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B: Liquids, Chemical and Dynamical Processes in Solution, Spectroscopy in Solution

On the Solute-Induced Structure-Making/Breaking Effect Rigorous Links Between Microscopic Behavior, Solvation Properties, and Solution Non-Ideality Ariel A. Chialvo J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.9b00364 • Publication Date (Web): 22 Feb 2019 Downloaded from http://pubs.acs.org on February 23, 2019

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The Journal of Physical Chemistry

On the Solute-induced Structure-Making/Breaking Effect Rigorous Links between Microscopic Behavior, Solvation Properties, and Solution Non-ideality Ariel A. Chialvo * Knoxville, TN 37922-3108, U.S.A. email: [email protected] ORCID # 0000-0002-6091-4563

February 20, 2019

ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ABSTRACT We studied the solute-induced perturbation of the solvent environment around a solute species from a microscopic viewpoint, and propose a novel approach to the understanding of the structure-making/breaking process, regardless of the type and nature of the solute-solvent interactions. Based on the Kirkwood-Buff fluctuation formalism we present a rigorous statistical mechanics description of the evolution of the solvent structure around the solute, analyze its response to small perturbations of the (TP ) state conditions and composition of the system, and make direct connections between a few equivalent micro- and macroscopic manifestations as probes for, and targets of, experimental measurements. We illustrate the analysis with theoretical results from integral equation calculations of model fluids, and experimental evidence from available data for a variety of aqueous electrolyte and non-electrolyte real fluid solutions. Finally, we provide a critical discussion about the inadequacy underlying a widely used de facto criterion for the classification of structure-making/breaking solutes. 1. INTRODUCTION There is an established consensus about the solvation of a solute in a solvent as a soluteinduced reorganization of the solvent environment toward the formation of the solvation structure.1-7 This reorganization can be microstructurally described as a solvent-density local perturbation resulting from the mutation of a solvent particle to a solute particle, followed by the propagation of such perturbation across the system up to a (correlation length) distance as dictated by the magnitude of the medium isothermal compressibility.8-10 The local density perturbation stems from the asymmetry between the strength of the original solvent-solvent and the resulting solute–solvent interactions whose obvious macroscopic outcome manifests as the thermodynamic non-ideality of the solution. 11-14 Significant effort has been placed on the development of tools for the extraction of structural information of aqueous electrolyte and non-electrolyte solutions from scattering experiments, mainly by x-ray and neutron scattering methods. 15-27 The meaningful interpretation of the experimental microstructural evidence has frequently been facilitated by molecular-based tools, either as molecular simulations of model systems to test the consistency of the hypothesis underlying the scattering methodology, 28, 29 or as empirical microstructural refinement of scattering data. 30, 31

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Furthermore, molecular-based simulation has been supporting the characterization of the solute-induced effects on the structure of fluid mixtures by means of microscopically-defined geometric parameters to assess the perturbation of the solvent structure and its orientational order within the species’ coordination shells. This characterization has typically involved quantitative measures of the deviations of molecular configurations from perfect tetra- and octahedral geometries as order descriptors including the tetrahedral parameters qT

32

and T ′ ,33 the

octahedral parameter Q′ , 33 all for aqueous solutions, as well as the bond-order parameter Qℓ

34

for liquids and solids in general. While all these order descriptors are widely used to provide meaningful interpretations at the molecular level, 35-42 we are unaware of any attempt to link the evolution of their magnitude to thermodynamic properties characterizing the solution nonideality. Likewise, we find a variety of experimental methods to probe locally the solvation environment, including infrared, 43, 44 Raman, 45, 46 x-ray absorption,47, 48 and femtosecond pumpprobe spectroscopies, 49 as well as neutron and x-ray diffraction.50-52 Yet, we find no clear link between the experimental observations and any metrics of either induced structural perturbation or solution non-ideality. In fact, the research effort in solution chemistry has typically focused on the solvation behavior of solutes, which has been traditionally classified as structure makers and structure breakers. 53-57 Alternatively, the terms kosmotropes (order-makers) and chaotropes (disordermakers) have also been frequently used in the specialized literature 58 as an interchangeable notation for structure makers and structure breakers, respectively. However, and despite the generalized use as interchangeable, 58-63 these terminologies are not, a fact that introduces additional ambiguities into the interpretation of the reported solvation behavior of solutes. 64 Given that the vast majority of the studied mixtures in the literature involves water as the solvent, the interpretation of the structure-making/breaking behavior has typically been linked to the evolution of the hydrogen-bond network around the species (solute) of interest. 55, 58, 65, 66 Nonetheless, Marcus 67, 68 introduced the concept of structuredness as a way to “relate to more subtle interactions characterizing bulk properties of a liquid...” Unfortunately, this new concept requires the description of three additional terms, namely: stiffness, openness, and ordering of a liquid, 69 and an additional measure of the hydrogen-bonding strength for the case of aqueous solutions. 55, 66 Consequently, it becomes difficult to envision a microscopic meaning for the

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solvent structuredness when the concept is completely blurred by the conjectured nature and macroscopic roots of the stiffness, openness, and ordering quantities. Likewise, a recurrent factor contributing to the confusion around the concept of structure maker/breaker solutes is the lack of a clear definition or identification of what constitutes a structure making/breaking event, i.e., the absence of an unambiguous relationship between the microstructural impact of the solute on the surrounding solvent environment ¾ the induced local density perturbation ¾ and the corresponding macroscopic manifestation that can be experimentally measured. 55, 64 This situation becomes abundantly clear in the contrasting outcome highlighted by Kundu and Kishore 70 who noted that “… glucose was classified as a structure-maker by Franks et al. 71 on the basis of V2,φ , by Kawaizumi et al. 72 on the basis of o C p,2 , by Miyajima et al. 73 on the basis of V2,φ , Gm , and viscosity measurements, and by Tait et

al. 74 on the basis of nuclear and dielectric relaxation studies. On the contrary, glucose has also

(

o ∂T been classified as structure-breaker by Neal and Goring 75 based upon ∂V2,m

)

P

, by Ben-

(

Naim 76 because of its effect on argon solubility, and by Hepler 77 from the sign of T ∂ 2V ∂T 2

)

P

.” In this contribution we present a simple, yet rigorous, account of the solute-induced perturbation of the solvent environment, precisely and unambiguously characterized by the radial pair correlation functions and their corresponding pair correlation function integrals (i.e., Kirkwood-Buff integrals) to identify the links alluded to in the subtitle of this manuscript. The main outcome of this study becomes a rigorous molecular thermodynamic tool able to predict the ability of a solute species to perturb ¾ structure make/break ¾ the surrounding solvent environment based on the behavior of its thermodynamic properties available experimentally. For that purpose, we address a few relevant issues including: (i) why are we concerned with the evolution of solvent microstructure around any species, be it solute or solvent?, (ii) how do we measure it?, (iii) does it matter whether the solute is enhancing or weakening the microstructure of its solvent surroundings?, (iv) could we derive a supporting statistical mechanics foundation to make the microscopic-to-macroscopic connection rigorous?, and (v) what experimental probe could provide an unambiguous measure of such phenomenon?

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To analyze the above-mentioned issues, we propose in §2 a novel approach to the study of structure-making/breaking processes starting with a simple binary mixture of an i − solute in a

j − solvent, i.e., the case of an infinitely dilute solute in an otherwise pure solvent. By invoking the Kirkwood-Buff fluctuation formalism, 78 we present a rigorous microscopic description of the evolution of the solvent structure around the solute, Sij∞ (TP ) , and make a direct connection with a few equivalent micro- and macroscopic manifestations as probes for and targets of experimental measurements. After discussing the main ideas behind the definition of the function Sij∞ (TP ) , we analyze in §3 the response of this function to small perturbations of the (TP ) state conditions and composition of the system. The responses are formally written in terms of the solvation thermodynamics of the infinite dilute solution, and consequently, fully accessible by theory, simulation, and experiment. As an instructive illustration, we report in §4 the behavior of

Sij∞ (TP ) and its response functions for infinitely dilute Lennard-Jones binary mixtures over a wide range of molecular asymmetries. The theoretical results are obtained by integral equations calculations based on the Percus-Yevick approximation ( IE − PY ), according to McGuiganMonson protocol.79 Then, in §5 we present a comparison between the experimental evidence based on the proposed approach and the literature suggested behavior for a variety of mixtures. This analysis is followed in §6 by a critical discussion about a widely used de facto criterion for the classification of structure-making/breaking solutes, and the extension of the analysis to systems of finite composition. Finally, we provide a short summary of the findings and outlook.

2. RIGOROUS DEFINITION OF SOLUTE-INDUCED MICROSTRUCTURAL PERTURBATION OF SOLVENT ENVIRONMENT AROUND A SPECIES As frequently discussed,3, 80, 81 the Kirkwood-Buff fluctuation formalism of mixtures 78 provides a general and rigorous framework to link the microstructure of the mixture of interest, as volume integrals over pair correlation functions, and its macroscopic manifestation in terms of chemical and mechanical partial molar properties. Moreover, this formalism is not restricted by either the nature of the intermolecular interactions, species molecular size and shape, or the number of components in the system. However, we must be mindful that for dissociable solutes,

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we need to account properly for the electroneutrality condition for ionic salts 82, 83 or the stoichiometric balance for neutral species. 84 For that purpose, we can describe the system of interest in terms of the solvation of N i − solute particles in a pure solvent at fixed (TP ) state conditions, i.e., as the formation of dilute system in a thought experiment involving N − solvent molecules in which N i ≪ N of them can be distinguished by their solute labels. Thus, the initial conditions of the system represent a special case of an ideal solution according to the Lewis–Randall rule, 85, 86 for which the residual properties of the N − N i = N j solvent-labeled particles and those of the N i − solute labeled particles are identical. 87 Then, the formation of the desired dilute non-ideal solution proceeds as a so-called alchemical (à la Kirkwood’s coupling-parameter 88) mutation of the N i − distinguishable solvent particles into the final N i − solute particles as depicted in the schematic representation of Figure 1 for N i = 2 . This solvation process entails a density perturbation of the surrounding solvent around the solute species, whose magnitude is the manifestation of the microstructural changes of the system and the focus of this analysis. 2.1. Non-dissociable solutes Of particular interest here is the Kirkwood-Buff total correlation function integral (TCFI), 78

Gαβ (TPx ) ≡ ∫ hαβ (TPx,r) d r 3N

(1)

where hαβ (TPx,r ) = gαβ (TPx,r ) − 1 denotes the radial pair correlation function for the αβ − pair interactions at the (TPx ) state conditions and composition. Moreover, the radial distribution function gαβ (TPx,r ) defines the β − species microstructure around the α − species.89 In that sense, we can think of Gαβ (TPx ) in Eqn. (1) as an isobaric-isothermal residual property,90 i.e.,

6


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IG Gαβ (TPx ) ≡ ∫ gαβ (TPx,r ) d r 3N − ∫ gαβ (TPx,r ) d r 3N

= Gαβ (TPx ) − ∫ d r 3N

(2)

= Gαβ (TPx ) − GαβIG (TPx ) res = Gαβ (TPx )

IG where gαβ (TPx,r ) = 1 represents the ideal gas uniform distribution so that, 91

N ijex (TPxi ) = 4πρ x j ∫ ⎡⎣ g ij (TPx,r ) − g jj (TPx,r ) ⎤⎦ r 2 dr 0 = N ij (TPxi ) − N jj (TPxi ) ∞

(3)

= N ijres (TPxi ) − N jjres (TPxi )

The quantity N ij (TPxi ) describes the absolute average number of j − species around any i − species at the prevailing state conditions and system composition, with its residual quantity counterpart N ijres (TPxi ) = N ij (TPxi ) − N j . In other words, N ijex (TPxi ) represents the average number of j − solvent molecules around the i − solute in excess or deficit to that around any j − solvent. This excess/deficit results from the strength asymmetry between the solvent-solvent and the solute-solvent interactions. Consequently, N ijex (TPxi ) provides an unambiguous and direct descriptor of the magnitude of the i − solute induced-perturbation of the surrounding j − solvent environment, regardless of the nature of the prevailing inter- and intra-molecular interactions. Running excess coordination numbers are frequently determined by molecular simulation, usually as an attempt to adjust the force-fields of a solute-solvent intermolecular potential, through the comparison with the corresponding experimentally extracted KirkwoodBuff integrals. 92, 93 However, these excess coordination numbers are not strictly equivalent to

Sij (TPx ) given their intrinsic link to function typically determined by simulation of close (isochoric-canonical or isothermal-isobaric), rather than the required open (grand canonical) ensembles. The resulting finite-range of the radial correlation functions precludes the proper calculation of the asymptotic (long-range) behavior, and consequently, prevents its proper integration as discussed by Ben-Naim elsewhere. 84

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According to Eqn. (3) we can adopt Sij (TPxi ) = N ijex (TPxi ) as the property that discrimate unambiguously the structure-making/breaking of the i − species in solution with j − species where we purposefully avoid identifying which species describes the solute or the solvent. Moreover, from the Kirkwood-Buff formalism we obtain,78

(

) = 1− ⎡1+ ρ x x ( G ⎣

Sij (TPxi ) ≡ ρ x j Gij − G jj i

(

with ρ (TPxi ) = xiυˆ i + x jυˆ j

)

−1

j

TPx

)

+ Gii − 2Gij ⎤ ( ρυˆ i )TPx jj ⎦TPx

(4)

while υˆ i (TPxi ) denotes the partial molar volume of the i −

species. If the i − solute induces a local density perturbation that modifies the original solvent environment from that described by G jj (TPxi ) , to that portrayed by

G jj (TPxi ) → Gij (TPxi ) > G jj (TPxi ) , then the net effect N ijex (TPxi ) > 0 is an excess or strengthening of the solvent environment around the solute. We identify this case as a soluteinduced structure-making effect. Conversely, if the i − solute-induced perturbation translates into G jj (TPxi ) → Gij (TPxi ) < G jj (TPxi ) process, then the net effect N ijex (TPxi ) < 0 is a deficit or weakening of the solvent environment around the solute, identified as a solute-induced structure breaking effect. We must also consider the case when the solute leaves the solvent environment unperturbed, G jj (TPxi ) → Gij (TPxi ) = G jj (TPxi ) , and consequently, N ijex (TPxi ) = 0 , whose implications and links to solution ideality will be discussed below in §3.1. The significance behind the Sij (TPxi ) definition becomes more obvious when we analyze the case of a single i − solute in a j − solvent, i.e., the case of an infinitely dilute solution. In fact, Eqn. (4) for this case reads, 94

(

)

Sij∞ (TP ) = ρ oj Gij∞ − G ojj = N ijex,∞ (TP )

(5)

so that, the magnitude of Sij∞ (TP ) tells us about how different the microstructure of the j − solvent is around the i − solute, relative to that around the j − solvent itself. Moreover, the sign

8


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of Sij∞ (TP ) qualifies the i − solute as either a structure-maker, Sij∞ (TP ) > 0 , or a structurebreaker, Sij∞ (TP ) < 0 , based on the corresponding behavior of the solvation shells of the solvent around the solute and around the solvent itself. The ability of Eqn. (5) to identify unambiguously structure-maker/breaker behavior becomes more useful after we link it to macroscopically defined quantities that can be measured experimentally. For instance, from Eqn. (4) we can define Sij∞ (TP ) in terms of the partial molar volumes of the pure solvent υ oj = 1 ρ oj , and of the solute at infinite dilution υˆ i∞ (TP ) as follows, 8, 10

Sij∞ (TP ) = 1− ρ ojυˆ i∞ (TP )

(6)

where the partial molar volume of the solute at infinite dilution reads,

(

υˆ i∞ (TP ) = υ oj (TP ) + G ojj − Gij∞

)

(7)

TP

Likewise, we can relate these quantities through the isothermal-isochoric rate of change of pressure induced by the solute in the pure solvent, ( ∂P ∂xi )T ρ o , as follows 83, 95 ∞

j

Sij∞ (TP ) = −κ oj ( ∂P ∂xi )T ρ o ∞

(8)

j

(G

∞ ij

)

− G ojj = −υ ojκ oj ( ∂P ∂xi )T ρ o ∞

(9)

j

where κ oj denotes the isothermal compressibility of the pure j − solvent. Thus, Eqns. (4)-(9) provide the highly desired microscopic to macroscopic connection that allows the unambiguous representation of the solute’s ability to perturb (to structure-make/break) the solvent environment around it, and affords the chance to measure its magnitude based on experimentally available thermodynamic data regardless of either the type of solute or the nature of the intermolecular interactions. 2.2. Dissociable solutes

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c

a

When the i − solute of molar density ρi is an electroneutral dissociable solute, Cνzc Aνza , following the dissociation process, c

a

c

Cνzc Aνza → ν cC z + ν a Az

a

(10)

we define ν = ν a + ν c , the molar density of dissociated species ρ diss = νρi , and the total molar density ρ = νρi + ρ j . Thus, we characterize the system composition in terms of either densities or mole fractions as follows, xdiss = ρ diss

(

(ρ

diss

+ ρj

= νρi νρi + ρ j

)

)

(11)

where x j = 1− xdiss is the mole/molecular fraction of the j − solvent. The dissociation scheme depicted in Eqn. (10) indicates that the composition of the dissociated species cannot change independently, and consequently, we must exercise some caution when applying the Kirkwood-Buff formalism.84 For that reason, in Appendix A, we illustrate the equivalence between the Kirkwood-Buff expressions for dissociable, comprising electrolytes and non-electrolytes, and non-dissociable solutes. According to the definition of Sij (TPxi ) , Eqns. (4)-(6), and the condition A6, it becomes clear that Sij (TPxi ) = Saj (TPxi ) = Scj (TPxi ) , i.e., the net solute-induced effect on the microstructure of the surrounding solvent is the same for either dissociated species. In fact, for the dissociative i − solute Cνzc Aνza , the mutation process portrayed in Figure 1 would comprise c

a

ν = ν c + ν a particles, such that the electroneutrality condition ν c z c + ν a z a = 0 is enforced. Therefore, the solvation process would involve two solvent density perturbations; one around the cations and denoted S+ j (TPxi ) , the other around the anions and identified as S− j (TPxi ) , i.e.,

10


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(

) x (G

S+ j (TPxi ) ≡ ρ x j G+ j − G jj = N +exj (TPxi ) = 1− ⎡1+ ρ xdiss ⎣ = S− j (TPxi )

j

jj

)

+ G+− − 2G+ j ⎤ ρυˆ diss ⎦

(12)

whose infinite dilution limit becomes,

( = ρ (G

Sij∞ (TP ) = ρ oj G+∞j − G ojj ∞ −j

o j

− G ojj

) )

(13)

∞ = 1− ρ ojυˆ diss

Moreover, the corresponding isothermal-isochoric rate of change of pressure induced by the dissociative i − solute, ( ∂P ∂xi )T ρ o = ν ( ∂P ∂xdiss )T ρ o , becomes ∞

∞

j

( ∂P ∂x )

∞

diss T ρ o j

(

j

)

∞ = ρ ojυˆ diss − 1 κ oj

(14)

or, as given by the alternative expression Eqn. B3, derived previously in Ref. 83, i.e.,

( ∂P ∂x )

∞

i T ρo j

(

)

= ρ ojυˆ i∞ − ν κ oj

(15)

∞ after invoking the identities υˆ i∞ (TP ) = νυˆ diss (TP ) and xdiss = ν xi .

3. MICROSTRUCTURAL RESPONSES TO INTERMOLECULAR ASYMMETRY AND ENVIRONMENTAL STATE CONDITIONS The solute-induced effect on the solution microstructure embodies the solvation process at the prevailing state conditions and composition; therefore, it becomes useful 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. Effect of solute-solvent molecular asymmetry A quick analysis of Eqn. (6) indicates that Sij∞ (TP ) = N ijex,∞ (TP ) becomes a linear function of the solute’s partial molar volume at infinite dilution at the corresponding state

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conditions, and consequently, it scales with the magnitude of the solute-solvent intermolecular interaction asymmetry. This behavior provides a venue to assess how the degree of dissimilarity between the solute and the solvent, i.e., the source of non-ideality of the resulting mixture, translates into a microstructural perturbation described as either structure-making or structurebreaking effect. In fact, according to Eqn. (8) an i − solute behaves as a structure-maker if

( ∂P ∂x )

∞

i T ρo j

< 0 , i.e., when the solute is characterized as non-volatile 96 or attractive. 11

Conversely, an i − solute behaves as a structure-breaker if ( ∂P ∂xi )T ρ o > 0 , i.e., when the solute ∞

j

is characterized as volatile 96 or weakly-attractive and repulsive 11 in the jargon of supercritical fluid solutions. 10, 97 To aid the interpretation of such observations, we discuss useful extreme types of molecular interaction asymmetries between the i − solute and the j − solvent. For that purpose, we can start with the case of an ideal gas solute, IG _ i , where we have null solute-solvent interactions in a real solvent and for which we have already shown that, 86

υˆ i∞,IG _ i (TP ) = kTκ oj (TP )

( ∂P ∂x )

∞,IG _ i

i T ρo j

(16)

( )

= kT ρ oj − κ oj

−1

so that υˆ iIG _ i (TP ) > 0 and ( ∂P ∂xi )T ρ o

∞,IG _ i j

(17) < 0 at conditions away from the solvent’s critical

_i conditions, i.e., as long as κ oj < κ o,IG . Under these conditions, Eqn. (7)-(8) indicate that an ideal j

gas solute would behave as a structure-maker, i.e., Sij∞,IG _ i (TP ) = 1− κ oj kT ρ oj > 0 . This, perhaps unexpected, result becomes more obvious when considering the mutation process depicted in Figure 1, where an original solvent molecule becomes less interactive with its surrounding and finally vanishes as a non-interacting point. This process is characterized by

Gij∞ (TP ) → 0 , i.e., Sij∞ (TP ) → − ρ oj G ojj = 1− ρ ojκ oj kT > 0 . Now, if we turn on the solute-solvent interactions to the level of those we might find in imperfect gas interactions, B _ i , characterized by the second virial coefficients Bii (T ) and

12


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Bij (T ) , while keeping the solvent-solvent interactions unchanged, then we have that (see B1 in Appendix B for details),

υˆ i∞,B _ i (TP ) = kTκ Jo (TP ) + 2Bij (T )

(18)

( ∂P ∂x )

(19)

i

∞,B _ i

T ρ oj

= ( ∂P ∂xi )T ρ o

∞,IG _ i j

+ 2 ρ oj Bij κ oj

Gij∞,B = −2Bij (T ) z (TP )

(

) (

(20)

)

with P kT ρ oj = ρ o,IG ρ oj = z , and thus, j

Sij∞,B _ i (TP ) = Sij∞,IG _ i (TP ) − 2 ρ oj Bij (T )

(21)

The resulting expression indicates that Sij∞,B _ i (TP ) > Sij∞,IG _ i (TP ) > 0 as long as the unlike-pair interaction second virial coefficient Bij (T ) < 0 . Under those conditions the imperfect gas solute will induce an additional microstructural perturbation on the surrounding solvent, over that of the ideal gas solute, and becomes a structure-making solute. Otherwise, if Bij (T ) > 0 we have two possible outcomes namely, either 0.5υ oj Sij∞,IG _ i > Bij (T ) so that the solute behaves as structuremaker with 0 < Sij∞,B _ i < Sij∞,IG _ i , or 0.5υ oj Sij∞,IG _ i < Bij (T ) , i.e., the solute becomes a structurebreaker. A further increase of the solute-solvent interactions, up to the point that the solute-solvent interactions become equal to the corresponding solvent-solvent interactions creates a special case of Lewis-Randall ideal solution ( IS ). 86 Thus, the i − solute behaves identically as the j − solvent, then G ojj (TP ) = Gij∞ (TP ) → Sij∞ (TP ) = 0 , i.e., null solute-solvent asymmetry, and consequently, the solute does not perturb the surrounding solvent environment. Finally, the most frequent scenario would be for systems with solute-solvent interactions different (either weaker or stronger) than the solvent-solvent interactions. The first case is characterized by Gij∞ (TP ) < G ojj (TP ) , such as in the case of non-polar gases in water, then from

13



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Eqns. (6)-(9) we have that ( ∂P ∂xi )T ρ o > 0 and υˆ i∞ (TP ) > 0 , consequently, Sij∞ (TP ) < 0 , i.e., the ∞

j

solute behaves as a structure-breaker species. Conversely, the second case is characterized by

Gij∞ (TP ) > G ojj (TP ) , such as in the case of highly interactive (large) solutes including strong electrolytes, for which ( ∂P ∂xi )T ρ o < 0 , υˆ i∞ (TP ) − υ oj (TP ) < 0 , so that Sij∞ (TP ) > 0 ; consequently, ∞

j

the solute behaves as a structure-maker compound. Note, however, that in this case we have two possible scenarios: the first case occurs when υˆ i∞ (TP ) < 0 , while the second case comprises

0 < υˆ i∞ (TP ) < υ oj (TP ) . For illustration purposes in Figure 2, we provide a schematic representation of the links between Sij∞ (TP ) , υˆ i∞ (TP ) , and ( ∂P ∂xi )Tυ o = − Sij∞ (TP ) κ oj (TP ) , and ∞

j

highlight some relevant boundaries. Moreover, if the system is a very low-density solution of imperfect gases, we have from Eqn. (6) that,

(

lim Sij∞ (TP ) = −2 ρ oj Bij − B jj

ρ oj →0

)

(22)

T

⊕ where Bαβ (T ) = −0.5 lim Gαβ (TP ) describes the second virial coefficient for the αβ − pair o

ρ j →0

interactions, and ⊕ denotes either a pure or an infinitely dilute species. According to Eqn. (22),

(

an imperfect gas solute will behave as a structure-maker when Bij − B jj

)

T

< 0 and vice versa. In

addition, if the i − solute behaved as an ideal gas solute, then lim Sij∞,IG _ i (TP ) = 2 ρ oj B jj (T )

(23)

ρ oj →0

Consequently, an ideal gas solute in an imperfect gas solvent will be a structure-maker whenever

B jj (T ) > 0 , otherwise, at lower temperatures the ideal gas in an imperfect gas solvent will behave as a structure-breaker. These two cases indicate that, even for mixtures involving imperfect gases, i.e., at conditions where they can be described accurately by z (TPxi ) = 1+ β PB (Txi ) the presence of an

14


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i − solute introduces a microstructural perturbation on the surrounding j − solvent environment

that can be rigorously characterized in terms of thermodynamic quantities. 3.2. Effect of changes in state conditions For that purpose, we determine the following temperature and pressure derivatives of Eqn. (5), i.e.,

( ∂S

∞ ij

∂T

)

P

= ρ ojυˆ i∞ βTjo − ρ ojυˆ i∞ βTi∞

(

)(

= 1− Sij∞ βTjo − βTi∞

(24)

)

and,

( ∂S

∞ ij

∂P

)

T

= − ρ ojυˆ i∞κ oj + ρ ojυˆ i∞κ i∞

(

)(

= − 1− Sij∞ κ oj − κ i∞

(

where κ α⊕ = − ∂ln υα⊕ ∂P

)

T

(25)

)

(

and βT⊕α = ∂ln υα⊕ ∂T

)

P

denote the isothermal compressibility and

isobaric thermal expansivity of the species α at the limiting composition condition ⊕ , i.e., either pure component ⊕ ≡ o or infinite dilution ⊕ ≡ ∞ . Consequently, the Sij∞ (TP ) − response to perturbation of temperature and pressure will depend on the sign of the product of two

(

)(

) (

)(

)

differences, i.e., either 1− Sij∞ βTjo − βTi∞ or Sij∞ − 1 κ Tjo − κ Ti∞ , respectively. According to the analysis in §3.1 a structure-maker ( SM ) solute, Sij∞ (TP ) > 0 , is characterized by,

1 > ρ oj υˆ i∞ → υˆ i∞ (TP ) < υ oj (TP )

(26)

therefore,

15



( ∂S

∞ ij

∂T

)

SM P

(

)(

= 1− S β − β !" # # $ ∞ ij

o Tj

∞ Ti

)

⎧⎪> 0 if βTjo > βTi∞ →⎨ o ∞ ⎪⎩< 0 if βTj < βTi

)

⎧⎪> 0 if βTjo < βTi∞ →⎨ o ∞ ⎩⎪< 0 if βTj > βTi

0 0 if κ oj < κ i∞ →⎨ o ∞ ⎪⎩< 0 if κ j > κ i

)

⎧⎪> 0 if κ oj > κ i∞ →⎨ o ∞ ⎪⎩< 0 if κ j < κ i

0

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