On the Solute-Induced Structure-Making/Breaking Phenomena: Myths, Verities, and Misuses in Solvation Thermodynamics

Abstract

We review the statistical mechanic foundations of the fundamental structure-making/breaking functions, leading to the rigorous description of the solute-induced perturbation of the solvent environment for the understanding of the solvation process of any species regardless of the type and nature of the solute–solvent interactions. Then, we highlight how these functions are linked to unambiguous thermodynamic responses resulting from changes in state conditions, composition, and solute–solvent intermolecular interaction asymmetries. Finally, we identify and illustrate the pitfalls behind the use of surrogate approaches to structure-making/breaking markers, including those based on Jones–Dole’s B-coefficient and Hepler’s isobaric-thermal expansivity, while highlighting their ambiguities and lack of consistency and the sources of misinterpretations.

1. Introduction

Since the early days of the molecular modeling of fluid mixtures, the main effort has focused on drawing formal (albeit rigorous) connections between the microstructural features of the system of interest and their macroscopic manifestation as thermodynamic properties. This approach follows logically from a pair of fundamental facts, namely: (a) the microstructure of a system arises from the presence of interparticle interactions, where an ideal gas represents the complete absence of them so that non-zero (statistical mechanic) pair correlation functions reflect the deviations of the actual pair distribution functions from the (ideal gas) uniform distribution counterpart triggered by the strength of the interactions; and (b) the microstructure of the mixture stems from the differences (e.g., asymmetries) between the unlike-pair interactions, giving rise to contrasting pair correlation functions [1]. Consequently, the macroscopic manifestations of these microscopic behaviors become described by the thermodynamic residual properties as deviations from those of the ideal gas model in the (a)-case and by the thermodynamic excess properties as deviation from a precisely-defined ideal solution reference in the (b)-case [2,3]. These two fundamental concepts become the blueprint for building the rigorous microscopic-to-macroscopic understanding of solvation processes, one based on the analysis of the solute-induced perturbation of the surrounding environment, leading to the unambiguous definition of the fundamental structure-making/breaking functions [4].

In fact, the isobaric–isothermal solute-induced microstructural distortion described as a local solvent-density perturbation (direct effect) caused by the intermolecular interaction asymmetry between the strength of the original solvent–solvent and the resulting solute–solvent interactions and propagating across the system up to a (correlation length) distance from the perturbing center (indirect effect) [5,6] leads to its macroscopic interpretation in terms of thermodynamics of solution non-ideality [7,8,9,10,11,12,13,14]. Thus, a great deal of effort in solution thermodynamics has been placed on the development of experimental methods to extract the structural information of aqueous electrolyte and non-electrolyte solutions from scattering experiments, such as X-ray (including XANES and EXAFS) [15,16,17,18,19,20,21,22], neutron diffraction with isotope substitution (NDIS) [20,23,24,25,26], hybrid empirical microstructural refinement techniques [27,28,29,30,31,32,33], and the molecular simulation-based interpretation of NDIS raw data [34,35,36,37,38,39,40], frequently complemented with infrared [41,42], Raman [43,44,45,46,47], NMR [48,49], and femtosecond pump-probe spectroscopies [50,51].

This research effort has often led to loose interpretations of the solvation behavior of solutes as either structure-makers or structure-breakers [52,53,54,55,56], followed by the more pretentious kosmotrope (order-maker) and chaotrope (disorder-maker) descriptors [57], usually used as an interchangeable alternative notation for structure-makers and structure-breakers, respectively [57,58,59,60,61,62], i.e., an unfortunate circumstance that introduced additional ambiguities and confusion into the reported interpretation of the solvation behavior of solutes. In fact, this sentiment was vented earlier as “Concepts like ‘structure-breaking’ and ‘structure-promoting’ solutes appear almost in every discussion of the physicochemical properties of aqueous solutions…Yet in spite of the great number of articles written on this subject, the very basic questions of how to define the ‘structure of water’ and which experiment should be done to measure the extent of ‘structural changes in the solvent’ were left unanswered. Almost invariably, these concepts slip into the discussion in a somewhat qualitative manner, i.e., it is assumed that they are already understood intuitively and no further clarification or elaboration is needed.” [63].

The common source of ambiguity around the concept of structure-maker/breaker solutes (or kosmotropes/chaotropes, for that matter) has been the lack of an explicit identification of the structure-making/breaking (kosmotrope/chaotrope) signature, and consequently, the absence of any definite formal link between the microstructural effect of the solute on the surrounding solvent environment—the solute-induced local density perturbation—and the corresponding macroscopic response whose magnitude can be measured experimentally [54,64]. Moreover, the alluded restructuring process highlights a few essential requirements to avoid the frequently used vague language while facilitating the interpretation of solvation phenomena, e.g., the need for (i) a meaningful unambiguous account for the microstructural perturbation of the environment around any species, (ii) the development of precisely defined descriptors of the solvation phenomena, and (iii) the identification of the corresponding macroscopic responses as thermodynamic signatures of the species solvation leading to the solution of thermodynamic non-ideality.

Based on the discussed scenario, the main goal of this manuscript is twofold: first, review the fundamentals underlying the rigorous and unambiguous interpretation of the solute-induced structure-making/breaking phenomena indicated in the title, and second, identify the myths, verities, and misuses of the structure-making/breaking markers in solvation thermodynamics, as emphasized in the subtitle of this manuscript. Given the scope of the current manuscript, and to keep the discussion on focus, this review mainly comprises our own developments on the matter while being illustrated with the analysis of data from numerous sources from the literature and aims to raise awareness on what can and cannot be construed from frequently invoked markers of structure-making/breaking phenomena.

For that purpose, we start reviewing the statistical mechanic foundations underlying the solute-induced distortion of the solvent environment in Section 2, where we discuss the interplay of two length scales in the solvation process associated with the microstructural perturbation and its propagation, leading to the explicit unambiguous and natural definition of the fundamental structure-making/breaking functions. In the process, we show how these functions become rigorously linked to thermodynamic responses, including the isochoric–isothermal pressure change upon solvation, the osmotic second virial coefficient, and a number of frequently invoked thermodynamic preferential solvation parameters. In Section 3, we discuss the responses of the fundamental structure-making/breaking functions to changes in state conditions, composition, and solute–solvent intermolecular interaction asymmetries. Then, in Section 4, we argue and illustrate the pitfalls behind the use of surrogate approaches, including those based on Jones–Dole’s B coefficient and Hepler’s isobaric-thermal expansivity, while highlighting their ambiguities and lack of consistency and fulfillment of required constraints, as well as identifying the sources of misinterpretations.

2. Rigorous Definition of Solute-Induced Microstructural Perturbation of the Solvent Environment

As indicated in the introduction, we must fulfill a few essential requirements in the microscopic characterization of the solvation process, including the unambiguous and precise description of the perturbation of the solvent environment upon solute solvation at the prevailing state conditions of the solution, and the rigorous linking of the microscopic behavior to the corresponding macroscopic manifestations leading to the experimental measurements [4]. For that purpose, we review the statistical mechanical-based foundations for the fundamental structure-making/breaking concept as a thermodynamic state function, link it to the pressure perturbation counterparts and preferential solvation phenomena, and discuss its attributes, including the universality of the definition.

2.1. Fundamental Split between Short- and Long-Range Contributions to the Solvation Behavior of Solutes

The solvation of any solute (electrolyte or otherwise) entails a disruption of the system microstructure resulting from the solute–solvent intermolecular interaction asymmetry, described formally and unambiguously as a finite local density perturbation (i.e., a short-range or direct effect) whose propagation across the system (i.e., a longer-range or indirect effect) is felt up to a distance given by the correlation length of the environment and scales with its isothermal compressibility [65]. Note that the local density perturbation associated with the solvation process remains finite even under special thermodynamic conditions, such as in the solvation of a solute at infinite dilution in a solvent at its critical conditions where the correlation length becomes divergent [12,66]. More importantly, the alluded split is rigorously described by the Ornstein–Zernike equation [67], and therefore, the local density perturbation becomes naturally linked to the direct correlation functions, while the ensuing propagation becomes linked to the indirect correlation functions [5,68,69]. This feature provides the means to unravel the rigorous connections between the species-induced microstructural perturbation in a solution, its propagation across the system, and the driving force underlying the solvation of any species at any state condition or composition. This formal characterization becomes more powerful through their reinterpretation according to the species’ molecular-based fundamental structure-making/breaking functions [4].

2.2. Definition and Attributes of the Fundamental Structure-Making/Breaking Functions ${\mathcal{S}}_{\alpha \beta }$

To dissipate the controversy underlying the meaning of the structure-making/breaking concept, we must (i) identify the precise link between the microstructural impact of the solute species on its surrounding environment, i.e., the induced local density perturbation and the ensuing macroscopic manifestation to be ultimately measured, (ii) provide a direct path for the calculation of the magnitude of the microstructural perturbation of the system according to an explicit, general, and rigorous molecular-based thermodynamic formalism, and (iii) be able to apply the concept equally and consistently to any solute–solvent intermolecular interaction asymmetry regardless of either the state conditions, number of components, or system composition [4].

The magnitude of the local density perturbation of the j-solvent environment around the i-solute can be rigorously measured in terms of Kirkwood–Buff [70] total correlation function integrals (TCFI) as follows [4,71]:

$$\begin{array}{cc}\hfill {\mathcal{S}}_{ij}\left(T,P,x_i\right)& =\rho x_j\left(G_{ij}-G_{jj}\right)\hfill \\ & =\left({\mathcal{N}}_{ij}-{\mathcal{N}}_{jj}\right)\hfill \end{array}$$

(1)

where ${\mathcal{N}}_{ij}\left(T,P,x_i\right)=\rho x_j{G}_{ij}$ describes the absolute average number of j-solvent molecules around any central i-solute at the prevailing state conditions and system composition. Note that the TCFIs in Equation (1), $G_{\alpha \beta }=4\pi {\displaystyle {\int }_0^{\infty }{h}_{\alpha \beta }\left(r_{\alpha \beta }\right)r_{\alpha \beta }^{2}d{r}_{\alpha \beta }}$, are volume integrals over the pair correlation function $h\left(r_{\alpha \beta }\right)={〈g_{\alpha \beta }^{\infty }\left(r_{\alpha \beta },{\omega }_{\alpha },{\omega }_{\beta }\right)〉}_{{\omega }_{\alpha }{\omega }_{\beta }}-1$, where $g_{\alpha \beta }^{\infty }\left(r_{\alpha \beta },{\omega }_{\alpha },{\omega }_{\beta }\right)$ depicts the spatial pair distribution function as the statistical mechanical microstructural descriptor, ${〈\cdots 〉}_{{\omega }_{\alpha }{\omega }_{\beta }}$ represents an orientational average, ${\omega }_{\alpha }$ describes the orientation of the $\alpha$-species, and $r_{\alpha \beta }$ defines the distance between the centers of mass of the two species. In other words, ${\mathcal{S}}_{ij}\left(T,P,x_i\right)$ characterizes the average number of j-solvent molecules around the central i-solute in excess (deficit) to that around any j-solvent and comprises the key microscopic information required to assess the magnitude of the i-induced microstructural perturbation of the j-solvent environment, i.e.,

$$G_{jj}\left(T,P,x_i\right)\underset{j\text{-}solventperturbation}{\overset{i\text{-}solutesolvation}{\to }}{G}_{ij}\left(T,P,x_i\right)$$

(2)

where the arrow highlights the microstructural evolution of the solvent environment from the original TCFI, $G_{jj}\left(T,P,x_i\right)$, to the final TCFI, $G_{ij}\left(T,P,x_i\right)$. This microstructural perturbation becomes characterized by three possible outcomes according to the sign of the difference ${\left(G_{ij}-G_{jj}\right)}_{TPx}$ as follows:

$${\left(G_{ij}-G_{jj}\right)}_{TPx}\Rightarrow \left\{\begin{array}{l}{G}_{ij}>G_{jj}\to {\mathcal{S}}_{ij}>0\mathrm{s}tructure\text{-}maker\\ G_{ij}\cong G_{jj}\to {\mathcal{S}}_{ij}\cong 0neithermaker\text{-}norbreaker\\ G_{ij}<G_{jj}\to {\mathcal{S}}_{ij}<0\mathrm{s}tructure\text{-}breaker\end{array}\right.$$

(3)

Therefore, according to Equations (1)–(3), the solute-induced solvent perturbation becomes unambiguously described as a strengthening of the solvent environment around the i-solute when $G_{ij}>G_{jj}$, i.e., ${\mathcal{S}}_{ij}\left(T,P,x_i\right)>0$, as the i-solute behaves as a structure-maker species. Conversely, the solvent perturbation weakens the solvent environment around the i-solute when $G_{ij}<G_{jj}$, i.e., ${\mathcal{S}}_{ij}\left(T,P,x_i\right)<0$, as the i-solute exhibits a structure-breaking effect. As we will discuss below, the j-solvent environment might remain unperturbed despite the presence of the i-solute as ${\left(G_{jj}\cong G_{ij}\right)}_{TPx}$, i.e., ${\mathcal{S}}_{ij}\left(T,P,x_i\right)\cong 0$. Note that the unperturbed condition represents also the signature of a special case of Lewis–Randall solution ideality because $G_{jj}\left(T,P,x_i\right)=G_{ij}\left(T,P,x_i\right)\ne 0$ describes the scenario where the i-solute behaves as the j-solvent, as well as the more general solution ideality condition ${\Delta }_{ij}\left(T,P,x_j\right)\equiv \left(G_{ii}+G_{jj}-2G_{ij}\right)=0$, leading to ${\left(G_{ii}-G_{ij}\right)}_{TPx}=-{\left(G_{jj}-G_{ij}\right)}_{TPx}⪋0$ [72,73].

At this point, we fulfilled the first requirement (i), and now we must provide a direct path for the calculation of the magnitude of the microstructural perturbation described by the three conditions in Equation (3), i.e., requirement (ii). For that purpose, we have invoked Kirkwood–Buff fluctuation formalism [70] and derived the following expressions for the general type of (electrolyte or non-electrolyte) i-solute at any system composition including infinite dilution [1,4]:

$${\mathcal{S}}_{ij}\left(T,P,x_j\right)=1-\rho {\widehat{\upsilon }}_i/\nu D\Rightarrow \left\{\begin{array}{l}\rho {\widehat{\upsilon }}_i<\nu D\to {\mathcal{S}}_{ij}>0\mathrm{s}tructure\text{-}maker\\ \rho {\widehat{\upsilon }}_i\cong \nu D\to {\mathcal{S}}_{ij}\cong 0neithermaker\text{-}norbreaker\\ \rho {\widehat{\upsilon }}_i>\nu D\to {\mathcal{S}}_{ij}<0\mathrm{s}tructure\text{-}breaker\end{array}\right.$$

(4)

where $\nu$ is the stoichiometric coefficient of the electrolyte i-solute, ${\widehat{\upsilon }}_i$ denotes its partial molar volume, $\rho$ is the molar density of the solution, and $D=\left[1+{\left(\partial \ln {\gamma }_j/\partial \ln x_j\right)}_{TP}\right]$ describes the material stability coefficient in terms of the composition slope of the activity coefficient of the j-solvent [74]. In other words, given the pair of volumetric properties $\left({\widehat{\upsilon }}_i,\rho \right)$ and the stability coefficient $D$, Equation (4) describes the thermodynamic response to the solute-induced effects on the microstructure of the fluid mixture and rigorously and unambiguously reveals the possible behaviors of the i-solute without requiring any explicit microstructural information of the system.

Finally, requirement (iii) is automatically satisfied given the absence of any constraint on the type of interactions, nature of the species, and number of components in the solution underlying the generality of the Kirkwood–Buff formalism. In fact, a universal and fundamentally based structure-making/breaking function (aka criterion or marker) must apply equally to species such as a non-interacting ideal gas i-solute [14,75], any i-species behaving identically to the j-solvent [12], and either weakly or strongly interacting i-electrolyte solutes, regardless of the type and description of the solute–solvent interactions. Therefore, the structure-making/breaking criterion must be able to identify the null solute-induced effect when the solute–solvent interaction asymmetry crosses the transition boundary between the solute-induced structure-making and the structure-breaking perturbations. Note that this transition condition might portray two significantly different scenarios, namely: (i) one characterized as a special cases of Lewis–Randall solution ideality, either when the interacting solute species behaves just as another solvent species and therefore keeps the solvent environment unperturbed [76], or (ii) when a non-interacting ideal gas species ($IG\_i$) is solvated by a real j-solvent along the thermodynamic path $\rho kT\kappa \left(T,P,x_j\right)=1$ [14,71]. In fact, from (4) and the fact that ${\widehat{\upsilon }}_i{\left(T,P,x_j\right)}^{G_{ii}=G_{ij}=G_{jj}^o}=\nu {\upsilon }_i^o\left(T,P\right)=\nu {\rho }^{-1}$ with $G_{jj}^o=kT{\kappa }_j^o-{\upsilon }_j^o$, we find that ${\mathcal{S}}_{ij}^{G_{ii}=G_{ij}=G_{jj}^o}\left(T,P,x_j\right)=1-\rho \upsilon =0$. Likewise, because ${\widehat{\upsilon }}_i^{IG\_i}\left(T,P,x_j\right)=kT\kappa$ [14] it follows immediately that ${\mathcal{S}}_{ij}^{IG\_i}\left(T,P,x_j\right)=1-\rho kT\kappa =0$, i.e., when the actual compressibility of the system becomes equal to that of the system behaving as an ideal gas environment, $\kappa \left(T,P,x_j\right)={\kappa }^{IG}={\left(\rho kT\right)}^{-1}$ (e.g., see Figure 6 and Appendix A of Ref. [71]).

2.3. Link between ${\mathcal{S}}_{\alpha \beta }$, the Isochoric-Isothermal Pressure Perturbation upon Solvation and Krichevskii’s Parameter

The isobaric–isothermal change of residual ($R$) partial molar Gibbs free energy for the solvation of a $\nu$-dissociable i-solute in a j-solvent can be expressed as follows,

$$\begin{array}{cc}\hfill {\mu }_i^R\left(T,P,x_i\right)-\nu {\mu }_j^R\left(T,P,x_i\right)& =\nu kT\ln {\left({\widehat{\varphi }}_i/{\widehat{\varphi }}_j\right)}_{TP}\hfill \\ & ={\displaystyle {\int }_0^{\rho }{\left(\partial P/\partial x_i\right)}_{T\rho }\left(d\rho /{\rho }^2\right)}\hfill \end{array}$$

(5)

where $\ln {\widehat{\varphi }}_{\alpha }\left(T,P,x_{\alpha }\right)$ denotes the partial molar fugacity coefficient of the $\alpha$-species in solution at the prevailing state conditions and composition, while the isochoric-isothermal pressure perturbation ${\left(\partial P/\partial x_i\right)}_{T,\rho }$ becomes the thermodynamic response to the microstructural reorganization of the system upon solvation, which in turn can be expressed in terms of the species partial molar volumes and the isothermal compressibility of the system as follows [4,12],

$${\left(\partial P/\partial x_i\right)}_{T\rho }=\rho \left({\widehat{\upsilon }}_i-\nu {\widehat{\upsilon }}_j\right)/\kappa$$

(6)

Equations (5) and (6) provide the macroscopic-to-microscopic rigorous link between the solvation-induced pressure perturbation and the resulting microstructural response of the system,

$${\left(\partial P/\partial x_i\right)}_{T\rho }=\nu D{\kappa }^{-1}\left[{\rho }_j\left(G_{jj}-G_{ij}\right)-{\rho }_{diss}\left(G_{ii}-G_{ij}\right)\right]$$

(7)

where we have invoked the following identities,

$$\begin{array}{c}\rho {\widehat{\upsilon }}_{diss}\left(T,P,x_i\right)=D\left[1+{\rho }_j\left(G_{jj}-G_{ij}\right)\right]\\ =\rho {\widehat{\upsilon }}_i/\nu \end{array}$$

(8)

With the diffusional stability coefficient $D=1+{\left(\partial \ln {\gamma }_i/\partial \ln x_i\right)}_{TP}$ [74], and

$$\rho {\widehat{\upsilon }}_j\left(T,P,x_i\right)=D\left[1+{\rho }_{diss}\left(G_{ii}-G_{ij}\right)\right]$$

(9)

As well as,

$$\kappa \left(T,P,x_i\right)=\left(D/kT\rho \right)\left[1+\rho \left(x_j{G}_{jj}+x_{diss}{G}_{ii}\right)+{\rho }^2{x}_{diss}{x}_j\left(G_{jj}{G}_{ii}-G_{ij}^2\right)\right]$$

(10)

Which define the partial molar volumes of the species, and the corresponding isothermal compressibility of the system with $x_{diss}=\nu x_i$ and ${\widehat{\upsilon }}_i=\nu {\widehat{\upsilon }}_{diss}$, according to the inversion of the Kirkwood-Buff formalism [1].

From Equations (1) and (6) we immediately find the explicit connection between the pressure perturbation upon solvation and the microstructural responses, i.e.,

$${\left(\partial P/\partial x_i\right)}_{T\rho }=\nu D{\kappa }^{-1}{\left({\mathcal{S}}_{ji}-{\mathcal{S}}_{ij}\right)}_{TPx}$$

(11)

where the phase stability requires $\kappa \left(T,P,x_i\right)>0$ and $D\left(T,P,x_i\right)>0$ [74] so that,

$$D\left(T,P,x_i\right)={\left(1-x_{diss}{\mathcal{S}}_{ij}-\nu x_j{\mathcal{S}}_{ji}\right)}^{-1}$$

(12)

Leading finally to,

$${\left(\partial P/\partial x_i\right)}_{T\rho }={\nu {\left({\mathcal{S}}_{ji}-{\mathcal{S}}_{ij}\right)}_{TPx}/\left[\kappa \left(1-x_{diss}{\mathcal{S}}_{ij}-\nu x_j{\mathcal{S}}_{ji}\right)\right]}_{TPx}$$

(13)

Note that the pressure perturbation upon solvation ${\left(\partial P/\partial x_i\right)}_{T,\rho }$ in a stable phase is always finite regardless of the state condition and composition, even though the right-hand-side of Equation (11) involves potentially divergent compressibility-driven contributions [12,77]. In fact, for the binary system comprising an infinitely dilute i-solute at the critical point of the pure j-solvent, the partial molar volume of the solute at infinite dilution will diverge at the same rate as the isothermal compressibility of the pure solvent, yet,

$$\begin{array}{cc}\hfill \underset{T\rho \to critical}{\lim }{\left(\partial P/\partial x_i\right)}_{T\rho }^{\infty }& =\underset{T\rho \to critical}{\lim }\left({\rho }_j^o{\widehat{\upsilon }}_i^{\infty }-\nu \right)/{\kappa }_j^o\hfill \\ & ={\mathcal{A}}_{Kr}\to finite\hfill \end{array}$$

(14)

where the so-called Krichevskii parameter ${\mathcal{A}}_{Kr}$ [77] becomes defined in terms of the limiting behavior of the structure-making/breaking function of the i-solute, i.e., [78]

$${\mathcal{A}}_{Kr}=-\nu \underset{T,{\rho }_j^o\to critical}{\lim }\left({\mathcal{S}}_{ij}^{\infty }/{\kappa }_j^o\right)$$

(15)

The finiteness of the alluded limiting function becomes more obvious after recalling that $\underset{T,{\rho }_j^o\to critical}{\lim }\left({\mathcal{S}}_{ij}^{\infty }/{\kappa }_j^o\right)={\mathcal{A}}_{Kr}^{IG\_i}\underset{T,{\rho }_j^o\to critical}{\lim }{\mathcal{S}}_{ij}^{\infty }\left(SR\right)$, with ${\mathcal{A}}_{Kr}^{IG\_i}=k{T}_c{\rho }_c$ where $SR$ denotes the short-range or solvation contribution to the property [79].

2.4. Link between ${\mathcal{S}}_{\alpha \beta }$ and the Second Osmotic Virial Coefficient of the Solute

The second osmotic virial coefficients derive from the isothermal solute-density expansion of the solute activity along different thermodynamic $\Theta$-paths including $\Theta \equiv \left({\mu }_j,P,{\rho }_j\right)$ (see e.g., Figure 4 and discussed in section SI-A of the Supplementary Information of Ref. [71]). According to the Lewis-Randall (LR) solution ideality reference, the structure-making/breaking function ${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)$ can be rigorously linked to the three distinctive types of second osmotic virial coefficients [71]. In fact, after invoking the limiting composition slope ${\left(\partial \ln {\gamma }_i^{LR}/\partial x_i\right)}_{TP}^{\infty }=-{\rho }_j^o{\left(G_{jj}^o+G_{ii}^{\infty }-2G_{ij}^{\infty }\right)}_{TP}$ [80], and recalling the definition of the isobaric–isothermal second osmotic virial coefficient, $B_i^{\prime }\left(T,P\right)=-0.5\left(G_{ii}^{\infty }-G_{ij}^{\infty }\right)$, it has been demonstrated that

$${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)={\left(\partial \ln {\gamma }_i^{LR}/\partial x_i\right)}_{TP}^{\infty }-2{\rho }_j^o{B}_i^{\prime }\left(T,P\right)$$

(16)

Likewise, from the definition of the isothermal iso $-{\mu }_j$ osmotic second virial coefficient, $B_i^{*}\left(T,P\right)=-0.5G_{ii}^{\infty }$ [81], follows immediately that,

$${\left(\partial \ln {\gamma }_i^{LR}/\partial x_i\right)}_{TP}^{\infty }={\rho }_j^o\left(kT{\kappa }_j^o-{\upsilon }_j^o\right)+2{\rho }_j^o\left(B_i^{*}-{\widehat{\upsilon }}_i^{\infty }\right)$$

(17)

And therefore,

$$\begin{array}{cc}\hfill {\mathcal{S}}_{ij}^{\infty }\left(T,P\right)& =-2{\rho }_j^o{B}_i^{*,LR-IS}+2{\rho }_j^o\left(B_i^{*}-{\widehat{\upsilon }}_i^{\infty }\right)-2{\rho }_j^o{B}_i^{\prime }\hfill \\ & =2{\rho }_j^o\left(B_i^{*,E}-B^{\prime }-{\widehat{\upsilon }}_i^{\infty }\right)\hfill \\ & =2{\rho }_j^o\left(B_i^{*}-B^{\prime }-{\widehat{\upsilon }}_i^{\infty }\right)-{\mathcal{S}}_{ij}^{\infty ,IG\_i}\hfill \end{array}$$

(18)

where

$$B_i^{*,LR-IS}\left(T,P\right)=0.5\left({\upsilon }_j^o-kT{\kappa }_j^o\right)=0.5{\upsilon }_j^o{\mathcal{S}}_{ij}^{\infty ,IG\_i}$$

(19)

And,

$$\begin{array}{cc}\hfill B_i^{*,E}\left(T,P\right)& =B_i^{*}-B_i^{*,LR-IS}\hfill \\ & =B_i^{*}-0.5{\upsilon }_j^o{\mathcal{S}}_{ij}^{\infty ,IG\_i}\hfill \end{array}$$

(20)

Moreover, from the definition of the isochoric-isothermal second osmotic virial coefficient [81], and subsequent interpretation in terms of Kirkwood-Buff integrals $B^{″}\left(T,P\right)=0.5\left[G_{ij}^{\infty ,2}/\left({\upsilon }_j^o+G_{jj}^o\right)-G_{ii}^{\infty }\right]$ [75], follows that [71],

$${\left({\mathcal{S}}_{ij}^{\infty }-1\right)}^2={\rho }_j^{o}kT{\kappa }_j^o\left[1+{\rho }_j^o{B}_i^{″}-{\left(\partial \ln {\gamma }_i^{LR}/\partial x_i\right)}_{TP}^{\infty }\right]$$

(21)

We must highlight that the derived ${\mathcal{S}}_{ij}^{\infty }=\Im \left(B_i^{\otimes }\right)$ relationships are all rigorous expressions involving neither approximations nor constraints on the type of solute, nature of the solvent, and magnitude of the intermolecular interactions asymmetries. However, for i-solutes whose non-idealities follows the $\ln {\gamma }_i\left(T,P,x_i\right)\sim {\left(1-x_i\right)}^2$ functionality, as in the case of gases in aqueous and organic solutions, [75,82] the limiting composition slope becomes ${\left(\partial \ln {\gamma }_i^{LR}/\partial x_i\right)}_{TP}^{\infty }=-2\ln {\gamma }_i^{LR}\left(T,P\right)$. Consequently, the fundamental structure-making/breaking function and the three osmotic second virial coefficients become related as follows,

$${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)=-{\left(\ln {\gamma }_i^{LR}+{\rho }_j^o{B}_i^{\ast ,E}\right)}_{TP}$$

(22)

$${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)=-2{\left(\ln {\gamma }_i^{LR}+{\rho }_j^o{B}_i^{\prime }\right)}_{TP}$$

(23)

$${\left({\mathcal{S}}_{ij}^{\infty }-1\right)}^2={\rho }_j^{o}kT{\kappa }_j^o\left[1+{\rho }_j^o{B}_i^{″}+2\ln {\gamma }_i^{LR}\right]$$

(24)

While we have illustrated the links for a non-electrolyte solute, similar connections can be drawn for a dissociative solute as described explicitly for electrolyte species in Ref. [83].

2.5. Preferential Solvation Phenomenon as a Balance among Structure-Making/Breaking Effects

Preferential solvation $\mathcal{P}\mathcal{S}$ usually refers to the effect of a third species (aka cosolvent) on the solubility of a solute, a phenomenon of great interest for the understanding of the thermodynamics underlying the solubility enhancement of low-solubility species in highly compressible environments [10,84,85,86,87], and the manipulation of the relative volatility of species to improve separations [88,89]. The behavior of these systems has been frequently interpreted by a diversity of thermodynamic preferential interaction parameters leading to a number of non-equivalent microscopic markers for the preferential solvation phenomenon [90,91]. The common denominator underlying this scenario is again the lack of an explicit microscopic definition of the underlying $\mathcal{P}\mathcal{S}$ marker as previously highlighted, i.e., “how do we measure the $\mathcal{P}\mathcal{S}$ of a given solute in a given mixture?” leading to “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?” [92].

To confront this issue, we have recently introduced a universal preferential solvation (state) function, $\mathcal{P}\mathcal{S}\left(T,P,x_j\right)={\rho }_o\left(G_{ik}^{\infty }-G_{ij}^{\infty }\right)$ [91], as unique microscopic measure of how much different the j-solvent environment around the i-solute is in comparison to the k-solvent environment counterpart, where ${\rho }_o={\rho }_j+{\rho }_k$. Thus, $\mathcal{P}\mathcal{S}\left(T,P,x_j\right)$ accounts unambiguously for the solvation of an infinitely dilute i-solute species in a mixed-solvent (i.e., a j-solvent plus a k-cosolvent) environment at fixed state conditions in terms of the solute-induced microstructural perturbations of the solvent and cosolvent environments resulting from the species intermolecular interaction asymmetries. This solute-induced differential microstructural perturbation, caused by the disparity of affinities between the i-solute and each of the solvents species, becomes more revealing when written explicitly written in terms of Kirkwood-Buff integrals [70] as follows,

$$\begin{array}{cc}\hfill \mathcal{P}\mathcal{S}\left(T,P,x_j\right)& ={\rho }_o\left(G_{ik}^{\infty }-G_{ij}^{\infty }\right)\hfill \\ & =N\left[\left({\mathcal{N}}_{ik}^{res,\infty }/N_k\right)-\left({\mathcal{N}}_{ij}^{res,\infty }/N_j\right)\right]\hfill \end{array}$$

(25)

where ${\rho }_{\beta }={\rho }_o{x}_{\beta }$, and ${\mathcal{N}}_{\alpha \beta }^{res,\infty }\equiv {\rho }_{\beta }{G}_{\alpha \beta }^{\infty }$ represents the average number of $\beta$-solvent molecules around the infinitely dilute $\alpha$-solute in either excess or deficit to that of the $\alpha \beta$-ideal gas uniform distribution. From Equation (25) follows immediately that $\mathcal{P}\mathcal{S}\left(T,P,x_j\right)=0$ for two precisely-defined scenarios; namely: (a) ${\left(G_{ik}^{\infty }=G_{ij}^{\infty }\right)}_{TPx}\ne 0$ for any interacting i-solute species at infinite dilution, where the j-solvent and the k-cosolvent behave identically as a single pure solvent leading to $G_{jj}=G_{kk}=G_{jk}=\left(kT{\kappa }_j^o-{\upsilon }_j^o\right)$, and (b) ${\left(G_{ik}^{\infty }=G_{ij}^{\infty }\right)}_{TPx}^{IG\_i}=0$ for the solvation of a non-interacting i-solute (ideal gas) species at infinite dilution in a mixed $\left(j+k\right)$-real solvent.

On the one hand, note that $\mathcal{P}\mathcal{S}\left(T,P,x_j\right)=0$ implies that $\left({\mathcal{N}}_{ik}^{res,\infty }/{\mathcal{N}}_{ij}^{res,\infty }\right)=\left(N_k/N_j\right)$, in other words, $\mathcal{P}\mathcal{S}\left(T,P,x_j\right)$ describes the average deviation of the mixed-solvent distribution around the i-solute at infinite dilution from that of the corresponding uniform bulk-phase value. On the other hand, $\mathcal{P}\mathcal{S}\left(T,P,x_j\right)\ne 0$ highlights the microstructural response of the system to the solute–solvent and solute-cosolvent interaction asymmetries, leading to the differential affinities of the solute with the solvent, whose macroscopic responses translate into thermodynamic excess properties describing the solution thermodynamic non-idealities [74]. Moreover, given the fact that the ternary system under consideration becomes fully characterized either by five Kirkwood-Buff integrals ($G_{ij}^{\infty }$, $G_{ik}^{\infty }$, $G_{jk}$, $G_{jj}$, $G_{kk}$), or four fundamental structure-making/breaking functions, (${\mathcal{S}}_{ij}^{\infty }$, ${\mathcal{S}}_{ik}^{\infty }$, ${\mathcal{S}}_{jk}$, ${\mathcal{S}}_{kj}$), where ${\mathcal{S}}_{kj}\left({\mathcal{S}}_{ij}^{\infty }\right)$ and ${\mathcal{S}}_{jk}\left({\mathcal{S}}_{ik}^{\infty }\right)$ with ${\mathcal{S}}_{kj}\ne {\mathcal{S}}_{jk}$. In other words, this ternary system becomes fully described by two independent fundamental structure-making/breaking functions, and consequently, the interpretation of preferential solvation phenomena based on $\mathcal{P}\mathcal{S}\left(T,P,x_j\right)$, Equation (25), becomes entirely described in terms of a linear combination of four fundamental structure-making/breaking functions ${\mathcal{S}}_{\alpha \beta }$ as follows [82,91],

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

(26)

We must emphasize that Equations (25) and (26) are exact statistical mechanical expressions for the alluded ternary system regardless of the type or nature of the species involved, while Equation (26) explicitly identifies the formal contribution of the relevant fundamental structure-making/breaking functions. Moreover, this expression becomes rigorously connected to the thermodynamic preferential solvation parameter in the isothermal–isobaric closed system ${\Gamma }_{i\alpha }^{({\chi }_i\to 0)}\left(x_j\right)\equiv {\left(\partial {\mu }_i^{\ominus }/\partial {\mu }_{\alpha }\right)}_{TP}$ (often identified as ${\nu }_{i\alpha }$, i.e., in Refs. ([93,94]) for $\alpha =\left(j,k\right)$, as follows:

$$\begin{array}{cc}\hfill {\Gamma }_{ij}^{(x_i\to 0)}\left(x_j\right)& \equiv {\left(\partial {\mu }_i^{r,\infty }/\partial {\mu }_j\right)}_{TP}\hfill \\ & ={\left[\partial \ln {\left(z{\widehat{\varphi }}_i\right)}^{\infty }/\partial \ln x_j\right]}_{TP}/\left[1+{\left(\partial \ln {\gamma }_j^x/\partial \ln x_j\right)}_{TP}\right]\hfill \\ & =x_j\mathcal{P}\mathcal{S}\left(T,P,x_j\right)\hfill \end{array}$$

(27)

where ${\mu }_i^{r,\infty }\left(T,P,x_j\right)=kT\underset{{\rho }_i\to 0}{\lim }\ln {\left(z{\widehat{\varphi }}_i\right)}_{T,P,{\rho }_i,x_j}$. Thus, by invoking the Gibbs–Duhem relation for the $\left(j+k\right)$-binary mixture, we find the complementary thermodynamic link,

$$\begin{array}{cc}\hfill {\Gamma }_{ik}^{(x_i\to 0)}\left(x_j\right)& \equiv {\left(\partial {\mu }_i^{r,\infty }/\partial {\mu }_k\right)}_{TP}\hfill \\ & =x_k\mathcal{P}\mathcal{S}\left(T,P,x_j\right)\hfill \end{array}$$

(28)

indicating that the isothermal–isobaric closed system ${\Gamma }_{i\alpha }^{({\chi }_i\to 0)}\left(x_j\right)\equiv {\left(\partial {\mu }_i^{\ominus }/\partial {\mu }_{\alpha }\right)}_{TP}$ comprises formally similar behaviors for the diffusible $\alpha$-species, resulting from the combined i-solute-induced perturbations of the j-solvent and k-cosolvent, as described by a linear combination of the four structure-making/breaking functions ${\mathcal{S}}_{\alpha \beta }$.

3. Microstructural Responses to Intermolecular Interaction Asymmetries, Environmental State Conditions, and Composition

Because the solvation process comprises the solute-induced effect on the solution microstructure at the prevailing state conditions and composition, it becomes instructive to evaluate how the solute–solvent molecular asymmetry, as well as individual perturbations of pressure, temperature, and composition, can modify the solvent microstructure around a solute species.

3.1. Response to Solute–Solvent Molecular Asymmetry

As a state function, the behavior of the fundamental structure-making/breaking function ${\mathcal{S}}_{ij}\left(T,P,x_j\right)$, Equation (1) of a stable fluid mixture becomes a continuous function of the magnitude of the solute–solvent interaction asymmetry, as well as the state conditions and system composition. Typically, the idea of structure-making/breaking behavior has been applied to solutes at infinite dilution [57,95], and therefore, we review first this limiting behavior for a binary fluid mixture, i.e., ${\mathcal{S}}_{ij}^{\infty }={\mathcal{S}}_{ij}\left(T,P,x_j\to 1\right)$. For the sake of generality, we start with Equation (1), written as follows:

$${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)=1-{\rho }_j^o{\widehat{\upsilon }}_i^{\infty }/\nu$$

(29)

whose isochoric-isothermal pressure perturbation upon solvation becomes [4]

$${\mathcal{S}}_{ij}^{\infty }=-\left({\kappa }_j^o/\nu \right){\left(\partial P/\partial x_i\right)}_{T{\rho }_j^o}^{\infty }$$

(30)

Equations (29) and (30) comprise two remarkable attributes: (a) the first indicates that the structure-making/breaking functions scale with the partial molar volume of the solute at infinite dilution, and consequently, with the magnitude of the solute–solvent intermolecular interaction asymmetry, and (b) the second provides an alternative avenue to experimentally measure the degree of dissimilarity between the solute and the solvent, i.e., the source of thermodynamic non-ideality of the resulting mixture.

In fact, Equation (30) indicates that an i-solute behaves as a structure-maker when ${\left(\partial P/\partial x_i\right)}_{T{\rho }_j^o}^{\infty }<0$, i.e., conventionally known as non-volatile [77] or attractive solutes [7]. Conversely, an i-solute behaves as a structure-breaker if ${\left(\partial P/\partial x_i\right)}_{T{\rho }_j^o}^{\infty }>0$, i.e., characterized as volatile [77] or weakly-attractive and repulsive [7] in the jargon of supercritical fluid solutions [12,85]. Obviously, nature presents a wide and continuous range of solute–solvent interaction asymmetries in binary fluid solutions. For instance, for a particular real j-solvent, we might consider a non-interaction (ideal gas) i-solute, $IG\_i$, characterized by $\nu =1$ so that [96]

$${\widehat{\upsilon }}_i^{\infty ,IG\_i}\left(T,P\right)=kT{\kappa }_j^o>0$$

(31)

$${\mathcal{S}}_{ij}^{\infty ,IG\_i}\left(T,P\right)=1-{\kappa }_j^{o}kT{\rho }_j^o⪋0$$

(32)

and a finite ${\left(\partial P/\partial x_i\right)}_{T{\rho }_j^o}^{\infty ,IG\_i}⪋0$. By cranking up the solute–solvent interactions, such as that of an inert gas solute, $B\_i$, the binary system can be characterized by the second virial coefficients $B_{ii}\left(T\right)$ and $B_{ij}\left(T\right)$ while keeping the solvent-solvent interactions unchanged, i.e., [4]

$${\widehat{\upsilon }}_i^{\infty ,B\_i}\left(T,P\right)=kT{\kappa }_j^o+2B_{ij}$$

(33)

and

$${\mathcal{S}}_{ij}^{\infty ,B\_i}\left(T,P\right)={\mathcal{S}}_{ij}^{\infty ,IG\_i}\left(T,P\right)-2{\rho }_j^o{B}_{ij}\left(T\right)$$

(34)

This expression indicates that ${\mathcal{S}}_{ij}^{\infty ,B\_i}\left(T,P\right)>{\mathcal{S}}_{ij}^{\infty ,IG\_i}\left(T,P\right)>0$ when the unlike-pair interaction second virial coefficient $B_{ij}\left(T\right)<0$, and consequently, the $B\_i$ solute induces an additional microstructural perturbation on the surrounding solvent, over that of the ideal gas solute, and becomes a structure-making solute. Otherwise, when $B_{ij}\left(T\right)>0$, there are two possible alternatives: either $0<B_{ij}\left(T\right)<0.5{\upsilon }_j^o{\mathcal{S}}_{ij}^{\infty ,IG\_i}$ giving rise to a structure-maker with $0<{\mathcal{S}}_{ij}^{\infty ,B\_i}<{\mathcal{S}}_{ij}^{\infty ,IG\_i}$ or $0<B_{ij}\left(T\right)>0.5{\upsilon }_j^o{\mathcal{S}}_{ij}^{\infty ,IG\_i}$ resulting in a structure-breaker. Note that at a specific $\left(T,P\right)$ state condition at which $B_{ij}\left(T\right)=- 0.5G_{jj}^o\left(T,P\right)$, Equation (34) indicates that the $B\_i$ solute does not perturb the microstructure of the j-solvent.

Should we keep increasing the magnitude solute–solvent interactions, up to the point that the solute–solvent interactions become equal to the corresponding solvent-solvent interactions, then we would have reached the null solute–solvent asymmetry, i.e., $G_{jj}^o\left(T,P\right)=G_{ij}^{\infty }\left(T,P\right) \to {\mathcal{S}}_{ij}^{\infty }\left(T,P\right)=0$, and the i-solute would not perturb the surrounding solvent environment. Not surprisingly, the most frequent scenario comprises systems with solute–solvent interactions different (either weaker or stronger) from the solvent–solvent interactions. When $G_{ij}^{\infty }\left(T,P\right) <G_{jj}^o\left(T,P\right)$, such as in the case of non-polar gases in water, we expect ${\widehat{\upsilon }}_i^{\infty }\left(T,P\right)>0$ and ${\left(\partial P/\partial x_i\right)}_{T{\rho }_j^o}^{\infty }>0$ so that ${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)<0$, i.e., the i-solute behaves as a structure-breaker species. Conversely, when $G_{ij}^{\infty }\left(T,P\right) >G_{jj}^o\left(T,P\right)$, such as in the case of strongly interactive solutes including strong electrolytes, we observe ${\widehat{\upsilon }}_i^{\infty }\left(T,P\right)-{\upsilon }_j^o\left(T,P\right)<0$ and ${\left(\partial P/\partial x_i\right)}_{T{\rho }_j^o}^{\infty }<0$ so that ${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)>0$, i.e., the i-solute becomes a structure-maker. Note, however, that the inequality ${\widehat{\upsilon }}_i^{\infty }\left(T,P\right)-{\upsilon }_j^o\left(T,P\right)<0$ can be satisfied by ${\widehat{\upsilon }}_i^{\infty }\left(T,P\right)\lessgtr 0$, i.e., by either ${\widehat{\upsilon }}_i^{\infty }\left(T,P\right)<0$ or $0<{\widehat{\upsilon }}_i^{\infty }\left(T,P\right)<{\upsilon }_j^o\left(T,P\right)$, as illustrated in Figure 1 below, where we provide a schematic with the discussed relationships among ${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)$, ${\widehat{\upsilon }}_i^{\infty }\left(T,P\right)$, and ${\left(\partial P/\partial x_i\right)}_{T\rho }^{\infty }=-{\mathcal{S}}_{ij}^{\infty }/{\kappa }_j^o$ while highlighting the relevant boundaries.

3.2. Response to Changes in State Conditions

From Equation (29), we immediately have the following temperature and pressure derivatives:

$$\begin{array}{cc}\hfill {\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial T\right)}_P& ={\rho }_j^o{\widehat{\upsilon }}_i^{\infty }\left({\beta }_{Tj}^o-{\beta }_{Ti}^{\infty }\right)/\nu \hfill \\ & =\left(1-{\mathcal{S}}_{ij}^{\infty }\right)\left({\beta }_{Tj}^o-{\beta }_{Ti}^{\infty }\right)\hfill \end{array}$$

(35)

and

$$\begin{array}{cc}\hfill {\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial P\right)}_T& ={\rho }_j^o{\widehat{\upsilon }}_i^{\infty }\left({\kappa }_i^{\infty }-{\kappa }_j^o\right)/\nu \hfill \\ & =\left({\mathcal{S}}_{ij}^{\infty }-1\right)\left({\kappa }_j^o-{\kappa }_i^{\infty }\right)\hfill \end{array}$$

(36)

where ${\kappa }_{\alpha }^{\oplus }=-{\left(\partial \ln {\upsilon }_{\alpha }^{\oplus }/\partial P\right)}_T$ and ${\beta }_{T\alpha }^{\oplus }={\left(\partial \ln {\upsilon }_{\alpha }^{\oplus }/\partial T\right)}_P$ identify the isothermal compressibility and isobaric thermal expansivity of the species $\alpha$ at the limiting composition condition $\oplus$, i.e., either pure component $\oplus =o$ or infinite dilution $\oplus =\infty$. Consequently, the magnitude and sign of the ${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)$ response to changes in either temperature or pressure depends on the product of two differences, i.e., either $\left(1-{\mathcal{S}}_{ij}^{\infty }\right)\left({\beta }_{Tj}^o-{\beta }_{Ti}^{\infty }\right)$ or $\left({\mathcal{S}}_{ij}^{\infty }-1\right)\left({\kappa }_j^o-{\kappa }_i^{\infty }\right)$, respectively. In fact, for a structure-making ($SM$) solute, ${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)>0$, characterized by the following condition:

$$\nu >{\rho }_j^o\left|{\widehat{\upsilon }}_i^{\infty }\right|\to \left|{\widehat{\upsilon }}_i^{\infty }\left(T,P\right)\right|<\nu {\upsilon }_j^o\left(T,P\right)$$

(37)

There are four distinctive responses to changes in temperature,

$$\begin{array}{l}{\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial T\right)}_P^{SM}=\left(1-{\mathcal{S}}_{ij}^{\infty }\right)\left({\beta }_{Tj}^o-{\beta }_{Ti}^{\infty }\right)\to \left\{\begin{array}{l}>0 if {\beta }_{Tj}^o>{\beta }_{Ti}^{\infty } \\ <0 if {\beta }_{Tj}^o<{\beta }_{Ti}^{\infty }\end{array}\right\} for 0<{\mathcal{S}}_{ij}^{\infty }<1\\ \\ {\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial T\right)}_P^{SM}=\left(1-{\mathcal{S}}_{ij}^{\infty }\right)\left({\beta }_{Tj}^o-{\beta }_{Ti}^{\infty }\right)\to \left\{\begin{array}{l}>0 if {\beta }_{Tj}^o<{\beta }_{Ti}^{\infty } \\ <0 if {\beta }_{Tj}^o>{\beta }_{Ti}^{\infty }\end{array}\right\} for {\mathcal{S}}_{ij}^{\infty }>1\end{array}$$

(38)

And another four distinctive responses to changes in pressure,

$$\begin{array}{l}{\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial P\right)}_T^{SM}=\left({\mathcal{S}}_{ij}^{\infty }-1\right)\left({\kappa }_j^o-{\kappa }_i^{\infty }\right)\to \left\{\begin{array}{l}>0 if {\kappa }_j^o<{\kappa }_i^{\infty } \\ <0 if {\kappa }_j^o>{\kappa }_i^{\infty }\end{array}\right\} for 0<{\mathcal{S}}_{ij}^{\infty }<1\\ \\ {\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial P\right)}_T^{SM}=\left({\mathcal{S}}_{ij}^{\infty }-1\right)\left({\kappa }_j^o-{\kappa }_i^{\infty }\right)\to \left\{\begin{array}{l}>0 if {\kappa }_j^o>{\kappa }_i^{\infty } \\ <0 if {\kappa }_j^o<{\kappa }_i^{\infty }\end{array}\right\} for {\mathcal{S}}_{ij}^{\infty }>1\end{array}$$

(39)

Otherwise, for a structure-breaking ($SB$) solute, ${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)<0$, is characterized by the following conditions:

$$\nu <{\rho }_j^o{\widehat{\upsilon }}_i^{\infty }\to {\widehat{\upsilon }}_i^{\infty }\left(T,P\right)>\nu {\upsilon }_j^o\left(T,P\right)>0$$

(40)

There are only two possible responses to changes in temperature

$${\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial T\right)}_P^{SB}=\left(1-{\mathcal{S}}_{ij}^{\infty }\right)\left({\beta }_{Tj}^o-{\beta }_{Ti}^{\infty }\right)\to \left\{\begin{array}{l}>0 if {\beta }_{Tj}^o>{\beta }_{Ti}^{\infty } \\ <0 if {\beta }_{Tj}^o<{\beta }_{Ti}^{\infty }\end{array}\right.$$

(41)

and another two possible responses to changes in pressure

$${\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial P\right)}_T^{SB}=\left({\mathcal{S}}_{ij}^{\infty }-1\right)\left({\kappa }_j^o-{\kappa }_i^{\infty }\right)\to \left\{\begin{array}{l}>0 if {\kappa }_j^o<{\kappa }_i^{\infty } \\ <0 if {\kappa }_j^o>{\kappa }_i^{\infty }\end{array}\right.$$

(42)

In summary, Equations (38)–(42) indicate contrasting solvent microstructural responses to changes in system pressure and temperature regardless of the structure-making/breaking nature of the i-solute. In fact, according to the thermodynamic relation ${\beta }_T=\kappa {\left(\partial P/\partial T\right)}_{\rho }$ with ${\left(\partial P/\partial T\right)}_{\rho }>0$, the microstructural response to either a pressure increase or temperature decrease accentuates both the intrinsic nature of the original structure-making when ${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)>1$ and structure-breaking solutes whenever the isothermal compressibility and the isobaric thermal expansivity of the pure solvent are both larger than those corresponding to the infinite dilute solute. Otherwise, when $0<{\mathcal{S}}_{ij}^{\infty }\left(T,P\right)<1$, the same changes in state conditions will weaken the intrinsic nature of the original structure-making and structure-breaking solutes whenever the isothermal compressibility and isobaric thermal expansivity of the pure solvent are both smaller than those of the infinite dilute solute.

Note that, according to the analysis of Section 3.1, we immediately find that

$${\mathcal{S}}_{ij}^{\infty ,G_{jj}^o=G_{ij}^{\infty }=G_{ii}^o}\left(T,P\right)={\left(\partial {\mathcal{S}}_{ij}^{\infty ,G_{jj}^o=G_{ij}^{\infty }=G_{ii}^o}/\partial T\right)}_P={\left(\partial {\mathcal{S}}_{ij}^{\infty ,I{G}_{jj}^o=G_{ij}^{\infty }=G_{ii}^o}/\partial P\right)}_T=0$$

(43)

For which $G_{jj}^o\left(T,P\right)=G_{ij}^{\infty }\left(T,P\right)=G_{ii}^{\infty }\left(T,P\right)=kT{\kappa }_j^o-{\upsilon }_j^o$, i.e., when a solute behaves identically as a solvent species, the structure-making/breaking function and its responses to $\left(T,P\right)$-perturbations are all zero. Moreover, for the non-interacting ideal gas solute, for which we have ${\mathcal{S}}_{ij}^{\infty ,IG\_i}\left(T,P\right)=1-{\kappa }_j^{o}kT{\rho }_j^o$, we have

$${\left(\partial {\mathcal{S}}_{ij}^{\infty ,IG\_i}/\partial T\right)}_P=-{\kappa }_j^{o}kT{\rho }_j^o\left({\beta }_{Tj}^o-{\beta }_{Ti}^{\infty ,IG\_i}\right)\to \left\{\begin{array}{l}<0 if {\beta }_{Tj}^o>{\beta }_{Ti}^{\infty ,IG\_i} \\ >0 if {\beta }_{Tj}^o<{\beta }_{Ti}^{\infty ,IG\_i}\end{array}\right.$$

(44)

$${\left(\partial {\mathcal{S}}_{ij}^{\infty ,IG\_i}/\partial P\right)}_T={\kappa }_j^{o}kT{\rho }_j^o\left({\kappa }_j^o-{\kappa }_i^{\infty ,IG\_i}\right)\to \left\{\begin{array}{l}>0 if {\kappa }_j^o>{\kappa }_i^{\infty ,IG\_i} \\ <0 if {\kappa }_j^o<{\kappa }_i^{\infty ,IG\_i}\end{array}\right.$$

(45)

The sign and magnitude of the responses to $\left(T,P\right)$-perturbations caused by a non-interacting solute in real solvents depend on how different ${\beta }_{Ti}^{\infty }$ and ${\kappa }_i^{\infty }$ are from its ideal gas values. Note that the authors have erroneously assumed in Ref. [4] that ${\beta }_{Ti}^{\infty ,IG\_i}=T^{-1}$ and ${\kappa }_i^{\infty ,IG\_i}=P^{-1}$, i.e., equivalent to assume that ${\left(\partial \ln {\kappa }_i^{IG\_i}/\partial T\right)}_P^{\infty }={\left(\partial \ln {\kappa }_i^{IG\_i}/\partial P\right)}_T^{\infty }=0$.

3.3. Response to Changes of State Conditions at Finite Composition

For the binary system forming a thermodynamic stable phase at finite compositions, i.e., exhibiting a finite $D$-coefficient, we deal with two fundamental structure-making/breaking functions, i.e., ${\mathcal{S}}_{ij}\left(x_j\right)\ne {\mathcal{S}}_{ji}\left(x_j\right)$, whose isothermal–isobaric composition dependence has been derived elsewhere [1]

$${\mathcal{S}}_{ij}\left(T,P,x_j\right)=1-\rho {\widehat{\upsilon }}_i/\nu \left[1+{\left(\partial \ln {\gamma }_j/\partial \ln x_j\right)}_{TP}\right]$$

(46)

and

$${\mathcal{S}}_{ji}\left(T,P,x_j\right)=1-\rho {\widehat{\upsilon }}_j/\left[1+{\left(\partial \ln {\gamma }_j/\partial \ln x_j\right)}_{TP}\right]$$

(47)

Therefore,

$${\left(\partial {\mathcal{S}}_{ij}/\partial T\right)}_{Px}=\left(1-{\mathcal{S}}_{ij}\right)\left[{\beta }_T-{\beta }_{Ti}+{\left(\partial \ln D/\partial T\right)}_{Px}\right]$$

(48)

with ${\beta }_T={\left(\partial \ln \upsilon /\partial T\right)}_P$, ${\beta }_{T\alpha }={\left(\partial \ln {\widehat{\upsilon }}_{\alpha }/\partial T\right)}_P$, while ${\widehat{\upsilon }}_{\alpha }$ identifies the partial molar volume of the $\alpha$-species, and

$${\left(\partial {\mathcal{S}}_{ji}/\partial T\right)}_{Px}=\left(1-{\mathcal{S}}_{ji}\right)\left[{\beta }_T-{\beta }_{Tj}+{\left(\partial \ln D/\partial T\right)}_{Px}\right]$$

(49)

with $D=\left[1+{\left(\partial \ln {\gamma }_j/\partial \ln x_j\right)}_{TP}\right]$ so that [74]

$${\left(\partial \ln D/\partial T\right)}_{Px}=-{\left(k{T}^{2}D\right)}^{-1}{\left(\partial {\widehat{h}}_j/\partial \ln x_j\right)}_{Px}$$

(50)

where ${\widehat{h}}_j\left(T,P,x_j\right)$ denotes the partial molar enthalpy of the j-species at the prevailing state conditions and composition.

Likewise,

$${\left(\partial {\mathcal{S}}_{ij}/\partial P\right)}_{Tx}=\left({\mathcal{S}}_{ij}-1\right)\left[\kappa -{\kappa }_i+{\left(\partial \ln D/\partial P\right)}_{Tx}\right]$$

(51)

where $\kappa =-{\left(\partial \ln \upsilon /\partial P\right)}_T$, ${\kappa }_{\alpha }=-{\left(\partial \ln {\widehat{\upsilon }}_{\alpha }/\partial P\right)}_T$, and

$${\left(\partial {\mathcal{S}}_{ji}/\partial P\right)}_{Tx}=\left({\mathcal{S}}_{ji}-1\right)\left[\kappa -{\kappa }_j+{\left(\partial \ln D/\partial P\right)}_{Tx}\right]$$

(52)

with

$${\left(\partial \ln D/\partial P\right)}_{Tx}={\left(kTD\right)}^{-1}{\left(\partial {\widehat{\upsilon }}_j/\partial \ln x_j\right)}_{Px}$$

(53)

while ${\widehat{\upsilon }}_j\left(T,P,x_j\right)$ denotes the partial molar volume of the j-species at the prevailing state conditions and composition.

As expected, these expressions converge into the corresponding ones derived for the case of the i-species becoming infinitely dilute in the j-species. In fact, considering that $\underset{x_j\to 1}{\lim }D=1$, the limiting behavior of Equation (46) becomes Equation (29) while $\underset{x_j\to 1}{\lim }{\mathcal{S}}_{ji}=1$. Similarly, Equations (48) and (51) become identical to Equations (35) and (36) given that $\underset{x_j\to 1}{\lim }{\beta }_T={\beta }_{Tj}^o$, $\underset{x_j\to 1}{\lim }\kappa ={\kappa }_j^o$, $\underset{x_j\to 1}{\lim }{\beta }_{Ti}={\beta }_{Ti}^{\infty }$, and $\underset{x_j\to 1}{\lim }{\kappa }_i={\kappa }_i^{\infty }$, while Equations (49) and (52) become null because $\underset{x_j\to 1}{\lim }{\mathcal{S}}_{ji}=1$. Consequently, as for the case of the infinite dilution, the magnitude and sign of the ${\mathcal{S}}_{\alpha \beta }\left(T,P,x_{\beta }\right)$ response to changes in either temperature or pressure depend on two factors, $\left(1-{\mathcal{S}}_{\alpha \beta }\right)\left[{\beta }_T-{\beta }_{T\alpha }+{\left(\partial \ln D/\partial T\right)}_{Px}\right]$ and $\left({\mathcal{S}}_{\alpha \beta }-1\right)\left[\kappa -{\kappa }_{\alpha }+{\left(\partial \ln D/\partial P\right)}_{Tx}\right]$, and give rise to either two sets of four possible outcomes for ${\mathcal{S}}_{\alpha \beta }\left(T,P,x_{\beta }\right)>0$ or two sets for ${\mathcal{S}}_{\alpha \beta }\left(T,P,x_{\beta }\right)<0$. For example, for ${\mathcal{S}}_{\alpha \beta }\left(T,P,x_{\beta }\right)>0$, we have that $\nu {\upsilon }_{\beta }^{o}D>\left|{\widehat{\upsilon }}_{\alpha }^{\infty }\left(T,P,x_{\beta }\right)\right|>0$ so that

$$\begin{array}{l}{\left(\partial {\mathcal{S}}_{\alpha \beta }/\partial T\right)}_{Px}^{SM}\to \left\{\begin{array}{l}>0 if \left[{\beta }_T-{\beta }_{T\alpha }+{\left(\partial \ln D/\partial T\right)}_{Px}\right]>0 \\ <0 if \left[{\beta }_T-{\beta }_{T\alpha }+{\left(\partial \ln D/\partial T\right)}_{Px}\right]<0\end{array}\right\} for 0<{\mathcal{S}}_{\alpha \beta }<1\\ \\ {\left(\partial {\mathcal{S}}_{\alpha \beta }/\partial T\right)}_{Px}^{SM}\to \left\{\begin{array}{l}>0 if \left[{\beta }_T-{\beta }_{T\alpha }+{\left(\partial \ln D/\partial T\right)}_{Px}\right]<0 \\ <0 if \left[{\beta }_T-{\beta }_{T\alpha }+{\left(\partial \ln D/\partial T\right)}_{Px}\right]>0\end{array}\right\} for {\mathcal{S}}_{\alpha \beta }>1\end{array}$$

(54)

$$\begin{array}{l}{\left(\partial {\mathcal{S}}_{\alpha \beta }/\partial P\right)}_{Tx}^{SM}\to \left\{\begin{array}{l}<0 if\left[\kappa -{\kappa }_{\alpha }+{\left(\partial \ln D/\partial P\right)}_{Tx}\right]>0 \\ >0 if \left[\kappa -{\kappa }_{\alpha }+{\left(\partial \ln D/\partial P\right)}_{Tx}\right]<0\end{array}\right\} for 0<{\mathcal{S}}_{\alpha \beta }<1\\ \\ {\left(\partial {\mathcal{S}}_{\alpha \beta }/\partial P\right)}_{Tx}^{SM}\to \left\{\begin{array}{l}>0 if \left[\kappa -{\kappa }_{\alpha }+{\left(\partial \ln D/\partial P\right)}_{Tx}\right]>0 \\ <0 if \left[\kappa -{\kappa }_{\alpha }+{\left(\partial \ln D/\partial P\right)}_{Tx}\right]<0\end{array}\right\} for {\mathcal{S}}_{\alpha \beta }>1\end{array}$$

(55)

Otherwise, for ${\mathcal{S}}_{\alpha \beta }\left(T,P,x_{\beta }\right)<0$ we have that ${\widehat{\upsilon }}_{\alpha }^{\infty }\left(T,P,x_{\beta }\right)>\nu {\upsilon }_{\beta }^{o}D>0$ so that there are only two possible responses to isobaric changes in temperature

$${\left(\partial {\mathcal{S}}_{\alpha \beta }/\partial T\right)}_P^{SB}\to \left\{\begin{array}{l}>0 if \left[{\beta }_T-{\beta }_{T\alpha }+{\left(\partial \ln D/\partial T\right)}_{Px}\right]>0 \\ <0 if \left[{\beta }_T-{\beta }_{T\alpha }+{\left(\partial \ln D/\partial T\right)}_{Px}\right]<0\end{array}\right.$$

(56)

and another two possible responses to changes in pressure

$${\left(\partial {\mathcal{S}}_{\alpha \beta }/\partial P\right)}_T^{SB}\to \left\{\begin{array}{l}>0 if \left[\kappa -{\kappa }_{\alpha }+{\left(\partial \ln D/\partial P\right)}_{Tx}\right]<0 \\ <0 if \left[\kappa -{\kappa }_{\alpha }+{\left(\partial \ln D/\partial P\right)}_{Tx}\right]>0\end{array}\right.$$

(57)

Note that, according to the analysis of Section 3.1, we immediately find that

$${\mathcal{S}}_{\alpha \beta }^{G_{\beta \beta }=G_{\alpha \beta }=G_{\alpha \alpha }}\left(T,P,x_{\alpha }\right)={\left(\partial {\mathcal{S}}_{\alpha \beta }^{G_{\beta \beta }=G_{\alpha \beta }=G_{\alpha \alpha }}/\partial T\right)}_{Px}={\left(\partial {\mathcal{S}}_{\alpha \beta }^{G_{\beta \beta }=G_{\alpha \beta }=G_{\alpha \alpha }}/\partial P\right)}_T=0$$

(58)

for which $G_{\beta \beta }\left(T,P,x_{\alpha }\right)=G_{\alpha \beta }\left(T,P,x_{\alpha }\right)=G_{\alpha \alpha }\left(T,P,x_{\alpha }\right)=kT{\kappa }_{\beta }^o-{\upsilon }_{\beta }^o$, i.e., when a solute behaves identically as a solvent species, the fundamental structure-making/breaking function and its responses to $\left(T,P,x_{\alpha }\right)$-perturbations are all zero. Moreover, for the system comprising a non-interacting finite composition ideal gas solute in a real j-solvent, there is only one structure-making/breaking function, ${\mathcal{S}}_{ij}^{IG\_i}\left(T,P,x_j\right)$ (see Appendix A), i.e.,

$${\mathcal{S}}_{ij}^{IG\_i}\left(T,P,x_j\right)=-{\rho }_j{G}_{jj}$$

(59)

And therefore,

$$\upsilon \left(T,P,x_j\right)={\widehat{\upsilon }}_j\left(1-x_i{\mathcal{S}}_{ij}^{IG\_i}\right)$$

(60)

whose isobaric temperature derivative results in the following expression:

$${\left(\partial {\mathcal{S}}_{ij}^{IG\_i}/\partial T\right)}_P=\left(1-{\mathcal{S}}_{ij}^{IG\_i}\right)\left({\beta }_{Tj}-{\beta }_{Ti}^{IG\_i}\right)\to \left\{\begin{array}{l}\left.\begin{array}{l}>0 if {\beta }_{Tj}>{\beta }_{Ti}^{IG\_i} \\ <0 if {\beta }_{Tj}<{\beta }_{Ti}^{IG\_i}\end{array}\right\} for {\mathcal{S}}_{ij}^{IG\_i}<1\\ \left.\begin{array}{l}<0 if {\beta }_{Tj}>{\beta }_{Ti}^{IG\_i} \\ >0 if {\beta }_{Tj}<{\beta }_{Ti}^{IG\_i}\end{array}\right\} for {\mathcal{S}}_{ij}^{IG\_i}>1\end{array}\right.$$

(61)

Moreover, for the isothermal pressure derivative, we find

$${\left(\partial {\mathcal{S}}_{ij}^{IG\_i}/\partial P\right)}_T=\left(1-{\mathcal{S}}_{ij}^{IG\_i}\right)\left({\kappa }_i^{IG\_i}-{\kappa }_j\right)\to \left\{\begin{array}{l}\left.\begin{array}{l}>0 if {\kappa }_i^{IG\_i}>{\kappa }_j \\ <0 if {\kappa }_i^{IG\_i}>{\kappa }_j\end{array}\right\} for {\mathcal{S}}_{ij}^{IG\_i}<1\\ \left.\begin{array}{l}<0 if {\kappa }_i^{IG\_i}>{\kappa }_j \\ >0 if {\kappa }_i^{IG\_i}>{\kappa }_j\end{array}\right\} for {\mathcal{S}}_{ij}^{IG\_i}>1\end{array}\right.$$

(62)

Note that (61)–(62) converge into (44)–(45) when the i-solute becomes infinitely dilute in the j-solvent.

4. Pitfalls from Surrogate Experimental Techniques

Researchers have often invoked surrogate experimental techniques to investigate the solvent environment around a solute and searched for correlations between the observed thermodynamic manifestations and the corresponding responses from different microscopic probes hoping to gain some understanding of the effect of a solute on the microstructure of the solution. Unfortunately, these research tools frequently provide a rather limited (i.e., short-ranged and/or orientational) view of the solute-induced effects, might invoke solvent-specific (e.g., water or other hydrogen-bonding species) approaches, or would depend on other markers to conjecture microscopic-to-macroscopic correspondence while overlooking that correspondence does not necessarily means cause-effect connection [97,98].

In what follows, we review two examples of these surrogate approaches, i.e., the conjectured markers associated with the behavior of the B-coefficient in Jones–Dole’s equation [99] together with its isobaric temperature derivative, and the isobaric-thermal expansivity of the solute in solution [95], whose misguided use has perpetuated their involvement as structure-making/breaking markers as if they had already been proven adequate for this purpose.

4.1. Myths and Verities about the Use of Jones–Dole’s B- and ${\left(\partial B/\partial T\right)}_P$-Coefficients as Structure Markers and Their Connection to the Fundamental Structure-Making/Breaking Function ${\mathcal{S}}_{ij}^{\infty }$

The empirical Jones–Dole’s equation for the relative viscosity of the dilute solution reads as follows:

$$\begin{array}{cc}\hfill {\eta }_r\left(T,P,c_i\right)-1& =A{c}_i^{0.5}+B{c}_i\mathrm{as}{c}_i\to 0\hfill \\ & \cong \ln {\eta }_r\left(T,P,c_i\right)\hfill \end{array}$$

(63)

where ${\eta }_r=\eta /{\eta }_j^o$ with ${\eta }_j^o\equiv \eta \left(c_i=0\right)$, $c_i$ denotes the molar concentration of the i-solute, and the coefficients $A$ and $B$ account for the direct ion-ion and the (ion) solute–solvent interactions, respectively [99]. If we assume that Jones–Dole’s equation provides an accurate representation of the shear viscosity of a dilute solution, the B-coefficient becomes determined from the limiting isothermal–isobaric composition slope, i.e.,

$$\begin{array}{cc}\hfill B& =\underset{c_i\to 0}{\lim }\left[\left({\eta }_r-1-A{c}_i^{0.5}\right)/c_i\right]\hfill \\ & \cong \underset{c_i\to 0}{\lim }\left[\left(\ln {\eta }_r-A{c}_i^{0.5}\right)/c_i\right]\hfill \end{array}$$

(64)

leading to $B\gtrless 0$. Because the B-coefficient is the coefficient of the linear concentration term in Jones–Dole’s equation, this coefficient has been associated with the solvent-mediated solute–solute interactions, and therefore, with the conjectured $B>0$ for a structure-maker species and $B<0$ for a structure-breaker species [53,100,101,102], later supplemented with the alternative isobaric temperature derivative ${\left(\partial B/\partial T\right)}_P\gtrless 0$ [57,103,104,105,106,107,108,109,110].

Despite early warnings, e.g., “This type of coefficient is therefore probably related to the disturbance of the structure which is present in such liquids…No general theory has yet been developed for the B-coefficient but important qualitative explanations have been advanced in terms of ion-solvent interactions…Interesting as these various correlations are, it should be emphasized that they are, as yet, qualitative in nature and a quantitative theory for the B-coefficient still awaits development” [111] that would have prompted legitimate questions, as identified below, there was instead a proliferation of rather convoluted claims involving unproven connections between the signs of either the B-coefficient or its temperature derivative ${\left(\partial B/\partial T\right)}_P$ and the solute-induced perturbation of the solvent microstructure, as if the alluded links were either self-evident or already established and validated. In other words, instead of addressing some genuine issues including whether Jones–Dole’s B-coefficient contains any embedded microstructural information, whether we could assign any definite foundation to the widely used assumption about the sign of the B-coefficient and its structure-making/breaking trend, and whether we could find a rational justification for the use of the sign of ${\left(\partial B/\partial T\right)}_P$ as a structure-making/breaking marker, unfortunately, the literature has been flooded with conjectures born on dicey narratives, including “A positive B-coefficient indicates that the ions tend to order the solvent structure and increase the viscosity of the solution while a negative B-coefficient indicates disordering and a decrease of viscosity” [102] and “the values of temperature coefficient of $B$, i.e., $\left(dB/dT\right)$ for the solute can provide direct evidence regarding their structure-making or breaking effect in solution. The values of $\left(dB/dT\right)$ are positive for hydrophilic structure-breaking groups. For hydrophobic structure-making groups, the value of $\left(dB/dT\right)$ is negative” [112], and a number of variations, as quoted in the Supporting Information of Ref. [113].

In a series of recent publications [4,71,113,114], we have addressed the issues identified above to test the validity and consistency of the outcomes from the application of the B-coefficient and/or its temperature derivative ${\left(\partial B/\partial T\right)}_P$ as structure-making/breaking markers. While all outcomes pointed to the lack of uniqueness of these markers, as clearly illustrated in

Table 1. Representative systems illustrating the eight $\left[B,{\left(\partial B/\partial T\right)}_P\right]$ sign-pair combinations and their resulting structure-making/breaking parameter ${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)$ for infinitely dilute solutes at ambient conditions.

Mixture	$\mathit{B}$	${\left(\partial \mathit{B}/\partial \mathit{T}\right)}_{\mathit{P}}$	${\mathcal{S}}_{\mathit{i}\mathit{j}}^{\mathit{\infty }}\left(\mathit{T},\mathit{P}\right)$	Ref.
NaCl + H2O	>0	>0	>0 (maker)	[104]
Me4NBr + H2O	>0	>0	<0 (breaker)	[104]
Ni(NO3)2 + H2O	>0	<0	>0 (maker)	[115]
Et4NBr + H2O	>0	<0	<0 (breaker)	[104]
KBr + H2O	<0	>0	>0 (maker)	[116]
KI + H2O	<0	>0	<0 (breaker)	[104]
IG_i+ H2O (a)	<0	<0	>0 (maker)	[113]
[Emim][NTf2] + EG	<0	<0	<0 (breaker)	[117]

^(a) Supercritical conditions where ${\kappa }_j^{o,IG}<{\kappa }_j^o$.

, where we observe that the same $\left[B,{\left(\partial B/\partial T\right)}_P\right]$ sign-pair combination could represent opposite structure-making/breaking parameter ${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)$ and, consequently, confirm the lack of uniqueness of the $\left[B,{\left(\partial B/\partial T\right)}_P\right]$ sign-pair combination as a structure-making/breaking marker.

To trace back the microscopic origin for such behavior according to explicit statistical mechanics arguments, we have interpreted the plausible links between the behavior of the B-coefficient (and/or its temperature derivative ${\left(\partial B/\partial T\right)}_P$) and the (electrolyte or non-electrolyte) solute-induced perturbation of the solvent environment. For that purpose, we invoked Feakins et al.’s [118,119] rigorous expression for the B-coefficient, derived according to the transition state theory [120], as follows:

$$B=\beta {\upsilon }_j^o\left(\Delta {\mu }_i^{♮,\infty }-\nu \Delta {\mu }_j^{♮,o}\right)-\left({\widehat{\upsilon }}_i^{\infty }-\nu {\upsilon }_j^o\right)$$

(65)

for the infinitely dilute i-solute where $\nu ={\nu }^{+}+{\nu }^{-}$ with $\nu =1$ for non-electrolyte solutes. $\Delta {\mu }_{\alpha }^{♮,\oplus }$ and ${\widehat{\upsilon }}_{\alpha }^{\oplus }$ denote the molar Gibbs free energy of activation of viscous flow and the corresponding partial molar volume for the $\alpha$-species, respectively, at the $\oplus$-composition condition, i.e., either infinite dilution or pure component. Equation (65) contributes two terms to the B-coefficient, namely the change of Gibbs free energy of activation for the viscous flow, $\left(\Delta {\mu }_i^{♮,\infty }-\nu \Delta {\mu }_j^{♮,o}\right)$ and the volumetric change when $\nu$-solvent molecules are alchemically mutated into a solute during the formation of an i-solute at infinite dilution in an otherwise pure j-solvent, $\left({\widehat{\upsilon }}_i^{\infty }-\nu {\upsilon }_j^o\right)$. This volumetric term provides direct access to the microstructural perturbation during the solvation process since, according to the rigorous definition of the structure-making/breaking parameter ${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)$, as described by Equation (4), $\left({\widehat{\upsilon }}_i^{\infty }-\nu {\upsilon }_j^o\right)=-\nu {\upsilon }_j^o{\mathcal{S}}_{ij}^{\infty }$ and Equation (65) becomes

$$B={\upsilon }_j^o\left[\nu {\mathcal{S}}_{ij}^{\infty }+\beta \left(\Delta {\mu }_i^{♮,\infty }-\nu \Delta {\mu }_j^{♮,o}\right)\right]$$

(66)

Equation (66) warrants the following important observations: (i) the B-coefficient must be defined by the limiting condition of infinite dilution, vide supra Equation (64), to be consistent with its TS-based interpretation, and (ii) the TS-based interpretation clearly illustrates that the B-coefficient results from the combination of the solute-induced perturbation of the solvent structure and the corresponding induced difference between the Gibbs free energy of activation for the viscous flow of the solute and the solvent. Consequently, Equation (66) not only confirms that the B-coefficient comprises some information about the solute-induced effect on the solvent microstructure but also suggests that the sign of the B-coefficient cannot unambiguously describe the structure-making/breaking nature of the dilute i-solute, contrary to the widespread practice in the literature [53,57,100,102,112,121,122,123,124,125,126,127,128].

To illustrate the lack of uniqueness of the B-coefficient and/or its temperature derivative ${\left(\partial B/\partial T\right)}_P$ as markers for the structure-making/breaking behavior, we first test the behavior of a prototypical solute (see Equation (15) of [113]) and then introduce a novel statistical mechanics proof. (Equation (15) of [114]). For example, in Ref. [113], we have determined the conditions at which the B-coefficient for the non-interacting ideal gas solute ($IG\_i$) might predict either a structure-making, ${\mathcal{S}}_{ij}^{\infty ,IG\_i}>0$, a structure-breaking, ${\mathcal{S}}_{ij}^{\infty ,IG\_i}<0$, or an unperturbed microstructure, ${\mathcal{S}}_{ij}^{\infty ,IG\_i}=0$, as illustrated below.

This straightforward and rigorous thermodynamic analysis illustrates the lack of uniqueness of the B-coefficient as a structure-making/breaking marker, in that either $B^{IG\_i}<0$ or $B^{IG\_i}>0$ could represent a structure-making, ${\mathcal{S}}_{ij}^{\infty ,IG\_i}>0$, and simultaneously a structure-breaking, ${\mathcal{S}}_{ij}^{\infty ,IG\_i}<0$, behavior depending on the state conditions of the pure solvent (see Table 1).

$$B^{IG\_i}\to \left\{\begin{array}{l}{\mathcal{S}}_{ij}^{\infty ,IG\_i}>0 \to \left\{\begin{array}{l}{B}^{IG\_i}>0 \to \left\{\begin{array}{l}\Delta {\mu }_j^{♮,o}>0 \mathrm{if}{\mathcal{S}}_{ij}^{\infty ,IG\_i}>{\rho }_j^o{B}^{IG\_i}\\ \Delta {\mu }_j^{♮,o}<0\mathrm{otherwise}\end{array}\right. \\ B^{IG\_i}<0 \to \Delta {\mu }_j^{♮,o}>0\end{array}\right.\\ \\ {\mathcal{S}}_{ij}^{\infty ,IG\_i}=0\to \left\{\begin{array}{l}{B}^{IG\_i}>0 \to \Delta {\mu }_j^{♮,o}<0\\ B^{IG\_i}<0 \to \Delta {\mu }_j^{♮,o}>0\end{array}\right\}\leftarrow {\kappa }_j^o=kT{\rho }_j^o\\ \\ {\mathcal{S}}_{ij}^{\infty ,IG\_i}<0 \to \left\{\begin{array}{l}{B}^{IG\_i}>0 \to \Delta {\mu }_j^{♮,o}<0 \\ B^{IG\_i}<0 \to \left\{\begin{array}{l}\Delta {\mu }_j^{♮,o}>0\mathrm{if}\left|{\mathcal{S}}_{ij}^{\infty ,IG\_i}\right|<{\rho }_j^o\left|B^{IG\_i}\right|\\ \Delta {\mu }_j^{♮,o}<0\mathrm{otherwise}\end{array}\right.\end{array}\right.\end{array}\right.$$

(67)

Moreover, for the non-interacting ideal gas solute, the system comprises $\nu =1$, $\Delta {\mu }_i^{♮,\infty ,IG\_i}=0$ [120], and ${\mathcal{S}}_{ij}^{\infty ,IG\_i}=1-kT{\rho }_j^o{\kappa }_j^o$ [4] so that its $B^{IG\_i}$-coefficient would read as follows:

$$\begin{array}{cc}\hfill B^{IG\_i}& ={\upsilon }_j^o\left({\mathcal{S}}_{ij}^{\infty ,IG\_i}-\beta \Delta {\mu }_j^{♮,o}\right)\hfill \\ & ={\upsilon }_j^o\left[1-\beta \Delta {\mu }_j^{♮,o}-\left({\kappa }_j^o/{\kappa }_j^{IG}\right)\right]\hfill \end{array}$$

(68)

with $\beta \Delta {\mu }_j^{♮,o}=\ln \left({\eta }_j^o{\upsilon }_j^o/\mathit{h}\mathcal{N}\right)$ [120], where ${\eta }_j^o$ and ${\upsilon }_j^o$ denote the shear viscosity and molar volume of the pure j-solvent, while $\mathit{h}$ and $\mathcal{N}$ identify Planck and Avogadro constants, respectively. Therefore, from Equation (68), we immediately find that

$$B^{IG\_i}=0\to \left\{\begin{array}{l}\beta \Delta {\mu }_j^{♮,o}+kT{\rho }_j^o{\kappa }_j^o=1 \to {\mathcal{S}}_{ij}^{\infty ,IG\_i}=\beta \Delta {\mu }_j^{♮,o}\ne 0 \\ \\ \beta \Delta {\mu }_j^{♮,o}=0\to {\mathcal{S}}_{ij}^{\infty ,IG\_i}=0\end{array}\right.$$

(69)

emphasizing that the condition of the null solute-induced effect on the viscosity of the dilute solution, $B^{IG\_i}=0$, represents two contrasting structure-making/breaking scenarios. The first one, the top line of Equation (69), indicates a non-zero solute-induced effect on the structure of a real solvent, ${\mathcal{S}}_{ij}^{\infty ,IG\_i}\ne 0$, whose sign depends on the $\left({\eta }_j^o{\upsilon }_j^o/\mathit{h}\mathcal{N}\right)\lessgtr 1$ condition. The second one, the bottom line of Equation (69), predicts an unperturbed structure of the solvent $\beta \Delta {\mu }_j^{♮,o}=0$ or $\left({\eta }_j^o{\upsilon }_j^o/\mathit{h}\mathcal{N}\right)=1$, a condition highly unlikely to occur in real solvents, as illustrated in

Table 2. Representative systems illustrating the $\left[B,{\left(\partial B/\partial T\right)}_P\right]$ sign-pair combinations and their resulting structure-making/breaking parameter ${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)$ for and infinitely dilute ideal gas solute in a variety of solvents mostly at ambient conditions (according to tabulated data from Ref. [129]).

j-Solvent	${\mathcal{S}}_{\mathit{i}\mathit{j}}^{\mathit{\infty },\mathit{I}\mathit{G}\_\mathit{i}}\left(\mathit{T},\mathit{P}\right)$	${\left(\partial {\mathit{B}}^{\mathit{I}\mathit{G}\_\mathit{i}}/\partial \mathit{T}\right)}_{\mathit{P}}$	${\mathit{B}}^{\mathit{I}\mathit{G}\_\mathit{i}}$	$\mathit{\beta }\mathbf{\Delta }{\mathit{\mu }}_{\mathit{j}}^{♮,\mathit{o}}$
ambient water	>0	>0	<0	3.7
supercritical water	<0	<0	<0	14.2
subcritical water	>0	<0	<0	8.1
dimethyl sulfoxide	>0	>0	<0	5.8
N-N dimethyl formamide	>0	>0	<0	5.0
Tetrahydrofuran	>0	>0	<0	4.6
NN-Dimethylacetamide	>0	>0	<0	5.4
acetonitrile	>0	>0	<0	3.9
ethylene glycol	>0	>0	<0	7.3
1,4 dioxane	>0	>0	<0	5.5
methanol	>0	>0	<0	4.0
ethanol	>0	>0	<0	5.1
1-propanol	>0	>0	<0	5.9
t-butanol	>0	>0	<0	6.9
trifluoroethanol	>0	>0	<0	5.7
acetone	>0	>0	<0	4.0
propylene carbonate???	>0	>0	<0	6.3
alfa-butyrolactonate	>0	>0	<0	5.8
formamide	>0	>0	<0	5.8
n-methyl formamide	>0	>0	<0	5.5
tetramethylurea	>0	>0	<0	6.1
n-methylpyrrolidinone	>0	>0	<0	5.9
pyridine	>0	>0	<0	5.7
benzonitrile	>0	>0	<0	5.7
nitromethane	>0	>0	<0	4.7
nitrobenzene	>0	>0	<0	6.1
chloroform	>0	>0	<0	4.7
1-1 dichloroethane	>0	>0	<0	4.6
1-2 dichloroethane	>0	>0	<0	5.0
tetramethylenesulfone	>0	>0	<0	7.8
tetrahydrothiophene	>0	>0	<0	5.3
trimethylphosphate	>0	>0	<0	6.4

for a series of condensed solvents.

According to the isobaric temperature derivative of Equation (66), Ref. [113] discusses the ability of the ${\left(\partial B/\partial T\right)}_P$ coefficient to discriminate between structure-making and structure-breaking solutes, under the four categories introduced by Tsangaris and Martin [103]. The resulting expression is

$$\begin{array}{cc}\hfill {\left(\partial B/\partial T\right)}_P& ={\beta }_{T,j}^{o}B+{\mathrm{\upsilon }}_j^o\left[\mathrm{\nu }{\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial T\right)}_P-\left(\Delta {\widehat{h}}_i^{♮,\infty }-\mathrm{\nu }\Delta h_j^{♮,o}\right)/k{T}^2\right]\hfill \\ & ={\beta }_{T,j}^{o}B+{\mathrm{\upsilon }}_j^o\left[\mathrm{\nu }\left(1-{\mathcal{S}}_{ij}^{\infty }\right)\left({\beta }_{T,j}^o-{\beta }_{T,i}^{\infty }\right)-\left(\Delta {\widehat{h}}_i^{♮,\infty }-\mathrm{\nu }\Delta h_j^{♮,o}\right)/k{T}^2\right]\hfill \end{array}$$

(70)

where the second line of Equation (70) highlights the replacement of ${\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial T\right)}_P$ with its corresponding explicit form previously derived [4]. Equation (70) indicates that the behavior (i.e., magnitude and sign) of ${\left(\partial B/\partial T\right)}_P$ depends on those of $B$, ${\mathcal{S}}_{ij}^{\infty }$, and ${\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial T\right)}_P$, as well as the differences $\left(\Delta {\mu }_i^{♮,\infty }-\mathrm{\nu }\Delta {\mu }_j^{♮,o}\right)$, $\left(\Delta {\widehat{h}}_i^{♮,\infty }-\mathrm{\nu }\Delta h_j^{♮,o}\right)$, and $\left({\beta }_{T,j}^o-{\beta }_{T,i}^{\infty }\right)$ giving rise to sixteen possible structure-making/breaking outcomes, as discussed in the Supporting Information of Ref. [113].

In brief, there are eight possible $\left[B,{\left(\partial B/\partial T\right)}_P\right]$ sign-pair combinations that could lead to a structure-making outcome ${\mathcal{S}}_{ij}^{\infty }>0$, with four exhibiting ${\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial T\right)}_P>0$ and another four exhibiting ${\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial T\right)}_P<0$, while the remaining eight combinations could result in a structure-breaking outcome ${\mathcal{S}}_{ij}^{\infty }<0$, with four exhibiting ${\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial T\right)}_P>0$ and another four exhibiting ${\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial T\right)}_P<0$. In other words, the sign of ${\left(\partial B/\partial T\right)}_P$ as a structure-making/breaking marker cannot distinguish between structure-making (${\mathcal{S}}_{ij}^{\infty }>0$) solutes and structure-breaking (${\mathcal{S}}_{ij}^{\infty }<0$). In summary, neither the ${\left(\partial B/\partial T\right)}_P$ alone nor the $\left[B,{\left(\partial B/\partial T\right)}_P\right]$ sign-pair combination can provide a unique one-to-one correspondence between their signs and the structure-making/breaking nature of the i-solute described here by the molecular-based parameter ${\mathcal{S}}_{ij}^{\infty }$, rendering these criteria deceptive markers of structure-making/breaking trends despite their widespread use in the literature, as illustrated in

Table 3. Experimentally based structure-making/breaking parameter ${\mathcal{S}}_{i\text{-}{H}_{2}O}^{\infty }\left(T,P\right)$ for aqueous infinitely dilute solutes at ambient conditions in comparison with predictions based on Hepler’s isobaric-thermal expansivity marker as well as Jones–Dole’s B-coefficient and ${\left(\partial B/\partial T\right)}_P$ derivative criteria.

i-Solute	${\mathcal{S}}_{\mathit{i}\text{-}{\mathit{H}}_2\mathit{O}}^{\mathit{\infty }}\left(\mathit{T},\mathit{P}\right)$	${\left(\partial \mathit{B}/\partial \mathit{T}\right)}_{\mathit{P}}$	$\mathit{B}$	$-\mathit{T}{\left({\partial }^2{\widehat{\mathit{\upsilon }}}_{\mathit{i}}^{\mathit{\infty }}/\partial {\mathit{T}}^2\right)}_{\mathit{P}}$	Ref.
water (a)	unperturbed	0	0	maker	[4]
urea	breaker	>0	>0	breaker [130]	[71,131,132]
N-methyl urea	breaker	>0	>0	breaker [133]	[71,132,134]
(1,3)-dimethyl urea	breaker	>0	>0	maker [130]	[71,132,134]
creatine	breaker	<0	>0	maker	[135]
creatinine	breaker	<0	>0	maker	[135]
nicotinic acid	breaker	<0	>0	maker	[136]
l-ascorbic acid	breaker	>0	>0	breaker	[71,136,137]
glycine	breaker	>0	>0	breaker	[138]
alanine	breaker	>0	>0	breaker	[138]
DTAB	breaker	>0	<0	breaker	[138]
caffeine	breaker	>0	>0	maker	[139]
theophilline	breaker	>0	>0	maker	[139]
theobromine	breaker	>0	>0	maker	[139]
l-serine	breaker	<0	>0	maker	[109]
l-arginine	breaker	<0	>0	maker	[109]
choline-biotinate	breaker	<0	>0	maker	[140]
choline-nicotinate	breaker	<0	>0	maker	[140]
choline-ascorbate	breaker	>0	>0	maker	[140]
LiCy	breaker	<0	>0	breaker	[141,142]
NaCy	breaker	~0	>0	maker	[141,142]
KCy	breaker	>0	>0	maker	[141,142]
CaCl2	maker	>0	>0	breaker	[143,144]
CdCl2	maker	<0	>0	breaker	[145,146]
NiCl2	maker	>0	<0	breaker	[146,147]
NH4NO3	breaker	>0	<0	breaker	[148,149]
MgCl2	maker	<0	>0	breaker	[146,150]

^(a) solute identical to the solvent.

below.

Table 2 and Table 3 highlight the fact that, despite the lack of uniqueness, the structure-making/breaking marker based on the signs of $\left[B,{\left(\partial B/\partial T\right)}_P\right]$ have been regularly invoked in combination with, or as an alternative to, the isobaric expansivity-based criterion to be discussed in Section 4.3 [107,109,110,137,139,151,152,153,154,155,156,157,158,159,160,161,162,163].

4.2. Sources of Ambiguities Underlying the Use of Jones–Dole’s B- and ${\left(\partial B/\partial T\right)}_P$-Coefficients as Structure Markers

Having illustrated the lack of uniqueness of the B-, ${\left(\partial B/\partial T\right)}_P$-, or any $\left[B,{\left(\partial B/\partial T\right)}_P\right]$ sign-pair combination as structure-making/breaking markers, the last piece of the puzzle to be solved was the origin of such behavior. For that purpose, Ref. [114] presented an alternative, yet equivalent, representation of Feakins’ TST approach by rewriting the first term of Equation (65) as follows:

$$\beta {\upsilon }_j^o{\left(\Delta {\mu }_i^{♮,\infty }-\nu \Delta {\mu }_j^{♮,o}\right)}_{TP}=\beta {\upsilon }_j^o\nu {\displaystyle \underset{0}{\overset{P}{\int }}\left({\upsilon }_j^o{\mathcal{S}}_{ij}^{\infty }-{\upsilon }_j^{♮,o}{\mathcal{S}}_{ij}^{♮,\infty }\right)d{P}^{\prime }}$$

(71)

which, after invoking the mean-value theorem for integrals [164], provided a more informative version of the link between the B-coefficient and the microstructure of the solvent around the solute as described by the fundamental structure-making/breaking function ${\mathcal{S}}_{ij}^{\infty }$,

$$B\left(T,P\right)=\nu z_j^o\overline{\left({\upsilon }_j^o{\mathcal{S}}_{ij}^{\infty }-{\upsilon }_j^{♮,o}{\mathcal{S}}_{ij}^{♮,\infty }\right)}-\nu {\upsilon }_j^o{\mathcal{S}}_{ij}^{\infty }$$

(72)

where $z_j^o=\beta P{\upsilon }_j^o$ identifies the compressibility factor of the pure j-solvent at the prevailing $\left(T,P\right)$ state conditions, and the horizontal bar on the RHS of Equation (72) describes the mean value of the function between parenthesis within the $\left[0,P\right]$ range.

Equation (72) highlights the interplay between microstructural perturbations at two different levels, the ground state and the transition state, where the perturbation associated with the activated state for the viscous flow is incorporated as a pressure integral, which defines the activation molar Gibbs free energy. Consequently, Feakins’ TST-based analysis of the B-coefficient describes a combination of disparate structure-making/breaking quantities, i.e., ground state versus transition state, which becomes the most obvious and direct theoretical attributes giving rise to the unavoidable drawback of the B-coefficient as a structure-making/breaking marker, beyond the ones previously discussed in detail elsewhere [78,113]. Likewise, under the same approach, the isobaric temperature derivative of Equation (72) led to

$$\begin{array}{cc}\hfill {\left(\partial B/\partial T\right)}_P=& \nu z_j^o\left\{{\beta }_{T,j}^{o,R}\overline{\left({\upsilon }_j^o{\mathcal{S}}_{ij}^{\infty }-{\upsilon }_j^{♮,o}{\mathcal{S}}_{ij}^{♮,\infty }\right)}+{\overline{\left[\partial \left({\upsilon }_j^o{\mathcal{S}}_{ij}^{\infty }-{\upsilon }_j^{♮,o}{\mathcal{S}}_{ij}^{♮,\infty }\right)/\partial T\right]}}_P\right\}-\hfill \\ & \nu {\left[\partial \left({\upsilon }_j^o{\mathcal{S}}_{ij}^{\infty }\right)/\partial T\right]}_P\hfill \end{array}$$

(73)

With the isobaric–isothermal residual thermal expansivity ${\beta }_{T,j}^{o,R}\equiv {\left({\beta }_{T,j}^o-{\beta }_{T,j}^{o,IG}\right)}_{TP}$, where the isobaric temperature derivatives ${\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial T\right)}_P$ and ${\left[\partial \left({\upsilon }_j^o{\mathcal{S}}_{ij}^{\infty }\right)/\partial T\right]}_P$ are simple functions of ${\mathcal{S}}_{ij}^{\infty }\left(T,P\right)$ by the exact relations [4]

$${\left(\partial {\mathcal{S}}_{ij}^{\infty }/\partial T\right)}_P=\left(1-{\mathcal{S}}_{ij}^{\infty }\right)\left({\beta }_{T,j}^o-{\beta }_{T,i}^{\infty }\right)$$

(74)

and

$$\begin{array}{cc}\hfill {\left[\partial \left({\upsilon }_j^o{\mathcal{S}}_{ij}^{\infty }\right)/\partial T\right]}_P& ={\upsilon }_j^o\left({\beta }_{T,j}^o-{\beta }_{T,i}^{\infty }\right)+{\upsilon }_j^o{\beta }_{T,i}^{\infty }{\mathcal{S}}_{ij}^{\infty }\hfill \\ & ={\upsilon }_j^o{\beta }_{T,j}^o-{\widehat{\upsilon }}_i^{\infty }{\beta }_{T,i}^{\infty }\hfill \end{array}$$

(75)

In other words, as in the case of the B-coefficient, ${\left(\partial B/\partial T\right)}_P$ involves solute-induced microstructural perturbations, and their temperature derivative, at not only the ground- but also the activated-state conditions. Consequently, neither the B-coefficient, ${\left(\partial B/\partial T\right)}_P$, nor any of its sign-pair combinations can unequivocally describe the actual solute-induced perturbation of the solvent microstructure at the ground state conditions, a conclusion that applies to any value of the stoichiometric coefficient $\nu$.

4.3. Frequent Pitfalls in the Application of Feakins’ TST Approach to the Study of B-Coefficients for Electrolytes

A persisting oversight in the application of Feakins’ TST approach to the B-coefficients involving electrolyte systems is the omission of the stoichiometric coefficient $\nu \ge 2$ [165] in Equation (66) and related expressions associated with the transition-state properties. This omission amounts to using $\nu =1$ in Equation (66) and leads to the corruption of the reported transition state thermodynamic quantities and their interpretation, given that it introduces significant errors in the values of the molar Gibbs free energy of activation for aqueous electrolytes involving $\nu =2$ [109,141,166,167,168,169,170,171,172,173,174] depending on the magnitude of the $\left({\upsilon }_j^o/{\widehat{\upsilon }}_i^{\infty }\right)$ ratio and becoming even larger with increasing $\nu$ [175,176,177,178,179]. As a matter of fact, according to the tabulated data from Ref. [124], it becomes evident the significantly large error introduced when assuming $\nu =1$ rather than the correct $2\le \nu \le 5$ as illustrated in

Table 4. Magnitude of the error in the calculated Gibbs free energy of activation when wrongly assuming $\nu =1$ in Equation (66) instead of the correct $2\le \nu \le 5$.

i-Solute	$\mathbf{\Delta }{\mathit{\mu }}_{\mathit{i}}^{♮,\mathit{\infty }}$ (kJ/mol)	$\mathit{\delta }\mathbf{\Delta }{\mathit{\mu }}_{\mathit{i}}^{♮,\infty }$ (%)
$HCl$	24.5	−12
$NCl$	26.7	−16
$RbCl$	12.6	38
$LiCl$	35.1	−24
$CsCl$	11.7	53
$E{t}_{4}NCl$	90.9	−14
$P{r}_{4}NCl$	172.7	−26
$NaF$	39.5	−33
$NaI$	19.9	8
$NaOH$	41.4	−35
$MgC{l}_2$	76.0	−41
$CuC{l}_2$	73.9	−42
$CaC{l}_2$	62.6	−34
$NiC{l}_2$	74.8	−41
$CrC{l}_3$	130.3	−54
$NdC{l}_3$	113.8	−51
$LuC{l}_3$	108.3	−50
$ThC{l}_4$	152.6	−57

, which translates into $- 57\lesssim \delta \Delta {\mu }_i^{♮,\infty }\left(\%\right)\lesssim 53$. In other words, the erroneous estimations of the molar Gibbs free energy of activation when assuming $\nu =1$ are carried over to the representation of the i-solute-induced perturbation of the j-solvent microstructure and its corresponding $\left(T,P\right)$-derivatives, i.e.,

$$\overline{\left[{\left(G_{ij}^{♮,\infty }-G_{jj}^{♮,o}\right)}_{TP}^{\nu \ge 2}\right]}\ne \overline{\left[{\left(G_{ij}^{♮,\infty }-G_{jj}^{♮,o}\right)}_{TP}^{\nu =1}\right]}$$

(76)

$${\overline{\left[\partial {\left(G_{ij}^{♮,\infty }-G_{jj}^{♮,o}\right)}^{\nu \ge 2}/\partial T\right]}}_P\ne {\overline{\left[\partial {\left(G_{ij}^{♮,\infty }-G_{jj}^{♮,o}\right)}^{\nu =1}/\partial T\right]}}_P$$

(77)

$${\overline{\left[\partial {\left(G_{ij}^{♮,\infty }-G_{jj}^{♮,o}\right)}^{\nu \ge 2}/\partial P\right]}}_T\ne {\overline{\left[\partial {\left(G_{ij}^{♮,\infty }-G_{jj}^{♮,o}\right)}^{\nu =1}/\partial P\right]}}_T$$

(78)

4.4. Myths and Verities about Hepler’s Isobaric-Thermal Expansivity as Structure-Making/Breaking Marker

Another surrogate tool frequently invoked for the assessment of the structure-making/breaking ability of a solute has been proposed by Hepler [95], who suggested a marker based on the conjecture that “since increasing pressure would also break up the bulky aggregates, the same reasoning suggests that the heat capacity of pure water should decrease with increasing pressure” and that this behavior could be extended to species in solution at infinite dilution. Hepler’s criterion is based on the sign of the isothermal pressure dependence of its partial molar heat capacity ${\left(\partial {\widehat{C}}_{Pi}^{\infty }/\partial P\right)}_T$, or its more easily accessible Maxwell equivalent expression $-T{\left({\partial }^2{\widehat{\upsilon }}_i^{\infty }/\partial T^2\right)}_P$. More specifically, this criterion states that whenever $-T{\left({\partial }^2{\widehat{\upsilon }}_i^{\infty }/\partial T^2\right)}_P<0$ or ${\left({\partial }^2{\widehat{\upsilon }}_i^{\infty }/\partial T^2\right)}_P>0$, the i-solute behaves as a structure-maker, and conversely, if $-T{\left({\partial }^2{\widehat{\upsilon }}_i^{\infty }/\partial T^2\right)}_P>0$ or ${\left({\partial }^2{\widehat{\upsilon }}_i^{\infty }/\partial T^2\right)}_P<0$, the i-solute behaves as a structure-breaker.

Unfortunately, as for the case of the B-coefficients, the lack of an explicit unambiguous cause-effect connection between either ${\left(\partial {\widehat{C}}_{Pi}^{\infty }/\partial P\right)}_T$ or $-T{\left({\partial }^2{\widehat{\upsilon }}_i^{\infty }/\partial T^2\right)}_P$ and the solute-induced perturbation of the solvent microstructure [1,4,71,78] makes any attempt to determine either the latter from volumetric data or the former from pressure perturbation calorimetry pointless [180,181,182,183]. Moreover, this so-called Hepler’s criterion immediately fails the compliance of two simple constraints associated with the behavior of a system comprising either the largest or the null solute–solvent intermolecular asymmetry.

For the largest solute–solvent intermolecular asymmetry we have the $IG\_i$ solute, characterized by ${\widehat{\upsilon }}_i^{\infty ,IG\_i}\left(T,P\right)=\nu kT{\kappa }_j^o$ [96,184], leading to the exact expression for its structure-making/breaking function [4]

$${\mathcal{S}}_{ij}^{\infty ,IG\_i}\left(T,P\right)=1-kT{\rho }_j^o{\kappa }_j^o$$

(79)

Consequently, from Equation (4), we expect to observe

$${\mathcal{S}}_{ij}^{\infty ,IG\_i}\left(T,P\right)\left\{\begin{array}{l}>0\to structuremaker\to {\kappa }_i^{IG\_i}>{\kappa }_j^o\\ =0\to unperturbedstructure\to {\kappa }_i^{IG\_i}={\kappa }_j^o\\ <0\to structurebreaker\to {\kappa }_i^{IG\_i}<{\kappa }_j^o\end{array}\right.$$

(80)

The significance of Equation (80) can be summarized as follows: (i) at ambient conditions, the $IG\_i$ in a real j-solvent exhibits a structure-making behavior, (ii) the microstructure of the pure j-solvent remains unperturbed along the states where the solvent behavior follows the ${\kappa }_j^o={\kappa }_j^{o,IG}={\left[kT{\rho }_j^o\right]}^{-1}$ condition, and (iii) after crossing this boundary, the ideal gas solute becomes a structure-breaker (e.g., see Figures 8 and 9 of Ref. [14] for the solvent water and the Lennard-Jones fluid, respectively).

Hepler’s criterion cannot account for the behavior described by Equation (80), in particular, the inability of the isobaric-thermal expansivity-based criterion to cross the ${\mathcal{S}}_{\beta \alpha }^{\infty ,IG\_\beta }\left(T,P\right)=0$ line, as described by $-T{\left({\partial }^2{\widehat{\upsilon }}_{\beta }^{\infty ,IG\_\beta }/\partial T^2\right)}_P=0$ in Hepler’s marker. In fact, it has been shown that ${\left({\partial }^2{\widehat{\upsilon }}_i^{\infty ,IG\_i}/\partial T^2\right)}_P^{{\kappa }_i^o={\kappa }_i^{IG\_i}}\ne 0$; in other words, Hepler’s criterion cannot describe the structure-making-to-breaking transition for the ideal gas solute $IG\_i$ in a real j-solvent along the $\left({\kappa }_j^o={\kappa }_j^{o,IG}\right)$-line. Moreover, for the null solute–solvent intermolecular asymmetry, as characterized by an i-solute behaving as a j-solvent and regardless of the definition of structure-making/breaking, the outcome must be an unperturbed solvent structure depicted by ${\mathcal{S}}_{ij}^{\infty ,G_{jj}^o=G_{ii}^{\infty }=G_{ij}^{\infty }\ne 0}\left(T,P\right)=0$ in the schematic Equation (3). While the physical representation for the null solute–solvent intermolecular asymmetry is characterized by ${\left(G_{jj}^o+G_{ii}^{\infty }-2G_{ij}^{\infty }\right)}_{TP}=0$ [72], its simplest and most advantageous case occurs when the Kirkwood-Buff integrals obey the ${\left(G_{jj}^o=G_{ii}^{\infty }=G_{ij}^{\infty }=kT{\kappa }_j^o-{\upsilon }_j^o\ne 0\right)}_{TP}$ condition. In other words, this condition is not only the microstructural signature of a pure j-solvent [96] but also the prototypical fingerprint for the microstructural transition between structure-making and structure-breaking processes driven by the solute–solvent intermolecular interaction asymmetry. This general and rigorous condition for the structure-making/breaking transition provides another opportunity to test the validity of the thermal expansivity-based criterion, which in Hepler’s terms becomes described by $-T{\left({\partial }^2{\widehat{\upsilon }}_i^{\infty }/\partial T^2\right)}_P=-T{\left({\partial }^2{\upsilon }_j^o/\partial T^2\right)}_P$.

It becomes immediately obvious that, given the condition $T{\left({\partial }^2{\upsilon }_j^o/\partial T^2\right)}_P>0$ for pure j-solvents at normal (around ambient), these fluids would describe the boundary $T{\left({\partial }^2{\upsilon }_j^o/\partial T^2\right)}_P=0$ and its crossing only if they meet at least two rather unlikely properties, namely an isobaric thermal expansivity ${\beta }_{Tj}^o$ independent of the state conditions, i.e., ${\left(\partial \ln {\upsilon }_j^o/\partial T\right)}_P\ne \mathcal{F}\left(T,P\right)$, and an isobaric-linear temperature-dependent isothermal compressibility, i.e., ${\kappa }_j^o\left(T,P\right)\equiv \mathcal{G}\left(P\right)+\mathcal{K}\left(P\right)T$. We are not aware of any real pure solvent following this bizarre behavior, and consequently, we must conclude that Hepler’s criterion cannot possibly describe and/or predict the transition between structure-making and structure-breaking events involving common solvents (in particular water which was the target of Hepler’s analysis). In summary, according to the above discussion and the evidence in Table 3, Hepler’s conjectured marker cannot account for the solute-induced perturbation of the solvent microstructure of infinitely dilute solution comprising any solute–solvent intermolecular asymmetries, i.e., either the smallest (zero, for the special case of $LR-IS$) or the largest (for the $IG\_i$ solute).

5. Concluding Remarks and Outlook

The usefulness of the structure-making/breaking concept has been previously challenged due to its lack of uniqueness [64], i.e., forged by the absence of a precise definition for what is meant by structure, and how to measure its perturbation. In addressing this challenge, we have argued that any criterion or marker for the structure-making/breaking ability of a solute must represent a state function and apply to any type and magnitude of solute–solvent molecular interaction asymmetry and should comply with some fundamental molecular thermodynamic constraints. More importantly, we have emphasized that the uniqueness does not (and cannot) depend on the criterion or probe used for the test of the type of solute-induced perturbation of the solvent microstructure.

On the one hand, we have reviewed the approaches to rigorously describe the ability of a solute species to perturb the solvent environment, to assign a precise statistical mechanics meaning to the structure-making/breaking event, and to rigorously link it to its thermodynamic responses, including quantitative measures of solution thermodynamic non-ideality which can be assessed experimentally. On the other hand, we have lent true molecular thermodynamic support for the testability, and eventual falsifiability, of a pair of popular conjectured structure-making/breaking markers whose validity has never been established but assumed in the current literature.

Supported by rigorous statistical thermodynamics, we discussed the proper interpretation of the role played by the species-induced microstructural perturbations of the solvent, their response to environmental state conditions and composition, as well as our ability to probe the system and test all conjectures in the development of a causative explanation for a variety of solvation phenomena. In other words, “…if a theory is testable, then it implies that events of a certain kind cannot happen; and so, it asserts something about reality. This is why we demand that the more conjectural a theory is, the higher should be its degree of testability.” [185].

In closing, we must stress that often times, authors attempt to add value to their experimental data by introducing hand-waving interpretations of the trends in the observed thermodynamic properties based on umbrella concepts of structure-making/breaking as if, by placing names to unsupported or conjectured ideas, they provide additional understanding. In fact, in the words of Richard Feynman, [186] we should always recognize “… the difference between knowing the name of something and knowing something”.