Nicolas Brodu

Sabine Sochard

Caroline Andriantsiferana

Jean-Stéphane Pic

Marie-Hélène Manero

Fixed-bed adsorption of toluene on high silica zeolites: experiments and mathematical modelling using LDF approximation and a multisite model

Keywords: volatile organic compounds, adsorption, breakthrough curve modelling, hydrophobic zeolite, compensating cation

The adsorption of toluene (TOL) as a target volatile organic compound has been studied experimentally and modelled on various hydrophobic zeolites: Faujasite (FAU), ZSM-5 (Z) and Mordenite (MOR). The influence of the nature of the compensating cation (H + or Na + ) has also been investigated for ZSM-5 zeolite, which is known to possess three kinds of adsorption sites (sinusoidal channels, straight channels and intersections). Type I isotherms observed on FAU, Na-Z and MOR fitted well with the Langmuir model. A deviation from a type I isotherm was observed for H-Z, because of the structure of this zeolite. The Successive Langmuir Model was more successful to fit the 'bump' of the experimental curve than the Double Langmuir. Classical shapes were found for MOR, FAU and Na-Z breakthrough curves that were fitted with good accuracy using the Linear Driving Force (LDF) approximation. In the case of H-Z, a change of profile was observed during the dynamic adsorption and the differences seen between the Na-Z and H-Z behaviours were explained by the strong interactions between Na + and adsorbed TOL at the intersection sites. The Na + cations prevented reorientation of TOL molecules at the intersection and thereby avoided the filling of the sinusoidal channel segments. Thus, a specific model was developed for fitting the breakthrough curve of H-Z. The model developed took into account these two types of adsorption sites with the overall uptake for each site being given by an LDF approximation.

Introduction

Volatile organic compounds (VOCs) are used extensively in many industrial processes. The main anthropogenic sources of emissions of VOCs are oil and gas refineries, and the surface treatment or the evaporative emissions due to the use of solvents. [START_REF] Agueda | Effect of channel geometry, degree of activation, relative humidity and temperature on the performance of binderless activated carbon monoliths in the removal of dichloromethane from air[END_REF][START_REF] Finne | Gas-phase adsorption of butan-1-ol: integrated procedure for assessing the performances of adsorbent materials[END_REF] Emissions of VOCs are responsible for different problems of indoor and outdoor air pollution [START_REF] Huang | Breakthrough of methyethylketone and benzene vapors in activated carbon fiber beds[END_REF] that may affect human health, as many of these compounds are toxic or even carcinogenic. [START_REF] Pires | Adsorption of volatile organic compounds in Y zeolites and pillared clays[END_REF] They have also been identified as playing a major role in serious environmental problems such as the photochemical smog that can cause damage to both humans and the environment. [START_REF] Nevers | Air pollution control engineering[END_REF][START_REF] Guillemot | Adsorption of tetrachloroethylene (PCE) in gas phase on zeolites of faujasite type: influence of water vapour and of Si/Al ratio[END_REF][START_REF] Dou | Adsorption performance of VOCs in ordered mesoporous silicas with different pore structures and surface chemistry[END_REF] Therefore, the removal of VOCs is of significant importance in air quality control and has triggered the interest in the development of abatement technologies to comply with the latest environmental regulations.

The usual way to reduce the harmful effects of VOCs is to reduce their concentration. Many processes are used in industry for controlling environmental pollution such as adsorption and catalytic oxidation. [START_REF] Boulinguiez | Adsorption on activated carbons of five selected volatile organic compounds present in biogas: comparison of granular and fiber cloth materials[END_REF][START_REF] Khan | Removal of volatile organic compounds from polluted air[END_REF][START_REF] Brodu | Removal of volatile organic compounds by heterogeneous ozonation on microporous synthetic alumino silicate[END_REF] Nowadays *Corresponding author. Email: nicolas.brodu@gmail.com among the treatments used for reducing VOCs, adsorption technology appears to be the best strategy as it removes pollutants economically in terms of simplicity of design, ease of operation and low energy requirements. [START_REF] Chica | Adsorption, desorption, and conversion of thiophene on H-ZSM5[END_REF][START_REF] Serrano | Characterization of adsorptive and hydrophobic properties of silicalite-1, ZSM-5, TS-1 and beta zeolites by TPD techniques[END_REF][START_REF] Serna-Guerrero | Applications of pore-expanded mesoporous silica: adsorption of volatile organic compounds[END_REF] Adsorption has also been found to be effective at low concentration levels. [START_REF] Das | Removal of volatile organic compound by activated carbon fiber[END_REF] The success of an adsorption process depends on the performance of adsorbents in terms of both equilibria and kinetics: a solid support that has a favourable adsorption isotherm as well as rapid kinetics is the ideal. Several adsorbents have been extensively studied for VOC removal from air streams, such as silica gel, zeolites, mesoporous materials and activated carbons. Activated carbon is the most widely used adsorbent thanks to its high surface area, pore structure [START_REF] Brosillon | Mass transfer in VOC adsorption on zeolite: experimental and theoretical breakthrough curves[END_REF][START_REF] Lee | Adsorption characteristics of binary vapors among acetone, MEK, benzene and toluene[END_REF] and low price. [START_REF] Bhatia | Adsorption of butyl acetate in air over silver-loaded Y and ZSM-5 zeolites: experimental and modelling studies[END_REF] However, there is increasing interest in developing new adsorbents as an alternative to activated carbon because of its humidity dependence [START_REF] Monneyron | Measurement and modeling of single-and multi-component adsorption equilibria of VOC on high-silica zeolites[END_REF] and the difficulty in regenerating it. [START_REF] Dragoi | Adsorption of organic pollutants over microporous solids investigated by microcalorimetry techniques[END_REF] The use of hydrophobic zeolites is interesting because it is reported that these materials have chemical and thermal stability, are not flammable and retain good adsorption capacities at relatively high humidity. [START_REF] Bhatia | Adsorption of butyl acetate in air over silver-loaded Y and ZSM-5 zeolites: experimental and modelling studies[END_REF] Moreover, they can easily be regenerated. [START_REF] Khan | Removal of volatile organic compounds from polluted air[END_REF] Zeolites are crystalline aluminosilicates with unique microporosity properties. The crystalline framework structure, high specific area and their structural and compositional flexibility make zeolites potential candidates for VOC adsorption. [START_REF] Chica | Adsorption, desorption, and conversion of thiophene on H-ZSM5[END_REF][START_REF] Monneyron | Measurement and modeling of single-and multi-component adsorption equilibria of VOC on high-silica zeolites[END_REF] Adsorption on zeolite has been reported to depend on the adsorbent crystalline framework, pore structure, compensating cation and chemical properties. [START_REF] Kim | The effect of pore structure of zeolite on the adsorption of VOCs and their desorption properties by microwave heating[END_REF] Numerous models have been published to describe the phenomena taking place in fixed-bed adsorption. Indeed, adsorption is composed of several steps: mass transfer in the boundary layer surrounding the adsorbent particles (external transfer), diffusion inside the pores (internal transfer), adsorption onto the pore walls and surface diffusion along the internal pore surface. The drawback of sophisticated models taking both external and internal transfer into account is that they require the determination of a large number of parameters and that they are difficult to solve numerically. [START_REF] Lucas | Mathematical model of supercritical CO2 adsorption on activated carbon -effect of operating conditions and adsorption scale-up[END_REF][START_REF] Carrère | Whey proteins extraction by fluidized ion exchange chromatography: isotherms determination and process modelling[END_REF] Therefore, in most studies, the adsorption process is modelled using simplistic mathematical expressions allowing significant savings in computation time. The most simplistic approach is to assume equilibrium, and to neglect resistance to mass transfer [START_REF] Carratala-Abril | Activated carbon for the removal of low-concentration gaseous toluene at the semipilot scale[END_REF]; obviously, this approach can be only used in certain conditions or to obtain an order of magnitude of the adsorption parameters. The Thomas model, which is a lumped parameter model, can give good results. [START_REF] Hashim | Prediction of protein breakthrough behavior using simplified analytical solution[END_REF][START_REF] Réguer | Measurement and modelling of adsorption equilibrium, adsorption kinetics and breakthrough curve of toluene at very low concentrations onto activated carbon[END_REF] In this model, a pseudo adsorption kinetic coefficient is determined by fitting the calculated breakthrough curve to experimental data. This parameter includes external and internal transfer as well as adsorption kinetics. Another well-known simple model is the Linear Driving Force (LDF) approximation [START_REF] Brosillon | Mass transfer in VOC adsorption on zeolite: experimental and theoretical breakthrough curves[END_REF][START_REF] Lorimier | Adsorption of toluene onto activated carbon fiber cloths and felts: application in indoor air treatment[END_REF][START_REF] Tantet | Breakthrough study of the adsorption and separation of sulphur dioxide from wet gas using hydrophobic zeolites[END_REF][START_REF] Tien | Adsorption calculations and modeling[END_REF]: the mass balance equation within the particle is replaced by an expression giving the overall uptake rate in the particle. This adsorption rate is assumed to be linearly proportional to a driving force defined as the difference between the surface concentration and the average adsorbed-phase concentration. Several authors have proposed a relationship linking the global transfer coefficient k p to the effective intrapellet diffusivity. Tien [START_REF] Tien | Adsorption calculations and modeling[END_REF] noticed that although this relationship has been obtained on a theoretical basis, k p should above all be considered as an adjustable parameter in modelling adsorption calculations. However, all these models assume only one single type of adsorption site and dynamic models that consider adsorption in multi-compartment porous media are rare. [START_REF] Lesage | Analysis and modelling of non-equilibrium sorption of aromatic micropollutants on GAC with a multi-compartment dynamic model[END_REF][START_REF] Ding | Kinetics of adsorption on activated carbon: application of heterogeneous vacancy solution theory[END_REF] Lesage et al. [START_REF] Lesage | Analysis and modelling of non-equilibrium sorption of aromatic micropollutants on GAC with a multi-compartment dynamic model[END_REF] assumed that the different types of pores are parallel, while Ding et al. [START_REF] Ding | Kinetics of adsorption on activated carbon: application of heterogeneous vacancy solution theory[END_REF] proposed a model where the two types of pores are interconnected, leading to a complex numerical resolution.

Our current research encompasses the development of a process combining VOC adsorption and catalytic ozonation on hydrophobic zeolites. The study presented here focused on the adsorption, 'that is, the first step of this combined process, and its aims were: (i) to investigate the effects of zeolite channel size and pore structure on the adsorption using three commercial hydrophobic zeolites: mordenite (MOR), ZSM-5 and faujasite (FAU); (ii) to analyse the effect of the compensating cation and (iii) to obtain and model the breakthrough curves for toluene (TOL). For these purposes, gas-solid equilibrium experiments and dynamic adsorption experiments were carried out. The breakthrough curves obtained by dynamic adsorption were compared to the theoretical prediction of the models. The influence of the pore structure of the zeolites on mass transfer was studied to understand the adsorption mechanism. In this study, two models are used: the LDF model and a two-adsorption sites model that was specially developed for this work.

Materials and methods

Adsorbents

Four kinds of hydrophobic microporous zeolites were used as adsorbents: MOR, FAU and two ZSM-5 (Z). For the latter, the nature of the compensating cation is different: Na + and H + . The schemes of the structure of these three zeolites are shown in Figure 1. Mordenite type zeolites consist mainly of 12-membered straight channels and apertures of 6.5 × 7 Å with the presence of side pockets (2.6 × 5.7 Å). FAU-type zeolites consist of cubic-octahedrons called sodalite cages or β-cages. [START_REF] Breck | Zeolite molecular sieves: structure, chemistry and use[END_REF] The assembly of β-cages linked together by hexagonal prisms gives 13 Å diameter supercages interconnected by 12 oxygen atom apertures (7.4 Å diameter). ZSM-5 zeolite has a three-dimensional pore network with 10-membered straight and sinusoidal ring channels and apertures of 5.3 × 5.6 Å and 5.1 × 5.5 Å, respectively.

The four zeolites were provided by Tosoh and Zeochem. Zeolite pellets consist of microcrystals fixed together as cylindrical pellets with a binder (clay, alumina), creating a meso/macroporous structure. A thermal degassing procedure was applied prior to the experiments. The zeolite samples were oven-dried at 500°C for 6 h, and stored in desiccators until further use.

Characterization of zeolite samples

Specific surface areas and pore volumes were obtained by nitrogen adsorption-desorption at -196°C using a Micromeritics ASAP 2010 instrument. Prior to nitrogen adsorption, the zeolite samples were degassed at 90°C for 1 h and then at 350°C for 4 h. Specific surface areas (S BET ) were calculated from nitrogen adsorption isotherms, using the Brunauer-Emmett-Teller (BET) equation. [START_REF] Brunauer | Adsorption of gases in multimolecular layers[END_REF] Micropore volumes (V micro ) were determined by applying the Horvath-Kawazoe method. [START_REF] Horvath | Method for the calculation of effective pore size distribution inn molecular sieve[END_REF] Note: / -Not given by the provider. a Equivalent spherical particle diameter of the cylindric pellets: V pellets /S pellets. b Equivalent spherical specific area (6/d p ). c Given by the provider. and the name of the zeolite. As it can be seen, the physical properties of H-Z and Na-Z are similar although the Si/Al 2 ratio and the nature of the compensating cation are different.

Adsorbate / Analytical procedure

TOL was used as a model VOC (Sigma-Aldrich, purity more than 99%). The kinetic diameter of this molecule is 5.8 Å. [START_REF] Lee | Adsorption characteristics of binary vapors among acetone, MEK, benzene and toluene[END_REF] The concentration of TOL was analysed using a GAS chromatograph Varian 3800 GC equipped with Flame Ionization Detectors. Separation was achieved using a CP-SIL 8 capillary column (30 m × 0.53 mm ID) with a 1.0 μm film thickness.

Adsorption isotherms

Adsorption isotherms were determined at 25°C ( ± 1°C) using a volumetric method detailed in a previous publication. [START_REF] Brosillon | Mass transfer in VOC adsorption on zeolite: experimental and theoretical breakthrough curves[END_REF] All the points were obtained in triplicate. The adsorbed quantity (Q e ) was calculated from a mass balance in the gas phase:

Q e = V m (C o -C e ), ( 1 )
where C o and C e are the initial and equilibrium concentrations (mol m -3 ), respectively , V is the volume of the reactor (m 3 ) and m is the mass of solid (kg).

Experimental breakthrough curves

The experimental set-up used is shown in Figure 2.

Dynamic adsorption experiments were conducted in a fixed-bed flow reactor (4.5 cm ID; length 15 cm), at 21°C ( ± 1°C) and 101 kPa. The dry air/VOC mixture was generated using a bubbling system and diluted by adding dry air. The flow rate was fixed at 1.83 m 3 h -1 and the VOC concentration was fixed at 0.011 mol m -3 . The VOC concentrations at the inlet and at the outlet of the reactor were analysed on-line.

The experimental conditions are summarized in Table 2. Total adsorbed TOL Q (mol kg -1 ) was calculated from a mass balance using the following equation:

Q = FC TOLin m t s 0 1 - C C TOLin dt, ( 2 )
where C TOLin is the inlet concentration of TOL (mol m -3 ); F is the volumetric flow rate (m 3 h -1 ) and t s is the time needed to reach zeolite saturation (h); C is the outlet TOL concentration as function of time (mol m -3 ).

Results and discussions

Isotherms

The adsorption isotherms of TOL are shown in Figure 3.

It can be noted that the H-FAU, H-MOR and Na-Z curves are type I isotherms. A higher adsorption capacity can be observed for H-FAU than for H-MOR and for Na-Z, probably due to a higher specific surface area (S BET ) for H-FAU than for H-MOR and for Na-Z as shown on Table 1. However, in the case of zeolites, the value of S BET is not the only important parameter since the kinetic diameters of the pollutants are often of the same order of magnitude than the pore aperture. In the present study, it can be noted that H-FAU possesses larger pore aperture, avoiding steric hindrance. Although the micropore volume of H-MOR is similar to H-FAU, the side pocket (2.6 × 5.7 Å) is not accessible to the molecules of TOL (5.8 Å) and the available adsorption volume is then reduced by 53%. [START_REF] Breck | Zeolite molecular sieves: structure, chemistry and use[END_REF] The usual Langmuir model gives a very good representation of the experimental H-FAU, Na-Z and H-MOR isotherms as expected for type I isotherms. However, the Langmuir model does not fit well with the experimental adsorption isotherm of H-Z, as shown in Figure 3(b). In fact, the isotherm of H-Z shows two steps: the first step is observed for Q e around 0.6 mol kg -1 ; then, a second step seems to start when C e is superior to 0.051 mol m -3 . This leads to a higher capacity for H-Z than for Na-Z although its specific surface is the same than Na-Z. S-shaped isotherms usually describe multilayer adsorption. However, the kinetic diameter of TOL (5.8 Å) is of the same order of magnitude than the pore aperture so that a second layer is not possible. A similar isotherm profile has been reported in the case of the adsorption isotherm of aromatic compounds over silicalite-1 zeolite by Song et al. [START_REF] Song | Adsorption and diffusion properties of hydrocarbons in zeolites[END_REF]. The authors clearly pointed that this behaviour is not due to multilayer adsorption but it is suggested that two sites of adsorption are active for H-Z: the intersections and the channels. Therefore, two different variations of Langmuir models can be used: the Double Langmuir (DL) model and the Successive Langmuir (SL) model. Both consider two sites of adsorption, but not with the same assumptions. The DL model describes a simultaneous filling of the two adsorption sites (just like heterogeneous surface with many adsorption sites [START_REF] Tien | Adsorption calculations and modeling[END_REF]) whereas the SL model considers a successive filling of the two adsorption sites. The equations can be read in Table 3 and the results of the modelling can be seen in Figure 3(b). It can be shown that both the models give a good accuracy for the first part of the curve. As suggested by Wang et al. [START_REF] Wang | Experiment and modeling of pure and binary adsorption of n-butane and butane-1 on ZSM-5 zeolites with different Si/Al ratios[END_REF], low-loading adsorptions take place at the energetically preferred site and thus, for low loadings, the DL model is as accurate as the SL model. This is not the case for higher loadings. Despite the high degree of accuracy with the experimental data, compared to a simple Langmuir model, the DL model does not represent the second step very well. This is probably due to the fact that this model describes a filling up of both sites at the same time (just like for a heterogeneous surface). The better result obtained by the SL model seems to show that the two sites do not fill up at the same time. It can be supposed that the intersections are the preferential adsorption site of the zeolite H-Z and in the case of high loadings, and after the complete loading of the TOL

Q e = b Q max C e 1+b C e . b Q e = b i Q maxi C e 1+b i C e if C e ≤ C lim = 0.051 mol m -3 , Q e = b i Q maxi C lim 1+b i C lim + b c Q maxc (C e -C lim ) 1+b c (C e -C lim ) if C e > C lim . c Q e = b i Q maxi C e 1+b i C e + b c Q maxc C e 1+b c C e .
molecules on these sites, the adsorption goes on inside the sinusoidal channels. This assumption is reinforced by the fact that the second step appears when four molecules per unit cell is sorbed corresponding to the number of the intersections per unit cell. [START_REF] Song | Adsorption and diffusion properties of hydrocarbons in zeolites[END_REF] Song et al [START_REF] Song | Adsorption and diffusion properties of hydrocarbons in zeolites[END_REF] suggested that there is redistribution and/or reorientation of molecules sorbed in the framework when the sorbed molecules start to populate the sinusoidal channels. Hence, we suggest that some of the molecules sorbed on the intersections can migrate to the sinusoidal channels, so that new molecules can again be adsorbed at the intersections. Then the two SL models seem to be more adapted to fit the isotherm of TOL on H-Z.

The identified values of the parameters of the models are given in Table 3. In the cases of the DL model and of the SL model, it should be noted that the value of Q maxi (0.66 or 0.69 mol kg -1 ) is close to the value corresponding to a loading of four molecules per unit cell (0.56 mol kg -1 ). Moreover, b i is greater than b c , indicating that the adsorption at the intersections is easier than the adsorption in the channels. Indeed, at low loadings, the TOL molecules only move along the straight channels up to the intersections. However, they have to reorient at the level of the intersections in order to populate the sinusoidal channels [START_REF] Song | Adsorption and diffusion properties of hydrocarbons in zeolites[END_REF][START_REF] Szabelski | Coverage dependence of aromatic hydrocarbon diffusion in silicalite: predictions of the three-site lattice model[END_REF] because of the strong adsorbate-adsorbate interactions, leading to a lower coefficient b c .

These results are consistent with other studies. [START_REF] Monneyron | Measurement and modeling of single-and multi-component adsorption equilibria of VOC on high-silica zeolites[END_REF][START_REF] Song | Adsorption and diffusion properties of hydrocarbons in zeolites[END_REF] The difference between the Na-Z and H-Z isotherms is probably related to the nature of compensating cation. A steric hindrance can be induced by the larger size of Na + (1.16 Å) compared to the diameter of H + (1.6.10 -5 Å). Since the kinetic diameter of TOL and the dimensions of the sinusoidal channels are 5.8 Å and 5.1 × 5.5 Å, respectively, Na + can block the entry of the sinusoidal channels. The strength of the interaction between the cation and TOL can be another explanation. Indeed, the cation Na + acts as a Lewis acid to form a strong interaction with π electrons in the case of TOL adsorption. [START_REF] Serra | Experimental and theoretical studies about the adsorption of toluene on ZSM5 and mordenite zeolites modified with Cs[END_REF] The interaction between Na + and TOL is greater than that between H + and TOL. This characteristic may affect the adsorption of the hydrocarbon on ZSM-5. Indeed, as explained by Song et al., [START_REF] Song | Adsorption and diffusion properties of hydrocarbons in zeolites[END_REF] the strong adsorbate-adsorbate interactions for aromatic adsorbates leads to redistribution and/or reorientation of the molecules adsorbed in the framework when the adsorbed molecules start to populate the channel segments. Thus, we can postulate that, in the case of Na-Z, the reorientation of TOL is not possible due to stronger interactions between Na + and TOL and thus TOL cannot enter the sinusoidal channels. Therefore the adsorption cannot continue inside the channels and is probably limited to the intersections for Na-Z, contrary to H-Z.

Breakthrough curves

Figure 4(a) and 4(b) shows the breakthrough curves of TOL for the different zeolites. The H-MOR gives a better breakthrough curve with a longer breakthrough time and a steeper slope than the other zeolites. The mass transfer of TOL into the pores of MOR is favoured compared to the other zeolites. The breakthrough of H-FAU appears earlier but the slope of the curve is more extended, leading to a later complete loading. Despite the similar physical properties of ZSM-5, the appearance of the breakthrough curve is different between Na-Z and H-Z as was observed for the adsorption isotherms. A change of slope is observed in the course of the breakthrough curve for H-Z whereas a classical shape is observed for Na-Z. This can be due to the strong interaction between the adsorbed molecules [START_REF] Zhang | Comparison of dynamic adsorption/desorption characteristics of toluene on different porous materials[END_REF] and to the presence of two sites of adsorption for the H-Z as mentioned earlier.

Modelling

According to the previous results, two models were developed. One kind of site is considered in the first model whereas the second model considers two kinds of adsorption sites. The following assumptions are used for both the models: (i) the pressure drop is negligible; (ii) the system is isothermal for low concentrations; (iii) the plug flow is assumed and the effect of the axial dispersion is considered; (iv) the volumetric flow rate is constant; (v) the adsorbent particles are spherical. In the gas phase, an external film around the particle is postulated. The second model assumes that TOL uptake at the i-sites and c-sites occurs simultaneously. This assumption is consistent with the DL model. However, this second model will be used with the SL model with an adaptation which will be discussed later.

Model with one adsorption site

According to the hypotheses stated earlier, the dynamic adsorption

∂C ∂t = - u ∂C ε ∂z + D ax ∂ 2 C ∂z 2 - ρ ∂ Q ∂t , ( 3 )
with the boundary conditions:

z = 0, D ax ∂C u ∂z = C -C TOLin , ( 4 )
z = H , ∂C ∂z = 0. ( 5
)
Q is the mean quantity adsorbed and its derivative is given by the following LDF approximation [START_REF] Brosillon | Mass transfer in VOC adsorption on zeolite: experimental and theoretical breakthrough curves[END_REF] where k p is the internal mass transfer coefficient taking into account inter and intracrystalline diffusion (i.e. diffusion through the binder and surface diffusion into the crystals).

∂ Q ∂t = k p (Q s -Q). ( 6 )
At the surface of the particle, the concentration of adsorbate in the solid phase Q s is linked to the concentration in the gas phase, C s by the Langmuir equation (Table 3).

The equality of flux through the surface leads to the following equation:

k c a(C -C s ) = ρk p (1 -) (Q s -Q), ( 7 )
where k c is the external mass transfer coefficient. It can be determined by the semi-empirical equation of [START_REF] Wakao | Effect of fluid dispersion coefficients on particle-to-fluid mass transfer coefficients in packed beds[END_REF]. [START_REF] Wakao | Effect of fluid dispersion coefficients on particle-to-fluid mass transfer coefficients in packed beds[END_REF] The initial conditions of the system were:

z ≥ 0, q = 0, ( 8
)
z = 0, C = C TOLin , ( 9
)
z > 0, C = 0. ( 10
)

Model with two adsorption sites

Two kinds of sites are considered with their transport features. Thus, two terms of consumption are present (∂Q i /∂t and ∂Q c /∂t).

The first and second adsorption sites are respectively the intersections and the channels. According to the hypotheses stated earlier, the dynamic adsorption model consists of the following equations:

• Mass balance in the fixed-bed is given by the following equation:

∂C ∂t = - u ∂C ε ∂z + D ax ∂ 2 C ∂z 2 - ρ ∂Q i ∂t + ∂Q c ∂t . (11)
• For each site, a system of equations similar to those of the first model is used.

∂Q x ∂t = k px (Q sx -Q x ) ( 12
)
k px is the internal mass transfer coefficient for the site x (x being i for the intersections and c for the channels).

k c a(C -C sx ) = ρk px (1 -) (Q sx -Q x ), ( 13
)
Q sx = b x Q maxx C sx 1 + b x C sx . ( 14
)
For both the sites, k c is the same external mass transfer coefficient. As already mentioned, k c is determined by the correlation of Wakao and Funazkri. [START_REF] Wakao | Effect of fluid dispersion coefficients on particle-to-fluid mass transfer coefficients in packed beds[END_REF] Yang has pointed out that this correlation should be used when axial dispersion is included in modelling fixed-bed adsorbers and that other correlations have to be used if the dispersion is neglected. [START_REF] Yang | Gas separation by adsorption processes[END_REF] In order to estimate the axial dispersion coefficient D ax , several correlations exist and give a wide range of values for each zeolite. However, Tien [START_REF] Tien | Adsorption calculations and modeling[END_REF] has pointed out that some of these correlations (like the wellknown correlation of Edwards and Richardson) have only limited accuracy if the particulate diameter is lower than 3 mm (that is the case for the four zeolites selected in this study), because in this case the limiting value of the Peclet number Pe can deviate significantly from the theoretical value of 2 which appears in these correlations. This problem is taken into account in the correlation used which is given by the following equation [START_REF] Delgado | Longitudinal and transverse dispersion in porous media[END_REF]:

D ax = D m τ + 0.65d P (u/ε) 1 + 7 D m (d P u/ε) , (15)
which is valid for Re < 100 (Re is about 70 for the four zeolites).

The initial and boundary conditions are similar to those of the first model. The total quantity adsorbed is the sum of the quantity at both types of site.

Determination of parameters and numerical solution

The two models contain a set of parameters available from experimental data and measurements. The two models are solved in the same way: the space variable is discretized in

N values. Each variable (C, Q, Q s , C s for the first model and C, Q i , Q si , C si , Q c , Q sc , C sc
for the second) is replaced by N variables leading to a set of 4N variables for the first model and 7N variables for the second one. The massbalance partial differential equation (Equation ( 3) or ( 11)) is then transformed into a system of N ordinary differential equations (time) using finite differences of order 2 to represent the space derivatives. Each ordinary differential equation is replaced by a system of N ordinary differential equations, and each algebraic equation is replaced by a system of N algebraic equations. Then the global algebrodifferential system (of 4N equations for the first model, 7N equations for the second model) was numerically solved using the ode15s function in MATLAB ® . The model parameters are determined by fitting the modelled results to the experimental data. An optimization procedure based on the Levenberg-Marquardt algorithm was used to fit the parameters to the experimental data. The following LS criterion is minimized:

LS = 1 n exp n exp j =1 (Cout j ,mod -Cout j ,exp ) 2 (Cout j ,exp ) 2 ( 16
)
n exp is the number of experimental data and C out is the concentration of the TOL in the gas phase at the outlet of the fixed-bed.

The identified parameter of the first model is k p . The identified parameters of the second model are: k pi , k pc .

Results of the simulation

As shown in Figure 4(a), the mathematical model with one site fitted well the experimental breakthrough of the three zeolites characterized by the isotherms well-described by the Langmuir model. However, among these three zeolites, the result for H-FAU was less accurate. The second model with two sites of adsorption also failed to represent the H-FAU breakthrough curve with good accuracy. Another model with one site of adsorption has been tested: the Thomas model. [START_REF] Hashim | Prediction of protein breakthrough behavior using simplified analytical solution[END_REF][START_REF] Réguer | Measurement and modelling of adsorption equilibrium, adsorption kinetics and breakthrough curve of toluene at very low concentrations onto activated carbon[END_REF] It led to a slightly better representation suggesting that the adsorption kinetics might be taken into account, though it lacked accuracy in the intermediate zone of the breakthrough curve. This could be explained by the lack of precision of the isotherm modelling in the area of inlet concentration as mentioned earlier.

The adsorbed quantities and the parameters of the one site model are reported in Table 4. The quantities are close to those obtained at equilibrium and estimated using adsorption isotherms. It appears that the intrapellet mass transfer coefficients k p is greater for H-MOR than for Na-Z and H-FAU. The mass transfer depends on several parameters: hydrodynamics, the affinity between the adsorbent and adsorbate, the pore size and so on. The closer to one the ratio between the kinetic diameter of TOL and the pore size of zeolites is, the higher the affinity between the surface and the adsorbate, thanks to Van Der Walls forces. This ratio is lower for FAU (0.78) than for MOR (0.88). Thus, the breakthrough is steeper as observed for H-MOR. The size of the pore for TOL is similar in the case of H-Z and Na-Z (1.02), but the ratio is slightly higher than 1. In fact when the ratio is identical or slightly higher than 1, the transport of molecules is more difficult. These results are consistent with other studies. [START_REF] Meininghaus | Sorption of volatile organic compounds on hydrophobic zeolites[END_REF] As far as H-Z is concerned, neither the first model with one site of adsorption nor the Thomas model described the profile of the experimental breakthrough curve of H-Z. Three breakthrough curves predicted by the second model are shown in Figure 4(b). First, the parameters Q maxi , Q maxc , b i , b c of the DL model are used. Hence, a parallel adsorption is simulated, and internal mass transfer coefficient is determined for each site (reported in Table 5). As shown in Figure 4(b), this model does not fit well with the experimental data. Moreover, surprisingly, k pc is greater than k pi although the molecules that reach the sinusoidal channels have to pass through the straight channels and the intersections before being adsorbed in the sinusoidal channels. This result reinforces our conviction that adsorption at intersections and adsorption in sinusoidal channels takes place successively rather than simultaneously. In Figure 4(b), it can be seen that the use of the parameters of the SL model instead of the parameters of the DL model gives better results but fails to represent the end of the breakthrough curve. Indeed, since the model does not interconnect the two sites, it seems to us that the use of the parameters of the SL model could not be suitable, particularly b i , because some of the molecules adsorbed at the intersections will migrate to the channels. So if it is only considered that the molecules finally stay at the intersections, the parameter b i would be lower. That is why in order to simulate the successive adsorptions, b i and b c have been identified together with k pi and k pc . This simulation fits the experimental breakthrough curve very well, as shown in Figure 4(b). Initially, the adsorption of TOL occurs in the intersections of the framework and then continues inside the sinusoidal channels after a reorientation of the molecules adsorbed at the intersections. The global mass transfer then becomes more limiting leading to a change in the slope, which explains why the identified mass transfer coefficient for the intersections is lower for H-Z than for Na-Z (Table 4). Indeed, for Na-Z, the intersections are probably the only adsorption site, as reported before, and no change in diffusion rate can then be observed. The identified values of the parameters are reported in Table 5. It should be noted that k pi is lower than k pc . Indeed, for intersections, intracrystalline diffusion is important since the molecules move along the straight channels to reach the intersections. While for the sinusoidal channels, intracrystalline diffusion is lower since this adsorption results from a reorientation of the molecules already adsorbed at the intersections. It can also be seen that the value of b c is more-or-less the same as for the isotherm. However, the value of b i is much lower as expected and previously explained.

Conclusion

Thanks to the results of experimental and modelling studies, a better understanding of adsorption phenomena is proposed according to the nature of the zeolites.

(i) It is obvious that adsorption is dependent on the way the zeolites are structured. According to the size of the pores, TOL can enter easily into the network (FAU) or can only enter with difficulty (Mordenite or Na-Z). The Langmuir model fits these 'type I' isotherms with a good accuracy and the LDF approximation gives a good simulation of breakthrough curves for these three adsorptions. (ii) However, some chemical phenomena can occur and change the adsorption characteristics. For example, with ZSM-5, the compensating cation can play a role. Because of some strong interactions, the solute can be blocked in one site of adsorption (in the case of Na-Z) whereas it can spread to a secondary internal network if no interaction occurs (in the case of H-Z). (iii) In that latter case, two sites of adsorption are highlighted and two SL models fit the experimental isotherm much better than a DL model suggesting that adsorption on the two sites occur in series rather than in parallel. (iv) In the case of the breakthrough curves, the development of a model is proposed, taking into account two different sites of adsorption, sinusoidal channels and intersections. This model fails to represent the breakthrough curve if parallel adsorption is assumed but can give a high level of accuracy with the hypothesis of successive adsorption.

Figure 1 .

 1 Figure 1. Zeolite structure: MOR (a), FAU (b), ZSM-5 (c).

Figure 2 .

 2 Figure 2. Experimental set-up for dynamic adsorption. (1) Dry air inlet; (2) flowmeter; (3) bubbler; (4) thermostated tank; (5) adsorption bed; (6) gas sample outlet; (7) three-way valve to gas analyser; (8) exhaust gas.

Figure 3 .

 3 Figure 3. (a) Adsorption isotherm of zeolites at 25°C and 101 kPa: ( ) Na-Z; ( ) H-MOR; ( + ) H-FAU; (-) simulated with Langmuir model. (b) Adsorption isotherm of zeolites at 25°C and 101 kPa: (♦) H-Z; (-••) simulated with Langmuir model; (•••) simulated with two successive Langmuir models; (-) simulated with Double Langmuir model.

Figure 4 .

 4 Figure 4. (a): Influence of zeolites on TOL breakthrough: ( ) Na-Z; ( ) H-MOR; ( + ) H-FAU; (-) simulated with Model-1. (b) TOL breakthrough curve on H-Z: (♦) Experimental data; (-••) adsorption on the two sites using the parameter of the Double Langmuir; (---) adsorption on the two sites using the parameter of the Successive Langmuir model; (-) adsorption using Q maxi and Q maxc of the Successive Langmuir model but identifying b i and b c to compensate for the lack of the interconnection of the two sites. Operating conditions: 21°C, 101 kPa, C TOLin : 0.01 mol m -3 . model consists of the following equations.

  Nomenclature a specific area (m² m -3 ) b Langmuir constant (m 3 mol -1 ) C concentration of the TOL in the gas phase at the outlet of the fixed-bed (mol m -3 ) C s gas concentration at the surface of the pellet (mol m -3 ) C e equilibrium concentration in gas phase (mol m -3 ) C 0 initial concentration (mol m -3 ) C TOLin inlet concentration (mol m -3 ) D ax axial dispersion coefficient (m² s -1 ) D m molecular diffusivity (m² s -1 ) d p particle diameter (m) DL Double Langmuir F fluid flow (m 3 h -1 ) H height of bed (m) k c external mass transfer coefficient (m s -1 ) k p intrapellet mass transfer coefficient (s -1 ) k pi intrapellet mass transfer coefficient in the first adsorption site for model 2 (s -1 ) k pc intrapellet mass transfer coefficient in the second adsorption site for model 2 (s -1

Table 1

 1

	summarizes

Table 1 . Physical-chemical properties of the zeolites.

 1

	Sample	Zeolite	Compensating cation	Pore size aperture (Å)	d p (mm) a	Binder weight (%)	a b (m² m -3 )	S BET (m² g -1 )	V micro (cm 3 g -1 )	Si/Al 2 ratio c
	Na-Z	ZSM-5	Na +	5.1 × 5.5	1.7	/	3529	309	0.12	360
	H-Z	ZSM-5	H +	5.3 × 5.6 5.1 × 5.5	1.6	20	3750	308	0.11	2100
	H-FAU FAU H-MOR MOR	H + H +	5.3 × 5.6 7.4 2.6 × 5.7	1.8 1.8	20 20	3333 3333	608 508	0.23 0.21	13.8 230
				(pocket)
				6.5 × 7 (channel)

Table 2 .

 2 Experimental conditions used to carry out the breakthrough curves.

	Sample	H (m)	m (g)	ρ (kg m -3 )	(-)
	Na-Z	0.09	132	845	0.36
	H-Z	0.09	98	680	0.39
	H-FAU	0.09	70	490	0.35
	H-MOR	0.09	91	658	0.34

Note: With H, m, ρ, being the length, the mass, the density and the void fraction of the fixed-bed of zeolite pellets.

Table 3 .

 3 Identified parameters fitting Langmuir, DL or SL models to the experimental isotherm data. Q maxi , b c , Q maxc are the parameters of the Successive Langmuir model or DL model with subscripts i and c denoting the intersections and channels, respectively.

		Isotherm model	Q max (mol kg -1 ) b (m 3 mol -1 )
	H-MOR	Langmuir a	0.75	552
	H-FAU	Langmuir	1.70	227
	Na-Z	Langmuir	0.57	810
	H-Z	Langmuir	0.75	309
		Successive Langmuir b	0.69/0.12	426/264
		Double Langmuir c	0.66/0.43	437/3.35
	Note: b i ,

a

Table 4 .

 4 Adsorbed amounts and mass transfer coefficients. k c Obtained from the relationship of Wakao and Funazkri: (k c d p /D m ) = 2.0 + 1.1Re 0.6 Sc 0.3 .

		Q (mol kg -1 )	Q e	a (mol kg -1 )	k p ( × 10 4 s -1 )	k c	b ( × 10 2 m s -1 )
	H-MOR	0.66		0.67	5.8		5.9
	H-FAU	1.22		1.18	1.7		5.9
	H-Z	0.51		0.50	see Table 5		6.2
	Na-Z	0.53		0.51	2.1		6.0
	a From isotherm models.

b

Table 5 .

 5 Equilibrium parameters and mass transfer coefficients for the two site models.

		Q maxi / Q maxc (mol kg -1 )	b i /b c (m 3 mol -1 )	k pi ( × 10 4 s -1 )	k pc ( × 10 4 s -1 )	k c ( × 10 2 m s -1 )
	Parallel adsorption	0.66 / 0.43	437 / 3.35	1.7	25.0	5.9
	Successive adsorption	0.69 / 0.12	426 / 264	0.07	19.0	5.9
		0.69 / 0.12	162 / 307	1.1	16.0	5.9
	Note: Identified values are in bold.

Acknowledgements

N. Brodu gratefully acknowledges Mr. Khairul Nizam SAIFUL EMPON for his valuable collaboration.

Funding

This work was supported by the French Ministry of Higher Education and Research.

Disclosure statement

No potential conflict of interest was reported by the authors.


Benjamin Rotenberg
email: benjamin.rotenberg@upmc.fr

Jean-Pierre Morel

Virginie Marry

Pierre Turq

Nicole Morel-Desrosiers

On the driving force of cation exchange in clays : Insights from combined microcalorimetry experiments and molecular simulation

Keywords: numbers clay, ionic exchange, thermodynamics, molecular simulation, microcalorimetry

We study the origin of the ionic exchange enthalpy in montmorillonite clays using microcalorimetry measurements and molecular simulation. We first determine the standard reaction enthalpy for well-defined interlayer water contents. We then show by a detailed analysis based on thermodynamic cycles that replacing Na + ions by Cs + in the interlayer of montmorillonite clays is an endothermic process, and that the overall exchange is exothermic only because it is dominated by the exothermic replacement of Cs + by Na + in the aqueous phase. This conclusion from ionic exchange enthalpies supports the one of a recent study of the ionic exchange free energy by Teppen and Miller [Soil. Sci. Soc. Am. J., 70, 31 (2006)] and contradicts long-held views on the role of ion-clay interactions in determining the ionic exchange thermodynamics. This calls for a paradigm shift for the origin of this exchange: The driving force is the "hydrophobicity" of Cs + compared to Na + and not its affinity for clay surfaces.

INTRODUCTION

Clays are layered aluminosilicate minerals found in large amount in the Earth's upper crust and are involved in many industrial and environmental processes. In particular, their retention properties have made clayey materials possible candidates to act as barriers for the disposal of toxic and radioactive waste. In smectite clays such as montmorillonite, the layers bear a permanent negative charge compensated by counterions located between them (interlayer space). These counterions are the origin of two interesting features : clay swelling and ionic exchange. The former refers to the entrance of water into the interlayer, while the later involves the replacement of natural counterions like Na + in the clay interlayer by other ions initially in the aqueous solution in contact with the mineral, and the concomitant release of Na + in the solution [START_REF] Slabaugh | Cation exchange properties of bentonite[END_REF][START_REF] Barrer | Thermochemistry and thermodynamics of ion exchange in a crystalline exchange medium[END_REF][START_REF] Martin | Thermodynamique de l'échange des cations alcalins dans les argiles[END_REF][START_REF] Fripiat | Comparaison entre les propriétés d'échange de la montmorillonite et d'une résine vis-à-vis des cations alcalins et alcalino-terreux I. réversibilité des processus[END_REF][START_REF] Robeyns | Thermodynamics of singly charged ion exchanges in trace regions on Camp Berteau montmorillonite[END_REF][START_REF] Maes | Charge density effects in ion exchange. part 2. Homovalent exchange equilibria[END_REF][START_REF] Ewin | Clay chemistry -investigation of thermodynamics of ion-exchange reactions by titration calorimetry[END_REF][START_REF] Dyer | The uptake of caesium and strontium radioisotopes onto clays[END_REF]. The swelling properties of clays have been studied extensively, both experimentally [START_REF] Cases | Mechanism of adsorption and desorption of water vapour by homoionic montmorillonite 1. The sodium exchanged form[END_REF][START_REF] Bérend | Mechanism of adsorption and desorption of water vapour by homoionic montmorillonite 2. The Li + , Na + , K + etc. exchanged forms[END_REF][START_REF] Michot | Water organisation at the solid-aqueous solution interface[END_REF][START_REF] Ferrage | New insights on the distribution of interlayer water in bi-hydrated smectite from X-ray diffraction profile modeling of 00l reflections[END_REF][START_REF] Weiss | Variations in interlayer cation sites of clay minerals as studied by 133 Cs mas nuclear magnetic resonance spectroscopy[END_REF][START_REF] Trausch | Separation of components of a 1 H NMR composite signal by nutation experiments under low amplitude radio-frequency fields. application to the water signal in clays[END_REF][START_REF] Rinnert | Hydration of a synthetic clay with tetrahedral charges: A multidisciplinary experimental and numerical study[END_REF][START_REF] Salles | A calorimetric study of mesoscopic swelling and hydration sequence in solid na-montmorillonite[END_REF] and theoretically [START_REF] Boek | Molecular modeling of clay hydration: a study of hysteresis loops in the swelling curves of sodium montmorillonites[END_REF][START_REF] Boek | Monte Carlo molecular modelling studies of hydrated Li-, Na-and K-smectites: Understanding the role of potassium as a clay swelling inhibitor[END_REF][START_REF] De Siqueira | Computer simulation evidence for enthalpy driven dehydration of smectite clays at elevated pressures and temperatures[END_REF][START_REF] Young | Simulations of clay mineral swelling and hydration: Dependence upon interlayer ion size and charge[END_REF][START_REF] Whitley | Free energy, energy, and entropy of swelling in Cs-, Na-, and Sr-montmorillonite clays[END_REF][START_REF] Smith | Molecular simulations, of the pressure, temperature, and chemical potential dependencies of clay swelling[END_REF][START_REF] Liu | A thermodynamic understanding of clay-swelling inhibition by potassium ions[END_REF][START_REF] Hensen | Adsorption isotherms of water in Li-, Na-and K-montmorillonite by molecular simulation[END_REF][START_REF] Hensen | Why clays swell[END_REF][START_REF] Tambach | A molecular mechanism of hysteresis in clay swelling[END_REF][START_REF] Tambach | Hysteresis in clay swelling induced by hydrogen bonding: Accurate prediction of swelling states[END_REF] and are now relatively well understood.

In the context of the geological disposal of nuclear waste, it is particularly important to understand the retention of radioactive traces by the geological medium. Although several mechanisms are at play for the retention of ions by clays [START_REF] Schlegel | Sorption of metal ions on clay minerals I. Polarized EXAFS evidence for the adsorption of Co on the edges of hectorite particles[END_REF][START_REF] Schlegel | Adsorption mechanisms of Zn on hectorite as a function of time, pH, and ionic strength[END_REF][START_REF] Dahn | Structural evidence for the sorption of Ni(II) atoms on the edges of montmorillonite clay[END_REF], ionic exchange is predominant for some of them, in particular Cs + , a long-lived radioisotope of which ( 137 Cs + ) is expected to be one of the main contributors to the long term radioactivity [START_REF] Andra | Évaluation de la faisabilité du stockage géologique en formation argileuse[END_REF]. In order to predict how cationic radioactive species are retained by clays, it is therefore important to have a good understanding of the thermodynamic properties of ionic exchange, i.e. the associated free energy, which controls the thermodynamic constant of the reaction, and enthalpy, which controls its variations with temperature via the van't Hoff equation. This is particularly important in the context of geological disposal of nuclear waste, since the radioactive parcels will heat the surrounding medium.

Several measurements of the equilibrium constants for ionic exchange in clay minerals, particularly in those of the montmorillonite type, have been reported [START_REF] Martin | Thermodynamique de l'échange des cations alcalins dans les argiles[END_REF][START_REF] Fripiat | Comparaison entre les propriétés d'échange de la montmorillonite et d'une résine vis-à-vis des cations alcalins et alcalino-terreux I. réversibilité des processus[END_REF][START_REF] Robeyns | Thermodynamics of singly charged ion exchanges in trace regions on Camp Berteau montmorillonite[END_REF][START_REF] Maes | Charge density effects in ion exchange. part 2. Homovalent exchange equilibria[END_REF][START_REF] Ewin | Clay chemistry -investigation of thermodynamics of ion-exchange reactions by titration calorimetry[END_REF][START_REF] Dyer | The uptake of caesium and strontium radioisotopes onto clays[END_REF], but the enthalpies of exchange were much less frequently determined [START_REF] Slabaugh | Cation exchange properties of bentonite[END_REF][START_REF] Barrer | Thermochemistry and thermodynamics of ion exchange in a crystalline exchange medium[END_REF][START_REF] Maes | Charge density effects in ion exchange. part 2. Homovalent exchange equilibria[END_REF][START_REF] Ewin | Clay chemistry -investigation of thermodynamics of ion-exchange reactions by titration calorimetry[END_REF][START_REF] Morel | Effect of temperature on the retention of Cs + by Na-montmorillonite: microcalorimetric investigation[END_REF]. One difficulty for rigorously defining thermodynamic quantities of reaction in the case of ionic exchange is that the water content in the interlayer may vary between the initial and final states. This is one of the reasons why the exchange enthalpy and free energy depend on the degree of exchange α.

It is therefore important to distinguish the exchange of traces (α → 0 for traces of Cs + in a Na-clay) from the complete exchange (α → 1 for the exchange from homoionic Na-clay to homoionic Cs-clay) [START_REF] Morel | Effect of temperature on the retention of Cs + by Na-montmorillonite: microcalorimetric investigation[END_REF]. Furthermore, the ionic selectivity, from which ionic exchange free energies are computed, is measured for clay suspensions. In many practical situations, however, clays constitute a solid phase (with a fixed, low water content) and the ions (e.g. traces of radioactive ions in the underground) are dissolved in the aqueous phase in contact with this solid. One thus needs to consider the water content in the clay phase when discussing the thermodynamics of ionic exchange.

According to the current paradigm for ionic exchange in clays, specific interactions between the ions and the clay surface, or the hydration properties of ions in the clay interlayer, are the driving force for the ionic exchange. The origin of clay selectivity for alkaline ions according to the series Cs + > Rb + > K + > Na + > Li + has been explained by a number of arguments : the size of hydrated cations, compared to the interlayer spacing [START_REF] Gast | Standard free energy of exchange for alkali metal cations on wyoming bentonite[END_REF][START_REF] Gast | Alkali metal cation exchange on chambers montmorillonite[END_REF][START_REF] Shainberg | Ion exchange equilibria on montmorillonite[END_REF], their ability to lose a water molecule at the clay surface, thus forming a stronger inner-sphere complex [START_REF] Maes | Charge density effects in ion exchange. part 2. Homovalent exchange equilibria[END_REF][START_REF] Eisenmann | Cation selective glass electrodes and their mode of operation[END_REF], their hydration state in the interlayer [START_REF] Laird | Relationship between cation exchange selectivity and crystalline swelling in 2:1 expanding phyllosilicates[END_REF] or their polarizability that influences the formation of surface complexes [START_REF] Shainberg | Ion exchange equilibria on montmorillonite[END_REF][START_REF] Maes | Highly selective ion exchanger in clay minerals and zeolites, j. a. david and k[END_REF][START_REF] Sposito | The surface chemistry of soils[END_REF]. These microscopic features are then thought to weight in favour of the interactions of clay interlayers with larger alkaline cations.

However, considering only the clay phase to infer properties of the ionic exchange can lead to erroneous conclusions. This fact was emphasized by Teppen and Miller, who analysed the free energy of ionic exchange of small alkaline ions for larger ones, using molecular simulation supported by the analysis of experimental data from the literature [START_REF] Teppen | Hydration energy determines isovalent cation exchange selectivity by clay minerals[END_REF]. They showed that the contribution of the clay phase to the ionic exchange free energy is in fact very unfavourable (∆G 0) and that it is the hydration free energy difference, i.e. the contribution of the water phase, that leads to an overall exchange of small alkaline ions for larger ones (e.g. Na + for Cs + ).

In the present paper, we investigate (a) the influence of the interlayer water content on the exchange enthalpy and (b) the origin of this enthalpy. Both issues are addressed by considering thermodynamic cycles involving exchange enthalpies in solution, immersion enthalpies of clays, and hydration enthalpies of cations. This strategy is particularly efficient to investigate the ionic exchange process, because all these quantities are directly measurable by microcalorimetry experiments. It should be noted that, contrary to immersion enthalpies, immersion free energies are not measurable directly, which means that a similar analysis of the exchange free energy for well-defined interlayer water contents would be difficult, if possible at all. We analyse the contribution of the clay phase (at well defined interlayer water contents) and that of the aqueous solution to the ionic exchange enthalpy. We support our experimental study by directly computing by molecular simulation the contribution of the clay phase to the exchange enthalpy. From this combined experimental and theoretical analysis we demonstrate that replacing Na + ions by Cs + in the interlayer of montmorillonite clays is an endothermic process, and that the overall exchange is exothermic only because it is dominated by the exothermic replacement of Cs + by Na + in the aqueous phase.

The paper is organized as follows. We first introduce the thermodynamic quantities associated with the ionic exchange and the thermodynamic cycles allowing us (a) to determine well-defined standard reaction enthalpies and (b) to characterize the relative contribution of the clay and aqueous phase to the overall exchange thermodynamics. We subsequently describe the experimental systems and methods, and present the results of microcalorimetry measurements. Then the molecular simulation systems and methods are presented. After validating their choice by studying the free energy associated with the exchange of traces, we present the simulation results on the complete exchange enthalpy. Finally, we discuss the experimental and simulation results and demonstrate how their combination supports our conclusions.

IONIC EXCHANGE THERMODYNAMICS

Ionic exchange reaction

The reaction of ionic exchange between clays at a given water content and ions in a dilute aqueous solution can be written as:

Na + nH 2 O + Cs + (aq) -→ Cs + nH 2 O + Na + (aq) (1)
As the reaction proceeds, the degree of exchange α varies from 0 (exchange of traces of Cs + in a Na-clay) to 1 (complete exchange from homoionic Na-clay to homoionic Cs-clay). In order to define reaction quantities, it is necessary to carefully define the initial and final reference states and ensure that mass (including water) is conserved during the reaction.

The standard reaction enthalpy, for the complete exchange from MNa to MCs with the same water content (n water molecules per cations), is ∆ r H 0 (n).

Na + nH 2 O ∆ i H Na + Cs + (aq) ∆ r H 0 (n) / / Cs + nH 2 O ∆ i H Cs + Na + (aq) Na + ∞H 2 O + Cs + (aq) ∆H • / / Cs + ∞H 2 O + Na + (aq) (2)
where ∆ i H is the enthalpy released upon immersion of a clay with a known water content into liquid water. We measure this immersion enthalpy by microcalorimetry as a function of counterion and water content, and combine it with ∆H • for the exchange in solution to obtain the ionic exchange enthalpy for well-defined water content as:

∆ r H 0 (n) = ∆H • -∆∆ i H (3)
where ∆∆ i H = ∆ i H Cs -∆ i H Na . We will explain in section 3 how we determine experimentally the immersion enthalpies for clays with n water molecule per cations.

Origin of the ionic exchange enthalpy

According to the current paradigm for ionic exchange in clays, specific interactions between the ions and the clay surface or/and the hydration properties of ions in the clay interlayer are the driving force for the ionic exchange. Only a single analysis of the ionic exchange free energy by Teppen and Miller [START_REF] Teppen | Hydration energy determines isovalent cation exchange selectivity by clay minerals[END_REF] emphasized the importance of also considering the contribution of the aqueous phase. There are indeed two contributions to the standard reaction enthalpy: on the one hand the ion-clay and ion-water interactions in the clay phase, on the other hand the ion-water interactions in the aqueous phase. This can be understood in terms of the following thermodynamic cycle, in which we introduce the exchange reaction with ions in the gas phase, where there are no interactions, used as a reference state.

Na + nH 2 O + Cs + (g) ∆ h H Cs ∆ r H 0 g (n) / / Cs + nH 2 O + Na + (g) ∆ h H Na Na + nH 2 O + Cs + (aq) ∆ r H 0 (n) / / Cs + nH 2 O + Na + (aq) (4)
This thermodynamic cycle involves the hydration enthalpy of both cations ∆ h H Na and ∆ h H Cs , the standard ionic exchange reaction with the gas phase ∆ r H 0 g (n) and that with the aqueous phase ∆ r H 0 (n). From this cycle we deduce the relation :

∆ r H 0 g (n) = ∆ r H 0 (n) + ∆∆ h H (5)
where ∆∆ h H = ∆ h H Cs -∆ h H Na . After determining ∆ r H 0 (n) by microcalorimetry (see section 3), we will be in a position to estimate the clay contribution ∆ r H 0 g (n) to the enthalpy for the exchange of ions between clays of fixed water content, and thus to investigate this quantity for well-defined reference states. In parallel to this indirect experimental determination we will also compute directly ∆ r H 0 g (n) using molecular simulation (see section 4).

MICROCALORIMETRY EXPERIMENTS

Materials

Homoionic Na-and Cs-montmorillonite clays (MNa and MCs, respectively) were prepared from ANDRA's reference Wyoming bentonite (MX80). The bentonite was first purified :

after removal of impurities (quartz, feldspars, etc.) by sedimentation of a suspension in deionized water, the clay solution was centrifuged several times and the upper part containing particles < 2µm was collected. Then homoionic MNa was obtained by ion-exchange :

the clay was suspended in 1 M aqueous NaCl and dialyzed to remove the excess salt. Conductivity measurements were used to check the presence or absence of salt in the dialyte.

This ion-exchange procedure was repeated twice. The absence of Cl -ions in the dialyte was confirmed by the AgNO 3 test. The homoionic Cs-montmorillonite was obtained from the Na-montmorillonite following the same procedure with 1 M aqueous CsCl. The purified clays were then dried under vacuum at 100 • C, and placed in a silicagel desiccator where their mass remained constant. Their purity was checked by X-ray diffraction. Dried samples were then placed in a desiccator containing saturated saline solutions, which control the relative humidity (RH). With saturated solutions of K 2 CO 3 and KCl, the relative humidity is of 43% and 85%, respectively. The mass of the samples is stabilized within a few days. II. Three to six immersions were carried out for each montmorillonite sample, with given counterion and hydration state. The reproducibility of the measurements is very good for RH=43% and 85%, however there is more variability for the samples prepared at RH=0%. This is due to the fact that the immersion enthalpy varies significantly with the water content when the latter is small [START_REF] Cases | Mechanism of adsorption and desorption of water vapour by homoionic montmorillonite 1. The sodium exchanged form[END_REF][START_REF] Bérend | Mechanism of adsorption and desorption of water vapour by homoionic montmorillonite 2. The Li + , Na + , K + etc. exchanged forms[END_REF], and that the water content of the driest samples can increase during the preparation of the microcalorimetry experiment. Exchange enthalpies for clays dispersed in water at 298 K were obtained as explained in ref. [START_REF] Morel | Effect of temperature on the retention of Cs + by Na-montmorillonite: microcalorimetric investigation[END_REF]. The exchange enthalpy depends on the degree of exchange α, with values ranging from -15 ± 0.5 kJ/mol of cation for α → 0 (which corresponds to traces of Cs + in a MNa) to -4.1 ± 0.2 kJ/mol for α → 1. The standard reaction enthalpy is obtained by integrating these values between α = 0 and 1, with the result ∆H 0 = -7.5 ± 1.0 kJ/mol. The standard state corresponds to clay in solution at infinite dilution, where the interlayer in equilibrium with the solution has an unknown water content.

HR (%) ∆ i H Na (kJ.mol -1 ) ∆ i H Cs (kJ.mol -1 ) 0 -37±4 -21±4 43 -25±1 -5±1 85 -8±1 -4±1

Immersion enthalpy as a function of water content

We have shown using the thermodynamic cycle (2) how the exchange enthalpy ∆ r H 0 (n) can be deduced from the exchange enthalpy in solution ∆H Then we can deduce the values for n = 6 (monolayer), since we have ∆ i H Na (RH = 43%) = 0.5∆ i H Na (n = 1) + 0.5∆ i H Na (n = 6). This yields ∆ i H Na (n = 6) = -13 ± 6 kJ/mol, and a similar calculation gives ∆ i H Cs (n = 6) = +6 ± 4 kJ/mol. Note that the main contribution to the uncertainty comes from the estimation of the water content, and not from the microcalorimetric measurements of immersion enthalpies. The positive sign for the immersion of MCs in the monohydrated state is consistent with the fact that MCs does not swell. This also holds for K-montmorillonite and a positive sign was found in that case by molecular simulation results [START_REF] Liu | A thermodynamic understanding of clay-swelling inhibition by potassium ions[END_REF].

Complete exchange enthalpy

From the immersion enthalpies for n = 1 and 6, which are stable states for both counterions, and the exchange enthalpy in solution ∆H • we now obtain the standard reaction enthalpy using Eq. ( 3). This yields ∆ r H 0

MOLECULAR SIMULATION

Molecular simulation allows to compute the energy and enthalpy of a system from the interactions between the atoms, and thus gives access to differences in thermodynamic quantities that are not easily accessible by direct measurement. For example, the reaction energy ∆ r U 0 g (n) of the ion-exchange reaction between a) the clay at a well-defined water content of n water molecules per cations, and b) ions in the gas phase (which can be used as a reference state for the ions) is obtained by substracting the energy of the MNa system to that of the MCs system (there are no interactions for isolated ions in the gas phase). Similarly, the hydration enthalpy difference ∆∆ h H for Na + and Cs + is obtained by computing the enthalpy of the ions in water (again, there is no contribution of the gas phase). Ionic exchange free energies can also be computed using the thermodynamic integration method [START_REF] Kirkwood | Statistical mechanics of fluid mixtures[END_REF][START_REF] Frenkel | Understanding Molecular Simulations, From Algorithms to Applications[END_REF], as will be explained in section 4 4.2. thickness of 6.54 Å (distance between surface oxygen atoms from the same layer). Their atomic structure was taken from X-ray diffraction measurements [START_REF] Maegdefrau | Die Kristallstruktur des Montmorillonits[END_REF]. The interlayer space contains the counterions and n water molecules per cation. Periodic boundary conditions in all directions are used so that the simulation box corresponds to infinite clay layers (no edges). A snapshot of the simulation box is represented in Fig. 2.

Interactions between atoms are modelled using pairwise potentials :

V ij = q i q j 4π 0 r ij + 4ε ij σ ij r ij 12 - σ ij r ij 6 (6)
with q i the (partial) charge of atom i, and the pair Lennard-Jones (LJ) parameters ε ij and σ ij are calculated from the individual ε ii and σ ii using Lorentz-Berthelot mixing rules. V ij contains an electrostatic (Coulomb) term, and a 6-12 Lennard-Jones term. This force field is completely determined by the atomic parameters {q i , ε ii , σ ii }. Water molecules were described using the rigid SPC/E model [START_REF] Berendsen | The missing term in effective pair potentials[END_REF]. The ions LJ parameters are taken from Åqvist [START_REF] Åqvist | Ion-water interaction potentials derived from free energy perturbation simulations[END_REF].

They were developped to reproduce the experimental hydration energy when used in combination with the SPC water model (SPC/E is an extension of this model). The clay layers are treated as rigid bodies. The partial charges and LJ parameters for clay atoms are the one from Smith [START_REF] Smith | Molecular computer simulations of the swelling properties and interlayer structure of cesium montmorillonite[END_REF]. In this force field the LJ parameters for surface oxygen atoms are identical to that of SPC/E water oxygens. All partial charges and individual Lennard-Jones parameters are summarized in Electronic Annex 1. Long-range electrostatic interactions were computed using Ewald summation, and a cut-off of 8 Å was used for the short-range LJ interactions.

We performed Monte-Carlo simulations in the N P T ensemble, i.e. at fixed number of particles, pressure P = 1 bar, and temperature T = 298 K. Thermodynamic quantities such as energy and enthalpy were computed by averaging their mechanical expressions U = i<j V ij and H = U +P V over microscopic configurations sampled from the N P T ensemble. For the clay systems we did not allow the rotation of the layers relative to each other, but only translational moves in the direction perpendicular to the surfaces, and in both directions along the surfaces. Only the pressure in the direction perpendicular to the clay surfaces is imposed (N P z T ensemble). This is necessary to preserve the symmetry of the clay particle.

We performed simulations for both Na + and Cs + counterions, and water contents of 1 (dry state), 6 (monolayer) and 12 (bilayer) water molecules per cations. Ions and solvent molecules are inserted randomly in the interlayers, and the system is first equilibrated at 1000 K for 10 5 steps. The temperature is then gradually decreased to 298 K. After 10 7 equilibration steps at this temperature, sampling of the energy and enthalpy is achieved for 5 10 7 to 10 8 steps. For the lowest hydration states, simulated annealings (i.e. repeating the above procedure via high temperatures) are used to ensure that the simulation correctly samples the whole phase space. The equilibrium interlayer distances obtained by N P z T simulation are ≈ 10.8, 12.3, and 15.3 Å for Na + with n = 1, 6 and 12, respectively, and 12.2 and 12.9 Å for Cs + with n = 1 and 6, respectively. These results are in good agreement with X-ray diffraction data, except for the dryest states for which smaller distances have been reported (see e.g. [START_REF] Bérend | Mechanism of adsorption and desorption of water vapour by homoionic montmorillonite 2. The Li + , Na + , K + etc. exchanged forms[END_REF][START_REF] Ferrage | New insights on the distribution of interlayer water in bi-hydrated smectite from X-ray diffraction profile modeling of 00l reflections[END_REF]). In that case, even a single water molecule leads to a larger distance than for a completely dry state.

Simulations of aqueous ions consisted of a single ion and 252 water molecules in a cubic box. The equilibrium size of the box corresponds to a concentration of about 0.2 M. The fact that the simulation box is charged (there is no anion) is not a problem : it has been shown that it is equivalent to the same system with a uniform neutralizing background, and that there is only a shift in the electrostatic energy [START_REF] Sprik | Electronic structure and solvation of copper and silver ions: A theoretical picture of a model aqueous reaction[END_REF]. For a cubic box of size L, ∆E = -αQ 2 /8π 0 L, with Q the total charge of the box and α a (Madelung) constant. Thus the shift is identical for the simulations with Na + or Cs + and compensates when considering differences in energy (or enthalpy).

In addition to Monte-Carlo simulations in the N P T ensemble, we also performed molec-ular dynamics simulations of the same systems in the N V T ensemble (at the equilibrium volume determined by the N P T simulations), using a modified version of the DLPOLY software package. These simulations were used to compute Helmholtz free energy differences by the thermodynamic integration technique [START_REF] Kirkwood | Statistical mechanics of fluid mixtures[END_REF][START_REF] Frenkel | Understanding Molecular Simulations, From Algorithms to Applications[END_REF]. It consists in gradually turning one type of ion into another using a coupling parameter λ ∈ [0, 1] which interpolates between the Hamiltonian V(λ = 0) = V Na of the system with Na + and that V(λ = 1) = V Cs with Cs + . Then the Helmholtz free energy difference between the two states is :

∆F Na→Cs = 1 0 ∂V ∂λ λ dλ (7)
The term in the integral is an ensemble average for a system at a fixed, intermediate value of λ. We have chosen a thermodynamic path in which all the pair LJ parameters vary linearly between that of Na and that of Cs :

ε λ,j = ε Na,j + λ(ε Cs,j -ε Na,j ) (8a) σ λ,j = σ Na,j + λ(σ Cs,j -σ Na,j ) (8b)
where j corresponds to any other atom, and where the extreme pair parameters ε Na,j , ε Cs,j , σ Na,j , and σ Cs,j are obtained from the Lorentz-Berthelot rules. We considered thermodynamic paths in which a single ion is changed from Na + to Cs + , Na + to K + , and K + to Cs + , both in pure water and in clay at a water content of 6 water molecules per cation, corresponding to a water monolayer. For each thermodynamic integration we performed a set of 11 simulations at different λ corresponding to a step ∆λ = 0.1, in which we sampled ∂V/∂λ for at least 100 ps. Fig. 3 shows the variation of the average ∂V/∂λ as a function of λ for the thermodynamic integration from Na + to Cs + in aqueous solution (solid line)

and in clay with n H2O = 6 water molecules per cation (dashed line). Numerical integration between λ = 0 and 1 yields the difference in hydration free energy ∆∆ h F and the enthalpy for the exchange of ions between clay and the gas phase ∆ r F 0 (6), respectively.

Free energy for the exchange of traces

We first present the results of the thermodynamic integration approach, and use them to validate our choice of parameters to describe the energetic properties of clay, ions and water. In the aqueous ions simulations, the resulting free energy difference corresponds to the hydration free energy difference ∆∆ h F for Na + , K + and Cs + . The results are reported in table IV.

∆∆ h F (kJ.mol -1 ) Simulation Burgess Marcus

Na + → Cs + +124±2 +127 +117 Na + → K + +72±2 +74 +72 K + → Cs + +53±2 +53 +45
TABLE IV: Hydration free energy difference as obtained by thermodynamic integration.

Our simulation results are compared to the values given in the thermodynamic databases compiled by Burgess [START_REF] Burgess | Metal Ions in Solution[END_REF] and Marcus [START_REF] Marcus | The thermodynamics of solvation of ions. part 2. The enthalpy of hydration at 298.15[END_REF].

We first notice that there is a good thermodynamic consistency between the results, i.e.

that ∆∆ h F (Na → Cs) = ∆∆ h F (Na → K) + ∆∆ h F (K → Cs). Furthermore, the simulation results are in good agreement with the experimental data for ∆∆ h G ∼ ∆∆ h F (the difference P ∆V between these two is only a few kJ/mol) reported in [START_REF] Marcus | Thermodynamics of solvation of ions. part 5. Gibbs free energy of hydration at 298[END_REF] : +117, +72 and +45 kJ/mol, and even better with the values of ref. [START_REF] Burgess | Metal Ions in Solution[END_REF] : +127, +74 and +53 kJ/mol. The excellent agreement with the latter is not surprizing, since the Lennard-Jones parameters for the cations were calibrated by Åqvist to reproduce these values when combined to the SPC water model [START_REF] Åqvist | Ion-water interaction potentials derived from free energy perturbation simulations[END_REF], a variant of which (SPC/E) is used in this study. Therefore the force field used for our simulations reasonably represents the interactions between ions and water.

Transformation ∆ r F 0 g (6) (kJ.mol -1 )

Na + → Cs + +120±2 Na + → K + +68±2 K + → Cs + +51±2
TABLE V: Free energy difference obtained by thermodynamic integration for ions in clay with 6 water molecules per cation.

The results for the thermodynamic integration for the exchange of a single ion in a MNa with 6 water molecules per counterions are reported in table V. We observe again a good thermodynamic consistency between the results. In addition, all values are positive and relatively large, meaning that the replacement of an alkaline ion by a larger one in a MNa is clearly thermodynamically unfavourable. More precisely, the value of ∆ r F 0 g (6) corresponds to the exchange of ions between a MNa and the gas phase (see below). The value for the Na + → Cs + exchange is in good agreement with the conclusion drawn by Teppen and Miller [START_REF] Teppen | Hydration energy determines isovalent cation exchange selectivity by clay minerals[END_REF] from experimental data (+117 kJ/mol), and the one for the K + → Cs + exchange is in good agreement with both their simulation results (+47 kJ/mol) and their conclusions drawn from experimental data (+54 kJ/mol). Since their simulation results were obtained using a different force field to represent ions and clay atoms, and a slightly different method to compute the free energy difference (free energy perturbation, see e.g. [START_REF] Frenkel | Understanding Molecular Simulations, From Algorithms to Applications[END_REF]), the similar conclusion to which we arrive, namely that the clay phase contributes unfavourably to the ionic exchange free energy, seems to be very robust. We can also conclude that the force field that we use gives a reasonable description of the thermodynamic properties of clay interlayers.

In the above-mentioned study, the authors considered the complete exchange from a Kclay to a Cs-clay, whereas here we consider the exchange of ionic traces in a MNa. This suggests that for a given water content, the short-range ion-clay and ion-water interactions are relatively independent of the nature of the other ions in the interlayer. Such an observation has already been made in simulations of Cs + in bihydrated Na-montmorillonite [START_REF] Marry | Microscopic simulations of interlayer structure and dynamics in bihydrated heteroionic montmorillonites[END_REF]: the structure of the inner-sphere Cs-surface complex was very similar to the one observed in monohydrated Cs-montmorillonite. It is therefore not surprising that energetic properties for the exchange of traces of Cs + in MNa show the same behaviour as for the complete exchange from homoionic MNa to homoionic MCs.

Complete exchange enthalpy

The enthalpy for the complete Na + → Cs + exchange is reported in table VI, as a function of the number of water molecules per cations: n H2O = 1 corresponds to the dry state, 6 to the monolayer state, and 12 to the bilayer state. Note that the latter is not experimentally stable in the Cs case [START_REF] Bérend | Mechanism of adsorption and desorption of water vapour by homoionic montmorillonite 2. The Li + , Na + , K + etc. exchanged forms[END_REF]. As for the free energy, ∆ r H 0 g (n) corresponds to the exchange of ions between clay and the gas phase. These values are all positive and large. Furthermore, they increase as the water content increases. All these observations will be discussed and compared to experimental results in the next section. The results for the contribution of the clay phase to the reaction enthalpy ∆ r H 0 g (n), computed from Eq. ( 5) using the values of ∆ r H 0 (n) determined in section 3 3.4 and the experimental value of ∆∆ h H = +133 kJ/mol [START_REF] Marcus | The thermodynamics of solvation of ions. part 2. The enthalpy of hydration at 298.15[END_REF], are reported in table VII. The same analysis using the experimental immersion enthalpies of Bérend et al. [START_REF] Bérend | Mechanism of adsorption and desorption of water vapour by homoionic montmorillonite 2. The Li + , Na + , K + etc. exchanged forms[END_REF] yields +93.5 ± 7 kJ/mol for n = 1, and +116.5 ± 15 kJ/mol for n = 6. VI. The latter can be seen as more direct estimates of ∆ r H 0 g (n) since they are based on the computation of the internal energy from the microscopic interactions between atoms. The agreement between experimental and simulation results, though not quantitative for the lowest water content, is good. The uncertainty on the experimental results reported in table VII does not allow to identify the variations of ∆ r H 0 g (n) with water content, but the results based on the experiments of ref. [START_REF] Bérend | Mechanism of adsorption and desorption of water vapour by homoionic montmorillonite 2. The Li + , Na + , K + etc. exchanged forms[END_REF] suggest an increase of ∆ r H 0 g (n) with increasing water content, as observed in simulations.

n H2O ∆ r H 0 g (kJ.mol -1 ) 1 +109.5 ± 9 6 +106.5 ± 11 ∞ +125.5 ± 1
As mentioned in the introduction, the current paradigm for ionic exchange considers that its driving force is to be found in specific interactions between the ions and the clay surface, or in the hydration properties of the ions in the clay interlayer, that would be more favourable to Cs + than to Na + . However, our simulation results for the free energy clearly support the conclusion of Teppen and Miller [START_REF] Teppen | Hydration energy determines isovalent cation exchange selectivity by clay minerals[END_REF] that the contribution of the clay phase to the ionic exchange free energy is in fact very unfavourable (∆G 0) and that it is the hydration free energy difference, i.e. the contribution of the water phase, that leads to an overall exchange of Na + for Cs + . The driving force for the Na + to Cs + exchange in montmorillonite is thus the "hydrophobicity" of Cs + compared to Na + , and not its affinity for the clay surfaces.

The same argument also explains the selectivity observed for alkaline earth cations in the order Ba 2+ > Sr 2+ > Ca 2+ > Mg 2+ , as was already suggested in ref. [START_REF] Teppen | Hydration energy determines isovalent cation exchange selectivity by clay minerals[END_REF] as an extension of their analysis of alkaline cations.

Both our experimental and simulation results support the conclusion that the same holds for the ionic exchange enthalpy: replacing Na + ions by Cs + in the interlayer of montmorillonite clays is an endothermic process, and the overall exchange is exothermic only because it is dominated by the exothermic replacement of Cs + by Na + in the aqueous phase. An advantage of considering enthalpies is that they are measured directly by microcalorimetry, whereas experimental free energies in solution are deduced from exchange constants, and immersion or swelling free energies are not easily measured. Therefore enthalpy lends itself better to the direct comparison between experimental and simulation results.

Implications for the exchange of traces

We mentioned in the experimental section that the exchange enthalpy depends on the degree of exchange α. In the trace regime (α → 0), the value measured in solution by microcalorimetry is -15 ± 0.5 kJ/mol [START_REF] Morel | Effect of temperature on the retention of Cs + by Na-montmorillonite: microcalorimetric investigation[END_REF]. We could not obtain a direct estimate by simulation using the same approach as before, because the change in enthalpy associated with the replacement of one Na + by one Cs + in a simulation box containing only Na + counterions is small compared to the absolute value of the enthalpy in both systems, and even to the equilibrium fluctuations of the enthalpy. Therefore the uncertainty on the difference obtained by this approach is too large for the result to be meaningful. The evaluation of the free energy did not suffer from this problem, since it was based on a quantity ( ∂V ∂λ ) involving only the exchanged ion.

However, the enthalpy for the exchange of traces can be estimated under the assumption that ions in the interlayer behave independently of the nature of the surrounding cations.

This assumption is reasonable and consistent with previous molecular simulation results.

In [START_REF] Marry | Microscopic simulations of interlayer structure and dynamics in bihydrated heteroionic montmorillonites[END_REF], it was shown that Cs + ions in a bihydrated heteroionic Na/Cs-clay adopt the same coordination mode as in the -experimentally unstable-bihydrated homoionic MCs : three surface oxygen atoms were included in the first coordination shell of the ion, as also observed on monohydrated homoionic MCs. The value of the enthalpy can then be approximated by ∆ r H 0 (12) = ∆ r H 0 g (12) -∆∆ h H = +113 -133 = -20 ± 4 kJ/mol. Since ∆ r H 0 g (n) increases with the water content, the choice of the -thermodynamically stable-bilayer state provides a lower bound on the value for suspended clays (for which the larger water content is unknown). This estimate is in good agreement with the value measured by microcalorimetry.

Finally, we can compare these results to the enthalpy deduced from the measurements of clay selectivities at various temperatures using the Van't Hoff relation. Liu et al. have reported a value of -18 ± 2 kJ/mol [START_REF] Liu | Effect of temperature on Cs + sorption and desorption in subsurface sediments at the Hanford site[END_REF], and Tertre et al. -19 ± 5 kJ/mol [START_REF] Tertre | Experimental sorption of Ni 2+ , Cs + and Ln 3+ onto a montmorillonite up to 150 • C. Geochim[END_REF]. The assumptions on Cs + in MNa and on the water content seem therefore to be reasonable.

The fact that independent routes (direct measurement or simulation on the one hand and selectivity measurements at different temperatures on the other hand) lead to the same value of the trace exchange enthalpy give further support to the above discussion. Since the ionic exchange reaction is exothermic, it is a priori unfavoured by an elevation of temperature.

However the relatively low value of the exchange enthalpy implies that temperature has only a weak influence on the retention of Cs + by clays, as was already concluded in [START_REF] Morel | Effect of temperature on the retention of Cs + by Na-montmorillonite: microcalorimetric investigation[END_REF].

CONCLUSION

We have studied the origin of the ionic exchange enthalpy in montmorillonite clays using microcalorimetry measurements and molecular simulation. We have first determined the standard reaction enthalpy for well-defined interlayer water contents. We have then shown by a detailed analysis based on thermodynamic cycles that replacing Na + ions by Cs + in the interlayer of montmorillonite clays is an endothermic process, and that the overall exchange is exothermic only because it is dominated by the exothermic replacement of Cs + by Na + in the aqueous phase. This conclusion supports the one of a recent study of the ionic exchange free energy [START_REF] Teppen | Hydration energy determines isovalent cation exchange selectivity by clay minerals[END_REF] and contradicts long-held views on the role of ion-clay interactions in determining the ionic exchange thermodynamics. This calls for a paradigm shift for the driving force of this exchange: The driving force is the "hydrophobicity" of Cs + compared to Na + and not its affinity for clay surfaces. This analysis has a broader range than the case of clays, and might explain the relative homogeneity of specific effects observed in many contexts according to "Hofmeister series" [START_REF] Kunz | The present state of affairs with Hofmeister effects[END_REF]: One should not forget the role of the hydration energy (i.e. the contribution of the water phase), even if specific interactions with the substrates come into play. The present study does not tell us about the role of the nature of the counterions on the kinetics of the exchange. This would require to take into account clay particle edges explicitely [START_REF] Rotenberg | Water and ions in clays : Unraveling the interlayer/micropore exchange using molecular dynamics[END_REF]. Moreover, particle edges provide other sites for the retention of cations [START_REF] Schlegel | Sorption of metal ions on clay minerals I. Polarized EXAFS evidence for the adsorption of Co on the edges of hectorite particles[END_REF][START_REF] Schlegel | Adsorption mechanisms of Zn on hectorite as a function of time, pH, and ionic strength[END_REF][START_REF] Dahn | Structural evidence for the sorption of Ni(II) atoms on the edges of montmorillonite clay[END_REF]. In particular, the understanding of the effect of pH on the sorption of cations on these edge sites might benefit from the combined theoretical and experimental approach followed in the present paper.

3. 2 .

 2 Microcalorimetric measurementsImmersion enthalpies of MNa and MCs at different relative humidities have been measured at 298 K using the Microsolution 2265 unit of the isothermal microcalorimeter TAM 2277 (Thermometric -TA Instruments). Samples of 5 to 25 mg are placed in three microcartridges which fit tightly in the measurement cell. The immersion process is started by the successive fall of the cartridges into 15 mL of water placed in a 20 mL vessel stirred by a propeller at 60 rpm. Figure1illustrates the typical shape of the resulting thermogram. The return to baseline after immersion of a clay sample takes a time reflecting the time constant of the microcalorimeter, of the order of 10 minutes, which is longer than the hydration process itself. Immersion enthalpies (in mJ/mg) are computed by integrating the thermogram peaks using the Digitam software (Thermometric). They are given in kJ/mol of cation in table

  (n = 1) = -37±4 kJ/mol and ∆ i H Cs (n = 1) = -21±4 kJ/mol.

  (1) = -7.5 -[-21 + 37] = -23.5 ± 9 kJ/mol and ∆ r H 0 (6) = -7.5 -[6 -(-13)] = -26.5 ± 11 kJ/mol. The main contribution to the uncertainty comes again from the estimate of the water content for a given relative humidity, and not from the measurements of enthalpies by microcalorimetry. We repeated the same analysis using the experimental immersion enthalpies of Bérend et al.[START_REF] Cases | Mechanism of adsorption and desorption of water vapour by homoionic montmorillonite 1. The sodium exchanged form[END_REF]. In that case Eq. (3) yields -39.5 ± 7 kJ/mol for n = 1, and -16.5 ± 15 kJ/mol for n = 6. Despite the relatively large uncertainties, both sets of data lead to consistent results : They clearly indicate that the ionic exchange is exothermic. Note that the ionic exchange enthalpy for n = 1 and 6 are more negative than the value obtained in solution (∆H • = -7.5 kJ/mol of cation). This underlines the need to consider carefully the water content in clays when analysing thermodynamic quantities.

1 .

 1 Origin of the ionic exchange enthalpy

FIG. 1 :FIG. 2 :

 12 FIG. 1: Thermogram obtained upon the successive immersions of three MNa (RH=43%) samples in water. Peak c: calibration. Peak 1: 13.0 mg yields ∆ i H Na = 23.9 mJ/mg. Peak 2: 17.0 mg yields ∆ i H Na = 23.2 mJ/mg. Peak 3: 23.4 mg yields ∆ i H Na = 23.5 mJ/mg.

Figure 2 29Figure 3

 23 Figure 2

Table I

 I

	i.e. in equilibrium with an aqueous phase, but in which the interlayer water content is
	unknown and might even differ between the initial and final states. Here we study ∆ r H 0 (n)
	(a) by combining thermodynamic data on ionic exchange for clays dispersed in solution and
	on immersion of clays with a known water content, as measured by microcalorimetry, and
	(b) by computing directly enthalpy differences using molecular simulation.
	2.2. Thermodynamic states and water content
	In order to define reaction quantities, it is necessary that the number of water molecules
	remains constant. Experimentally, however, the clay water content depends both on the
	nature of the counterion and on the water chemical potential. Well-defined water contents
	can be obtained by equilibrating dry clay powders with atmospheres of controlled relative
	humidity [9, 10]. It is well known that only a few hydration states are thermodynamically
	stable : they correspond approximately to 6-7 H 2 O per cation (monolayer), both for MNa-
	and MCs, and 12-14 H 2 O per cation (bilayer), only for MNa. Intermediate water contents
	correspond to mixtures of mono-and bilayers. For very low relative humidities, a lower
	water content can be obtained, down to 1.4 H 2 O per cation for MNa, and 1.0 H 2 O per
	cation for MCs.
	Moreover, the exchange enthalpy can be measured directly only in the case of a clay sus-
	pension, i.e. in equilibrium with an aqueous solution, symbolized by the ∞H 2 O superscript
	in the cycle below. In that case, the water content inside the clay phase is unknown and
	the measured ∆H • , expressed per mole of cation, is not a reaction quantity because the
	water content might be different for MNa and MCs. In order to obtain experimentally the
	reaction enthalpy ∆ r H 0 (n) corresponding to n H 2 O per cation, we introduce the following
	thermodynamic cycle:
	summarizes the different
	thermodynamic quantities and the notations used in this paper. All previous experimental
	studies of the thermodynamics of ionic exchange were conducted on clays dispersed in water,

TABLE I :

 I Notations used for thermodynamic quantities.

TABLE II :

 II Experimental immersion enthalpies for Na-and Cs-montmorillonite, as a function of the relative humidity (RH). Results are given per mole of cation.

  • and the immersion enthalpies ∆ i H Na and ∆ i H Cs via Eq. 3. These immersion enthalpies must be determined as a function of the water content n. As mentioned in section 2, the experimental hydration states correspond to clays in equilibrium with vapor at a given relative humidity. Only a few hydration states are thermodynamically stable and most conditions correspond to mixtures of these states. TableIIIreports the proportion of each state as a function of the relative humidity,

	RH (%)		Na			Cs
		dry (%) mono (%) bi (%) dry (%) mono (%) bi (%)
	0	100	0	0	100	0	0
	43	50	50	0	40	60	0
	85	0	20	80	20	80	0

as estimated by Bérend et al. for Na-and Cs-Wyoming montmorillonite (Clay Spur 26 from Ward's Natural Science)

[START_REF] Bérend | Mechanism of adsorption and desorption of water vapour by homoionic montmorillonite 2. The Li + , Na + , K + etc. exchanged forms[END_REF]

.

TABLE III :

 III Although obtained for samples of a different montmorillonite, prepared under slightly different conditions, these results can be used together with table II to estimate the value of the immersion enthalpy ∆ i H(n) for n = 1 and 6. The values for RH=0% correspond to the values for n = 1, thus ∆ i H Na

Proportion of hydration states: dry (1-2 water molecules per cation), monolayer (6-7) and bilayer

[START_REF] Ferrage | New insights on the distribution of interlayer water in bi-hydrated smectite from X-ray diffraction profile modeling of 00l reflections[END_REF][START_REF] Weiss | Variations in interlayer cation sites of clay minerals as studied by 133 Cs mas nuclear magnetic resonance spectroscopy[END_REF][START_REF] Trausch | Separation of components of a 1 H NMR composite signal by nutation experiments under low amplitude radio-frequency fields. application to the water signal in clays[END_REF]

as a function of the relative humidity (RH) of the atmosphere in equilibrium with the clay. These values are estimations from Bérend et al.

[START_REF] Bérend | Mechanism of adsorption and desorption of water vapour by homoionic montmorillonite 2. The Li + , Na + , K + etc. exchanged forms[END_REF]

.

  (Al 3.25 Mg 0.75 ) O 20 (OH) 4 X 0.75 with X the counterion Na + or Cs + . Although no experiments for K + montmorillonite were undertaken in the present study, we also performed molecular simulations for this counterion. The simulation box of lateral dimensions 20.72 × 17.94 Å2 contains two clay layers, both with 4 × 2 unit cells and a

	4.1. System
	The	simulated	clay	system	has	the	idealized	composition
	Si 8

TABLE VI :

 VI Clay contribution to the standard enthalpy for the exchange reaction between a Na-Clay with n water molecules per cation and a Cs-Clay with n water molecules per cation (see the thermodynamic cycle 4). The values for n = 1, 6 and 12 correspond to the dry, monolayer and bilayer states, respectively -even if the latter is not experimentally stable in the Cs case. These simulation results should be compared to the experimental ones of table VII.

TABLE VII :

 VII Clay contribution to the standard enthalpy for the exchange reaction between a Na-Clay with n water molecules per cation and a Cs-Clay with n water molecules per cation (see the thermodynamic cycle 4). Results are obtained from the experimental immersion enthalpies, enthalpy for the exchange in solution and the proportions of each hydration states. Experimental results therefore clearly indicate that interactions in the clay interlayer contribute unfavourably to the replacement of Na + cations by Cs + . This conclusion is further supported by the simulation results of table

Acknowledgments

B.R. acknowledges financial support of the Agence Nationale pour la Gestion des Déchets

Radioactifs (ANDRA). The authors are grateful to ANDRA for providing the MX80 bentonite samples and acknowledge financial support from the GNR PARIS. The authors thank