domain: chim phys

text:
O Bernard
email: olivier.bernard@sorbonne-universite.fr

Association in electrolyte solution: Implementing inner sphere ion pairing into the mean spherical approximation

Keywords: Electrolytes, Thermodynamic properties, Association

Association in electrolyte solution: Implementing inner sphere ion pairing into the mean spherical approximation

O.

Introduction

Debye and Hückel (DH) theory continues to play a vital role in the interpretation of the thermodynamic properties of electrolytes at low concentration [START_REF] Debye | Gefrierpunktserniedrigung und verwandte erscheinunge[END_REF][2][START_REF] Robinson | Electrolyte Solutions[END_REF]. It provided a justification for the variations in square root of the concentration observed for several experimental quantities in highly dilute solutions of strong electrolytes. In the concentration range where this limiting law is relevant, the ions can be characterized only by their charge to describe solution properties. In this model, the solvent is seen as a continuum surrounding the ions and characterized by its dielectric constant. At high concentration, a size must be assigned to the ions. This induces deviations from the limiting law and allows one to account for the variation of the thermodynamic and transport properties of dissociated electrolytes. In the DH theory, the size of the ions was introduced as a short-range boundary condition in the determination of the average electrostatic potential around the ions. Indeed, the screening length characterizing the range of this electrostatic potential does not depend on the size of the ions. This implies a limitation in the field of application of the DH theory. In order to provide a more satisfactory description of the properties at higher concentrations, many further developments have been made. In particular, the mean spherical approximation theory (MSA) constitutes an extension of DH [START_REF] Waisman | Exact solution of an integral equation for the structure of a primitive model of electrolytes[END_REF][START_REF] Waisman | Mean spherical model integral equation for charged hard spheres I. Method of solution[END_REF][START_REF] Waisman | Mean spherical model integral equation for charged hard spheres[END_REF][START_REF] Blum | Mean spherical model for asymmetric electrolytes I. Method of solution[END_REF][START_REF] Blum | Mean spherical model for asymmetric electrolytes. 2. Thermodynamic properties and the pair correlation function[END_REF][START_REF] Hiroike | Supplement to Blum's theory for asymmetric electrolytes[END_REF]. It provides also an analytical description of several properties such as activity and osmotic coefficients. It better takes into account the effect of the size of the ions on their distributions in space and on the screening length in concentrated solutions.

In applications, the size parameters of the ions are most often ajusted by fitting properties, such as the osmotic coefficients, is described as best as possible by the theoretical expressions adopted. This approach is particularly satisfactory in the case of alkali halides in aqueous solution [START_REF] Friedman | Theory of mixed electrolyte solutions and application to a model for aqueous lithium chloride-cesium chloride[END_REF][START_REF] Ramanathan | Models having the thermodynamic properties of aqueous solutions of tetraalkylammonium halides[END_REF][START_REF] Friedman | Models having the thermodynamic properties of aqueous alkaline earth halides and NaCl-MgCl2 mixtures[END_REF][START_REF] Triolo | Simple electrolytes in the mean spherical approximation[END_REF][START_REF] Wu | Vapor-liquid equilibria of mixed-solvent electrolyte solutions: ion-size effects based on the MSA theory[END_REF][START_REF] López-Pérez | Estimation of activity coefficients at different temperatures by using the mean spherical approximation[END_REF]. Furthermore, the sizes of the ions in solution determined in this way are usually greater than the crystallographic sizes. This difference is generally ascribed to the fact that ions in solution can have strong short-range interactions with water molecules that induce the formation of a solvation layer around them. The dimensions of the ions thus determined are representative of hydrated species and delineate the region beyond which the solvent can be seen as a continuum. Solvation and hydration are observed in microscopic measurements such as dielectric spectroscopy, low-frequency Raman spectra or NMR relaxation [START_REF] Haggis | The dielectric properties of water in solutions[END_REF][START_REF] Barthel | Dielectric relaxation of aqueous electrolyte solutions[END_REF][START_REF] Buchner | Dielectric relaxation of aqueous NaCl solutions[END_REF][START_REF] Ohtaki | Structure and dynamics of hydrated ions[END_REF][START_REF] Rudolph | Hydration of Lithium Ion in Aqueous Solutions[END_REF][START_REF] Rudolph | Hydration of the calcium(II) ion in an aqueous solution of common anions (ClO -4 , Cl -, Br -, and NO -3 )[END_REF][START_REF] Ansell | X-ray and neutron scattering studies of the hydration structure of alkali ions in concentrated aqueous solutions[END_REF][START_REF] Bagno | NMR techniques for the investigation of solvation phenomena and non-covalent interactions[END_REF]. Furthermore, in order to account for thermodynamic quantities such as compressibility or hydration heat capacity, it is often assumed that a firm layer of water molecules is present around the ions. These hydration layer are assumed to have properties different from free bulk water molecules. Hydration can also be highlighted in the structure factors of the ions in neutron or X-ray diffraction measurements or from radial distribution functions obtained from numerical simulations. When the solvation energy is high, the layers of water surrounding the ions repel each other at short distances. This feature appears to be particularly outstanding when the distance of least approach between cations and anions can be greater than the sum of their respective crystallographic radii. In this situation, it seems more relevant to consider hydrated sizes to characterize ions in solution.

Furthermore, to better describe the properties of electrolytes, association between unlike ions must be taken into consideration. When such ions are attracted towards each other, at short distance this attraction competes with the interaction with the solvent molecules surrounding each of the ions. If the solvation energy is large enough, the attraction between ions fails to break their hydration layers. When association occurs between strongly solvated ions, the ions in contact retain their respective solvation layers.

Then, to provide a simple picture of this situation, the anion-cation pair can be represented as a dumbbell consisting of two spheres bound together.

However, the competition between solvation and association does not necessarily leave the hydration layers unchanged. This may be due to the fact that the individual solvation energy of some ions is weaker than the attraction by counter-ions, as a result of electrostatic or more specific interactions, which can possibly lead to a true chemical bond. This overlapping of the solvation layers during association has been considered to interpret various experimental data from Raman spectroscopy, dielectric relaxation, ionic conductivity or sound absorption [START_REF] Rudolph | Raman and infrared spectroscopic investigation of contact ion pair formation in aqueous cadmium sulfate solutions[END_REF][START_REF] Rudolph | A Raman spectroscopic study of hydration and water-ligand replacement reaction in aqueous cadmium(II)-sulfate solution: Inner-sphere and outer-sphere complexes[END_REF][START_REF] Rudolph | Raman spectroscopic investigation of speciation in MgSO 4 (aq)[END_REF][START_REF] Roy | Restructuring of Hydration Shell Water due to Solvent-Shared Ion Pairing (SSIP): A Case Study of Aqueous MgCl 2 and LaCl 3 Solutions[END_REF][START_REF] Buchner | Ion-pair and solvent relaxation processes in aqueous Na 2 SO 4 solutions[END_REF][START_REF] Buchner | Complexity in "simple" electrolyte solutions: Ion pairing in MgSO 4 (aq)[END_REF][START_REF] Akilan | Temperature effects on ion association and hydration in MgSO 4 by dielectric spectroscopy[END_REF][START_REF] Mamatkulov | Waterseparated ion pairs cause the slow dielectric mode of magnesium sulfate solutions[END_REF][START_REF] Buchner | Systematic variations of ion hydration in aqueous alkali metal fluoride solutions[END_REF][START_REF] Barthel | The temperature dependence of the properties of electrolyte solutions. I. A semi-phenomenological approach to an electrolyte theory including short range forces[END_REF][START_REF] Yokoyama | Ion association of some 2:2 electrolytes in water at 25 °C. III. A new interpretation of experimental results and the determination of formation of inner-sphere ion-pairs[END_REF][START_REF] Atkinson | Ion association of magnesium sulfate in water at 25[END_REF][START_REF] Bechtler | Ultrasonic relaxation studies in aqueous solutions of 2-2 electrolytes[END_REF][START_REF] Marcus | Ion Pairing[END_REF][START_REF] Van Der Vegt | Water-Mediated Ion Pairing: Occurrence and Relevance[END_REF]. Different configurations of associated ions have been considered. In addition to ion pairs separated by solvent, ion pairs with shared solvent (with one water molecule between ions) and pairs in contact (without any water molecules between the two ions) have been taken into account to describe the dielectric relaxation or the absorption of sound in the electrolyte. Different association constants are assigned to these three types of configuration. Alternatively, the pairing ions were sometimes split into inner-sphere and outer-sphere complexes, with no distinction between contact and solvent-shared paired ions. Moreover the competing effects of solvation and association led to the formulation of the law of matching water affinities which states that ions of opposite charge are more likely to form inner sphere ion pairs when they have equal water affinities [START_REF] Collins | Charge density-dependent strength of hydration and biological structure[END_REF]. Additionally, hydration layers can also be strongly modified during an acid-base reaction. A hydronium ion can be considered as bulky when dissociated, due to its solvation. When forming an inner sphere complex with an anion a significant part of their hydration layers can be released.

To calculate the proportions of the various constituents, several association constants have been defined, depending on whether the solvation layers overlap between the paired ions. Moreover, in order to describe the deviations from ideality in the thermodynamics of the solutions, it is advisable to take into account the geometric shapes specific to each of the complexes assumed to be present in the medium.

Previously simple forms of complexes have been used in the literature. First, due to its simplicity, the anion-cation pair was modeled as a sphere. This representation has been used by different groups [START_REF] Ebeling | Mean spherical approximation-mass action law theory of equilibrium and conductance in ionic solutions[END_REF][START_REF] Barthel | Calculation of Osmotic Coefficients of Nonaqueous Electrolyte Solutions with the Help of Chemical Models[END_REF][START_REF] Cartailler | Thermodynamics of ion association in the mean spherical approximation[END_REF][START_REF] Vilariño | Protonation of Glycine in Saline Media: Evaluation of the Effect of Ionic Strength by Use of the Mean Spherical Approximation[END_REF][START_REF] Vilariño | Theoretical calculations of the ionic strength dependence of the ionic product of water based on a mean spherical approximation[END_REF][START_REF] Krienke | MSA Models of Ion Association in Electrolyte Solutions[END_REF][START_REF] Krienke | Osmotic and activity coefficients of strongly associated electrolytes over large concentration ranges from chemical model calculations[END_REF][START_REF] Villard | Non-additivity of ionic radii in electrolyte solutions: Hofmeister effect on mixtures modeled by an Associated MSA model[END_REF]. The radius of the spherical complex is an additional parameter in this model. During the association process of two ions, if one of them penetrates deep into the hydration layer of the other, it can replace one or more solvent molecules in its solvation shell. The complex formed can then be considered as roughly spherical. This situation is assumed to relate to contact association.

As an alternative, the anion-cation pair can be represented as a dumbbell consisting of the two spheres bound together. This picture may represent complexes formed by two spherical ions retaining their respective solvation layers. The corresponding thermodynamic properties can be described by using the associative or binding MSA (AMSA or BiMSA) theory [START_REF] Holovko | On the effects of association in the statistical theory of ionic systems. Analytic solution of the PY-MSA version of the Wertheim theory[END_REF][START_REF] Yu | An analytical study of the effects of association in a 2-2 electrolyte solution I. Associative mean spherical approximation[END_REF][START_REF] Blum | The general solution of the binding mean spherical approximation for pairing ions[END_REF][START_REF] Bernard | Binding mean spherical approximation for pairing ions: An exponential approximation and thermodynamics[END_REF][START_REF] Kalyuzhnyi | Thermodynamics of the associative mean spherical approximation for the fluid of dimerizing particles[END_REF]. This paper is dedicated to Myroslav Holovko, a well-known expert in statistical mechanics, especially in integral equation theories. He is one of the founders of the AMSA or BiMSA theory.

Finally, it is desirable to describe complexes having an intermediate structure between these two configurations. Various models of the thermodynamic contributions due to these dimers, similar to dumbbells having a overlapping of spheres, have been presented for uncharged hard spheres [START_REF] Boublik | Hard convex body equation of state[END_REF][START_REF] Tildesley | An equation of state for hard dumbell fluids[END_REF][START_REF] Rosenfeld | Scaled field particle theory of the structure and the thermodynamics of isotropic hard particle fluids[END_REF][START_REF] Boublik | Equation of state of linear fused hard-sphere models[END_REF][START_REF] Walsh | A modified thermodynamic perturbation theory equation for molecules with fused hard sphere cores[END_REF][START_REF] Boublik | Equation of state of chain molecules[END_REF][START_REF] Amos | Bonded hard-sphere (BHS) theory for the equation of state of fused hard-sphere polyatomic molecules and their mixtures[END_REF][START_REF] Oversteegen | General methods for free-volume theory[END_REF][START_REF] Bernard | Relationship between thermodynamic perturbation and scaled particle theories for fused dimers fluids[END_REF]. The influence of the charges carried by these objects has been less studied [START_REF] Zhou | Chemical association in simple models of molecular and ionic fluids. III. The cavity function[END_REF][START_REF] Zhou | Chemical association in simple models of molecular and ionic fluids. IV. New approximation for the cavity function and an application to the theory of weak electrolytes[END_REF][START_REF] Duda | Solution of the associative mean spherical approximation for the shielded sticky point electrolyte model[END_REF][START_REF] Hu | Thermodynamic properties of aqueous solutions: Nonsymmetric sticky electrolytes with overlap between ions in the meanspherical approximation[END_REF][START_REF] Jiang | Thermodynamic properties of unsymmetrical sticky electrolytes with overlap between ions from Ornstein-Zernike equation[END_REF][START_REF] Stell | Chemical association in simple models of molecular and ionic fluids[END_REF]. It should be noted that the AMSA approximation has been applied to the fluid of charged hard spheres having a bonding distance less than the size of the hard spheres [START_REF] Duda | Solution of the associative mean spherical approximation for the shielded sticky point electrolyte model[END_REF]. However, the description of the thermodynamics is not as explicit as when the bond occurs at the contact distance. Then, in this work, an approximate description of the electrostatic contributions is presented to account for dimers having a overlapping of spheres when they are charged. Different thermodynamic formalisms have been used previously to describe association in solution. Some have been referred to as chemical models, when they explicitly consider paired solutes as a third species. Others ascribe the pairing from very directional attractive interactions and are said to come from a physical framework. Many studies have presented relationships between these two formalisms [START_REF] Stell | Chemical association in simple models of molecular and ionic fluids[END_REF][START_REF] Ebeling | An Analytical Calculation of the Equation of State and the Critical Point in a Dense Classical Fluid of Charged Hard Spheres Ann[END_REF][START_REF] Olaussen | New microscopic approach to the statistical mechanics of chemical association[END_REF][START_REF] Fisher | Exact thermodynamic formulation of chemical association[END_REF][START_REF] Jiang | Criticality and phase behavior in the restricted-primitive model electrolyte: Description of ion association[END_REF][START_REF] Schröer | On the chemical and the physical approaches to ion association[END_REF][START_REF] Molina | Primitive models of ions in solution from molecular descriptions: A perturbation approach[END_REF][START_REF] Schröer | Molecular association in statistical thermodynamics[END_REF]. These relations are used here more explicitly to describe ionic solutions within the framework of MSA theory. They are also generalized to describe ion pairs of intermediate geometries. As a starting point, for simplicity it was assumed that in a binary electrolyte the dissociated anions and cations have the same diameter. Previously, this simple model with a single size parameter for a given salt, led to a satisfactory description of dissociated electrolytes in water. In this article, only one type of ion pair was considered at a time. The two extreme geometries, namely the spherical and dumbbelllike complexes without overlapping, are reviewed first. A common formalism is presented in order to highlight the common features and the differences in the thermodynamic expressions related to the charges carried by the various constituents present. The use of a common formalism makes it possible to take into account subsequently several types of complexes simultaneously. Then alternative thermodynamic expressions are established to describe ion pairs having intermediate configurations. Extensions, called variational MSA, of MSA type models have been set up before [START_REF] Velázquez | Variational mean spherical scaling approximation for nonspherical molecules: The case of dimers[END_REF][START_REF] Blum | The solvation shell in ionic solutions: variational mean spherical scaling approximation[END_REF][START_REF] Enriquez | Scaling in complex systems: analytical theory of charged pores[END_REF]. This approach was established previously to consider ions having non-spherical shapes such as ellipsoids. This theoretical tool has been used here to represent dumbbell-shaped charged complexes. As a result, the theoretical expressions obtained interpolate between MSA relations for spherical ions and those from BiMSA in which the complexes are dumbbells without overlap.

2 Thermodynamic and association for two particular configurations Before presenting the method used to establish the thermodynamics of an electrolyte forming dimers, with overlapping of the spheres representing the cation and the anion, two particular configurations can be recalled.

• The first is that obtained when the larger of the two spheres completely covers the other. Then, the dimer is a sphere of the same size as the larger ion and the electrolyte is made up of a mixture of charged hard spheres. The MSA theory seems suitable to describe this case.

• The second case is one in which the two ions are in contact without overlap. It is the largest distance between the two constituents of the pair that will be considered in this study. This configuration is the one previously described by the BiMSA theory.

These two configurations represent bounds to the equations determined subsequently for configurations comprised in between. However, these two extreme configurations have been described within different theoretical frameworks. In order to be able to pass continuously from one configuration to another it is necessary to determine how to describe intermediate complexes in a unique theoretical framework. The chemical model, initially used to describe spherical ion pairs, is recalled first. In this model the complex is seen as a third charged species which is added to the two free ions. It has been named a three-solute (3S) model. In particular the free energy is split into ideal and excess parts.

The ideal contribution applies to a mixture of three different species.

In contrast, the BiMSA theory was established within the framework of thermodynamic perturbation theory (TPT) and Wertheim's integral equations [START_REF] Blum | The general solution of the binding mean spherical approximation for pairing ions[END_REF][START_REF] Bernard | Binding mean spherical approximation for pairing ions: An exponential approximation and thermodynamics[END_REF]. There is no mention of free ions. The ideal free energy describes the mixture of two species only by not distinguishing free and bound ions. To distinguish it from the previous one, this approach has sometimes been called a physical model. Each of the thermodynamic frameworks is presented first, without presupposing the geometry of the complexes for which each framework has been chosen as most appropriate.

Then, explicit expressions adapted to a given complex geometry are given.

Thermodynamic framework of the chemical model 2.1.1 General relations

In the most compact configuration, where the pair is a sphere, the mixture of free ions and dimers is most simply described by considering three charged species, denoted with the indices 1, 2 and 3. The species 1 and 2 are the two free ions

ρ 1 =ρ C α C ρ 2 =ρ A α A (1)
where ρ C and ρ A , denote the total number densities of species C and A, and where α C and α A are the fractions of free C and A, respectively. The third particle is the pair such that ρ 3 = ρ P . Conservation of the total quantities of C and A requires

ρ C =ρ 1 + ρ 3 ρ A =ρ 2 + ρ 3 (2)
For this mixture of three solutes, the Helmholtz free energy F 3S is split into two terms according

βF 3S = βF id 3S + βF ex 3S (3)
where βF id 3S is the free energy of the ideal mixture given by

βF id 3S = 3 i=1 ρ i βµ o i + ln ρ i -1 (4)
and βF ex 3S is the excess contribution due to interactions in solution. Since the three constituents are charged spheres, these excess terms can be described taking into account a hard sphere contribution βF HS 3S and an electrostatic contribution like that of the MSA theory βF MSA 3S .

βF ex 3S = βF HS 3S + βF el 3S (5)
The 3S subscript indicates a mixture of 3 species.

The chemical potential µ i of a constituent can be determined by differentiation of the free energy,

βµ i = ∂βF 3S ∂ρ i ρ k̸ =i = βµ o i + ln ρ i + βµ ex i ( 6
)
where the excess part βµ ex i of the chemical potential can be written using γ i , the individual activity coefficient of the constituent i

βµ ex i = ln γ i (7)
Moreover, the chemical equilibrium condition between the free ions and the pair may be written as

µ 1 + µ 2 = µ 3 (8)
In addition, the Helmholtz free energy F 3S must also represent the thermodynamics of the solution by considering only the two total densities ρ A and ρ C . Indeed these densities are the real variables known experimentally. If, on the one hand, the chemical species noted here 1, 2 and 3 can have a real existence, on the other hand they may only serve to better take into account attractive interactions at short distance (physical framework). In particular, this latter point of view can be put forward when the association is only due to short-range electrostatic interactions, as in the approach of Bjerrum. In this situation ion pairs do not necessarily have a long lifetime and are used to more properly describe the short-range presence probabilities between anions and cations. Consequently, whether one chooses to use a chemical model, or a more physical description of the type presented in the following subsection, the calculated free energy must be the same for given total densities ρ A and ρ C .

Accordingly, the quantities ρ 1 , ρ 2 and ρ 3 are seen as functions of ρ A and ρ C . Then, the chemical potentials µ A and µ C are determined by differentiation with respect to ρ A or ρ C respectively, taking into account the conservation of the total quantities of A and C. Using the relationship

βF 3S ∂ρ M = 3 k=1 βF 3S ∂ρ k ∂ρ k ∂ρ M (9)
for M = A or C. (In summations or enumerations, to avoid confusion with densities of free and bound ions presented previously, capital letters are used for subscripts relating to total densities). Then, by using also the conservation of A and C given by eq. ( 2), it is found that

βµ C = ∂βF 3S ∂ρ C = βµ 1 + βµ 3 -βµ 1 -βµ 2 ∂ρ 3 ∂ρ C (10a) βµ A = ∂βF 3S ∂ρ A = βµ 2 + βµ 3 -βµ 1 -βµ 2 ∂ρ 3 ∂ρ A (10b)
If the condition [START_REF] Blum | Mean spherical model for asymmetric electrolytes. 2. Thermodynamic properties and the pair correlation function[END_REF] of chemical equilibrium between the free ions and the pair is fulfilled, the term in brackets in the previous equations is zero, then it is not necessary to evaluate the terms proportional to ∂ρ 3 /∂ρ C or ∂ρ 3 /∂ρ A . So, it is deduced that µ C = µ 1 . Similarly it is found that µ A = µ 2 [START_REF] Stell | Chemical association in simple models of molecular and ionic fluids[END_REF][START_REF] Olaussen | New microscopic approach to the statistical mechanics of chemical association[END_REF].

Moreover, eq. ( 8) is equivalent to the following chemical equilibrium relation

K o p = ρ 3 ρ 1 ρ 2 γ 3 γ 1 γ 2 ( 11
)
where the association constant K o p is related to the standard chemical potential µ o i of the three constituents by

K o p = exp [βµ o 1 + βµ o 2 -βµ o 3 ] (12)

Application to the spherical dimers

As said before, these relations are directly applicable to the association model in which the ion pair is represented by a sphere. In this situation, the excess free energy F ex 3S is the sum of two contributions, according to eq. ( 5). These two contributions, βF HS 3S and βF MSA 3S , are both functions of the three densities ρ 1 , ρ 2 and ρ 3 . The hard sphere contribution can be evaluated using the expressions given in [START_REF] Mansoori | Equilibrium thermodynamic properties of the mixture of hard spheres[END_REF][START_REF] Boublik | Hard-sphere equation of state[END_REF], when the three species have different diameters. When the ions all have the same diameter the expressions given in [START_REF] Carnahan | Equation of State for Nonattracting Rigid Spheres[END_REF] are recovered,

βF HS 3S = ρ 4ϕ -3ϕ 2 (1 -ϕ) 2 (13)
where ρ = ρ 1 + ρ 2 + ρ 3 and ϕ = (π/6)ρσ 3 .

In the same way the electrostatic term is also a function of the three densities. The charge of the third species is z 3 = z 1 + z 2 . From the MSA theory, the electrostatic free energy term is given by

βF el 3S = βF MSA 3S = βE MSA 3S + Γ 3 3π (14)
where βE MSA

3S

is the internal energy which is given by

βE MSA 3S = -λ 3 k=1 ρ k Γz 2 k 1 + Γσ ( 15
)
We used the notation

λ = βe 2 4πε o ε r (16)
for the Bjerrum distance, with β = 1/k B T (k B being the Boltzmann constant and T the temperature), e is the elementary charge, ε o is the permitivity of a vacuum, and ε r is the dielectric constant of pure solvent. The MSA parameter Γ is given by

Γ 2 = πλ 3 k=1 ρ k z 2 k (1 + Γσ) 2 (17)
By differentiation the electrostatic contribution to the activity coefficient ln γ i of ions i is deduced

ln γ MSA i = -λ Γz 2 i 1 + Γσ (18)
2.2 The physical thermodynamic framework

General relations

The thermodynamic description of stretched dimers has been established by considering only two species, namely the monomers A and C, which are able to bond together due to highly directional attractive interactions. In particular, for uncharged species the Helmoltz free energy has been obtained using the Wertheim's thermodynamic perturbation theory (TPT) [START_REF] Wertheim | Fluids with highly directional attractive forces. I. Statistical thermodynamics[END_REF][START_REF] Wertheim | Fluids with highly directional attractive forces. II. Thermodynamic perturbation theory and integral equations[END_REF][START_REF] Wertheim | Fluids of dimerizing hard spheres, and fluid mixtures of hard spheres and dispheres[END_REF][START_REF] Wertheim | Thermodynamic perturbation theory of polymerization[END_REF]. The AMSA or BiMSA theories, based on integral equations, have provided a means of evaluating additional electrostatic contributions [START_REF] Bernard | Binding mean spherical approximation for pairing ions: An exponential approximation and thermodynamics[END_REF][START_REF] Kalyuzhnyi | Thermodynamics of the associative mean spherical approximation for the fluid of dimerizing particles[END_REF]. In the physical framework, the thermodynamic quantities are not described with the densities of the three previous species but rather with ρ A and ρ C . As before, the Helmholtz free energy

F 2S is split into two terms [86] βF 2S = βF id 2S + βF ex 2S (19)
where βF id 2S is the free energy of the ideal mixture given by

βF id 2S = ρ A βµ o A + ln ρ A -1 + ρ C βµ o C + ln ρ C -1 (20)
and βF ex 2S is the excess contribution due to interactions in solution. As an alternative, in the physical thermodynamic framework, the excess free energy is split into three contributions

βF ex 2S = βF HS 2S + βF ass 2S + βF el 2S (21)
As before, a hard spheres term βF HS 2S and an electrostatic term βF el 2S are used. The hard sphere contribution can be evaluated using the equations coming from [START_REF] Mansoori | Equilibrium thermodynamic properties of the mixture of hard spheres[END_REF], namely by using [START_REF] Triolo | Simple electrolytes in the mean spherical approximation[END_REF] with ϕ = (π/6)ρ t σ 3 , which is now a function of the two densities ρ A and ρ C because ρ t = ρ A + ρ C . An associative term βF ass 2S , reflecting the effect of association, has been added. In addition, the concentration of ion pairs ρ P = ρ 3 appears in both the associative and electrostatic terms, as a result of attractive interactions. The associative contribution F ass 2S to the free energy is given by [START_REF] Wertheim | Fluids of dimerizing hard spheres, and fluid mixtures of hard spheres and dispheres[END_REF] βF ass

2S = C M =A ρ M ln α M + 1 2 (1 -α M ) (22a) = ρ A ln α A + ρ C ln α C + ρ P (22b)
where the fractions of free ions are defined in eq. (1).

Application to dumbbell-shaped dimers

These relations are directly applicable to the association model in which the ion pair is represented by two spheres bonded together. Additionally for this geometry, to specify the assessment of attractive interactions, the concentration of ion pairs ρ P is now defined by

ρ p = ρ C α C ρ A α A K p ( 23
)
where the term K p is related to g CA (σ) the pair distribution function at contact between

C and A [83] K p = K o p g CA (σ) (24)
When the species C and A are not charged, g CA (σ) only involves a contribution due to the hard spheres g HS CA (σ), for which explicit expressions are given in [START_REF] Mansoori | Equilibrium thermodynamic properties of the mixture of hard spheres[END_REF][START_REF] Boublik | Hard-sphere equation of state[END_REF]. When the constituents of the pair are charged, the pair distribution function g CA (σ CA ), used in equation [START_REF] Rudolph | Raman and infrared spectroscopic investigation of contact ion pair formation in aqueous cadmium sulfate solutions[END_REF], contains an extra electrostatic contribution. In this case, this function was defined by [START_REF] Bernard | Binding mean spherical approximation for pairing ions: An exponential approximation and thermodynamics[END_REF][START_REF] Simonin | Ionic solutions in the binding mean spherical approximation: thermodynamic properties of mixtures of associating electrolytes[END_REF]

g CA (σ) = g HS CA (σ) exp -βW el CA ( 25
)
where the electrostatic energy W el CA was given in the BiMSA by

βW el CA = λ σ z C z A (1 + Γ B σ) 2 -z C z A ( 26
)
The presence of the electrostatic term βW el CA in the function g CA (σ) affects the associative contribution to chemical potentials. By differentiation it is found

∂βF ass 2S ∂ρ M = ln α M -ρ p ∂ ln g HS CA (σ) ∂ρ M + ρ p ∂βW el CA ∂ρ M (27)
for M = A or C. The first two terms are those obtained for uncharged hard spheres with TPT [START_REF] Wertheim | Fluids of dimerizing hard spheres, and fluid mixtures of hard spheres and dispheres[END_REF]. The last term is another electrostatic contribution which complements those deduced from the differentiation of βF el 2S .

For ions of different diameters, the expression of the electrostatic contribution to the free energy βF el 2S , can be found in [START_REF] Bernard | Binding mean spherical approximation for pairing ions: An exponential approximation and thermodynamics[END_REF][START_REF] Simonin | Ionic solutions in the binding mean spherical approximation: thermodynamic properties of mixtures of associating electrolytes[END_REF][START_REF] Vilariño | Ionic solutions in the binding mean spherical approximation. Thermodynamics of associating electrolytes up to very high concentrations[END_REF][START_REF] Simonin | Organic electrolyte solutions: Modeling of deviations from ideality within the binding mean spherical approximation[END_REF]. In order not to make this development unnecessarily cumbersome, only the relations usable when the ions have the same diameter are now given. In this case, the term βF el 2S was given by BiMSA, as follows

βF el 2S = βF BiMSA 2S = βE el0 + Γ B 3 3π (28)
The internal energy E el used to obtain F el 2S includes two contributions

βE el = βE el 0 + βE el 1 ( 29
)
The term E el 0 , is similar to the usual expression for the internal energy in the MSA (without association) but in which the parameter Γ is replaced by the new parameter Γ B defined below in eq. [START_REF] Barthel | The temperature dependence of the properties of electrolyte solutions. I. A semi-phenomenological approach to an electrolyte theory including short range forces[END_REF].

βE el 0 = -λ ρ C z C Γ B z C 1 + Γ B σ + ρ A z A Γ B z A 1 + Γ B σ (30)
The second contribution βE el 1 , is the one due to anion-cation association. When the ions have the same diameter, βE el 1 is related to βW el CA , defined in eq. ( 26), by the simple relation

βE el 1 = ρ p βW el CA (31)
When the diameters of the ions are different, the contribution βE el 1 is not exactly equal to ρ p βW el CA . Additional terms were found in the expressions of βF BiMSA 2S [START_REF] Bernard | Binding mean spherical approximation for pairing ions: An exponential approximation and thermodynamics[END_REF]. Nevertheless it is important to present the two contributions of the internal energy. Indeed, the free energy was obtained by a charging process from the internal energy. If part of the latter is omitted, the deduced free energy will be modified. Moreover, the following general relation between the internal energy and the parameter Γ B was found [50]

∂βE el ∂Γ B ρp = - Γ B 2 π (32)
Accordingly, the parameter Γ B is defined by

Γ B 2 = πλ ρ C z 2 C (1 + Γ B σ) 2 + ρ A z 2 A (1 + Γ B σ) 2 + 2 ρ p z C z A (1 + Γ B σ) 3 (33)
Thereafter, the relation ( 32) is used in the variational extensions of the MSA theory.

In the physical framework the determination of the chemical potentials µ A and µ C , by differentiation of the free energy, appears to be more tedious. However, by summing the derivatives of βF ass

2S

and βF BiMSA

2S

with respect to the parameter Γ B , it was found that

∂F 2S ∂Γ B = 0 (34)
Consequently, this relation allows one to calculate the derivatives of the free energy with respect to the densities while keeping the parameter Γ B constant. This enables us to determine the electrostatic part of the individual activity coefficients lnγ el M of ions M as

ln γ el M = -λ Γ B z 2 M 1 + Γ B σ (35)
which is similar to the activity coefficient expression (18) obtained with the MSA, but with the parameter Γ B instead of Γ.

Transition between the two representations

From this point several improvements are possible. First, it is described below how to allow simultaneously for the two geometries of ion pairing. However either the chemical or the physical model is used for one or the other geometry. So, it is necessary to be able to describe all pairing configurations with a single approach. Previously, relations were established to describe dumbbell-like pairs within the chemical model framework [START_REF] Vilariño | Ionic solutions in the binding mean spherical approximation. Thermodynamics of associating electrolytes up to very high concentrations[END_REF].

In this framework, the chemical potential of the paired ions is explicitly defined. This enabled one to modify the size of the constituents of the complex without changing those of the free ions. Relations allowing the transition between the two frameworks are now taken up. First of all, the equivalence between the two theoretical frameworks is necessary to impose that the free energy F 2S is equal to F 3S . By comparing these two expressions for the free energy it is found that

βF id 3S = βF id 2S + βF ass + ρ p ln γ 1 γ 2 γ 3 (36)
Furthermore, to be in agreement with the relation [START_REF] Ramanathan | Models having the thermodynamic properties of aqueous solutions of tetraalkylammonium halides[END_REF], the function g CA (σ) must be related to the ratio of the activity coefficients according to

ln g CA (σ) = ln γ 1 γ 2 γ 3 ( 37
)
This relation is necessary to establish a connection between the two thermodynamic frameworks. It is obvious in the physical representation. It is not necessarily emphasised in the chemical description, which simply defines the activity coefficient of the ion pair. Then, the free energy of solutions containing dumbbell-shaped dimers can be rewritten within the framework of the chemical model, from eq.( 36), according to

βF 2S = βF id 3S + βF HS 2S + βF el 2S -ρ p ln g CA (σ) (38)
In addition, the electrostatic contribution to the chemical potential of dumbbell-shaped dimers can be deduced from eqs. ( 35) and ( 37)

ln γ el 3 = ln γ el A + ln γ el C + βW el CA (σ) (39)
Besides, when the dimers are considered as spherical, as seen before, this shape can result from the complete integration of the smaller ion into the bigger one. In this configuration the distance between the two ions can be taken equal to zero. The formula [START_REF] Marcus | Ion Pairing[END_REF], relating the ratios of activity coefficients to the pair distribution function, can be generalized for distances l, between the two ions, lower than the minimum distance of approach σ.

ln γ 3 = ln (γ 1 γ 2 ) -ln y CA (l) (40)
In this relation the pair distribution function has been replaced by the background or cavity correlation function y CA (l). This function can be seen as an analytic continuation, at distances less than σ, of the pair distribution function. It is connected to the latter and to the interaction potential V CA (l) by the following relationship: g CA (l) = y CA (l) exp -βV CA (l). For l = σ, y CA (σ) = g CA (σ). In continuity with the relation [START_REF] Rudolph | A Raman spectroscopic study of hydration and water-ligand replacement reaction in aqueous cadmium(II)-sulfate solution: Inner-sphere and outer-sphere complexes[END_REF], this function can also be splitted into two contributions according to

ln y CA (l) = ln y HS CA (l) -βW el CA (l) ( 41
)
For uncharged species, the description of the association for distances l ≤ σ has been studied. Explicit formulas for calculating the function y HS CA (l) have been established [START_REF] Boublik | Background correlation functions in the hard sphere systems[END_REF][START_REF] Labik | Cavity distribution functions of pure and mixed hardsphere systems[END_REF]. The electrostatic part is less well known. Various expressions describing W el CA (l) are presented in the next section.

When l = 0 the function ln y CA (l) can be determined explicitly from the various contributions to the logarithms of the activity coefficients, as they have been described in the part relating to the chemical model. In particular, when l = 0, the electrostatic contribution to the function y CA (0) can be calculated from the activity coefficients given in eq. ( 18). It is obtained

ln γ MSA 3 = ln γ MSA 1 + ln γ MSA 2 -λ 2Γ z A z C 1 + Γσ ( 42
)
As an alternative, by using eqs.( 40), ( 41) and ( 42), the following expression for the function W el CA (0) has been found within the MSA

βW el CA (0) = -λ 2Γ z A z C 1 + Γσ (43)
In the next section, an extension of the relations with a single diameter but with overlapping binding ions is presented.

Generalisation with overlapping ions in a dimer

Eq. ( 40) allows one to define the activity coefficient of the pair from those of the free ions and the background correlation function. It was seen in the previous section that this relation is applicable for the two extreme configurations of the pairing ions. Moreover, in view of eq. ( 18) and ( 35), the free ion activity coefficients are described by almost the same relationship for both configurations. The two equations differ only through the parameter Γ in [START_REF] Buchner | Dielectric relaxation of aqueous NaCl solutions[END_REF], being replaced by Γ B in [START_REF] Atkinson | Ion association of magnesium sulfate in water at 25[END_REF]. As a first step, it is assumed that the activity coefficients of free ions can be described with the same type of relation, when the pairing ions are separated by a distance l that is such that 0 ≤ l ≤ σ.

ln γ el M = -λ Γ v z 2 M 1 + Γ v σ ( 44
)
where the parameter Γ v is given by a relation of the same type as equations defining the parameters Γ and Γ B . This relationship can be determined using eq. ( 32). Then, having defined the activity coefficients of the free ions, it remains to determine the term ln y CA (l) in eq. ( 40) to evaluate ln γ p .

3.1 Closure relation for ln y CA (l)

From eq. ( 41), the function ln y CA (l) is split into a hard sphere and an electrostatic term.

The hard spheres contribution of this function was previously described very correctly [START_REF] Boublik | Background correlation functions in the hard sphere systems[END_REF][START_REF] Labik | Cavity distribution functions of pure and mixed hardsphere systems[END_REF]. So, in this study only the term βW el CA (l) was evaluated. Various closure relations have been used to relate pair correlation functions to interaction potentials. In particular for electrolytes, the MSA and the hypernetted chain (HNC) approximations are suitable.

Due to the presence of an exponential connecting g CA (σ) to the potential W el CA (σ) in eq. ( 25), the HNC relation has been preferred to obtain an expression for W el CA (l). The HNC relation is given by [START_REF] Van Leeuwen | New method for the calculation of the pair correlation function[END_REF][START_REF] Morita | A New approach to the theory of classical fluids[END_REF][START_REF] Verlet | On the theory of classical fluids[END_REF]

ln g CA (l) = -βV CA (l) + h CA (l) -c CA (l) (45)
The cavity correlation function y CA (l) is related to g CA (l) and to the interaction potential V CA (l) by the relation

ln g CA (l) = -βV CA (l) + ln y CA (l) (46)
which allows us to deduce y CA (l) from the HNC relation. If only the electrostatic part of ln y CA (l) is determined with the HNC closure, then only the corresponding parts of h CA (l) -c CA (l) are needed. With this restriction, the relation adopted is therefore

ln y el CA (l) = -βW el CA (l) = h el CA (l) -c el CA (l) (47)
Then expressions for the functions h el CA and c el CA are needed. Previously, they were determined in the BiMSA theory by using Wertheim integral equations. In particular, when l = σ, the electrostatic part of the total correlation functionis given by

h el CA (σ) = -λ z A z C σ (1 + Γ B σ) 2 (48)
and the direct correlation function c el CA (σ) = -λz A z C /σ. Then, eq. ( 47) is suitable to retrieve eq. ( 26). For distances l < σ, h el CA (l) = 0. Therefore in this range of distance, eq. ( 47) reduces to

ln y el CA (l < σ) = -c el CA (l) (49)
Equivalently, from the HNC relation, the activity coefficient of pairing ions is given by ln

γ el P = ln γ el A + ln γ el C + c el CA (l) (50)
where the activity coefficients of the free ions are given by eq. ( 44).

In order to apply these relations, analytical MSA expressions were used to evaluate the direct correlation functions c el CA (l) [START_REF] Hiroike | Supplement to Blum's theory for asymmetric electrolytes[END_REF][START_REF] Blum | Relation between the free energy and the direct correlation function in the mean spherical approximation[END_REF]. For ions of the same diameter, it has been found

ln y el CA (l ≤ σ) = λ Γ v z A z C (1 + Γ v σ) 2 2 + Γ v (2σ -l) (51)
In addition, the parameter Γ, initially present in the MSA formula, has been replaced by Γ v . Moreover, the strict inequality: l < σ, has been replaced by l ≤ σ, because there is continuity of the function h CA -c CA in the MSA approximation at l = σ.

Variational extension of the MSA theory

The parameter Γ v was determined using the variational generalization of MSA thermodynamics [START_REF] Velázquez | Variational mean spherical scaling approximation for nonspherical molecules: The case of dimers[END_REF][START_REF] Blum | The solvation shell in ionic solutions: variational mean spherical scaling approximation[END_REF][START_REF] Enriquez | Scaling in complex systems: analytical theory of charged pores[END_REF]. Application of this method is now described. One of the common properties of MSA and BiMSA is that the electrostatic free energy F el is related to E el by the relation

βF el = βE el + [Γ v ] 3 3π (52)
Here too, the parameter Γ v is used to designate both the MSA parameter, the BiMSA parameter and more generally the screening parameter corresponding to an intermediate geometry. Another common property of MSA and BiMSA given by eq. ( 32) is generalized as follows

∂βE el ∂Γ v ρp = - [Γ v ] 2 π ( 53
)
With this relation the electrostatic part of the individual activity coefficients γ el M of the ions M can be obtained by differentiation of the internal electrostatic energy E el according to

ln γ el M = ∂βE el ∂ρ M Γ v ,ρp (54)
The internal energy E el is the sum of the following two contributions

βE el 0 = -λ ρ C z C Γ v z C 1 + Γ v σ + ρ A z A Γ v z A 1 + Γ v σ ( 55
)
βE el 1 = ρ p βW el CA (l) = -λρ P Γ v z C z A (1 + Γ v σ) 2 2 + 2Γ v σ -Γ v l (56)
Then Γ v is deduced, by differentiating βE el with respect to Γ v , which yields

[Γ v ] 2 = πλ ρ C z 2 C (1 + Γ v σ) 2 + ρ A z 2 A (1 + Γ v σ) 2 + 2 ρ P z C z A (1 + Γ v σ) 3 1 + Γ v σ -l (57)
This relation completes the description of the free energy model for a binary electrolyte when association leads to ion pairs with overlapping spheres. As with the boundary configurations used previously, the equations can be solved iteratively. For example, from relation [START_REF] Ramanathan | Models having the thermodynamic properties of aqueous solutions of tetraalkylammonium halides[END_REF], giving the ratio of bound ions to free ions, neglecting initially the activity coefficients, the concentrations of free and bound ions can be calculated. Then, the parameter Γ v can then be determined with these concentrations for a given distance l.

The activity coefficients can be evaluated. The ratio of these coefficients can be taken into account in relation [START_REF] Ramanathan | Models having the thermodynamic properties of aqueous solutions of tetraalkylammonium halides[END_REF] and the concentrations of free and bound ions can be calculated again. These equations are thus solved iteratively until there are no significant differences between the concentrations determined between two successive iterations. Then, the chemical potential of the pair 3 is equal to the sum of the chemical potentials of the ions 1 and 2. When this relation is satisfied, βµ C = βµ 1 and βµ A = βµ 2 .

Discussion

Use has been made of the background correlation function to define the thermodynamics of electrolytes when overlapping occurs between pairing ions. In this study, this function has been used to represent the variation of the probability that two ions have to be at a distance less than σ CA . The use of the direct correlation function from MSA in the HNC relation has allowed to establish explicit expressions of the electrostatic part of the cavity correlation function for ions all having the same diameter. These expressions recover the explicit formulas previously established in the MSA for spherical pairs on the one hand and for two spheres in contact with the BiMSA on the other hand. In view of eqs. [START_REF] Yu | An analytical study of the effects of association in a 2-2 electrolyte solution I. Associative mean spherical approximation[END_REF] and [START_REF] Blum | The general solution of the binding mean spherical approximation for pairing ions[END_REF], ln γ el P (l) and ln y el CA (l) are linear functions of the distance l between the paired ions.

Based on previous works, this linear variation can be easily found. In order to determine the cavity correlation function, Zhou and Stell assumed that this function could be represented by an interpolation polynomial between two states. By applying this hypothesis to the electrostatic contribution of the cavity function, the potential W el CA (l) can be represented by the following relation

βW el CA (l) = βW el CA (0) (1 -x) + βW el CA (σ) x (58)
where W el CA (0) and W el CA (σ) are given by ( 43) and [START_REF] Rudolph | Raman spectroscopic investigation of speciation in MgSO 4 (aq)[END_REF], respectively, and x = l/σ. This definition restores eq. ( 47) deduced from the HNC relation, and eq. ( 57) for Γ v using the variational scheme presented previously. The use of an interpolation polynomial is simpler. However the HNC relation has more physical basis.

From the theoretical expressions presented in the previous section, the logarithm of the mean activity coefficient ln γ ± of a 1-1 electrolyte solution in a water-like solvent at 25 o C (ε r = 78.3) has been evaluated. The cations and anions in the electrolyte were assumed to have the same diameter σ = 4 nm. In order to highlight the influence of the association, increasing values of the association constant were considered. The association constant K o p , was assumed to be equal to: 0, 0.5 or 1 M -1 (where M denotes molarity). The distance l, between the components of the complex, was supposed equal to: zero, half or one σ diameter, to display the influence of its geometry. The results of these calculations are shown in Fig. 1. A comparison of the mean activity coefficients obtained with association and without association shows that the higher the association constant, the greater the decrease in this quantity. Furthermore, for a given association constant, below 0.5 M, the curves obtained with different distances are almost identical. On the other hand, at higher concentrations the activity coefficient ln γ ± is higher as the distance l tends towards σ. Then, it seems important to characterize the geometry of the complexes at high concentrations.

To go further, one can ask whether the approach developed so far for ions of the same size can still be used when their diameters are different. Starting from the MSA theory, a simplified expression of the ion activity coefficient is given by ln γ el i ≃ -λ

Γ v z 2 i 1 + Γ v σ i (59)
where the parameter Γ v is equal either to Γ when only spheres are present in solution, or to Γ B when the paired ions are in the form of stretched dumbbells. Moreover, the interaction energy W el CA (σ CA ) between two ions at the distance σ CA = (σ A + σ C )/2 is given by

βW el CA (σ CA ) ≃ 2λ (σ A + σ C ) z A z C (1 + Γ B σ A ) (1 + Γ B σ C ) -z A z C (60)
which is also a generalization of the equation ( 26) defined previously for ions having the same size.

On the other hand, assuming that σ A ≥ σ C , the ion pairs are spherical when l ≤ s AC , with: s AC = (σ A -σ C )/2. In this limit, the pairing ions still have a total charge z 3 = z A + z C , and a diameter equal to σ A . In order to determine the function W el CA (l = s AC ), it is necessary to subtract from the excess chemical potential of the pairing ions the excess chemical potentials of the free ions. By using eq. ( 59) to calculate the electrostatic contributions to these chemical potentials, it is obtained instead of eq. ( 43),

βW el CA (s AC ) = -λ 2Γz A z C 1 + Γσ A -λ Γz 2 C 1 + Γσ A + λ Γz 2 C 1 + Γσ C (61)
The first term, proportional to z A z C , is a generalization of relation [START_REF] Vilariño | Protonation of Glycine in Saline Media: Evaluation of the Effect of Ionic Strength by Use of the Mean Spherical Approximation[END_REF]. The other terms, proportional to z 2 C , are new. They come from the fact that the C ions do not have the same size when they are free or bound.

In order to describe intermediate configurations, such as s AC ≤ l ≤ σ CA , the HNC relation can be used again to determine the term proportional to z A z C , namely: ln

y el CA (l ≤ σ CA ) = -βW el CA (l ≤ σ CA ) ≃ -c el CA (l).
Then, to use eqs. ( 50) and ( 51), the direct correlation function c el CA (l) can be deduced from the MSA equations [9, 93]

c el CA (l) = -λ Γ v z A z C (1 + Γ v σ A ) (1 + Γ v σ C ) 2 + Γ v 2σ CA -L(l) (62)
where the function L(l) is given by

L(l) = l + (σ A -σ C ) 2 4 l (63)
These relations allow a generalization of eq. ( 51).

On the other hand, there are no terms proportional to z 2 C in eq. ( 62) for c el CA (l), in order to recover the corresponding terms present in (61) when l = s AC . The HNC relation therefore seems insufficient to account for these terms obtained when the diameters of the ions are different. An interpolation polynomial could be used again to represent this contribution, which is absent in the HNC relation. To simplify the notation, the terms proportional to z 2 C in eq. ( 61) can be named W el CC (s AC ), according to

βW el CC (s AC ) = -λ Γ v z 2 C 1 + Γ v σ A + λ Γ v z 2 C 1 + Γ v σ C (64)
So, to account for the change in this contribution as a function of the distance l, a simple linear interpolation may be assumed, for s AC ≤ l ≤ σ CA ,

βW el CC (l) = βW el CC (s AC ) (1 -x) (65)
with x = (l -s AC )/σ C . This function is zero when l = σ CA .

Other interpolation schemes are possible. In particular, it can be noted that the function c el CA (l) is no longer a linear function of l but rather depends on L(l) when the ions have different sizes. This variable allows one to consider the point l = s AC as an extremum of the function c el CA (l). Then it can be assumed that W CC (l) also depends on L(l). If the function L(l) is selected to define an interpolation polynomial, the change in W el CC (l), can be represented alternatively by

βW el CC (l) = βW el CC (s AC ) ∆P AC (l) (66)
with

∆P AC (l) = 1 2σ 2 C σ 2 A + σ 2 C -(σ A + σ C ) L(l) (67)
This function is proportional to the overlapping surface of the two spheres. It is involved in the description of the background correlation functions of hard spheres. Then, as before, from a definition of the function W CA (l), the internal energy E el 1 = ρ P W CA (l) can be made explicit. By using eq. ( 53), the screening parameter Γ v can be obtained by differentiation of the internal energy. At this point the description of the thermodynamics for associated ions of different diameters is limited to being only a mathematical problem.

Conclusion

The consideration of charged dimers, in the form of dumbbells with overlapping spheres, has been undertaken in the continuity of the MSA theory. The connection between various association models has been reviewed. This can facilitate the consideration of several types of complexes simultaneously. Starting from simple geometries studied previously, dimers with a distance 0 ≤ l ≤ σ have been considered. As a generalisation, the cavity correlation function was used to represent the probability variation in these intermediate geometries. The HNC relation was used to define a relation between this function and the electrostatic part of the direct correlation function deduced from the MSA theory. The deduced expressions, give back the explicit formulas previously obtained in the MSA for spherical pairs on the one hand and for two spheres in contact with the BiMSA on the other hand. A variational generalisation of the binding mean spherical approximation was used to establish an explicit expression for the screening parameter Γ v , which is required to evaluate the activity coefficients when the pairing ions are squeezed dumbbell shaped.

In most sections of this study, the ions were assumed to have the same diameter. The approach used in this work could be extended for ions of different sizes. Further theoretical modeling, including numerical simulations, of solutions containing such pairing ions would be necessary to improve understanding of association in electrolytes. The black continuous curve represents the result obtained with the MSA theory without association. The three red curves with circles represent the result obtained when a constant of association K o p = 0.5 M -1 between cations and anions is taken into account. The three lower blue curves were calculated considering an association constant K o p = 1 M -1 . For each of these two groups of three curves, the upper curve with dashes is determined with l = σ, the one with dashes and dots corresponds to l = 0.5 σ and the lowest with dots is l = 0.

Figure 1 :

 1 Figure 1: Mean activity coefficient for a 1-1 electrolyte as a function of the molar concentration C. Cations and anions have the same diameter σ = 0.4 nm. From top to bottom:


Naveen Srinivassane

Aïcha Saïd

Frédéric Hameau

Rémy Vauché

Florence Podevin

Sylvain Bourdel

Naveen Srinivassane1

Oscillator based bio-sensor for skin sweating detection: A feasibility survey

Keywords: voltage control oscillator, sweat sensor, wet skin, dry skin, oscillation frequency

diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

Nowadays, the progress in healthcare brought a fast evolution for the devices dedicated to physiological measurements. The major exploited solutions are based on magnetic resonance imaging (MRI), X-ray, and ultrasound. However, all those techniques are either invasive for the human body or consume a lot of energy. Thus, even if these techniques are effective, they are not applicable to new healthcare applications such as mobile monitoring or home diagnosis. For such a new application field, the challenge is to design compact, low-power, easy to use and robust devices.

The radiofrequency (RF) domain is an interesting solution to answer all these challenges. Indeed, RF solutions can be not invasive for the human body and low power. In particular, frequencies from 20 to 120 GHz show shallow penetration in the skin [START_REF] Kharkovsky | Near-field microwave and mm-wave noninvasive diagnosis of human skin[END_REF]. There is also no physical impact. In parallel, sweat only involves a superficial change in the dielectric behaviour of the skin. Thus, skin sweating detection using RF can be considered using a low power sensor since there is no need for deep penetration.

It has been shown that the water content in tissues differs from healthy and cancerous skin. Indeed, malignant lesions have around 26 % higher water content than healthy tissues [START_REF] Taeb | Millimetre-wave waveguide reflectometers for early detection of skin cancer[END_REF]. This change affects the dielectric parameters of different tissues so they can be electromagnetically characterized. Several approaches have been investigated to detect the variation of the reflected signals from healthy and cancerous skin. In [START_REF] Khokhar | Near-field tapered waveguide probe operating at millimeter waves for skin cancer detection[END_REF], they showed a difference of 0.3 dB in |𝑆11| and a 6° phase variation between normal and cancerous skin at 30 GHz using a simulated waveguide probe. Moreover, the use of a vector network analyser (VNA) to measure the reflection coefficient of a waveguide closed by the skin [START_REF] Kharkovsky | Near-field microwave and mm-wave noninvasive diagnosis of human skin[END_REF] or a coaxial sensor similarly closed [START_REF] Mehta | Microwave reflectometry as a novel diagnostic tool for detection of skin cancers[END_REF] have firstly been explored. An output voltage variation of 16 mV at 40 GHz is detected in [START_REF] Kharkovsky | Near-field microwave and mm-wave noninvasive diagnosis of human skin[END_REF] and a phase variation of 9 ° is detected at 1.5 GHz in [START_REF] Mehta | Microwave reflectometry as a novel diagnostic tool for detection of skin cancers[END_REF] between dry and wet skin. To improve the compactness of such systems, [START_REF] Laemmle | A 125-GHz Permittivity Sensor With Read-Out Circuit in a 250-nm SiGe BiCMOS Technology[END_REF] proposes an integrated 125 GHz reflectometer based on a 6-port structure. Instead of monitoring reflection coefficient, some authors propose the use of integrated radar systems [START_REF] Huang | A 5.8-GHz radar sensor chip in 0.18-μm CMOS for non-contact vital sign detection[END_REF]. However, both are complex architectures whereas oscillator-based sensors are much simpler to design and operate. For example, their use has been investigated to detect changes in the heartbeat [START_REF] Kim | Wireless RF Vital Signal Sensor Using Autoregressive Spectral Estimation Method[END_REF] [8]. The system operates in near-field around 2.4 GHz and measures the antenna impedance modification through an oscillator connected to the antenna. The system is quite simple and can use an autoregressive spectral estimation method [START_REF] Kim | Wireless RF Vital Signal Sensor Using Autoregressive Spectral Estimation Method[END_REF] or a PLL [START_REF] Hong | Advanced non-contact near-field proximity vital sign sensor using phase locked loop[END_REF] to retrieve the frequency shift which occurs when antenna impedance is modified.

In this study, we propose to investigate the use of an oscillator based sensor working around 30 GHz for which the dielectric behaviour of the skin is significantly modified by the sweat (+1.55% for conductivity and +14.4% for permittivity). The frequency deviation estimator is not studied here. The paper focuses on the design of the oscillator and especially the design of a cross-coupled pair (CCP) and its connection to a resonator designed and simulated using HFSS software. As the resonating frequency is different between the two environments (wet and dry skin), the oscillation frequency is modified according to the skin state. The sensitivity of the bio-sensor is directly related to the offset frequency (∆𝑓) between the two states and has to be maximized. The main objective is to design a miniaturized contactless system that can discriminate the two states (wet and dry) of the skin. This paper is organized as follows. Section II presents the different blocks of the oscillator: the resonator, the CCP, and the output buffers. The design methodology is detailed and simulation results are provided for each block. In Section III, the study is validated by simulations of the oscillation frequency and the phase noise for wet and dry skin. Section IV gives the conclusion and the perspectives.

II. SYSTEM DESIGN

A. Architecture

The studied system presented in Fig. 1 consists of a sensor designed on a Teflon substrate connected to a CCP integrated in 55 nm BiCMOS technology and assembled with flip chip process to minimize interconnection effects. The role of the CCP is to compensate for the losses of the resonator. These are a key parameter design because small CCP can compensate for low losses, which enables low power consumption and low parasitic capacitance. Indeed, any parasitic capacitance added to the resonator by the CCP 979-8-3503-0320-9/23/$31.00 ©2023 IEEE reduces the oscillator ∆𝑓 which reduces the sensor sensitivity. Thus, the resonator shall be optimized to achieve low losses at the resonant frequency and a large ∆𝑓 between dry, and wet skin. The CCP is composed of transistors 𝑁1 and 𝑁 which are biased through 𝜆 meandered lines to 4 isolate DC from RF as shown in Fig. 1. Two buffers at the outputs enable matching to perform measurements on a 50 Ω impedance environment. Both the CCP and the buffers, are biased with a current mirror embedding a noise filtering capacitor 𝐶𝑑𝑒𝑐. The simulation results of the sensor based on two-planar patches are shown in Fig. 3 where the resonant frequency is read at φ = 0°. The resonant frequency for the dry skin is 31.701 GHz whereas it is 31.830 GHz for the wet skin, which leads to a ∆𝑓 = 130 MHz. The sensitivity is relatively good between the two states of the sensor. This first observation suggests a large margin for the design of the CCP.

The real part of the sensor impedance can be extracted from Eq. ( 1) and is presented in Fig. 4. At the resonant frequency associated to each state, losses are extremely low since a high parallel resistance 𝑅𝑝, between 80 to 90 kΩ, has been found. This second observation definitely states a relaxed constraint on the CCP design which is favorable to ∆𝑓 maximization. In addition, the value of 𝑅𝑝 keeps almost constant around the resonance frequency, which ensures robustness in the design. The variation of 𝑅𝑝 is about 10 percent around the resonance frequency for wet and dry skin respectively (TABLE II. ). To keep the distance between the sensor and the skin, it will be considered to make a sensor with a cavity on its edges or add an interface layer on the patch for practical implementations.

B. Sensing resonator

The sensing resonator is based on two-planar resonating patches, which act as a unique differential near field sensor for sweat detection. One patch is presented in Fig. 2. It consists of a metallic square shape deposited on a grounded Teflon layer with a thickness equal to 200 𝜇𝑚 and dielectric permittivity equal to 2.1. Indeed, a substrate with a low relative permittivity allows to maximize the variation of the antenna impedance. The sensor is placed at 1 mm on top of the skin to ensure near-field radiation. Two access lines (not represented in Fig. 2) are used to feed the resonator (one access line per patch). The resonator is optimized to maximize ∆𝑓 at 30 GHz, which leads to a patch size of 2.2 mm by 1.3 mm. The resonator has been simulated and optimized with HFSS software. As shown in Fig. 2, an infinite box, where the permittivity and conductivity are chosen regarding the skin state, is placed at 1 mm away from the patch to emulate the skin. The dielectric parameters of the wet and dry skin are given in TABLE I. [START_REF] Andreuccetti | An Internet resource for the calculation of the dielectric properties of body tissues in the frequency range 10 Hz -100 GHz[END_REF].

C. Cross coupled pair transistors

The CCP has to compensate for the resonator losses. To ensure the oscillations, the system needs to respect the Barkhausen criteria [START_REF] Sharma | Design of a 77-GHz LC-VCO With a Slow-Wave Coplanar Stripline-Based Inductor[END_REF], which is equivalent in a LC oscillator to the following condition:

gives a compensation factor equal to 77 which is sufficiently large to compensate for extra-losses due to layout and interconnection between resonator and CCP (not taken into account in this feasibility study) and also starts oscillations. The parasitic capacitance of the CCP, associated to 𝜆 4 transmission lines, seen from the resonator side is about 4.34 fF. The characteristic of the CCP are given in TABLE III. where 𝑔𝑚 is the transconductance of the CCP, 𝑅𝑛𝑒g the negative resistance achieved by the CCP, 𝑅𝑝 the real part of the resonator at the resonant frequency, and A a real positive value greater than one (often named compensation factor).

The A parameter is typically equal to 3 or 4 for perfectly mastered integrated resonating tanks which ensures good loss compensation through the CCP. For bio-sensors, with flip-chipped die on the Teflon board, as the proposed one, it seems much more reasonable to aim at higher values of A, in the order of tens. Knowing 𝑅𝑝, providing A, it is then possible to size the CCP with the following relationship giving 𝑔𝑚 as a function of the biasing current (𝐼𝑝𝑜𝑙 ), the transistor width (W) and its length (L) where 𝐾𝑛depends on the mobility of the electrons and the oxide capacitance. A large 𝑅𝑝 is interesting as it enables large 𝑅𝑛𝑒g , or small 𝑔𝑚 achievable with low biasing current. To isolate RF signal from DC path, a 𝜆 transmission 4 line is preferable to a choke inductor in order to minimize the parasitic capacitance and then maximize ∆𝑓. The transmission line is first sized, before considering the CCP effect, as it is an important contributor of the resonator frequency shift. A 1 µm width has been chosen. The length To size the CCP, parametric simulations are run out to extract the negative resistance (Fig. 5) and the parasitic capacitance (Fig. 6) for different transistor widths in function of 𝐼𝑝𝑜𝑙. According to the curves in Fig. 5, a CCP designed with the minimum width of W = 3 μm achieves a

D. Buffers

In Fig. 1, the transistors 𝑁𝑏𝑢𝑓𝑓𝑒𝑟1 and 𝑁𝑏𝑢𝑓𝑓𝑒𝑟2 are source follower amplifiers acting as output buffers. They are biased by the 𝑉𝑑𝑑 voltage and a specific current source (𝐼𝑏𝑢𝑓𝑓𝑒𝑟). The goal is to synthesize an output impedance of 50 Ω. Knowing that [START_REF] Sharma | Design of a 77-GHz LC-VCO With a Slow-Wave Coplanar Stripline-Based Inductor[END_REF] of 𝐶 𝑝𝑎𝑟_𝑏𝑢𝑓𝑓 = 0.85 fF which leads to a total parasitic 0.4 capacitance of 6.04 fF. It is worth mentioning that such kind of buffer is mandatory only for measurement purpose. In the case of a full implementation with a frequency deviation estimator such as a PLL, the constraint on the required buffers might be relaxed.

III. SIMULATION OF THE OSCILLATOR

The complete oscillator has been simulated in the two states (wet and dry skin) using two different S-Parameter boxes obtained from electromagnetic (EM) simulations (HFSS) of the resonator. As shown in Fig. 7, two separated frequencies are observable at 27.876 GHz and 27.911 GHz for a dry and wet skin respectively, leading to a ∆𝑓= 35 MHz. Compared to the resonator performances, a frequency shift of 11.7 percent is observed due to the loading effect of the CCP and the buffers. A ∆𝑓 reduction down to 26 percent of the initial value is also observed. By the way, 35 MHz stays a frequency deviation that can be easily detected. As presented in TABLE V. the output power is 4.7 dBm which corresponds to a 1.1 V voltage dynamic at the buffer outputs. Output buffer voltages are shown in Fig. 8. The phase noise at 35 MHz of the carrier is equal to -129 dBc from simulations. The frequency deviation, the output dynamic and the phase noise show the potentiality of the system. These performances allow us to consider this biosensor architecture for future implementation. The study is performed on the basis of EM simulations for the resonator and design kit based electrical simulations for the active integrated parts of the sensor. Depending of the skin sweat (dry or wet) the bio-sensor provides an output frequency deviation of 35 MHz around a frequency value of 27.9 GHz, with an output power of 4.7 dBm and a phase noise of -129 dBc at 35 MHz. The CCP has been oversized to ensure oscillations even with additional losses due to interconnects and layout. This first study shows the potentiality of this architecture for bio-sensor and paves the way toward further implementation.

Fig. 2 .

 2 Fig. 2. Schematic of the sensor

Fig. 1 .

 1 Fig. 1. Proposed system architecture

Fig. 4 .

 4 Fig. 4. Real part of the sensor impedance with dry and wet skin

Fig. 5 .

 5 Fig. 5. Negative impedance variation

Fig. 6 .

 6 Fig. 6. Parasitic capacitance variation

  : 1 negative resistance of -4.030 kΩ for 𝐼𝑝𝑜𝑙 = 1 𝑚𝐴 which ℜ(𝑍𝑜𝑢𝑡) = 𝑚 (4)

  to be sized to provide 𝑔𝑚 = 20 mS. The main parameters of the buffer are listed in TABLE IV. The width W = 8 μm leads to a parasitic capacitance 0.6

Fig. 7 .

 7 Fig. 7. Harmonic balance simulation with the phase noise

Fig. 8 .

 8 Fig. 8. Transient simulation IV. CONCLUSION This paper has presented a feasibility survey of an oscillator based bio-sensor for sweat detection. The oscillator is composed of an on-skin resonator coupled to a CCP with buffers integrated in 55 nm BiCMOS technology.The study is performed on the basis of EM simulations for the resonator and design kit based electrical simulations for the active integrated parts of the sensor. Depending of the skin sweat (dry or wet) the bio-sensor provides an output frequency deviation of 35 MHz around a frequency value of 27.9 GHz, with an output power of 4.7 dBm and a phase noise of -129 dBc at 35 MHz. The CCP has been oversized to ensure oscillations even with additional losses due to interconnects and layout. This first study shows the potentiality of this architecture for bio-sensor and paves the way toward further implementation.

TABLE I .

 I DIELECTRIC SKIN PROPERTIES AT 30 GHZ[START_REF] Andreuccetti | An Internet resource for the calculation of the dielectric properties of body tissues in the frequency range 10 Hz -100 GHz[END_REF]

	100
	50
	0
	-50
	-100

TABLE II

 II

				.		SENSOR PARAMETERS
				State		Dry		Wet
			Resonance frequency		31.701 GHz		31.830 GHz
			Rp at resonance frequency		81.89 kΩ		90.77 kΩ
								Dry skin
								Wet skin
			30	30.5	31	31.5	32	32.5	33
	State	Dry	Wet		Frequency (GHz)
	σ	27.099 S/m	27.52 S/m
	ε𝑟	15.51	17.743

Fig. 3. Phase of the sensor with dry and wet skin

TABLE III .

 III CHARACTERISTICS OF THE CROSS COUPLED PAIR

						W	3 μm
						𝐿	55 nm
						𝐼 𝑝𝑜𝑙	1 mA
						𝑉 𝑑𝑑	1 V
						W 𝑙𝑖𝑛𝑒	1 μm
						𝐿 𝑙𝑖𝑛𝑒	1518 μm
						|𝑅 𝑛𝑒𝑔 |	4.03 kΩ
						𝐶 𝑝𝑎𝑟	4.34 fF
						0
	|𝑅 𝑛𝑒g	| =	2 𝑔𝑚	≤	𝑅 𝑝 𝐴	(2)

-2000

TABLE IV

 IV

		0.2
		0
		-0.2
	.	CHARACTERISTICS OF THE BUFFERS
		-0.4

TABLE V .

 V

	-0.6
	15	15.05	15.1	15.15	15.2	15.25	15.3
				Time (s)
	SIMULATION RESULTS