consequently, it provides the molecular-based framework for the proper interpretation, potential tuning of the species solvation behavior, and eventual thermodynamic modeling of mixed-solvent systems.

In this context, the main goals of this review include addressing the current state of affairs regarding the solvation thermodynamics of mixed-solvent systems, including (i) its rigorous molecular-based foundations, (ii) the underlying connections between so-called preferential interaction parameters and the system microstructure, (iii) the microstructural characterization of the behavior of the mixed-solvent environments around the dilute solute via fundamental structure-making/-breaking functions and the universal preferential solvation function, (iv) the discussion of potential drawbacks behind the molecular simulation-based determination of preferential interaction parameters, and (v) the examination of frequent modeling pitfalls behind the interpretation of preferential solvation from the experimental data of Gibbs free energy of solute transfer.

2. Fundamentals from Statistical Mechanics and Chemical Thermodynamics

The rational manipulation of these solvation phenomena requires a fundamental understanding of the molecular-based mechanism associated with the impact of the state conditions and composition of the mixed-solvent, the solute–solvent, solute–cosolvent, and solvent–cosolvent intermolecular asymmetries on the microstructure of the system as well as the resulting thermodynamic responses, including the diffusive stability of the homogeneous phase.

2.1. Molecular-Based Description of the Structure-Making/-Breaking Functions and Their Attributes

The key elements of interest in the solvation of a solute in a mixed-solvent are the solute-induced microstructural evolution of the surrounding environment and its associated thermodynamic responses [3,9]. In particular, we must recognize the key role played by the relative affinity of the solute with its mixed-solvent environment in the manipulation of the solvation behavior of any solute. In fact, the contrasting affinity represented by the differential perturbation of the solvent relative to that of the cosolvent around the solute becomes the typical microstructural signatures that can be rigorously linked to experimentally available macroscopic quantities [91,94,95,96,97]. For that purpose, we have recently developed an explicit formalism to achieve such microscopic-to-macroscopic connection according to the statistical mechanics-based fundamental structure-making/-breaking functions ${\mathcal{S}}_{\alpha \beta }\left(T,P,x_{\alpha }\right)$ associated with any species in solution [65,93], whose macroscopic interpretation would lead naturally to the thermodynamic modeling of solvation in mixed-solvents.

The magnitude of the local density perturbation of the $\beta$-species environment around any $\alpha$-species can be rigorously determined in terms of the total correlation function integrals (TCFIs), also known as Kirkwood–Buff integrals [4], as follows [65,98]:

$${\mathcal{S}}_{\alpha \beta }\left(T,P,x_{\alpha }\right)=\rho x_{\beta }\left(G_{\alpha \beta }-G_{\beta \beta }\right)$$

(1)

with ${\mathcal{S}}_{\alpha \beta }\left(T,P,x_{\alpha }\right)\ne {\mathcal{S}}_{\beta \alpha }\left(T,P,x_{\alpha }\right)$ where ${\mathcal{N}}_{\alpha \beta }=\rho x_{\beta }{G}_{\alpha \beta }$ describes the absolute average number of $\beta$-species molecules around any central $\alpha$-species (solute or solvent) at the environmental state conditions and composition, while $\rho$ identifies the number density of the system. Therefore, the function ${\mathcal{S}}_{\alpha \beta }\left(T,P,x_{\alpha }\right)$ characterizes the average number of $\beta$-species around the central $\alpha$-species in excess/deficit to that encountered around any $\beta$-species, and consequently, embodies the required microscopic information to measure the magnitude of the $\alpha$-species-induced microstructural perturbation of the $\beta$-species environment,

$$G_{\beta \beta }\left(T,P,x_{\alpha }\right)\underset{\beta -species perturbation}{\overset{\alpha -species solvation}{\to }}{G}_{\alpha \beta }\left(T,P,x_{\alpha }\right)$$

(2)

where the arrow describes the perturbation of the solvent environment, as a TCFI, from its original magnitude to the final value. In fact, this evolution, ${\left(G_{\alpha \beta }-G_{\beta \beta }\right)}_{TPx}$, characterizes three possible scenarios as follows:

$${\left(G_{\alpha \beta }-G_{\beta \beta }\right)}_{TPx}\Rightarrow \left\{\begin{array}{l}{G}_{\alpha \beta }>G_{\beta \beta }\to {\mathcal{S}}_{\alpha \beta }>0\mathrm{s}tructuremaker\\ G_{\alpha \beta }=G_{\beta \beta }\to {\mathcal{S}}_{\alpha \beta }=0neithermakernorbreaker\\ G_{\alpha \beta }<G_{\beta \beta }\to {\mathcal{S}}_{\alpha \beta }<0\mathrm{s}tructurebreaker\end{array}\right.$$

(3)

Therefore, according to Equations (1)–(3), the solute-induced perturbation of the solvent environment translates unambiguously as a strengthening of the $\beta$-species environment around the $\alpha$-species when $G_{\alpha \beta }>G_{\beta \beta }$, i.e., ${\mathcal{S}}_{\alpha \beta }>0$, and the $\alpha$-species behaves as a structure-maker. Otherwise, when $G_{\alpha \beta }<G_{\beta \beta }$, i.e., ${\mathcal{S}}_{\alpha \beta }<0$, the solvent perturbation weakens the $\beta$-species environment around the $\alpha$-species and the $\alpha$-species becomes a structure-breaker. Obviously, there is a natural transition between the two cases, i.e., when the $\alpha$-species behaves as the $\beta$-species for which $G_{\alpha \beta }=G_{\beta \beta }⪋0$, i.e., ${\mathcal{S}}_{\alpha \beta }=0$, and consequently, the solvent environment remains unperturbed.

While the three conditions in Equation (3) describe microscopically three potential structure-making/-breaking scenarios for any electrolyte or non-electrolyte binary solution, we need to be able to assess those conditions based on its thermodynamic properties. For that purpose, in Refs. [3,65] we have derived the following formal expressions for the fundamental structure-making/-breaking function ${\mathcal{S}}_{\alpha \beta }\left(T,P,x_{\alpha }\right)$ for any $\alpha$-species at any state condition and composition,

$${\mathcal{S}}_{\alpha \beta }\left(T,P,x_{\alpha }\right)=1-\rho {\widehat{\upsilon }}_{\alpha }/\nu D\Rightarrow \left\{\begin{array}{l}\rho {\widehat{\upsilon }}_{\alpha }<\nu D\to {\mathcal{S}}_{\alpha \beta }>0 structure maker\\ \rho {\widehat{\upsilon }}_{\alpha }=\nu D\to {\mathcal{S}}_{\alpha \beta }=0 neither maker nor breaker\\ \rho {\widehat{\upsilon }}_{\alpha }>\nu D\to {\mathcal{S}}_{\alpha \beta }<0 structure breaker\end{array}\right.$$

(4)

where $\nu$ is the stoichiometric coefficient of the electrolyte $\alpha$-species (i.e., $\nu =1$ for a non-electrolyte), ${\widehat{\upsilon }}_{\alpha }$ denotes its partial molar volume, and $D$ denotes the diffusive stability coefficient [7]. In other words, Equation (4) describes the thermodynamic response to the $\alpha$-species-induced effects on the microstructure of the $\beta$-species, and reveals rigorously and unambiguously which of the possible behaviors of the $\alpha$-species without requiring any explicit microstructural information of the system.

Note that, as hinted above by the inequality ${\mathcal{S}}_{\alpha \beta }\left(T,P,x_{\alpha }\right)\ne {\mathcal{S}}_{\beta \alpha }\left(T,P,x_{\alpha }\right)$, these structure-making/-breaking functions for finite concentrations are defined as pairs, and are formally connected to other solvation-related thermodynamic properties, including the isochoric–isothermal pressure effect upon solvation, ${\left(\partial P/\partial x_{\alpha }\right)}_{T\rho }$, the isothermal–isobaric diffusive stability coefficient, $D=\left[1+{\left(\partial \ln {\gamma }_j/\partial \ln x_j\right)}_{TP}\right]$, and the universal preferential solvation of a species at infinite dilution in mixed-solvents as lengthily discussed elsewhere [3,93]. However, as one of the species approaches infinite dilution, then

$$\begin{array}{l}\underset{x_{\alpha }\to 0}{\lim }{\mathcal{S}}_{\alpha \beta }\left(T,P,x_{\alpha }\right)\equiv {\mathcal{S}}_{\alpha \beta }^{\infty }=1-{\widehat{\upsilon }}_{\alpha }^{\infty }/\nu {\upsilon }_{\beta }^o\\ \underset{x_{\alpha }\to 0}{\lim }{\mathcal{S}}_{\beta \alpha }\left(T,P,x_{\alpha }\right)\equiv {\mathcal{S}}_{\beta \alpha }^{\infty }=1\end{array}$$

(5)

so that, the system becomes characterized by just one fundamental structure-making/-breaking function, ${\mathcal{S}}_{\alpha \beta }^{\infty }\left(T,P\right)$.

2.2. Thermodynamic Preferential Interaction Parameters in Open, Semi-Open, and Closed Systems

Thermodynamic preferential interaction (or binding) parameters in ternary systems are defined according to the number of diffusible species, and consequently, associated with a particular experimental approach including vapor pressure osmometry [99,100], isopiestic distillation [101,102], and equilibrium dialysis [103,104]. Thus, depending on which species in the ternary system is diffusible, either solvent, cosolvent, or both, these preferential interaction parameters are defined as ${\Gamma }_{{\mathcal{Q}}_j{\mathcal{Q}}_k}^{({\chi }_i)}\left(x_j\right)={\left(\partial {\chi }_k/\partial {\chi }_i\right)}_{T{\mathcal{Q}}_j{\mathcal{Q}}_k}$, i.e., the isothermal change in the diffusible solvent concentration in response to the dissolution of a solute when the chemical potential of the diffusible species is kept constant [105,106]. Unfortunately, according to the derivations by Smith [91] and others [90,92,93], these thermodynamic responses lead to a variety of composition-dependent microstructural manifestations depending on the (open/closed) nature of the system under study as illustrated in

Table 1. Relations between the thermodynamic preferential interaction parameters and the Kirkwood–Buff integrals according to the nature of the system [91].

$\mathit{S}\mathit{y}\mathit{s}\mathit{t}\mathit{e}\mathit{m}$	$\mathit{\rho }$	$\mathit{m}$	$\mathit{x}$
${\Gamma }_{{\mu }_j{\mu }_k}^{({\chi }_i\to 0)}\left(G_{\alpha \beta }\right)$ 1	${\rho }_k{G}_{ik}^{\infty }$	${\rho }_k\left(G_{ik}^{\infty }-G_{ij}^{\infty }\right)$	$-x_k+x_j{\rho }_k\left(G_{ik}^{\infty }-G_{ij}^{\infty }\right)$
${\Gamma }_{{\mu }_j}^{({\chi }_i\to 0)}\left(G_{\alpha \beta }\right)$ 2	$-1+{\rho }_k\left(G_{ik}^{\infty }-G_{kk}\right)$	$-1+{\rho }_k\left(G_{ik}^{\infty }+G_{kj}-G_{ij}^{\infty }-G_{kk}\right)$	$-1+x_j{\rho }_k\left(G_{ik}^{\infty }-G_{ij}^{\infty }+G_{jk}-G_{kk}\right)$
${\Gamma }_{{\mu }_k}^{({\chi }_i\to 0)}\left(G_{\alpha \beta }\right)$ 2	${\rho }_k\left(G_{ik}^{\infty }-G_{jk}\right)$	$\left({\rho }_k/{\rho }_j\right)+{\rho }_k\left(G_{ik}^{\infty }-G_{ij}^{\infty }+G_{jj}-G_{jk}\right)$	$x_j{\rho }_k\left(G_{ik}^{\infty }-G_{ij}^{\infty }+G_{jj}-G_{jk}\right)$

¹ open, ² semi-open.

. See Appendix B of Ref. [93] for a detailed derivation of these formal expressions.

We can additionally define the thermodynamic preferential solvation parameter in the isothermal–isobaric closed system, ${\Gamma }_{i\alpha }^{({\chi }_i\to 0)}\left(x_{\alpha }\right)\equiv {\left(\partial {\mu }_i^{\ominus }/\partial {\mu }_{\alpha }\right)}_{TP}$, sometimes identified in the literature by the symbol ${\nu }_{i\alpha }$ for $\alpha =\left(j,k\right)$, [86,107] as follows:

$${\Gamma }_{i\alpha }^{(x_i\to 0)}\left(x_{\alpha }\right)\equiv {\rho }_{\alpha }\left(G_{i\beta }^{\infty }-G_{i\alpha }^{\infty }\right)$$

(6)

where ${\mu }_i^{\ominus }\left(T,P,x_j\right)=\underset{x_i\to 0}{\lim }{\left({\mu }_i-kT\ln {\rho }_i\right)}_{TP}$ describes the standard chemical potential of the $i$-solute in the mixed-solvent environment. Therefore, for the $\left(\alpha +\beta \right)$-binary solvent, the two preferential solvation parameters are related as $x_{\alpha }/{\Gamma }_{i\alpha }^{(x_i\to 0)}\left(x_{\alpha }\right)+x_{\beta }/{\Gamma }_{i\beta }^{(x_i\to 0)}\left(x_{\alpha }\right)=0$ by the Gibbs–Duhem relation.

In summary, given the diversity of the microstructural outcomes as attested in Table 1, it becomes difficult to choose one of the thermodynamic preferential solvation parameters, and its microscopic analog, as the signature for the preferential solvation phenomenon ($\mathcal{P}\mathcal{S}$). In other words, at this juncture, we could not assert anything about the $\mathcal{P}\mathcal{S}$ behavior of a solute in a mixed-solvent system given the lack of a precise microstructural manifestation of the differential solute affinity of the solvent environment relative to that of the cosolvent around the solute.

2.3. Universal Preferential Solvation Function and Its Link to the Fundamental Structure-Making/-Breaking Functions

One of the first preferential solvation measures was introduced by Ben-Naim [83] and described as the first-order coefficient of the expansion of the local mole fraction deviation of the solvent around the infinitely dilute $i$-solute in a mixed $\left(j+k\right)$-solvent system, ${\delta }_{ji}=x_{ji}\left(local\right)-x_j$, leading to the following formal definition:

$${\delta }_{ji}^0=x_j{x}_k\left(G_{ij}^{\infty }-G_{ik}^{\infty }\right)$$

(7)

Equation (7) comprises at least two essential ingredients for the formal definition and assessment of $\mathcal{P}\mathcal{S}$; namely, it measures the difference in the $i$-solute affinities between the solvent and the cosolvent; $\left(G_{ij}^{\infty }-G_{ik}^{\infty }\right)$, and the alluded difference is explicitly and rigorously connected to the Gibbs free energy of transfer between a pure solvent and the desired mixed-solvent environment. Alternative definitions of $\mathcal{P}\mathcal{S}$ have been given in the literature; including the use of ${\nu }_{i\alpha }$, Equation (6) [108,109];

$$\begin{array}{cc}\hfill \mathcal{P}{\mathcal{S}}^{\prime }\left(x_j\right)& \equiv {\rho }_k\left(G_{ij}^{\infty }-G_{ik}^{\infty }\right)\hfill \\ & ={\rho }_o{\delta }_{ji}^0/x_j\hfill \end{array}$$

(8)

where ${\rho }_o=\underset{x_i\to 0}{\lim }\rho$ and [110],

$$\mathcal{P}{\mathcal{S}}^{″}\left(x_j\right)\equiv \left[\left(G_{ij}-G_{ik}\right)+\rho \left({\widehat{\upsilon }}_k-{\widehat{\upsilon }}_j\right)\right]x_i$$

(9)

Unfortunately, as discussed elsewhere [93], Equations (7) and (8) share some common drawbacks regarding the misrepresentation of the actual preferential solvation at the infinite dilution of either the $k$- or the $j$-species. In fact, according to Equation (7), ${\delta }_{ji}^0\left({\rho }_k\to 0\right)=0$ regardless of the relative affinity of the $i$-solute with either the $j$-solvent or the $k$-cosolvent, and represents a rather misleading characterization of the relative affinity of the $i$-solute between the pair of solvents. By considering the definition of relative affinity, $\mathcal{R}\mathcal{A}\left(x_{\alpha }\right)$, we find that

$$\mathcal{R}\mathcal{A}\left(x_k\to 0\right)\equiv \underset{x_k\to 0}{\lim }\left(G_{ik}^{\infty }-G_{ij}^{\infty }\right)\ne 0$$

(10)

Equation (10) implies that as $x_k\to 0$, the system behaves as an infinitely dilute $\left(i+j\right)$-binary system so that $\mathcal{R}\mathcal{A}\left(x_j\to 1\right)={\left[\partial {\mu }_i^{\ominus }\left(x_j\right)/\partial {\mu }_j\left(x_j\right)\right]}_{TP}\ne 0$ except for the unlikely case when $G_{ik}^{\infty }=G_{ij}^{\infty }$ at the actual state conditions of the mixed-solvents. Moreover, we should also highlight a yet unreported unfortunate feature of Equation (9) which predicts the unphysical outcome $\mathcal{P}{{\mathcal{S}}^{″}}^{,IG\_i}\left(x_i\ne 0\right)\ne 0$ for an ideal gas solute $IG\_i$, i.e.,

$$\mathcal{P}{{\mathcal{S}}^{″}}^{,IG\_i}\left(x_i\ne 0\right)={\rho }_i\left({\widehat{\upsilon }}_k-{\widehat{\upsilon }}_j\right)\ne 0$$

(11)

unless ${\left({\widehat{\upsilon }}_k={\widehat{\upsilon }}_j\right)}_{TPx}$.

The most promising alternative $\mathcal{P}\mathcal{S}$ definition was recently introduced by the author as a universal state function [93],

$$\mathcal{P}\mathcal{S}\left(T,P,x_j\right)={\rho }_o\left(G_{ik}^{\infty }-G_{ij}^{\infty }\right)$$

(12)

considering that any formal unambiguous definition must obey a pair of natural constraints in the description of an infinitely dilute $i$-solute in a $\left(j+k\right)$-mixed solvent at fixed state conditions. The alluded scenarios are as follows: (a) a non-interacting (ideal gas) $i$-species must depict a null preference for either solvent species, $\mathcal{P}{\mathcal{S}}^{IG\_i}\left(x_j\right)=0$, and (b) when a $j$-solvent and the $k$-cosolvent behave identically, the $i$-species prefers equally either species in the $\left(j+k\right)$-mixed environment, a condition that renders identical solute–solvent and solute–cosolvent intermolecular interaction asymmetries, i.e., $\mathcal{P}\mathcal{S}\left(x_j\right)=0$ with $G_{jj}=G_{jk}=G_{kk}=\left(kT{\kappa }_j^o-{\upsilon }_j^o\right)\ne 0$. Moreover, $\mathcal{P}\mathcal{S}\left(x_j\right)\ne 0$ represents the microstructural response of the $\left(j+k\right)$ mixed solvent to the solute–solvent and solute–cosolvent interaction asymmetries, i.e., differential affinities of the solute between the solvent and the cosolvent whose macroscopic manifestations are given by the thermodynamic excess properties describing the solution non-idealities [7].

2.4. Links between the Thermodynamic Preferential Interaction Parameters and the Universal Preferential Solvation Function

The existing concerns propelled a careful scrutiny of the thermodynamic definitions of preferential interaction parameters and the alternative microstructural signatures of preferential solvation. These concerns were spelled out as a pair of questions: “does preferential interaction mean preferential solvation?, or alternatively, has preferential interaction been unintentionally taken as an equivalent to preferential solvation?” [93]. The microstructural interpretation of the thermodynamic preferential interaction parameters in Table 1 implies that most of them cannot describe preferential solvation, ${\Gamma }_{{\mathcal{Q}}_j{\mathcal{Q}}_k}^{({\chi }_i\to 0)}\left(x_j\right)\leftrightarrow \mathcal{P}\mathcal{S}\left(x_j\right)$, because they cannot fulfill the null requirement discussed in the previous section. More problematic is the fact that, as addressed elsewhere, [93] the thermodynamic preferential interaction ${\Gamma }_{{\mathcal{Q}}_j{\mathcal{Q}}_k}^{({\chi }_i\to 0)}\left(x_j\right)$ parameters and the preferential solvation $\mathcal{P}\mathcal{S}\left(x_j\right)$ markers might plausibly be describing either the same microscopic marker, yet resulting from two different thermodynamic responses, or representing two different microstructural scenarios.

In fact, from a microstructural viewpoint, we established the ${\Gamma }_{{\mu }_j{\mu }_k}^{(m_i\to 0)}\left(x_j\right)={\Gamma }_{i\alpha }^{(x_i\to 0)}\left(x_j\right)$ identity, where the left- and right-hand sides of this expression are representations of two significantly different thermodynamic responses given the nature of the systems involved. To be more precise, the left-hand side describes an open system where ${\Gamma }_{{\mu }_j{\mu }_k}^{(m_i\to 0)}\left(x_j\right)={\left(\partial m_k/\partial m_i\right)}_{T{\mu }_j{\mu }_k}^{m_i\to 0}$ measures the rate of change in the molal concentration of the $k$-cosolvent when perturbing the molality of the $i$-solute around its infinite dilution while the chemical potential of the mixed-solute is kept constant. In contrast, the right-hand side describes a closed system where ${\Gamma }_{i\alpha }^{(x_i\to 0)}\left(x_j\right)={\left(\partial {\mu }_i^{\ominus }/\partial {\mu }_{\alpha }\right)}_{TP}$ defines the rate of change in the standard chemical potential of the $i$-solute caused by a perturbation of the composition of the $\alpha$-solvent. Likewise, ${\Gamma }_{{\mu }_j{\mu }_k}^{(m_i\to 0)}\left(x_j\right)={\Gamma }_{i\alpha }^{(x_i\to 0)}\left(x_j\right)=0$ might be describing two significantly different intermolecular interaction asymmetries, such as a system comprising a real $i$-solute in an ideal mixed-solvent whose species behave identically, in contrast to another system involving an ideal gas $i$-solute in a real mixed-solvent. Obviously, these systems represent two completely different microstructural environments, i.e., $G_{ij}^{\infty }=G_{ik}^{\infty }\ne 0$ for the first scenario, and $G_{ij}^{\infty }=G_{ik}^{\infty }=0$ for the second.

In summary, while the description of the thermodynamic preferential interaction parameters and the preferential solvation counterparts are all rigorous for the infinitely dilute solute in a binary mixed-solvent, we must be cautious when discussing preferential interaction and preferential solvation so that they are not taken as equivalent concepts, and avoid the misinterpretations of the experimental evidence, leading to the mischaracterization of the solvation behavior of species in mixed-solvent environments.

3. Exploring the Preferential Solvation Behavior of Solutes in Mixed-Solvents

The differential solvation behavior of an $i$-solute at infinite dilution in a $\left(j+k\right)$-binary solvent environment has been described in terms of precisely defined fundamental structure-making/-breaking functions ${\mathcal{S}}_{\alpha \beta }$ [93], and its macroscopic (thermodynamic) responses through the Kirkwood–Buff fluctuation formalism [111]. This depiction ensures a highly desirable uniqueness in the outcome while imposing no constraints on the type of solute, the nature of the mixed-solvents, or the solute–solvent and solute–cosolvent intermolecular interaction asymmetries.

To explore the differential solvation behavior of any $i$-solute at infinite dilution in a $\left(j+k\right)$-binary solvent, let us consider an isobaric–isothermal ternary system comprising $N_j$ solvent and $N_k$ cosolvent particles with $N_i$ solute species such that $\left(N_j+N_k\right)\gg N_i\to 0$. Under these conditions, we denote the number density of an $\alpha$-species as ${\rho }_{\alpha }=\rho x_{\alpha }$, where $\rho \equiv \left(N_i+N_j+N_k\right)/V$ with the mole fraction $x_{\alpha }=N_{\alpha }/N$. To connect the microscopic behavior of the solvents around the $i$-solute and the macroscopic (thermodynamic) responses, we recall that at isothermal conditions, the species densities in the phase are functions of the chemical potentials in the grand canonical (open) ensemble, so that for our system we have,

$$kT{\left(d\ln {\rho }_i\right)}_T=\left(1+{\rho }_i{G}_{ii}\right)d{\mu }_i+{\rho }_j{G}_{ij}d{\mu }_j+{\rho }_k{G}_{ik}d{\mu }_k$$

(13)

Consequently, after invoking the definition of the standard chemical potential ${\mu }_i^{\ominus }\left(T,P,x_j\right)$ and the ${\rho }_i\to 0$ limit, we have the following:

$$d{\mu }_i^{\ominus }{\left(x_j\right)}_{TP}=-{\rho }_j{G}_{ij}^{\infty }d{\mu }_j-{\rho }_k{G}_{ik}^{\infty }d{\mu }_k$$

(14)

where the system is now a binary $\left(j+k\right)$-mixed solvent whose Gibbs–Duhem relation ${\left({\rho }_{j}d{\mu }_j+{\rho }_{k}d{\mu }_k\right)}_{TP}=0$ leads to the desired expression linking thermodynamics, the left-hand side of Equation (15), and microstructure, the right-hand side of Equation (15), as follows:

$$\begin{array}{cc}\hfill {\left[\partial {\mu }_i^{\ominus }\left(T,P,x_j\right)/\partial {\mu }_j\right]}_{TP}& ={\rho }_j\left(G_{ik}^{\infty }-G_{ij}^{\infty }\right)\hfill \\ & =x_j\mathcal{P}\mathcal{S}\hfill \end{array}$$

(15)

Equation (15) embodies the mechanism behind the tuning of the solvation behavior of an $i$-solute in a (diffusively stable, i.e., finite $D\left(x_j\right)>0$) mixed $\left(j+k\right)$-solvent environment through the manipulation of the chemical potential of the $j$-solvent.

To draw a more explicit link, we need to formally determine the five $TCFI$’s (also known as Kirkwood–Buff integrals), $G_{\alpha \beta }$, through the inversion of the Kirkwood–Buff formalism [112]. Typically, we first extract the first three functional relations $G_{\alpha \beta }\left({\widehat{\upsilon }}_{\alpha },{\widehat{\upsilon }}_{\beta },{\kappa }_o,D\right)$ of the mixed-solvent phase given explicitly as follows (see Appendix C of Ref. [93] for details):

$$G_{jk}=kT{\kappa }_o-{\widehat{\upsilon }}_j{\widehat{\upsilon }}_k{\rho }_o{D}^{-1}$$

(16)

$$G_{jj}=G_{jk}+\left({\rho }_o{\widehat{\upsilon }}_k/D-1\right){\rho }_j^{-1}$$

(17)

$$G_{kk}=G_{jk}+\left({\rho }_o{\widehat{\upsilon }}_j/D-1\right){\rho }_k^{-1}$$

(18)

in terms of the species partial molar volumes $\left({\widehat{\upsilon }}_i^{\infty },{\widehat{\upsilon }}_j,{\widehat{\upsilon }}_k\right)$, the diffusive stability coefficient $D$, and the isothermal compressibility ${\kappa }_o$ of the mixed-solvent environment. For the remaining pair $G_{i\alpha }^{\infty }\left({\widehat{\upsilon }}_i^{\infty },{\widehat{\upsilon }}_{\beta },{\kappa }_o,D,{\left[\partial {\mu }_i^{\ominus }\left(x_j\right)/\partial x_j\right]}_{TP}\right)$ integrals, we need the composition dependence of the partial molar volume of the $i$-solute, ${\widehat{\upsilon }}_i^{\infty }\left(x_j\right)$, and the rate of change in the standard chemical potential of the $i$-solute in response to variations in the mixed-solvent composition, ${\left[\partial {\mu }_i^{\ominus }\left(x_j\right)/\partial x_j\right]}_{TP}$. In particular, this thermodynamic derivative is usually available in the following equivalent forms, whose choice might depend on either the type of solute or the nature of the experimental solubility data (see Appendix D of Ref. [113] for details)

$${\left[\partial {\mu }_i^{\ominus }\left(x_j\right)/\partial x_j\right]}_{TP}=\left\{\begin{array}{l}{\left[\partial {\mathsf{\Delta }}_{tr}{g}_i\left(x_j\right)/\partial x_j\right]}_{TP}\\ kT{\left[\partial \ln {\left({\widehat{\varphi }}_{i}z\right)}^{\infty }/\partial x_j\right]}_{TP}\\ {\left[\partial {\mu }_i^{\ast }\left(x_j\right)/\partial x_j\right]}_{TP}\end{array}\right.$$

(19)

where the first line on the right of Equation (19) involves the Gibbs free energy of the transfer of the $i$-solute between the pure $k$-cosolvent phase to the binary $\left(j+k\right)$-mixed solvent, and the second and third lines invoke two representations of the isothermal–isochoric residual chemical potential of the $i$-solute at infinite dilution, ${\mu }_i^r\left(T,{\rho }_o,x_j\right)$, with ${\mu }_i^{*}\left(T,{\rho }_o,x_j\right)$ identifying the pseudo chemical potential [114]. Then, from Equations (15) and (19), we complete the inversion, i.e.,

$$G_{ij}^{\infty }=kT{\kappa }_o-{\widehat{\upsilon }}_i^{\infty }-x_k{\widehat{\upsilon }}_k{D}^{-1}{\left[\partial {\mu }_i^{\ominus }\left(x_j\right)/\partial x_j\right]}_{TP}$$

(20)

and,

$$G_{ik}^{\infty }=kT{\kappa }_o-{\widehat{\upsilon }}_i^{\infty }+x_j{\widehat{\upsilon }}_j{D}^{-1}{\left[\partial {\mu }_i^{\ominus }\left(x_j\right)/\partial x_j\right]}_{TP}$$

(21)

Finally, from Equations (15)–(19), we find that either the microscopic quantity $\mathcal{P}\mathcal{S}\left(T,P,x_j\right)$,

$$\mathcal{P}\mathcal{S}\left(T,P,x_j\right)=\left({\mathcal{S}}_{ik}^{\infty }-{\mathcal{S}}_{jk}\right)/x_k-\left({\mathcal{S}}_{ij}^{\infty }-{\mathcal{S}}_{kj}\right)/x_j$$

(22)

or the thermodynamic property ${\left[\partial {\mu }_i^{\ominus }\left(T,P,x_j\right)/\partial {\mu }_j\right]}_{TP}$,

$${\left[\partial {\mu }_i^{\ominus }\left(T,P,x_j\right)/\partial {\mu }_j\right]}_{TP}=\left(x_j/x_k\right)\left({\mathcal{S}}_{ik}^{\infty }-{\mathcal{S}}_{jk}\right)-\left({\mathcal{S}}_{ij}^{\infty }-{\mathcal{S}}_{kj}\right)$$

(23)

is described entirely in terms of a linear combination of the same four fundamental structure-making/-breaking functions. These equations provide the formal and rigorous interpretation of the source of the differential solvation, i.e., it results from a linear combination of solute-induced and solvent-induced effects on the microstructure of the mixed-solvent environment, while highlighting the link to the non-ideality of the mixed-solvent ternary system.

3.1. Preferential Solvation of Non-Polar Gasses in Polar Mixed-Solvent Environments

The understanding of the solvation behavior of volatile solutes in mixed-solvent environments is of great theoretical and practical relevance [65,93,115,116]. In fact, while traditionally challenging to model properly given the large solute–solvent intermolecular interaction asymmetries [65,93,115,116], the availability of a complete experimental dataset provided the opportunity to apply the discussed formalism to study a series of non-polar gases in 2,2,2-trifluoroethanol–water mixed-solvent systems based on the solubility of $\mathrm{He}$, $\mathrm{Ne}$, $\mathrm{Ar}$, $\mathrm{Kr}$, $\mathrm{Xe}$, ${\mathrm{H}}_2$, ${\mathrm{N}}_2$, ${\mathrm{O}}_2$, ${\mathrm{CH}}_4$, ${\mathrm{C}}_2{\mathrm{H}}_4$, ${\mathrm{C}}_2{\mathrm{H}}_6$, ${\mathrm{CO}}_2$, ${\mathrm{CF}}_4$, and ${\mathrm{SF}}_6$ at ambient conditions in light water [117], 2,2,2-trifluoroethanol (TFE) [118], and the 2,2,2-trifluoroethanol–water mixed-solvent over the entire range of composition [119,120].

For the purpose of the Kirkwood–Buff inversion, the solubility data were represented by ${\left[\partial {\mu }_i^{\ominus }\left(x_j\right)/\partial x_j\right]}_{TP}={\left[\partial \ln {\left({\widehat{\varphi }}_{i}z\right)}^{\infty }/\partial x_j\right]}_{TP}$ in Equations (20) and (21), and complemented with the composition dependence of the partial molar volumes [121], the isothermal compressibility [122], and the activity coefficients [123] of the gases in 2,2,2-trifluoroethanol–water mixed-solvent. Moreover, for the partial molar volume of the $i$-solutes at infinite dilution in the mixed-solvents, we invoked the composition dependence resulting from the quasi-ideal solution (also known as Krichevskii’s relation [124,125]) behavior of Henry’s law constant of the solutes in the mixed-solvent environments, $\ln {\mathcal{H}}_{i,j+k}^E\left(x_j\right)\simeq 0$ [119,120], i.e.,

$$\ln {\mathcal{H}}_{i,j+k}^{Krichevskii-IS}\left(T,P,x_j\right)\cong x_j\ln {\mathcal{H}}_{i,j}\left(T,P\right)+\left(1-x_j\right)\ln {\mathcal{H}}_{i,k}\left(T,P\right)$$

(24)

where ${\mathcal{H}}_{i,\alpha }\left(T,P\right)$ identifies Henry’s law constant of the $i$-solute in a pure $\alpha$-solvent at the prevailing state conditions (e.g., see Figure 1 in Ref. [113]). This relation, written in the following alternative form,

$$\ln {\widehat{\varphi }}_{i,j+k}^{\infty }\left(T,P,x_j\right)\cong x_j\ln {\widehat{\varphi }}_{i,j}^{\infty }+\left(1-x_j\right)\ln {\widehat{\varphi }}_{i,k}^{\infty }$$

(25)

whose isothermal-pressure derivative, after assuming the validity of Equation (24) over the isothermal $\left(P\pm \mathsf{\Delta }P\right)$ range, leads to the sought expression for ${\widehat{\upsilon }}_{i,j+k}^{\infty }\left(T,P,x_j\right)$,

$${\widehat{\upsilon }}_{i,j+k}^{\infty }\left(T,P,x_j\right)\cong x_j{\widehat{\upsilon }}_{i,j}^{\infty }\left(T,P\right)+\left(1-x_j\right){\widehat{\upsilon }}_{i,k}^{\infty }\left(T,P\right)$$

(26)

where ${\widehat{\upsilon }}_{i,\alpha }^{\infty }\left(T,P\right)$ denotes the partial molar volume counterparts to ${\mathcal{H}}_{i,\alpha }\left(T,P\right)$.

The isobaric–isothermal composition-dependent microstructure of the $\left(\mathrm{TFE}+{\mathrm{H}}_2\mathrm{O}\right)$-system becomes represented by $G_{TFE TFE} \left(x_{TFE}<0.32\right)>0$ with a maximum at $x_{TFE}\simeq 0.16$ and $G_{TFE TFE} \left(x_{TFE}\gtrsim 0.32\right)<0$, as well as $G_{H_{2}O H_{2}O} \left(x_{TFE}<0.05\right)<0$ and $G_{H_{2}O H_{2}O} \left(x_{TFE}\gtrsim 0.05\right)>0$, resulting in a significantly large positive deviation from Lewis–Randall ideality, ${\mathsf{\Delta }}_{TFE H_{2}O}^{LR-IS}\left(T,P,x_{TFE}\right)=0$, [112,126] as depicted in Figure 1.

On the one hand, this picture shows a significant magnification of $G_{H_{2}O H_{2}O }\left(x_{TFE}\right)$ from its pure component counterpart $G_{H_{2}O H_{2}O}^o = kT{\kappa }_{H_{2}O}^o-{\upsilon }_{H_{2}O}^o$, with its maximum value at $x_{TFE}\simeq 0.25$, while $G_{TFE H_{2}O} \left(0\le x_{TFE}\le 1\right)<0$ and exhibits a minimum at $x_{TFE}\simeq 0.2$. This microstructural behavior translates also into a significant reduction in the diffusive stability coefficient, $D\left(x_{TFE}\simeq 0.2\right)\simeq 0.1$, from its Lewis-Randall ideality $D^{LR-IS}\left(x_{TFE}\right)=1$ counterpart as depicted in Figure 2a, which indicates a close approach to the incipient immiscibility condition [127]. On the other hand, the observed maxima in the total correlation function integrals $G_{H_{2}O H_{2}O }\left(x_{TFE}\simeq 0.25\right)$ and $G_{TFE TFE} \left(x_{TFE}\simeq 0.16\right)$ are the microstructural manifestations of like-species aggregation in Figure 1, which can be identified as a ${\mathrm{H}}_2\mathrm{O}$-induced structure-breaking of the $\mathrm{TFE}$-environment, ${\mathcal{S}}_{H_{2}O TFE }\left(x_{TFE}\right)<0$, and simultaneously, a $\mathrm{TFE}$-induced structure-breaking of the ${\mathrm{H}}_2\mathrm{O}$-environment, ${\mathcal{S}}_{TFE H_{2}O}\left(x_{TFE}\right)<0$, as evidenced in Figure 2b.

For the interpretation of the differential solvation behavior of the non-polar gases in the mixed $\left(\mathrm{TFE}+{\mathrm{H}}_2\mathrm{O}\right)$-solvent environment, we evaluate simultaneously the isobaric-isothermal behavior of the two relevant structure-making/-breaking functions, ${\mathcal{S}}_{i TFE}^{\infty }\left(x_{TFE}\right)$ and ${\mathcal{S}}_{i H_{2}O}^{\infty }\left(x_{TFE}\right)$, as well as the resulting preferential solvation function $\mathcal{P}\mathcal{S}\left(x_{TFE}\right)$, where the $i$-species identifies one of the fourteen non-polar gas solutes at infinite dilution and ambient conditions. In fact, from Figure 3a,b we can highlight some common features underlying the behavior of ${\mathcal{S}}_{i TFE}^{\infty }\left(x_{TFE}\right)$ and ${\mathcal{S}}_{i H_{2}O}^{\infty }\left(x_{TFE}\right)$, which translates into the final differential solvation of the infinitely dilute solutes between the solvent and the cosolvent.

On the one hand, the function ${\mathcal{S}}_{i TFE}^{\infty }\left(x_{TFE}\right)$ in Figure 3a starts at ${\mathcal{S}}_{i TFE}^{\infty }\left(x_{TFE}=0\right)=0$, follows through a negligibly small negative value at $x_{TFE}\simeq 0.03$, becomes quickly large and positive at increasing $\mathrm{TFE}$-concentration, goes through a maximum within $0.15<x_{TFE}<0.25$, to finally decrease to ${\mathcal{S}}_{i TFE}^{\infty }\left(x_{TFE}=1\right)={\rho }_{TFE}^o\left(G_{ i TFE}^{\infty }-G_{TFE TFE}^o\right)\lessgtr 0$. This behavior indicates that the solvation of the gas solutes enhances the microstructure of the $\mathrm{TFE}$-solvent, i.e., they behave as structure-making $i$-solutes, except for the negligibly small structure-breaking effect for $x_{TFE}\le 0.03$, and for $i=\left({\mathrm{CF}}_4,{\mathrm{SF}}_6\right)$ at $x_{TFE}>0.6$. On the other hand, in Figure 3b ${\mathcal{S}}_{i H_{2}O}^{\infty }\left(x_{TFE}\right)$ starts at ${\mathcal{S}}_{i H_{2}O}^{\infty }\left(x_{TFE}=0\right)={\rho }_{H_{2}O}^o\left(G_{ i H_{2}O}^{\infty }-G_{H_{2}O H_{2}O}^o\right)⪅0$, decreases significantly up to a minimum at $x_{TFE}\simeq 0.2$ (similar to that for the diffusive stability coefficient $D$), and converges at ${\mathcal{S}}_{i H_{2}O}^{\infty }\left(x_{TFE}=1\right)=0$. Thus, the solvation of these gas solutes strongly weakens the microstructure of the ${\mathrm{H}}_2\mathrm{O}$-solvent, i.e., they behave as structure-breaking $i$-solutes over the entire range of mixed-solvent composition.

The observed opposite structure-making/-breaking trends in Figure 3a,b give rise to a prominent change in the sign of the preferential solvation functions of the gas solutes, i.e., from $\mathcal{P}\mathcal{S}\left(x_{TFE}\to 0\right)\gg 0$ to $\mathcal{P}\mathcal{S}\left(x_{TFE}\to 1\right)⪅0$, through a significantly large minimum $\mathcal{P}\mathcal{S}\left(0.15\le x_{TFE}\le 0.19\right)\ll 0$, as depicted in Figure 4a. This behavior appears as a common feature of all these gas solutes in the mixed $\left(\mathrm{TFE}+{\mathrm{H}}_2\mathrm{O}\right)$-solvent environment, and becomes the signature of the fact that these gas solutes are better solvated by water than by the mixed $\left(\mathrm{TFE}+{\mathrm{H}}_2\mathrm{O}\right)$-environment when the solvent is highly dilute in $\mathrm{TFE}$, i.e., except for the narrow concentration range. Indeed, for $0.03\le x_{TFE}\le 0.05$, these gas solutes are equally solvated by the two solvents, while at $x_{TFE}⪅0.03$ these solutes become better solvated by $\mathrm{TFE}$ than water. Thus, if we wish to tune the solubility of these gas solutes, Figure 4b illustrates how the manipulation of the chemical potential of the $\mathrm{TFE}$-solvent provides the way to accomplish the task, i.e., ${\left[\partial {\mu }_i^{\ominus }\left(x_{TFE}\right)/\partial {\mu }_{TFE}\right]}_{TP}$ or its equivalent form,

$${\left[\partial {\mu }_i^{\ominus }\left(x_{TFE}\right)/\partial {\mu }_{TFE}\right]}_{TP}=kT{\left\{\partial \ln \left[{\mathcal{H}}_{i,TFE+H_{2}O}\left(x_{TFE}\right)/\rho kT\right]/\partial {\mu }_{TFE}\right\}}_{TP}$$

(27)

In other words, Equation (27) and Figure 4b indicate that by perturbing the chemical potential of the $\mathrm{TFE}$-solvent (through changes in the composition of the mixed-solvent), we can modify the standard chemical potential of any of the gas solutes, and consequently, their solubility, i.e., the magnitude of the $i$-solute’s Henry law constant.

3.2. Preferential Solvation of Pharmaceutical Species in Polar Mixed-Solvent Environments

The incorporation of a second solvent might have opposite effects on the resulting solute solubility; cosolvents might increase the solubility of non-polar drugs over that in aqueous environments, a desired goal for the formulations of concentrated solutions of non-polar drugs for systemic therapeutic treatments [128,129]; meanwhile, a co-non-solvent will drastically reduce the solubility of a solute in mixed-solvent environments whose individual components exhibit good solvency, to become a versatile tool for the manipulation of solute aggregation and/or crystallization toward specific pharmaceutical applications [130,131,132]. For example, we have recently explored the preferential solvation behavior of a few pharmaceutical solutes [133], for which the Kirkwood–Buff inversion involved the solubility data as ${\left[\partial {\mu }_i^{\ominus }\left(x_j\right)/\partial x_j\right]}_{TP}={\left[\partial {\mathsf{\Delta }}_{tr}{g}_i\left(x_j\right)/\partial x_j\right]}_{TP}$ in Equations (20) and (21), and comprising a common mixed ethyl acetate (j) + ethyl alcohol (k) environment, including indomethacin [134], naproxen [134], meloxicam [135], and piroxicam [136], used as nonsteroidal anti-inflammatory drugs, while mevastatin [137] prescribed as a cholesterol-lowering agent, and posaconazole [138], involved in the treatment of fungal infections [139].

In Figure 5a, we display the universal preferential solvation $\mathcal{P}\mathcal{S}\left(x_{ETAC}\right)$ behavior of these six pharmaceutical solutes in ethyl acetate–ethanol mixed-solvent systems, and contrast them with the corresponding first-order preferential solvation parameter ${\delta }_{ETAC i}^0\left(x_{ETAC}\right)$ in Figure 5b. The remarkably similar outcome of the universal preferential solvation behavior of this pharmaceutical species is their increasing trend with the increasing concentration of ethyl acetate, except for posaconazole which shows a significant increase up to $x_{ETAC}\simeq 0.6$, and then a decrease to slightly negative, when it reaches the condition of pure ethyl acetate. The fact that $\mathcal{P}\mathcal{S}\left(x_{ETAC}<x^0\right)<0$ in Figure 5a, where $x^0$ the composition of the mixed-solvent at which $\mathcal{P}\mathcal{S}\left(x_{ETAC}=x^0\right)=0$, confirms that these pharmaceutical solutes prefer ethyl acetate over ethanol as the solvation environment within the indicated range of ethyl acetate mole fraction; in other words, it manifests microscopically as $G_{i ETAC}^{\infty }\left(x_{ETAC}<x^0\right)>G_{i ETOH}^{\infty }\left(x_{ETAC}<x^0\right)$ and macroscopically as ${\mathsf{\Delta }}_{tr}{g}_i\left(0<x_{ETAC}<x^0\right)<0$, ${\left[\partial {\mathsf{\Delta }}_{tr}{g}_i\left(x_{ETAC}<x^0\right)/\partial x_{ETAC}\right]}_{TP}<0$ as well as ${\left[\partial {\mu }_i^{\ominus }\left(x_{ETAC}\right)/\partial {\mu }_{ETAC}\left(x_{ETAC}\right)\right]}_{TP}<0$. Note, however, that not all the observed trends are monotonous in mixed-solvent composition; in fact, they exhibit inflection points (e.g., piroxicam and meloxican) as well as relative extrema (e.g., mevastatin and posaconazole).

It is also useful to highlight the contrasting behavior between the universal $\mathcal{P}\mathcal{S}\left(x_{ETAC}\right)$ function and the always non-monotonous Ben-Naim’s first-order ${\delta }_{ETAC i}^0\left(x_{ETAC}\right)$ parameter comprising three roots, i.e., ${\delta }_{ETAC i}^0\left(x_{ETAC}\right)=0$ for $\left\{x_{ETAC}=0;x_{ETAC}=1;0<x_{ETAC}<1\right\}$. The contrasting behavior underscores the main drawback of ${\delta }_{ETAC i}^0\left(x_{ETAC}\right)$ as a marker for preferential solvation as extensively discussed elsewhere [93].

4. Discussion on Alternative Exploration Routes and Relevant Observations

Whenever we have the ability to either generate (e.g., by molecular simulations) or extract (e.g., from scattering experiments) the proper microstructure of a mixture, there is an opportunity for us to study a variety of solvation phenomena and attempt linking microstructural details with thermodynamic behavior. This approach is usually guided by statistical mechanics in conjunction with chemical thermodynamics as previously stated in the introduction. However, to obtain meaningful outcomes, we need to be aware of the proper foundations, the underlying hypothesis, and the potential limitations of the invoked formalisms as we discuss and illustrate below.

4.1. Molecular Simulation Approaches to Thermodynamic Preferential Interaction Parameters

Molecular-based simulation of model systems has been targeted as a source of information on the preferential interaction/solvation of infinitely dilute solutes in mixed-solvent has been molecular simulation of model ternary systems [80,140,141]. These studies typically invoke the molality-based thermodynamic preferential interaction parameter ${\Gamma }_{{\mu }_j{\mu }_k}^{(m_i\to 0)}$ (e.g., Table 1), i.e.,

$$\begin{array}{cc}\hfill {\Gamma }_{{\mu }_j{\mu }_k}^{(m_i\to 0)}& =x_k\mathcal{P}\mathcal{S}\left(T,P,x_j\right)\hfill \\ & ={\mathcal{N}}_{ik}^{res,\infty }-\left(N_k/N_j\right){\mathcal{N}}_{ij}^{res,\infty }\hfill \end{array}$$

(28)

where the above microstructural interpretation relies on the Kirkwood–Buff fluctuation formalism, and therefore, on the ability to extract a meaningful representation of ${\mathcal{N}}_{\alpha \beta }^{res,\infty }\equiv {\rho }_{\beta }{G}_{\alpha \beta }^{\infty }$ from simulation, i.e.,

$${\mathcal{N}}_{\alpha \beta }^{res,\infty }=4\pi {\rho }_{\beta }{\displaystyle {\int }_0^{\infty }\left[{〈g_{\alpha \beta }^{\infty }\left(r_{\alpha \beta },{\omega }_{\alpha },{\omega }_{\beta }\right)〉}_{{\omega }_{\alpha }{\omega }_{\beta }}-1\right]r_{\alpha \beta }^{2}d{r}_{\alpha \beta }}$$

(29)

with the orientational average ${〈\cdots 〉}_{{\omega }_{\alpha }{\omega }_{\beta }}$, where ${\omega }_{\alpha }$ describes the orientation of the $\alpha$-species and $r_{\alpha \beta }$ defines the distance between the centers of mass of the two species [1].

The second line of Equation (28), however, is actually approximated according to the truncated version of the corresponding volume integrals as follows,

$$\begin{array}{cc}\hfill {\Gamma }_{{\mu }_j{\mu }_k}^{(m_i\to 0)}& ={\mathcal{N}}_{ik}^{res,\infty }-\left(N_k/N_j\right){\mathcal{N}}_{ij}^{res,\infty }\hfill \\ & \approx {〈n_{ik}^{\infty }\left(r<r_c\right)〉}_{\mu VT}-\left({〈n_k〉}_{\mu VT}/{〈n_j〉}_{\mu VT}\right){〈n_{ij}^{\infty }\left(r<r_c\right)〉}_{\mu VT}\hfill \end{array}$$

(30)

where the ${〈\cdots 〉}_{\mu VT}$ denotes a grand-canonical ensemble average and $r_c$ describes the truncation/cutoff distance which defines the radius of the preferential interaction shell, also known as the correlation radius, while ${\mathcal{N}}_{i\alpha }^{res,\infty }={〈n_{i\alpha }^{\infty }\left(r_c\to \infty \right)〉}_{\mu VT}$ and $N_{\alpha }={〈n_{\alpha }\left(r_c\to \infty \right)〉}_{\mu VT}$. Therefore, the approximation in the second line of Equation (30) assumes implicitly that,

$${〈n_{ik}^{\infty }\left(r\ge r_c\right)〉}_{\mu VT}-\left({〈n_k〉}_{\mu VT}/{〈n_j〉}_{\mu VT}\right){〈n_{ij}^{\infty }\left(r\ge r_c\right)〉}_{\mu VT}\cong 0$$

(31)

In turn, the right-hand-side of Equation (28) is estimated in terms of the following molecular simulation time averages, i.e.,

$${\Gamma }_{{\mu }_j{\mu }_k}^{(m_i\to 0)}\cong {\displaystyle \sum _{\tau =1}^{{\tau }_{run}/{\tau }_{rec}}{\Gamma }_{{\mu }_j{\mu }_k}^{(m_i\to 0)}\left(\tau \right)}/\left({\tau }_{run}/{\tau }_{rec}\right)$$

(32)

with (e.g., S11 of Ref. [142])

$${\Gamma }_{{\mu }_j{\mu }_k}^{(m_i\to 0)}\left(\tau \right)=n_{ik}^{\infty }\left(r<r_c,\tau \right)-\left[{\displaystyle \frac{n_k-n_{ik}^{\infty }\left(r<r_c,\tau \right)}{n_j-n_{ij}^{\infty }\left(r<r_c,\tau \right)}}\right]n_{ij}^{\infty }\left(r<r_c,\tau \right)$$

(33)

where $n_{\alpha \beta }^{\infty }\left(r<r_c,\tau \right)$ denotes the number of $\alpha$-species separated by a center of mass-to-center of mass $r$-distance from the $\beta$-species, while $n_{\alpha }$ is the total number of $\alpha$-species in the simulation system. Given that $r_c<10\mathsf{\AA }$ is used in most cases [140,141,143], and considering the simulation evidence on the radial distribution functions of representative aqueous systems [79,142,144,145,146,147,148], it becomes difficult to expect an accurate representation of Equation (28) based on the simulation determination of Equation (30), i.e., the condition described by Equation (31) would not be fulfilled, especially for large $i$-solutes such as proteins.

As long as $r_c$ refers to the center of mass-to-center of mass, the shortcomings from this issue might be alleviated by increasing the cutoff distance; however, the simulation approach suffers from another significantly more serious issue that has been usually overlooked. This issue concerns the actual statistical mechanic meaning of ${\mathcal{N}}_{\alpha \beta }^{res,\infty }={\rho }_{\beta }{G}_{\alpha \beta }^{\infty }$. In fact, in the formulation of the preferential interaction parameter, Equation (28), ${\mathcal{N}}_{\alpha \beta }^{res,\infty }\left(r<r_c\right)$ must be evaluated as a volume integral over the $\alpha$-center of mass to $\beta$-center of mass radial distances. However, the reported simulation results from the application of Equation (29) frequently involve a radial distance other than the required center of mass-to-center of mass, including the “closest distance from the solvent or cosolvent to any atom of the protein solute ” [144], the “the distance from the solvent/cosolvent to the surface of the solute protein” [82,149,150], and “the radial distance from the solvent or cosolvent to the protein van der Waals surface” [141,151]. Obviously, these simulation prescriptions do not, and cannot, describe the truncated form ${\mathcal{N}}_{\alpha \beta }^{res,\infty }\left(r_c\right)={\rho }_{\beta }{G}_{\alpha \beta }^{\infty }\left(r_c\right)$ in that, the simulated ${\mathcal{N}}_{\alpha \beta }^{res,\infty }\left(r_c\right)$ becomes defined as a truncated integral over a radial correlation function between sites other than the species centers of mass, i.e., it neither represents Kirkwood-Buff integrals, nor can it fulfill any Kirkwood-Buff fluctuation theory relationship [4]. Any choice of pair distance other than the center-to-center of mass in the determination of the simulation averages introduces an implicit angular dependence into the pair correlation functions invalidating all further relationships, including the one used in the simulation approach, e.g., Equations (28) and (29).

4.2. Local Composition-Based Preferential Solvation Approach

As discussed in §3, the starting point of the microstructural-based analysis of preferential solvation is the inversion of the Kirkwood-Buff fluctuation theory to obtain the formal expressions for the calculation of the five relevant Kirkwood-Buff $G_{\alpha \beta }$ integrals. In addition to performing such formal inversion, Ben-Naim [83] proposed a means to measure the preferential solvation of a solute at infinite dilution in a binary $\left(\alpha +\beta \right)$-mixed solvent according to the idea of local composition and the deviation from its bulk counterpart, i.e.,

$$\begin{array}{cc}\hfill {\delta }_{\alpha i}\left(R_c\right)& =x_{\alpha i}\left(R_c\right)-x_{\alpha }\hfill \\ & =x_{\alpha }{x}_{\beta }\left(G_{\alpha i}^{\infty }-G_{\beta i}^{\infty }\right)/\left(x_{\alpha }{G}_{\alpha i}^{\infty }+x_{\beta }{G}_{\beta i}^{\infty }+V_c\right)\hfill \end{array}$$

(34)

where the local mole fraction of the $\alpha$-solvent around the $i$-solute is defined as follows:

$$x_{\alpha i}\left(R_c\right)={\mathcal{N}}_{i\alpha }^{\infty }\left(R_c\right)/\left[{\mathcal{N}}_{i\alpha }^{\infty }\left(R_c\right)+{\mathcal{N}}_{i\beta }^{\infty }\left(R_c\right)\right]$$

(35)

with ${\mathcal{N}}_{i\alpha }^{\infty }\left(R_c\right)$ describing the number of $\alpha$-solvent molecules in the correlation shell of radius $R_c$ centered at the $i$-solute. This number becomes formally given by the following statistical mechanic expression:

$$\begin{array}{cc}\hfill {\mathcal{N}}_{i\alpha }^{\infty }\left(R_c\right)& \equiv 4\pi {\rho }_{\alpha }{\displaystyle {\int }_0^{R_c}{g}_{i\alpha }^{\infty }\left(r_{i\alpha }\right)r_{i\alpha }^{2}d{r}_{i\alpha }}\hfill \\ & ={\rho }_{\alpha }{G}_{\alpha i}^{\infty }+{\rho }_{\alpha }{V}_c\left(R_c\right)\hfill \end{array}$$

(36)

which highlights its explicit dependence on the size of $R_c$, and consequently, that of the local composition $x_{\alpha i}\left(R_c\right)$.

The $R_c$-dependence $x_{\alpha i}\left(R_c\right)$ becomes an intrinsic drawback of the local composition definition, and consequently, the outcome of ${\delta }_{\alpha i}\left(R_c\right)$ and any idea of preferential solvation becomes dependent on the arbitrary choice of the coordination volume $V_c\left(R_c\right)$. To avoid such an issue, Ben-Naim suggested an $R_c$-independent preferential solvation parameter ${\delta }_{\alpha i}^0$, resulting from the first-order expansion of Equation (34) with respect to $V_c^{-1}$, i.e.,

$${\delta }_{\alpha i}^0=x_{\alpha }{x}_{\beta }\left(G_{\alpha i}^{\infty }-G_{\beta i}^{\infty }\right)$$

(37)

where $G_{\alpha \beta }^{\infty }\left(T,P\right)$ represents the Kirkwood-Buff integral for the $\alpha \beta$-pair of species. We should note that Equation (37) describes null preferential solvation at the two edges of the composition range, regardless of the solvent-to-cosolvent relative solute affinities, ${\left(G_{\alpha i}^{\infty }-G_{\beta i}^{\infty }\right)}_{TPx}={\rho }_o^{-1}\mathcal{P}\mathcal{S}$, where typically $G_{\alpha i}^{\infty }\left(T,P,x_j\right)\ne G_{\beta i}^{\infty }\left(T,P,x_j\right)$ as clearly illustrated in Figure 5a,b.

A variation around Ben-Naim’s ${\delta }_{\alpha i}^0$ was introduced by Marcus, [94] who took Equation (34) and invoked a conjectured expression for the estimation of the composition-dependent correlation volume $V_c\left({\widehat{\upsilon }}_i^{\infty },x_{\alpha i},x_{\alpha }\right)$ such as Equation 8a,b in Ref. [152]. Thus, the pair of Equations, ${\delta }_{\alpha i}\left(V_c,x_{\alpha i}\right)$ and $V_c\left({\widehat{\upsilon }}_i^{\infty },x_{\alpha i},x_{\alpha }\right)$, must be solved simultaneously by iteration to obtain the desired ${\delta }_{\alpha i}\left(V_c,x_{\alpha i}\right)$. It becomes immediately obvious that regardless of the procedure used in the estimation of $V_c\left({\widehat{\upsilon }}_i^{\infty },x_{\alpha i},x_{\alpha }\right)$, Equation (34) involves a potential source of trouble given the lack of explicit enforcement of the splitting requirement

$$G_{i\alpha }^{\infty }\left(T,P,x_{\alpha }\right)=4\pi {\displaystyle {\int }_0^{R_c>\zeta }{g}_{i\alpha }^{\infty }\left(r_{i\alpha }\right)r_{i\alpha }^{2}d{r}_{i\alpha }-V_c}\left(R_c>\zeta \right)$$

(38)

where $\zeta$ identifies the correlation length of the mixed-solvent environment, i.e., the distance traveled by the solute-induced perturbation of the microstructure of the solvent environment around it [3]. While this requirement is automatically fulfilled by the first-order preferential solvation parameter ${\delta }_{\alpha i}^0\equiv x_{\alpha }{x}_{\beta }\left(G_{i\alpha }^{\infty }-G_{i\beta }^{\infty }\right)$, we must also be mindful that the resulting $R_c=\sqrt[3]{3V_c\left({\widehat{\upsilon }}_i^{\infty },x_{\alpha i},x_{\alpha }\right)/4\pi }$ from the iterative solution of the pair, ${\delta }_{\alpha i}\left(V_c,x_{\alpha i}\right)$ and $V_c\left({\widehat{\upsilon }}_i^{\infty },x_{\alpha i},x_{\alpha }\right)$, might translate into an $R_c\ll \zeta$ as suggested by the microstructural information from the molecular simulation of aqueous systems [147,148].

Here, we should also bring to the reader’s attention a pair of frequently encountered modeling pitfalls when applying this approach. Note that, when discussing the Kirkwood-Buff formal inversion, we highlighted the condition of diffusive stability, as described by a finite positive stability coefficient $D\left(0\le x_j\le 1\right)$ [7]. In other words, the binary mixed-solvent system must exhibit full miscibility over the entire compositional range at the prevailing state conditions. Otherwise, the Kirkwood-Buff inversion cannot be achieved within the miscibility gap, and the failure to recognize this condition has led to a variety of bizarre interpretations of the faulty preferential solvation calculations as thoroughly discussed elsewhere [93,113]. Moreover, it should be obvious that no matter how we approach the iterative solution of ${\delta }_{\alpha i}\left(V_c,x_{\alpha i}\right)$ and $V_c\left({\widehat{\upsilon }}_i^{\infty },x_{\alpha i},x_{\alpha }\right)$, the calculated preferential solvation ${\delta }_{\alpha i}\left(V_c,x_{\alpha }\right)=x_{\alpha i}\left(R_c,x_{\alpha }\right)-x_{\alpha }$ must always result in a physically meaningful local mole fraction, i.e., $0<x_{\alpha i}\left(R_c,x_{\alpha }\right)<1$. Unfortunately, we have encountered a considerably large number of published research concerning the solvent effects on the solubility of pharmaceutical species (as discussed and sampled in SI.1 and SI.2 of the Supplementary Information of Ref. [133]) where the data comprised ${\delta }_{\alpha i}\left(V_c,x_{\alpha }\right)<0$ that led to physically meaningless $x_{\alpha i}\left(V_c,x_{\alpha }\right)={\delta }_{\alpha i}\left(V_c,x_{\alpha }\right)+x_{\alpha }<0$.

Given the number of publications exhibiting the same problem, we performed a forensic analysis of the available tabulated data to find that most of the affected publications invoking Marcus’ approach—the iterative solution of ${\delta }_{\alpha i}\left(V_c,x_{\alpha i}\right)$ given by Equation (34) and the correlation volume $V_c\left({\widehat{\upsilon }}_i^{\infty },x_{\alpha i},x_{\alpha }\right)$ as given in Ref. [152]—followed literally either Equations 5.33a,b of Ref. [67], Equation (6) of Ref. [153], Equations 2.51–2.52 of Ref. [154], or Equations 6.25a,b of Ref. [155]. A compilation of publications involving this Kirkwood-Buff inversion issue (about 165 publications to date) is given in the Supplementary Information of Ref. [133].

It turned out that the unfortunate misprint in the original sources, Refs. [67,153,154,155], written in our notation as follows:

$$\begin{array}{cc}\hfill G_{ij}^{\infty ,faulty}& =kT{\kappa }_o-{\widehat{\upsilon }}_i^{\infty }+x_k{\widehat{\upsilon }}_k{\left(kTD\right)}^{-1}{\left[\partial {\mu }_i^{\ominus }\left(x_j\right)/\partial x_j\right]}_{TP}\hfill \\ & =G_{ij}^{\infty }+2x_k{\widehat{\upsilon }}_k{\left(kTD\right)}^{-1}{\left[\partial {\mu }_i^{\ominus }\left(x_j\right)/\partial x_j\right]}_{TP}\hfill \end{array}$$

(39)

has gone unnoticed, where $G_{ij}^{\infty ,faulty}$ identifies the misprinted form of the correct $G_{ij}^{\infty }$ quantity given by Equation (20), after the equivalences represented by Equation (19). Equation (39) highlights the misprinted sign in the third term of its first line, an error that might have become immediately obvious should the users have checked the difference between Equations (20) and (21) to recover the original solubility expression, Equation (15), or its equivalent form,

$${\left[\partial {\mu }_i^{\ominus }\left(x_j\right)/\partial x_j\right]}_{TP}=kTD{\rho }_o\left(G_{ik}^{\infty }-G_{ij}^{\infty }\right)$$

(40)

instead of the faulty outcome,

$${\left[\partial {\mu }_i^{\ominus }\left(x_j\right)/\partial x_j\right]}_{TP}^{faulty}=kTD\left(G_{ik}^{\infty }-G_{ij}^{\infty ,faulty}\right){\left(x_j{\widehat{\upsilon }}_j-x_k{\widehat{\upsilon }}_k\right)}^{-1}$$

(41)

which satisfies the correct Equation (15) only when $x_j=1$. Moreover, the introduction of $\left(G_{ij}^{\infty ,faulty}-G_{ik}^{\infty }\right)$ into the numerator of the preferential solvation parameter, Equation (34), will generate a faulty preferential solvation parameter, i.e.,

$${\delta }_{ji}^{faulty}\left(x_j\right)={\displaystyle \frac{{\delta }_{ji}\left(x_j\right)\left(x_j{G}_{ij}^{\infty }+x_k{G}_{ik}^{\infty }+V_c\right)+2x_j{x}_k^2{\widehat{\upsilon }}_k{\left(kTD\right)}^{-1}{\left[\partial {\mu }_i^{\ominus }\left(x_j\right)/\partial x_j\right]}_{TP}}{\left(x_j{G}_{ij}^{\infty ,faulty}+x_k{G}_{ik}^{\infty }+V_c^{faulty}\right)}}$$

(42)

Moreover, as a matter of illustration in Figure 6a,b, we display side by side the comparison between ${\rho }_o{\delta }_{ji}^{0,faulty}=x_j{x}_k{\rho }_o\left(G_{ij}^{\infty ,faulty}-G_{ik}^{\infty }\right)$ against the properly inverted ${\rho }_o{\delta }_{ji}^0=-x_j{x}_k\mathcal{P}\mathcal{S}\equiv x_j{x}_k{\rho }_o\left(G_{ij}^{\infty }-G_{ik}^{\infty }\right)$ for the same set of systems in Figure 5a,b. Figure 6a,b shows clearly the contrasting behavior between ${\rho }_o{\delta }_{ETAC i}^{0,faulty}\left(x_{ETAC}\right)$ and ${\rho }_o{\delta }_{ETAC i}^0\left(x_{ETAC}\right)$, e.g., ${\rho }_o{\delta }_{ETAC i}^0\left(x_{ETAC}<0.37\right)>0$ while ${\rho }_o{\delta }_{ETAC i}^{0,faulty}\left(x_{ETAC}<0.37\right)<0$, where $x_{ETAC}\cong 0.37$ identifies the common middle root of ${\delta }_{ETAC i}^{0,faulty}\left(x_{ETAC}\right)=0$ for the mixed ethyl acetate (j) + ethanol (k) environment. This common middle root of ${\delta }_{ji}^{0,faulty}\left(x_j\right)=0$ for a mixed-solvent system becomes independent of the identity of the solute as demonstrated in Appendix A or Ref. [133].

In summary, the use of the incorrect $G_{ij}^{\infty ,faulty}$ rather than $G_{ij}^{\infty }$ Kirkwood–Buff integral as input in the simultaneous determination of the preferential solvation parameter ${\delta }_{ji}\left(V_c,x_{ji}\right)$ and corresponding correlation volume $V_c\left({\widehat{\upsilon }}_i^{\infty },x_{ji},x_j\right)$ leads to the iteration of Equation (42) rather than Equation (34) and steers always to the incorrect, and frequently, unphysical results as thoroughly discussed elsewhere [133].

4.3. Controversial Definitions and Claims about Preferential Solvation

We frequently find unfortunate misconceptions surrounding the differential solvation of a solute at infinite dilution in a binary $\left(j+k\right)$-solvent when the mixed-solvent behaves as a Lewis-Randall ideal solution, i.e., ${\mathsf{\Delta }}_{jk}^{LR-IS}\left(T,P,x_j\right)\equiv G_{jj}+G_{kk}-2G_{jk}=0$ [67,152,156,157,158]. For instance, in Ref. [152] the author indicates that “A feature to be noted in the application of Equation (7) is that even for ideal mixtures (where $G^E=0$, $V^E=0$ and $G_{ii}+G_{jj}=2G_{ij}$ the ${\delta }_{ij}$ are non-zero, as if preferential solvation existed in them”, and consequently, follows a suggestion that the three Kirkwood-Buff integrals “be corrected” to force ${\delta }_{ij}^{LR-IS}=0$. These rather controversial ideas have been proven misleading and possibly wrong, as discussed by Ben-Naim [159,160]. In fact, under the Lewis-Randall ideality condition, $G_{jk}-G_{jj}=-\left(G_{jk}-G_{kk}\right)$ [126], which indicates that there is an exact compensation of affinities between $jk$- and either $jj$- or $kk$-intermolecular type interactions. In other words, even when the mixed-solvent system (at some precise state conditions and composition) exhibits ${\mathsf{\Delta }}_{jk}^{LR-IS}\left(T,P,x_j\right)=0$, the average solvent (cosolvent) environment around any cosolvent (solvent) species differs from that around itself as formally described by the differences $G_{jk}-G_{jj}=-\left(G_{jk}-G_{kk}\right)\ne 0$. The only possible exception occurs when the $j$-solvent behaves identically as the $k$-cosolvent, as discussed in §2.3. Consequently, in terms of the fundamental structure-making/-breaking functions, we have that ${\mathcal{S}}_{kj}/{\rho }_j=-{\mathcal{S}}_{jk}/{\rho }_k\ne 0$, i.e., a non-zero preferential solvation for both species (for additional details, see Ref. [93]).

4.4. Thermodynamic Consistency of the Input Properties

For the correct implementation of the formalism, we also need to handle properly the inputs, i.e., ensuring their thermodynamic consistency, including the representations of ${\widehat{\upsilon }}_i^{\infty }\left(T,P,x_j\right)$, ${\kappa }_o\left(T,P,x_j\right)$, and any excess property ${\mathcal{P}}^E\left(T,P,x_j\right)$. In particular, all the regressed excess property should obey the ${\mathcal{P}}^E\left(T,P,x_j=1\right)=0$ and ${\mathcal{P}}^E\left(T,P,x_j=0\right)=0$ conditions. Unfortunately, these crucial requirements often go unfulfilled in the cited literature, when the regressed Gibbs free energy $g^E\left(T,P,x_j\right)$ for the mixed-solvent describes $g^E\left(x_j=0 \mathrm{and}/\mathrm{or} x_j=1\right)\ne 0$ [134,161,162,163,164,165,166], i.e., not only an obvious thermodynamic inconsistency, but also a source of corruption of the resulting composition profiles of the solvent–cosolvent intermolecular interaction asymmetry as described by ${\mathsf{\Delta }}_{jk}\left(T,P,x_j\right)$. Consequently, this thermodynamic inconsistency translates into an ill-behaved coefficient of diffusive stability $D\left(T,P,x_j\right)$, spills over the determination of all Kirkwood–Buff integrals, as well as distorts the calculation of $\mathcal{P}\mathcal{S}\left(T,P,x_j\right)$ and ${\mathcal{S}}_{\alpha \beta }\left(T,P,x_j\right)$.

5. Concluding Remarks, Recommendations, and Outlook

In this review, we have placed most emphasis on the rigorous formal connections between microstructural behavior and thermodynamic responses involving the solvation of infinitely dilute species in binary mixed-solvents. The explicit microscopic-to-macroscopic relationships capture the fundamental notion underlying the thermodynamic solvent effects and their applications in physico-chemical separation processes, while highlighting one of the main shortcomings in the existing studies of preferential solvation. Thus, the main goal behind this emphasis, regardless of the choice of modeling scheme for the solvent effects, is the rational guide to the selection of cosolvents for any specific application, e.g., to improve separation, enhance solubility, and control transport properties through the manipulation of the mixed-solvent environment. Indeed, the alluded shortcomings in recent studies have prevented revealing how the preferential solvation parameters (or any adequate relative affinity marker) might be quantitatively applied to manipulate the solvation power of a mixed-solvent environment for any desired application while avoiding the unsupported interpretations of solvation behaviors according to the handwaving mechanisms of intermolecular interactions [167,168,169,170,171].

In addition, the identified microscopic-to-macroscopic relationships in terms of rigorous fundamental structure-making/-breaking functions have offered us versatile tools to (a) study the actual preferential solvation behavior of a contrasting variety of solutes in binary mixed-solvents, (b) provide a rigorous interpretation of the differential affinity between the solute and the components of the solvent environment in terms of an unambiguous preferential solvation function, (c) discuss the ability of a popular local composition model of solvent effects and test the correctness of its underlying Equations, and (d) conduct a forensic analysis to detect the source of some thermodynamic pitfalls in a number of recently published work on the modeling of preferential solvation.