Keywords: adsorption, xenon, nanoporous materials, metal-organic frameworks, thermodynamics

Breathing Transitions in MIL-53(Al) Metal-Organic Framework Upon Xenon Adsorption

Anne Boutin, Marie-Anne Springuel-Huet, Andrei Nossov, Antoine Gédéon, Thierry Loiseau, Christophe Volkringer, Gérard Férey, François-Xavier Coudert and Alain H. Fuchs* Porous Metal-Organic Frameworks (MOFs) are a topical class of materials that display an extremely large range of crystal structures and host-guest properties, potentially giving them a major impact in in many areas of science and technology [1a-e] . A growing number of these materials show exceptional guest-responsive behaviors upon gas adsorption, due to the flexibility of their hybrid organicinorganic frameworks [1e,2a-d] . The MIL-53 materials family [3a,b] has recently attracted a lot of attention on account of its massive flexibility and the occurrence of a double structural transition ("breathing") upon adsorption of some gases (CO 2 , H 2 O) but not others (H 2 , CH 4 ) [3c] , at room temperature. It was also reported very recently that these transitions occurred upon gas phase adsorption of several n-alkanes [3d] or xylene isomers [3e] . It has been suggested that apolar species such as methane or rare gases could not induce breathing because of their too low adsorption enthalpies in MIL-53 materials [3c] . Here, we report a xenon adsorption study in MIL-53(Al) in the temperature range 195K-323K in which we clearly observe breathing transitions in the measured adsorption isotherms. With the use of a recently developed thermodynamic model, we predict for the first time a phase diagram for xenon adsorption in MIL-53. The present method is very general and could be applied to any other guest-flexible host system.

The MIL-53(Al) framework topology is formed of unidimensional chains of corner-sharing AlO 4 (OH) 2 octahedra linked by 1,4-benzenedicarboxylate (BDC) ligands, which results in linear lozenge-shaped channels large enough to accommodate small guest molecules [3b,f] . The breathing transitions mentioned above are taking place between two forms of this system: a large pore (lp) structure and a narrow pore (np) one (Figure S1 in the Supporting Information). The MIL-53 materials family has been the subject of numerous structural chemistry studies, but quite less of thorough thermodynamics analysis [1c,e] .

In Figure 1, we report experimental xenon adsorption isotherms in MIL-53(Al) at various temperatures. The low-temperature isotherms (195 and 220 K) display a first step at ~3 molecules per unit cell and reach a high-pressure plateau at ~8 to 9 molecules/uc. These values are consistent with the np and the lp structures respectively. The 220 K isotherm plotted on a log scale is particularly interesting (Figure S2 in the Supporting Information). It demonstrates the existence of two well-defined transitions which, given the large amount of previously published data on MIL-53, can be confidently ascribed to the lp-np and the np-lp transitions respectively, even though structural data for the {Xe,MIL-53(Al)} system are not yet available. In the 195 K adsorption isotherm, the step corresponding to the np-lp transition is clearly visible in Figure 1, but a careful examination of the low pressure regime shows no sign of the lp-np transition. This could either be due to the fact that it occurs at a very low and hardly detectable pressure, or to the fact that the starting empty material is already in the np structure at this temperature. A full thermodynamic analysis will show, further below, that the second guess is the right one.

The 273 and 292 K isotherms clearly display a phase transition, with a hysteresis loop, which can be attributed to the lp-np transition, given that the high pressure plateaus (~3 molec/uc) correspond to the filling of the np structure. The reopening of the MIL-53(Al) structure may take place at higher pressure, but was beyond the reach of the present experiments. Finally, the 323 K isotherm shows no sign of phase transition in the accessible pressure range, and the framework is thus believed to remain in the lp form, since it is known as the most stable state at room temperature and above. We conclude from this first part of the study that, unexpectedly, adsorption of a rare gas can induce structural breathing in MIL-53.

We now turn to the thermodynamic analysis of the data reported in Figure 1. We use the so-called osmotic thermodynamic ensemble which is the appropriate one to describe the adsorption of a fluid in a flexible porous material [4a] . For materials exhibiting clear structural transitions between different metastable framework structures (as opposed to the phenomenon of progressive, continuous swelling for instance), we demonstrated in an earlier work that the use of an "osmotic sub-ensemble" adequately describes the equilibrium between host structures upon fluid adsorption [4a,b] . This model was successfully applied to understand the presence or absence of breathing effects in MIL-53(Al) upon CO 2 , CH 4 or linear alkanes adsorption at room temperature [4b,c] .

As in our previous studies, we use Langmuir fits of the experimental isotherms as approximations to the "rigid host" isotherms in both lp and np structures. The Langmuir fits are shown in Figure S3 and S4 and the Langmuir parameters used are given in Table S1. We have used the 273 and 292 K stepped isotherms to determine the transition enthalpy and entropy of the empty host material (see reference 4b for calculation method), and found: ΔH host = H lp -H np = 15 kJ.mol -1 and ΔS host = S lp -S np = 74 J.K -1 .mol -1 . The transition enthalpy value is in fair agreement with the very recent estimation of ΔH host from Devautour-Vinot et al [5a] who combined TGA and DSC water desorption analysis of three MIL materials, and found a value of 20 kJ.mol -1 . Provided that the transition enthalpy and entropy can both be considered as constant in the temperature range investigated, one can simply compute the free energy difference between the empty lp and the np structures. Not unexpectedly, the lp form is predicted to be the most stable one at room temperature. We use the same values to predict that the transition free energy vanishes at T = 203 K. As pointed out earlier [5b] , one of the interests of the osmotic ensemble model lies in the fact that it enables to compute equilibrium thermodynamic data for the bare host material, using thermodynamic adsorption data only. Here we predict that the np structure becomes the most stable one below 203 K. This is fully consistent with the fact that no lp-np transition was observed in the 195 K experimental isotherm. At this temperature, the bare material is already in its np form.

Given the approximations of the model and the uncertainties associated with the different data fits, the predicted equilibrium lpnp transition in the empty MIL-53(Al) is obviously subject to a large uncertainty. A crude estimate is ±20 K, something that we will be able to test once adsorption isotherms will be available for other {guest,MIL-53(Al)} systems over a wide range of temperatures. Nevertheless, the present prediction is in keeping with the neutron scattering study of the bare MIL-53(Al) performed by Liu et al. [5b] who observed a reversible lp-np transition, accompanied by a large hysteresis in the range of 150 to 350 K. The present work thus provides another strong indication that the low-temperature stable form of MIL-53(Al) is the closed np form. There are qualitative (entropic) reasons to believe that the contracted form should be the most stable one at low temperature, but this is at variance with the recent DFT calculations of Coombes et al [5c] who predicted the open lp form to be the most stable one at zero Kelvin, although the authors noted that the computed potential energy surface was very flat. It should also be mentioned that MIL-53(Ga) and MIL-53(Al) were recently observed in the np structure 5d , upon dehydration at 353 and 333 K respectively.

We note in passing that the neutron powder diffraction data of Liu et al [5b] enable to compute, for the bare material, the "magnitude of breathing" that Llewellyn and coworkers [3c] correlated with the van der Waals volume of the guest molecule. This is shown in Figure S5 in the Supporting Information. The limiting value of 39% breathing for the empty material correlates well with the existing data for different adsorbate systems. We predict, from the value of its van der Waals volume, a magnitude of breathing for xenon of ~23.5%.

In the third part of this study, we use our model together with the fits performed on the experimental xenon adsorption isotherms in order to investigate the full temperature-loading phase diagram of MIL-53(Al). By solving the osmotic thermodynamic equations numerically, we determine for each temperature, whether breathing occurs and if so, what are the transition pressures. For this we need to know how the affinity K and the maximum of loading N max Langmuir parameters change with temperature, for each phase. These are obtained through a simple linear fit of the existing data (Figures S6 and S7 in the Supporting Information). All the parameters needed to compute the {Xe, MIL-53} phase diagram are given in Table S2 in the Supporting Information.

The predicted temperature-xenon pressure diagram is shown in Figure 2. The lp phase is found to be stable at high temperature and again at lower temperature, with the existence of an intermediate np phase stability domain for xenon pressure lower than a limiting pressure of ~1.6 bar. This is reminiscent of the re-entrant behavior observed in some liquid crystals [START_REF] De Gennes | The Physics of Liquid Crystals[END_REF] . As noted above, however, the low-temperature stable phase in absence of xenon (zero pressure) is the np phase. Before we turn into the discussion of this diagram, it must be stressed that our model predicts the thermodynamic stability of the phases of a material at full equilibrium and does not take into account hysteresis effects for the time being. Hysteresis was systematically encountered in all reported MIL-53 experiments. It often leads to some complicated mixtures of phases effects. In a recent structural study of MIL-53(Fe), Millange et al [START_REF] Millange | [END_REF] observed intermediate structures which they attributed to a heterogeneous mixtures of crystallites being either in the open or closed form, depending on their contact with the guest molecules. Such a behavior cannot be taken into account in our model, which only describes what would happen in an homogeneous system at complete equilibrium. In Figure S8 of the Supporting Information, we have sketched the same phase diagram with an arbitrary uncertainty of 1 kJ.mol -1 in the osmotic potential, in order to provide an idea of what this diagram would look like in presence of hysteresis effects. Some theoretical efforts are still needed to capture the mechanistic origin of these large hysteresis effects.

The main features of the phase diagram depicted in Figure 2 can be understood as follows. We start from the equilibrium np-lp temperature at zero pressure of 203 K. The initial slope of the transition curve is positive and rather steep. An examination of the osmotic thermodynamic equations shows that it is proportional to the logarithm of (K np /K lp ), the ratio of adsorption affinities in the two structures. This term is clearly positive, since the affinity of the guest adsorbate for the closed form of the framework is expected to be higher than for the open form [4b,c] . To put it very simply, an adsorbed species is interacting with both walls in the closed form of MIL-53(Al), while it mainly interacts with one of the walls in the open structure. The K np /K lp >1 condition thus favors the closed np phase and consequently the phase transition temperature increases with the xenon loading (i.e. the stability domain of the np phase increases with P Xe ). At higher temperature the transition free energy increases and it becomes more and more difficult to maintain the np form as the most stable one. This causes the observed bending of the transition line above 250 K. For obvious entropy reasons, the lp phase will eventually become more stable at high temperature, whatever the xenon loading. This is also true at high pressure. As the adsorbate pressure increases, at any temperature, the lp structure will eventually become more stable than the np one because it can accommodate a higher loading of guest molecules. Since the lp phase is the most stable one at high enough temperature as well as at high adsorbate pressure, one has to conclude that the stability domain of the np phase should be limited in adsorbate pressure (P lim ~1.6 bar in the case of xenon) as seen in Figure 2.

Finally, the loop in the transition line (i.e. the strong decrease in the transition pressure in the low temperature part of the transition line), is due to the fact that the np-lp transition in this domain is taking place at the maximum loading of the np structure. The xenon pressure needed to reach this maximum loading is strongly decreasing at low temperature. Another view of the {Xe,MIL-53(Al)} phase diagram is shown in Figure S9 in the Supporting Information. In this figure we have sketched the transition temperatures as a function of loading, i.e. the number of adsorbed xenon atoms in both the np and lp phase. It shows that the low temperature np-lp transition is indeed taking place at the maximum of loading of the np structure.

The above thermodynamic considerations are very general and are not limited to the special case of xenon adsorption. We expect the main features of the xenon phase diagram to hold for any other {guest,MIL-53(Al)} system. The K np /K lp >1 condition is expected to hold true for all the simple guest molecules that have been investigated so far. This means that there should be a range of temperatures above the equilibrium np-lp transition temperature of the bare MIL-53(Al) material (203 K in our model, subject to the uncertainties described above), where the initially (empty) open structure contracts upon any guest molecule adsorption. The fact that this has not been observed in some cases at room temperature (Ar, CH 4 , …) might simply mean that this temperature is above the transition line maximum. It is worth mentioning here that the breathing effect in {Xe,MIL-53(Al)} would not have been detected by a single room temperature adsorption experiment. The present predictions should now be confronted to experimental structural data.

The use of the osmotic thermodynamic model, combined for the first time with a series of gas adsorption experiments at various temperatures, has allowed to shed some new light on the fascinating phase behavior of the MIL-53(Al) flexible material. We derived a generic temperature-loading phase diagram, and we predict that the breathing effect in MIL-53 is a very general phenomenon, that should be observed in a limited temperature range, whatever the guest molecule, since it is expected that the affinity of an adsorbate for the closed np form of the framework will always be higher than for the open lp structure. We believe that this very general model will provide a useful tool for experimentalists to tune their experimental conditions. Work is in progress to apply this method to the transition behaviors of CO 2 and CH 4 in MIL-53, as well as to other flexible materials.

Experimental Section

The MIL-53(Al) sample was prepared as using the synthesis procedure previously described in the literature, following the DMF exchange intermediate route [3d] . The xenon adsorption-desorption isotherms were carried out at various temperatures on an automatic ASAP 2020 Micromeretics. Prior to isothermal experiments, the sample (about 150 mg) was outgassed at 423 K overnight at a pressure of 10 -7 bar. To obtain the isotherms at different temperatures, the following thermo baths were used: water for 292 K, ice/water for 273 K, dry ice/acetone for 195 K, dry ice/acetonitrile for 220 K. For the isotherm at 323 K, an oven was used. Typically about 30 to 50 data points were measured for each isotherm that took 20 to 40 hours to measure.

Figure 1 .

 1 Figure 1. Experimental xenon adsorption isotherms in MIL-53 (Al). Open symbols: adsorption branch; filled symbols: desorption branch. Left scale: measured adsorbed volumes. Right scale: corresponding number of adsorbed molecules per unit cell. The low-pressure desorption branch at 195 K (hysteresis loop) has not been recorded, for technical reasons.

  Physique CNRS and Univ. Paris-Sud, 91405 Orsay Cedex (France) Dr. M.-A. Springuel-Huet, Dr. A. Nossov, Prof. Dr. A. Gédéon Laboratoire Système Interfaciaux à l'Échelle Nanométrique Université Pierre et Marie Curie 75252 Paris Cedex 05 (France) Dr. T. Loiseau [+] , Dr C. Volkringer, Prof. Dr. G. Férey [++] Institut Lavoisier, UMR CNRS 8180 and Université de Versailles St-Quentin en Yvelines, 78035 Versailles (France) [ + ] Present address: Unité de Catalyse et de Chimie du Solide, USTL-ENSCL, 59652 Villeneuve d'Ascq (France) [ ++ ] and Institut Universitaire de France, Paris (France) Dr. F.-X. Coudert, Prof. Dr. A.H. Fuchs Ecole nationale supérieure de chimie de Paris (Chimie ParisTech), CNRS and Univ. Pierre et Marie Curie 11, rue Pierre et Marie Curie, 75005 Paris (France) E-mail: alain.fuchs@enscp.fr Fax: (+33)1-43292059

Figure 2 .

 2 Figure 2. Temperature-xenon pressure phase diagram of MIL-53(Al). Solid line: osmotic thermodynamic model. Open symbols: experimental transition steps in the isotherms. The dashed line corresponds to the experimental temperature (323 K) for which no transition took place.


C T Nzogo Metoule

S Delaby

J H Ferrasse

O Boutin

Hydrothermal process development for the treatment of asbestos containing waste

Keywords: Asbestos, Asbestos containing waste, Hydrothermal Process, Supercritical water, Chrysotile

In France, the maximum industrial production of asbestos was 150 000t.year -1 between 1973 and 1975. In France, there are currently two types of waste management for waste containing asbestos: landfilling and vitrification by plasma torch. With the storage of thousand tons waste by year, most sites will be soon saturated. Despite its performance, the vitrification process is energetically expensive.

Nowadays hydrothermal process is a mature technology used for the treatment of organics wastes, for instance by wet air oxidation. However, its application for mineral waste treatment is not developed, because technical difficulties. [START_REF] Ball | The dehydration of brucite[END_REF] realised the first study on chrysotile denaturation by hydrothermal process. [START_REF] Sigon | Process and plant for the hydrothermal treatment of asbestos and/or asbestoscontaining materials in supercritical water[END_REF] claimed a patent for degradation of asbestos in non-hazardous material by using supercritical water as only reagent. Finally, [START_REF] Anastasiadou | Hydrothermal conversion of chrysotile asbestos using near supercritical conditions[END_REF] established the hydrothermal conversion of chrysotile into forsterite in supercritical condition (temperature 700°C and pressure 2.4MPa) during 1h. Due to the presence of the mineral phase, it is necessary to reach high temperature. The objective is either to degradate the organic phase (glue for instance) and to change the morphologic form as well as the crystalline phase of asbestos. An example is given in equation 1 for chrysolite.

Mg3(OH)4Si2O5⇒Mg2SiO4 + MgSiO3 + 2H2O

(Eq. 1)

The aims of this work is to experimentally treat asbestos containing waste. First, an experimental assessment method is developed to qualify the performance of hydrothermal process. Then, a parametrical study is realized to determine influence of temperature, pressure, treatment duration, concentration and variety of asbestos (Table 2).

Figure 1. Experimental apparatus

The result obtained show a total degradation of chrysotile, crocidolite varieties, and fibrocement (containing chrysotile), under these hydrothermal conditions: T≥720°C, P≥25 MPa, t≥1h, C≥ 0.02 mg.ml -1 . Different analytical methods (X ray diffraction, particle morphology …) allow to characterise the final product obtained.

For instance, 120mg of chrysolite was treated at 740°C and 25MPa during 6h. The analysis (Figure 2) indicate that there is no more chrysolite and the formation of forsterite and clinoenstatite. A modification of the aspect of the fibers is also observed (Figure 3).

  X ray diffraction analysis before (a) and after (b) treatment (a) (b) Figure 3. MEB analysis before (a) and after (b) treatment


S Lefèvre

O Boutin

J H Ferrasse

C Chubilleau

A Viand

Wet air oxidation for the treatment of solid wastes generated on autarkic sites

Keywords: oxidation, hydrothermal, subcritical water, autarkic

Industrial effluents have variable compositions depending on the industry: refineries, pharmaceuticals, distilleries, food processing, paper mill… The most widespread treatment of effluents containing organic pollutants is the biological way, but microorganisms are unsuitable in the case of refractory or toxic products. Legislations regulate more and more severely the management of these wastes and favor the development of alternative processes allowing to treat effectively particular pollutions, as in pre-treatment before a biological process or for a complete degradation of organic matters in carbon dioxide and in water. Other technologies are under research and development, as for instance Wet Air Oxidation (WAO), especially for wastewaters that contains high chemical oxygen demand.

WAO or hydrothermal sub-critical oxidation was developed at first by the company ZIMPRO [START_REF] Zimmermann | Waste Disposal[END_REF] in the United States. The first WAO process allowed the synthesis of vanillin from liquid residues of paper mill industry. In the years 1960-1970, Sterling (which detained ZIMPRO patents) developed low temperature processes to treat various types of waste (urban, agricultural, regeneration of active carbon, etc.). It is in the 80s that WAO knows a renewed interest as treatment method for residual liquids, because of new regulations at the world level for environmental protection. WAO aims at oxidizing the organic fraction of an aqueous effluent by contacting organic pollutants with an oxidizing agent. WAO processes work mostly in conditions of temperature ranging from 100 to 320°C and a total pressure (of air or pure oxygen, according to the choice of the oxidizer) of 0,5 to 20MPa (Luck, 1979;[START_REF] Kolaczkowski | Wet air oxidation: a review of process technologies and aspects in reactor design[END_REF]. The oxidation yield is about 70 to 95% with a residence time of 30min to several hours. The 5 to 30% of organic matter which are not totally transformed into the liquid phase are mainly in the form of acetic acid, formic acid and other volatile fatty acids. That is why most of WAO units are followed by a biological treatment to degrade these acids. In these operating conditions, the solubility of oxygen in water is more important than in the ambient conditions and favors degradation kinetics of the organic fraction in carbon dioxide and water. [START_REF] Andreozzi | Catalysis Today, Advanced oxidation processes (AOP) for water purification and recovery[END_REF] suggest that this technology is the most adapted solution for the aqueous effluents having a COD included between 20 and 200g.L -1 what corresponds in approximately 1 to 10% of organic matter. From 20g.L -1 , the heat necessary for the functioning of the process is supplied by the exothermicity of the oxidation reaction and beyond 200g.L -1 , the amount of oxidizer becomes too important and generate higher costs (compression of the oxidizer, etc.) that makes incineration more attractive.

Autarkic sites (ships, oil rig, isolated communities,) generate liquid waste streams (black/gray water, cooking greases, oils, infirmary discharges) and solid (catering waste, paper, cardboard, plastics) containing organic matter, that are often stored for longer or shorter periods, then transported and incinerated by specialized companies. The recent scientific literature is dense about the experimental studies in laboratory on WAO of simple organic compounds or more complex effluents, on the use of catalysts allowing to lower the conditions of pressure and temperature. However, all these studies do not supply enough information to design this process to industrial scale. On reaction aspects, this work consists to run experiments with oxygen concentration conditions suitable for designing an industrial treatment unit. For that purpose, kinetic studies are conducted with an initial total organic carbon (TOC) concentration about 3g.L -1 of solid wastes. The different solid wastes are chosen for these experiments because it is often found in real wastes (canteen food wastes, can film packaging, corn flakes package, water bottle). The oxygen excess chosen for the experiments (70% excess of oxygen) is realistic contrary to 1000-2000% of excess found in literature [START_REF] Baillod | Wet Oxidation and Ozonation of Specific Organic Pollutants[END_REF]Portela et al, 1997).

The reactor used for this study is made of stainless steel 316L (Figure 1). The autoclave has the following characteristics: 30 MPa maximum pressure, 350°C maximum temperature, 200cm 3 internal volume. The temperature in the cell is kept stable by a hot (electric power)/cold (double jacket with air or water) regulating system. The stirring device is a Rushton type mixer with eight blades, with a hollow shaft allowing a recirculation by extraction of the gas phase into the liquid phase. A motorized pump allows the introduction of fluids inside the reactor. A dip tube at the bottom of the reactor allows to take samples at regular intervals during the experiments. For WAO experiments, the autoclave is charged with 60cm 3 of waste and first pressurized with nitrogen. When the required temperature is reached, the pump is filled with air (oxygen/nitrogen mixture) and pressurized to an exceeding pressure of 40MPa. Opening the supply valve of the reactor designates the beginning of the reaction. Preliminary calculations permit to fix an excess of oxygen of 70% and the initial phenol/oxygen stoichiometry is constant for each tested temperature. The mixing speed is set at 800rpm. The reaction evolution is monitored by analyzing samples by TOC measurement by a Shimadzu TOC analyzer VCPH. The calibration of total carbon and inorganic carbon is made from mother solutions at 1000ppm. Two experiments were carried out for each operating condition, to verify the reproducibility and to have for each waste TOC reduction curves as a function of time with points every 15min. The operating conditions and TOC reduction are synthesized in Table 1 for every waste.

Table 1. TOC reduction and operating conditions for WAO experiments on solid wastes

The results obtained show that WAO is a high-performance alternative for the destruction of organic solid and liquid waste generated on autarkic sites. The experimental data obtained (operating conditions, TOC reduction) will be used to carry out a techno-economic feasibility analysis for the implantation of WAO unit on this kind of site, with an environmental comparison (done by life cycle analysis methodology) with the conventional solution.

Figure 1 .

 1 Figure 1. WAO experimental apparatus


Jorge Peixinho
email: jorge.peixinho@univ-lehavre.fr

Grégory Lefèvre

François-Xavier Coudert

Olivier Hurisse

Water evaporation in silica colloidal deposits

Introduction

The boiling of water is an everyday experience and the boiling temperature of water at atmospheric pressure is a "fixed point" used to calibrate thermometers since the 18 th century [1]. However, boiling in confined spaces or in porous materials is a complex phenomenon due to the presence of walls.

In heat exchangers and steam generators, particles settle and form porous deposits. The deposition process of particles depends on multiple factors including pH and surface roughness [2,3]. Understanding the effects of the deposits on evaporation and boiling of water represents an important issue for Électricité de France (EDF) because it causes a severe reduction in efficiency particularly in steam generators [4]. Here we mimic deposits using a model system made of mono-disperse silica particles and study evaporation and boiling of water confined in the pores of micro-spheres colloidal silica.

The liquid-vapour phase transition of water confined in porous materials is not well documented whereas there are several reports on melting and freezing using calorimetry [5,6,7]. In these studies, melting point depression, hysteretic behaviour between freezing and melting, and modification of the bulk structure are quantified for a range of pores, chemical compositions, etc. However, a limited number of experimental studies deals with evaporation of water in a porous media at temperatures around the boiling point.

A discussion of the phase change of liquid-vapor systems in porous media including capillary effects, thickness of the porous layer and superheating can be found in [START_REF] Kaviany | Principles of heat transfer in porous media[END_REF]. Several numerical models and simulations based on continuous equations for mass conservation, heat conduction, Darcy's law and phase transition rules [START_REF] Stemmelen | [END_REF]10,11] are available. These models predict temperature and liquid saturation fields. Recently, a more detailed description of the interfaces has been obtained using lattice-gas dynamic mean field theory [12] and molecular dynamics [13]. Specifically, these studies reported the presence of microscopic liquid films moving ahead of the main flow depending on the dynamics of adsorption/desorption properties: viscosity ratio, wetting angle, chemical potential and morphology of the porous media. All previous models are qualitative and the purpose of this study is to provide quantitative data.

The laboratory experiments presented here exploit silica [START_REF] Iler | The chemistry of silica: solubility, polymerization, colloid and surface properties and biochemistry of silica[END_REF] because it is chemically inert, nontoxic, polar and stable for the range of temperatures considered in this study (T < 400 o C). At room temperature, the desorption of water out of films of colloidal silica is associated with the formation of drying cracks [START_REF] Singh | [END_REF]16]. At higher temperature, the evaporation rate is increased. Water droplets containing colloidal silica experience a complex evaporation kinetic [17]. Indeed, the droplets shrink. This indicates that some water flows outwards entraining particles. These particles accumulate close to the liquidvapour interface and form a shell. The subsequent behaviour depends on the mechanical properties of this shell. A loose shell favors water to evacuate. When water evacuates the shell accumulates particles, becomes elastic and buckles. However, if the shell becomes too tough, vapour develops under the heated crust and pressure builds up until large cracks form. The sharp temperature increase due to the formation of the shell can be predicted by heat and mass transfer models [17,18,19]. These models are based on diffusion and Darcy's law for the calculation of the pressure drop through the shell.

Here, we report experiments on the evaporation and boiling of water confined in the pores of colloidal silica. First, the model system is studied using scanning electronic microscopy and adsorption measurements. Secondly, calorimetry measurements are performed and the results are compared with water in membranes having cylindrical pores and heptane in the colloidal silica. Finally, it is suggested that the morphology of the material is essential to understand the boiling kinetics.

Materials

The deposits were made of mono-dispersed silica micro-spheres purchased as dilute particles suspensions from Granuloshop (France). Two sizes of particles: 179 ± 8 nm or 1.16 ± 0.05 µm in diameter were used. A deposit was obtained by settling a particle suspension column above an aluminum crucible with a inner diameter of 5 mm. After sufficient settling time, the stagnant water was removed. The crucibles were stored in the oven at 35 • C. The deposits were characterized using scanning electron microcopy (SEM), adsorption of nitrogen and adsorption of water through attenuated total reflection-infrared (ATR-IR) spectroscopy.

SEM. The deposits were imaged in order to assess the arrangement of the particles. A regular arrangement of the 1.16 µm silica micro-sphere was observed, as shown in Fig. 1, together with a number of faults. The images suggest that the packing fraction of the deposit is lower than the packing fraction of equal spheres, that is ≃ 0.74, and much larger that the random close packing (0.63).

Adsorption of nitrogen. The adsorption of nitrogen molecules N 2 (the adsorbate) to the surface of silica (the adsorbent) creates a film and the amount of adsorbate on the adsorbent as a function of its pressure at constant temperature is usually described as an isotherm. The isotherm, shown in Fig. 2, was carried out at 77 K (-196.15 • C) using a gas adsorption instrument (Belsorp-max from BEL Japan, Inc.) after outgassing the sample at 200 • C 1 μm 10 μm 20 μm for several hours. The sample was a 25.5 mg deposit of colloidal silica of 179 nm diameter. The density of the colloidal silica is 2.2 g.cm -3 [START_REF] Lide | Handbook of Chemistry and Physics[END_REF]. Globally, the isotherms are typical of mesoporous or macroporous materials. There is a slight hysteresis at high pressures, close to saturation. The adsorption and desorption isotherms are then analysed using several methods: Langmuir method, BET method, t-plot method, BJH method or NLDFT method. The specific surface area is 24 ± 2 m 2 .g -1 and the mean pore diameter is 60 ± 15 nm. The measured specific surface area is larger than expected from theoretical consideration of perfectly smooth mono-disperse spheres (≃15 m 2 .g -1 ), because of the faults described earlier and the roughness of the spheres.

Adsorption of water. Let's turn now to the adsorption of water at room temperature. At an atomistic level, an ice-like region forms close to the silica surface through roughly three monolayers. Above this ice-like region, there is a transition region whose structure is disordered as the liquid layers above it. A schematic is shown in Fig. 3(a). The adsorption-desorption of water molecules is studied using a specific technique, proposed by Asay and Kim [START_REF] Asay | [END_REF], using attenuated total reflection-infrared (ATR-IR) spectroscopy signals at different relative humidities. Practically, a water drop of 1 µL containing 5% silica micro-spheres of 179 nm is first dried for several hours on the diamond crystal surface of a spectrometer (Thermo Scientific Nicolet 6700 FT-IR with the Pike MIRacle ATR accessory). The dried deposit is enclosed in a box and there is a flow of nitrogen gas (2-3 mL.min -1 ). The relative humidity of the flow is controlled by the input of a mixture of dry nitrogen gas and water saturated nitrogen. The relative humidity was measured using a thermo-hydrometer (Hanna instruments) with a resolution of ±5%. The )

Figure 2: Nitrogen adsorption and desorption isotherms of a silica colloidal deposit made of mono-dispersed micro-spheres of 179 nm in diameter. The experiment was performed at 77 K and the lines represent NLDFT isotherm fits. temperature of the system was maintained at 25 ± 1 • C. In this analysis, the evanescent wave penetrates the deposited silica and the gas phase. The absorbed molecules are detected, enabling a vibrational spectroscopy study. There are two main adsorption bands: a first peak at 1650 cm -1 due to H-O-H bending vibration and a group of peaks at 3000-3700 cm -1 due to O-H stretching vibrations. The area below the group of stretching peaks changes more significantly with relative humidity than the area below the bending peak. The peaks at 3250 and 3400 cm -1 correspond to stretching vibration of ice-like and liquid water, respectively [START_REF] Asay | [END_REF]. Detailed information about the molecular structure of water can be obtained from the analysis of the absorbance peaks presented in Fig. 3(b). The basic assumption is that the adsorbed water on the silica surface grows proportionally to the peaks area and the absorbance results indicate that the stretching vibration of ice-like and liquid water are present at 30% relative humidity. As the relative humidity increases above 80%, the thickness of the absorbed layer increases exponentially and bulk condensation is taking place. The numerical integration of the peaks in Fig. 3(b) at different relative humidity leads to a water adsorption isotherm shown in Fig. 4. The type of isotherm is consistent with previous results [22] where the effect of silanization and water adsorption kinetic were taken into account.

Results and discussion

Water evaporation and boiling within silica deposits was studied using calorimetry. First a series of drying experiments at temperatures above the boiling point is presented together with photomicrographs of the surfaces of the deposits. For membranes having cylindrical pores, no patterns were observed, whereas cracks and craters are found in dried particle deposits. Then a series of differential scanning calorimetry (DSC) experiments [START_REF] Höne | Differential scanning calorimetry[END_REF] indicate that the boiling within silica deposits caused two endothermic peaks: one at 100 • C and a second at higher temperature. The second peak is confirmed by thermo-gravimetry analysis (TGA) and experiments using heptane as a working fluid.

Isothermal drying at high temperatures

Nesić and Vodnik [17] studied the evaporation of pendant water droplets containing colloidal silica. They measured simultaneously the weight and temperature of the colloidal silica droplets as a function of time in a steady stream of air at constant temperature. They confirmed the existence of a multi-stage temperature curve describing the evaporation kinetics.

In Fig. 5(a), a schematic of the evaporation kinetics of water from membranes having cylindrical pore is presented. The pores are associated with the diameter of the cylindrical holes. Clearly the evaporation in cylindrical pore is straightforward because it takes advantage of vent or chimney effect, whereas in deposits there are a number of restrictions or obstructions. Indeed, in the case of deposits, depicted in Fig. 5(b), the voids in the deposits have a range of dimensions associated to the complex morphology of the pores. Moreover, the rapid evaporation of liquid water is likely to entrain particles, which accumulate close to the liquid-vapour interface, and form aggregates. As water evacuates, vapour develops under the heated crust and pressure builds up until large cracks form.

In the following the preparation of the drying experiments is described. As explained before the silica micro-spheres sedimented into the crucible and the deposit slowly dried at low temperature forming a homogeneous surface. The experiment starts by adding water. The porous deposit absorbs the water in few seconds. Next, the crucible was closed and welded to a perforated aluminium lid with a top hole of 50 µm. The effect of the lid is to locally increase the vapor pressure over the sample and control the influence of the surrounding flow. The samples are weighed before and after the run in order to confirm the evaporation. The calorimeter (DSC 822e from Mettler-Toledo) was used under a steady stream of nitrogen (80 mL.min -1 ).

In addition to the tests with the deposits, experiments were also conducted using membranes having cylindrical pores and particle deposits. The membranes (Anodisc 25 from Whatman) are made of aluminum oxide and had cylindrical pores of 0.2 µm in diameter over a thickness of 60 µm. The membranes weighed 11±1 mg. The deposits described before were made of 1.16 µm diameter micro-spheres and the sample weighed 19.6±0.4 mg. Given the size of the crucible, the stack of micro-sphere is about 500 layers and the total volume available within the pores is 0.5 mL. Initially, the deposit is filled with 10 mL of water and consequently a liquid film covers the deposit. At t = 0, the sample is inserted in the pre-heated calorimeter. Isothermal measurements provide a direct insight into the kinetics of the drying. In presented. The temperature increases from 25 • C, reaches a plateau and then at a critical time t c there is a sharp increase in temperature until the final temperature of the calorimeter. For drying temperatures above 100 • C the critical time, t c , decreases rapidly, as indicated in Fig. 5(d).

During drying, the dominant stress is the capillary pressure from the tiny menisci between particles:

∆P = 2σ cos θ/r ( 1
)
where σ is the surface tension coefficient, θ is the solvent-particle contact angle and r is the radius of curvature which is typically r ≈ 0.1R for randomly close-packed spheres of radius R. The radius of the meniscus, r, can be obtained from the Kelvin equation provided that the saturation vapor pressure is known. Since σ ≃ 72 mN.m -1 and cos θ ≈ 1, the capillary pressure: ∆P ≈ 6 × 10 5 Pa or 6 atm for d = 1.16 µm particles. Relative to the ambient pressure surrounding the droplet, the liquid within the pores has a lower pressure. Therefore every meniscus acts like a low pressure pump trying to suck liquid from other places. Also liquid flow may be able to move particles around.

In figure 6, the dried silica deposits are observed under an optical microscope. The smooth surface observed on the deposit at 25 • C is the initial pattern before adding water and running the experiment. As the drying temperature increases (i.e. 95 • C), there is the formation of cracks. At higher temperature (105 and 175 • C), craters are observed.

There are several possible scenarios describing the drying. In a first scenario, the craters are due to the vapor accumulated underneath the deposit, which was able to form large bubbles. These vapor bubbles are about 1600 times less dense that the deposit. The vapor rapidly accumulated and grew below or within the deposit. Its expansion is important and the force associated to the buoyancy are sufficient to crack the deposit and create craters. In a second scenario, a drying front travels from the upper surface through the deposit towards the crucible. In this case the cracks would be formed by the capillary pressure.

It is important to note that in the case of membranes, no visual defects were observed.

Scanning experiments

Let's look now at the temperature scanning experiments. The usual experiment in DSC consists in measuring both the sample and a reference temperature. The difference of heat required to increase the temperature of the sample and the reference is measured as a function of temperature. More of less heat must flow into the sample depends on the nature of the process. For example, as water evaporates and boils from a sample it will require more heat flowing into the sample to maintain its temperature constant. This process is endothermic. A single evaporation peak is observed in Fig. 7 when increasing temperature with a rate of 0.75 and 10 K.min -1 and after adding various quantities of water to the cylindrical pore membrane. The width and the area of the peak is proportional the quantity of water added. The right side of the peak is sharp indicating the abrupt emptying of the pores.

In the case of a deposit of 1.16 µm silica micro-sphere, there is a first peak at 100 • C, as showed in Fig 8(a). More importantly, a second peak is observed around 120 • C, which seems to correspond to the desorption of confined water. This second paek was also observed for deposit of smaller particle of 127 nm. The position of the minimum of the second peak changes with the scanning rate (see inset in Fig 8(a)) suggesting a transient delay of the boiling of the water in the pores.

Another scanning experiment at a heating rate of 5 • K.min -1 was carried out using a TGA where the weight loss was measured using a microbalance. Again the 1.16 µm silica deposit experiences a second peak, which water content corresponds to 5% of the evaporated water. This measurement confirms that the secondary peak corresponds to a weight loss.

Additional experiments using heptane, C 7 H 16 , were conducted. Heptane was chosen because its boiling temperature of 98.61 • C close to the boiling temperature of water. However, the density of heptane is 680 kg.m -3 ; also the wetting properties are different from that of water: the surface tension coefficient is 20 mN.m -1 and the contact angle with silica is 40 • . In the case of heptane, DSC scanning runs indicate that the area under the secondary peak equals that of the first peak. In Fig. 9, the second peak is centered around 140 • C. The fact that the second peak is observed, both for water and heptane in colloidal deposits, whereas it is not seen in the case of membranes, suggests that the morphology of the deposits is responsible for the secondary desorption. The secondary peak for heptane is wider indicating that the water surface, the wetting and the buoyancy properties have an effect on the secondary release. As said before, the dominant stress during evaporation, is the capillary pressure from the tiny menisci between particles. These menisci move through the deposit at variable speeds depending on the properties of the fluids and the morphology of the deposit. These motions induce delays in the evaporation and it is suggested that the secondary peak is due to such kinetic effect. Another hypotheses is that the flow in the porous deposit induces particle motions and deformations of the material which delays the secondary desorption. The measure of temperature differences between the sample and the reference is proportional to the phase transition enthalpies. This allows to detect endo or exo-thermic transitions. For water ∆H vap = 1936 ± 210 J.g -1 (NIST ∆H vap = 2257 J.g -1 ). The uncertainty of the enthalpy was inferred from four repeated measurements. For heptane ∆H vap = 368 ± 60 J.g -1 (NIST ∆H vap = 334 J.g -1 ).

The liquid-vapour phase transition of water in porous materials has also been observed using DSC and TGA in order to obtain both boiling temperature and vaporization heat [START_REF] Etzler | [END_REF]25]. Two studies used DCS and TGA to investigate boiling on porous materials [26,27]. They found a double peak corresponding to the desorption of the fluid from the material and it is suggested the physicochemical properties of surface water are responsible from the peak. However no information is given on the final state of the material.

Eventually, boiling point elevation was not observed in the present calorimetric measurements because the pore here are large. Indeed, boiling point elevation was observed in a sample of silica MCM-41 mesoporous [START_REF] Hansen | Progress in zeolites and microporous materials[END_REF] where water in 2 nm diameter pore material experiences boiling at 390 K (116.8 • C).

Conclusion

The evaporation and boiling of water in deposits of silica micro-spheres were studied. The deposits were characterised using scanning electron microscopy, adsorption of nitrogen, and adsorption of water through attenuated total reflection-infrared spectroscopy. Initially the deposits made of monodispersed micro-sphere are regularly ordered together with a number of faults. A number of drying experiments were carried out using differential scanning calorimetry and thermal gravimetric analysis. It has been found that the desorption of water in deposits is a two step process. The bulk water evaporates first, then the desorption of water within the pore takes place at higher temperature (125 • C) when the heating rate is 1 to 10 K.min -1 .

Interestingly the final deposit pattern films were also observed using optical microscopy and cracks appear on the surface of the dried silica. The cracks were observed at drying temperature below the boiling temperature. At larger temperatures, craters are observed indicating that vapor bubbles formed underneath the deposit and travelled through the deposit.

The results are compared with water evaporating from membranes having cylindrical pores and for heptane. In the case of cylindrical capillaries, no second peak and no cracks or deformation of the membrane is observed. When the fluid is heptane, the secondary peak is wider and shifted meaning that the wetting and density of the liquid (Bond number) have also an effect on evaporation in deposits.

Figure 1 :

 1 Figure 1: SEM images of an arrangement of silica micro-spheres of 1.16 µm in diameter

Figure 3 :

 3 Figure 3: (a) Schematic, inspired from Asay and Kim [21], illustrating the structure of water molecules adsorbed on the silica surface. As the relative humidity increases, water adsorb in the liquid configuration ---hydrogen bonds, -covalent bonds. (b) ATR-IR spectra of water absorbed on silica at different relative humidities. From bottom to top, relative humidity equals 30, 80, 90, 94 and 96%. The O-H stretching vibration peak of ice-like water and liquid water are indicated by vertical dashed lines at 3200 and 3400 cm -1 .

Figure 4 :

 4 Figure 4: Adsorption isotherm of water on silica surface. Square symbols represent the adsorbed water from the area of the bending and stretching vibration peaks. The error bars represent the variation over two runs of the experiment. The solid line is drawn to guide the eyes. The experiment was performed at 25 • C.

Fig. 5 (Figure 5 :Figure 6 :

 556 Figure 5: (a) Schematics of the evaporation kinetics of a water imbibed membrane. (b) Schematics of the drying kinetics of colloidal silica deposit. (c) Evolution of the temperature of a drying membrane imbibed in water at 105 • C, t c defines the critical time and the different curves correspond to the results obtained for different runs. (d) t c versus the drying temperature. The error bars represent the standard deviation based on 3-4 runs of the experiment and the line is a power law fit of the data.

Figure 7 :

 7 Figure 7: DSC results for water and heptane imbibed membranes having cylindrical pores at different heating rates

Figure 8 :Figure 9 :

 89 Figure 8: (a) DSC curves of 1.16 µm deposits showing the energy versus the temperature for different scanning rates. The inset is a magnification of the second peaks. (b) TGA curves showing the energy and weight loss versus the temperature.

Acknowledgments

This research is supported by EDF. We are grateful to Alexander Neimark (Rutgers University) for discussions regarding adsorption measurements and Philippe Vermaut (LPCS, Chimie-ParisTech and CNRS) for discussions regarding the DSC experiments.


Franc ¸ois-Xavier Coudert

Systematic Investigation of the Mechanical Properties of Pure Silica Zeolites: Stiffness, Anisotropy, and Negative Linear Compressibility †

While the energetic stability of the large number of possible SiO 2 polymorphs has been widely addressed by both experimental and theoretical studies, there is a real dearth of information on their mechanical properties. We performed a systematic study of the elastic properties of 121 pure silica zeolites, including both experimentally synthesized and hypothetical structures, by means of density-functional theory calculations. We found that most frameworks exhibit high elastic anisotropy, and the experimentally synthesized structures are among the most mechanically stable ones. We propose to extend the "feasibility" criterion proposed in earlier literature to include elastic anisotropy, allowing to further reduce the number of possible targets for synthesis of SiO 2 polymorphs. We also predict that a small number of hypothetical pure silica zeolites present large negative linear compressibility (NLC), which we attribute to the wine-rack motif of their framework.

Introduction

Pure silica zeolites are metastable porous phases of SiO 2 , 1,2 also known as all-silica zeolites, siliceous zeolites, zeosils, etc. Their porous structure is formed upon crystallization in the presence of templating agents, and retained after a calcination process which removes the template but does not break the structural connectivity. In addition to the traditional applications of microporous materials, such as adsorption, separation and gas storage, they have a practical application that finds its root in the hydrophobicity of their pore surface: the combination of small pore size and nonwetting surface can be leveraged to adsorb, store, or dissipate energy. 3 There is also a growing enthusiasm for their use as thin films as low k materials, chemical sensors, membrane reactors, and microelectronic devices. [4][5][6] While more than 200 zeolite frameworks have been observed in natural or synthesized zeolitic materials, less than a quarter of that number (46 out of 206) can be experimentally obtained as pure silica compounds. The synthesis of new pure silica zeolites remains a considerable challenge. In order to evaluate the viability of the zeolite frameworks in their SiO 2 form, researchers have worked on characterizing the lattice energies of various pure silica zeolite structures and linking it to their structural features. In their seminal work in 1994, Gale et al. 7 used force field-based energy minimizations of † Electronic Supplementary Information (ESI) available: Energies of the 163 pure silica zeolites whose geometry was optimized. Full characterization of the 121 pure silica zeolites whose elastic constants were calculated.

a CNRS & Chimie ParisTech, UMR 7575 LECIME, 11 rue Pierre et Marie Curie, 75005 Paris, France; E-mail: fx.coudert@chimie-paristech.fr 10 silica frameworks, including 6 zeolite structures, and compared the energies obtained to that of the thermodynamically stable SiO 2 polymorph α-quartz. They linked the relative energies of the frameworks to their density, a correlation which was soon confirmed on a larger number of frameworks. 8 This correlation was turned into a feasibility criterion by the observation that all known pure silica zeolites synthesized to date fall within 30 kJ/mol (per SiO 2 unit) of the dense α-quartz phase, which was then used to assess the viability of hypothetical zeolite frameworks. 9 Finally, the growth of computational power available has made it possible to perform systematic studies of the energetics of pure silica zeolite frameworks. An example of such is the recent systematic work by Sastre et al. 10 based on well-established force fields, which showed that a combination of both lattice energy and density may be adopted as empirical criteria to assess the thermodynamic feasibility of the frameworks.

However, there has been no systematic exploration of the mechanical properties of pure silica zeolites to date. The elastic properties of a few specific zeolites have been explored, either experimentally [11][12][13] or by means of molecular simulation. [START_REF] Astala | [END_REF][14][15][16] The only large-scale study of pure silica zeolites' elastic properties was performed in 2000 by Grima et al. 14 , based on force field energy minimization, but it was exclusively focused on materials with negative Poisson's ratio (also called auxetic materials). Other properties, such as the zeolites' Young's and shear moduli, as well as their linear compressibility, were not discussed. Nonetheless, mechanical properties are of great importance for practical applications of these microporous materials, and in particular their Young's and shear moduli. There is currently a dire need for better understanding how widely they can vary in a single family of materials (SiO 2 polymorphs) and how they are linked to characteristic features of the inorganic framework. This need is rendered even more pressing by the recent development of new families of metal-organic frameworks, such as the Zeolitic Imidazolate Frameworks, 17 that present a large number of polymorphic crystalline structures 18,19 isomorphic to zeolites, and for which the links between framework topology, linker functionalization, and energetic and mechanical properties have only recently been addressed. 20,21 In the present paper, we report a large-scale study of the energetic and mechanical properties of synthesized and hypothetical pure silica zeolites. We performed quantum chemistry calculations at the density functional theory (DFT) level of theory, determining the relative energies of 163 different pure silica frameworks and the second-order elastic constants tensors of 121 of these frameworks. State-of-the-art DFT methods was shown on a series of pure silica structures to give good agreement with other methodologies and experimental data. 22 In such a large scale study, the use of parameter-free DFTbased calculations avoids a dependence of the results obtained on force field calibration, which may not be able to reproduce equally well all the different zeolite frameworks (especially those with different secondary building units).

Methodology

Energy calculations

We started from the full list of 206 structures (at the current date) of the Database of Zeolite Structures 23 maintained by the Structure Commission of the International Zeolite Association (IZA). These structures (atom coordinates and unit cell parameters) were optimized with DLS-76 24 assuming a pure SiO 2 composition. These initial structures were then fully relaxed by optimizing both atomic positions and unit cell parameters. 163 pure silica structures reached convergence within the limits of the computational available: this represents 79% of all the IZA-listed frameworks. The full list of reported in Figure 1. The same calculations were also performed on the α-quartz structure, for the purpose of comparison. All first principles calculations were performed in the density functional theory approach with periodic unit cell, full use of the crystals' symmetry elements and localized basis sets as implemented in the CRYSTAL09 code. [START_REF] Dovesi | [END_REF] We used the B3LYP hybrid exchange-correlation functional, 26 with empirical correction for the dispersive interactions following the "D2" scheme of Grimme. 27 All electron basis sets were used for oxygen and silicon. 28 The full results (optimized energies and structures in CIF format) are available as part of the Supporting Information.

Elastic tensor calculations

The tensorial Hooke's law establishes a general relationship between the strain ε and the stress σ in a solid in the linear elastic regime, through the fourth-order tensor C of secondorder elastic constants: 29 σ i j = ∑ kl C i jkl ε kl . Taking advantage of the symmetries of stress and strain matrices and using the Voigt notation, the elasticity tensor (or stiffness tensor) C i jkl can be expressed as a 6 × 6 symmetric matrix of 21 elastic constants C i j . The crystal system of the material considered yields additional symmetry constraints, further reducing the number of independent elastic constants: 4 for cubic crystals, 5 in the hexagonal case, 6 or 7 for the tetragonal classes, 9 for orthorhombic crystals and 13 for monoclinic crystals. [START_REF] Nye | Physical Properties of Crystals -Their Representation by Tensors and Matrices[END_REF] The calculation of second-order elastic constants of the materials studied were performed with the use of the CRYS-TAL09 code, [START_REF] Dovesi | [END_REF] whose implementation was detailed in Ref. 30. In short, starting from the optimized geometry of the relaxed crystal, each deformation mode of the crystal is scanned and the second-order elastic constants can be calculated by finite differences as second derivatives of the energy:

C i j = 1 V ∂ 2 E ∂ ε i ∂ ε j (1)
For each deformation mode, 5 different values of strain were sampled (including the relaxed structure), with a total amplitude of ±2%. The accuracy of this methodology is now well established for the calculation of elastic constants in dense inorganic crystals [START_REF] Perger | [END_REF] as well as porous materials: carbon nanotubes, 31 boron nitride nanotubes 32 and metal-organic frameworks. [33][34][35] From the second-order elastic constants of each material, a full tensorial analysis was performed 35,36 (see ref. 35 for details) and key quantities were derived that characterize the mechanical behavior of the structure in the elastic regime: Young's modulus, shear modulus and linear compressibility. In an anisotropic medium such as a crystal, these properties are directional; for each property, we calculated the minimal and maximal values reached.

These calculations of elastic tensor were performed on a total of 121 pure silica zeolite structures, i.e. 59% of the IZA database (see Figure 1). These included 13 cubic structures, 26 hexagonal structures, 21 tetragonal structures, 40 orthorhombic structures and 21 monoclinic structures. The full results are available as a spreadsheet as part of the Supporting Information.

Results and discussion

Energetics of pure silica zeolites

Following the methodology of earlier works 7,10 for studying the feasibility of pure silica zeolites, we show in Figure 2 a plot of lattice energy relative to α-quartz versus specific volume, for all 163 frameworks whose structure was optimized in this work. Overall, an inverse correlation between stability and specific volume is found, as is expected (i.e. lattice energy increases with volume, or decreases with density). We also see, despite this overall correlation, there is significant spread of the energies: in addition to the RWY framework, an outlier nearly 100 kJ/mol less stable than α-quartz, a large number of structures are very high in energy. The list of frameworks with ∆E > 30 kJ/mol include OSO, JST, OBW, BOZ, NPT, SOS, BSV, CZP, WEI, PUN and AFY. Moreover, we can see that frameworks types that have been experimentally synthesized as pure silica zeolites (red symbols in Fig. 2) all have relatively low lattice energy, the larger one being FAU at ∆E ≈ 22 kJ/mol. The left pane of Figure 2 shows a zoom on this low-energy region of the graph. There, it can be seen that all known pure silica frameworks fall in a gray band of width ±15 kJ/mol. In addition, a large number of hypothetical pure silica materials also fall in that region of the energy-density diagram, indicating that with the right choice of template and synthesis conditions, their synthesis should be achievable. Of particular interest on this diagram are the points of lowest energy at a given density, such as MEP, SFN, UFI and VFI.

In addition, we can use these results to find correlations between lattice energy and other properties of the zeolite frameworks. We start by noting that, among the synthesized pure silica zeolites, those with 0D and 1D pore networks are present in abundant numbers, while those with three-dimensional channel system are underrepresented (see Table 1). In order to understand this effect, we plot in Figure 3 the histograms of lattice energies for channels as a function of the dimensionality of their pore network. There is an overall increase of lattice energy from 0D to 3D, which can be attributed to the energy-density correlation: networks with a channel system of higher dimension are typically less dense. Furthermore, we note that all the pure silica zeolites with a lattice energy higher than 30 kJ/mol per SiO 2 have 3D channels, with the exception of the ASV framework.

These findings are overall in good agreement with the earlier study by Bushuev et al., 10 therefore confirming the conclusions of their work and validating our DFT-based approach. We refer the reader to the discussion in ref. 10 for a longer discussion of the structure/stability relationship in pure silica zeolites, and focus in the rest of this manuscript on their mechanical properties.

Young's and shear moduli

We calculated the second-order elastic constants of 121 pure silica zeolite structures, of which 92 are hypothetical structures and 29 have been experimentally synthesized. Among the various elastic properties we calculated from those elas-tic tensors, we focus in this section on the Young's and shear moduli, which characterizes the stiffness of the material in response to uniaxial compression and shear, respectively. Because of the anisotropic nature of the crystals, these elastic moduli are directional, i.e. they vary in space with the direction of the applied stress. Because one of the main interest in mechanical properties of zeolites is their usually high mechanical stability, we discuss here the minimum Young's and shear moduli for each framework; their anisotropy and its impact will be discussed in the next section.

Figure 4 provides a graphical representation of the 121 pure silica zeolites' mechanical properties in a Young's modulusshear modulus space. We first note the very impressive range of both moduli in this family of materials, despite all of them being formed of the same strong Si-O bonds: the minimal Young's modulus for pure silica zeolites ranges from 11.5 GPa (SOS) to 114.5 GPa (MTN), and the minimal shear modulus from 3.7 GPa (OWE) to 42.2 GPa (MTN). For each modulus, the variations span an order of magnitude, which is remarkable. We also see, as in most materials, shear modulus is usually weaker than Young's modulus; however, both moduli a strongly correlated. We thus propose to follow an alternative way to characterize the "softness" of the frameworks, by looking at the smallest eigenvalue λ 1 of the 6 × 6 matrix of elastic constants. This lowest eigenvalue corresponds to the softest elastic modulus, i.e. to the stiffness of the material in the softest deformation mode (the associated eigenvector), whether this deformation mode is pure compression, pure shear or sum of the two. This approach was recently proposed and validated in a study of the flexibility of metal-organic frameworks. 35 Another point that can be seen from Figure 4 is that, while there is a certain spread, most of the SiO 2 frameworks experimentally attested have medium to large stiffness. In fact, only 3 of the synthesized pure silica frameworks have a Young's modulus lower than 35 GPa, while it accounts for more than a third (36%) of the hypothetical frameworks. Just like there is a known correlation between experimental feasibility and lattice energy, there seem to be a correlation between feasibility and stiffness. In order to understand this correlation better and to check whether it might be an indirect correlation via a third quantity, we plotted in Figure 5 the correlations between density, lattice energy and stiffness (as measured by the smallest eigenvalue, λ 1 , of the elastic matrix). The results show little in the way of correlation between stiffness and density (or specific volume; left panel of Fig. 5): in that regard, mechanical stiffness is quite different from energetic feasibility, which presents a clear correlation, although with some spread. We can provide a quantitative confirmation of this conclusion by calculating Pearson's correlation coefficient for both sets of data: it is -0.21 for stiffness vs. density, compared to 0.72 for energy vs. density.

On the right panel of Fig. 5, we can check the correlation be-tween stiffness and lattice energy. There is now a modest correlation (Pearson's r of -0.29), which becomes quite strong if you focus on the most feasible frameworks: Pearson's r is -0.57 for the points with ∆E ≥ 25 kJ/mol; the corresponding linear regression is plotted as dashed line in Fig. 5). This clear correlation is somewhat unexpected, as it links the relative stability of a framework with its elasticity, which is a local property. However, we find no correlation between the stiffness of the pure silica zeolites and the crystal class (symbol color in Fig. 5) or symmetry of their framework. These two facts lead us to conclude that an important factor in the variations of stiffness within the family of pure silica zeolites is the deformation of the Si-O bond lengths and Si-O-Si angles in the frameworks: when bond lengths and bond angles are strained, the lattice energy of the framework increases and the framework stiffness diminishes, because its is weakened by the strain.

Anisotropy of the elastic properties

As indicated earlier, because of the anisotropic nature of crystals, monocrystalline elastic moduli are directional: Young's modulus E(u) depends on a direction, indicated by unit vector u, and the shear modulus G(u, n) depends on the shear direction u and shear plane normal n. We investigate here the anisotropy in pure silica zeolites's elastic properties by looking at the ratio of their maximal and minimal values. Figure 6 reports the (E max , E min ) and (G max , G min ) plots of the 121 pure silica zeolites studied in this work. While some frameworks have little anisotropy (i.e. correspond to a point near the y = x line in these diagrams), most pure silica zeolites exhibit a large anisotropy in both their Young's and shear moduli. Some frameworks exhibit almost an order of magnitude difference between their stiffest deformation mode and their softest one. The extent of the anisotropy of the two elastic moduli is an important quantity: very high anisotropy usually indicates limited mechanical stability of the material, which is in turn a limiting factor for practical applications. For this reason, we performed a more quantitative analysis of elastic anisotropy, we define the anisotropy η as:

η = min E max E min , G max G min ( 2
)
We then propose to plot the experimental and hypothetical pure silica zeolites in an energy-stability diagram (∆E, η).

It clearly appears, on that diagram (Figure 7), that indeed mechanical stability plays some role in the experimental feasibility of zeolite frameworks as pure SiO 2 materials: most of the experimentally synthesized frameworks are grouped in bottom-left area of the plot, corresponding to both low lattice energy and low elastic anisotropy. We thus propose to add mechanical stability, as characterized by low anisotropy of the elastic properties, as a new criterion to feasibility of pure silica frameworks (in addition to the established density and energy criteria). We suggest that materials satisfying the conditions of ∆E ≤ 20 kJ/mol per SiO 2 and η ≤ 4 may be the most promising pure silica frameworks yet undiscovered. From our calculations, the list of most feasible framework types would include MEP, CAS, LOS, LIO, AFG, ATO, CAN, EPI, DAC and BCT.

Zeolites with negative linear compressibility

Negative linear compressibility (NLC) is a counterintuitive property of some materials which, under hydrostatic compression, see one or more of their linear dimensions decrease. Thermodynamics imposes that the bulk compressibility of a material, which is the inverse of its bulk modulus, be positive: isotropic compression must lead to a decrease in volume.

For most materials, this decrease in volume stems from a decrease in all of their linear dimensions. However, a handful of materials show NLC, i.e. while their overall volume diminishes upon compression, one or more of their dimensions actually increase in the process. NLC is a rare and highly desirable property, and practical applications of materials exhibiting NLC include artificial muscles, actuators and pressure sensors. 37 In the search for materials with the largest possible NLC, researchers have looked for structures with specific structural motifs, and in particular the wine-rack 38 and honeycomb 37 topologies. Recent reports have evidenced NLC in metal-organic frameworks, 39 including several cases of colossal negative linear compressibility in compliant metal-organic frameworks based on the wine-rack motif. 34,35 However, to our knowledge, NLC has never been demonstrated in a zeolite yet. 16 of the 121 pure silica zeolites studied in this work exhibit negative linear compressibility, which corresponds to 13% of the frameworks studied. We thus predict that NLC is not a rare phenomenon in zeolites, though it has never been demonstrated experimentally, nor hypothesized from ab initio calculations. The extent of the NLC of pure silica zeolites, measured by the linear compressibility β min in the direction where it is the most negative, varies between β min = -0.2 TPa -1 (for the AFN framework) and β min = -13.7 TPa -1 (for the GIS framework). The later is a remarkably large value, in the same range as the strongest NLC effects discovered very recently for inorganic materials (-75 TPa -1 for Ag 3 [Co(CN 6 )]; [START_REF] Goodwin | Proc. Nat. Acad. Sci[END_REF] -42 TPa -1 for Zn[Au(CN 2 )] 2 ; 37 -12 TPa -1 for KMn[Ag(CN 2 )] 3 [START_REF] Cairns | [END_REF] ). Moreover, GIS is among the "feasible" pure silica zeolites, with a lattice energy of 17.4 kJ/mol per SiO 2 relative to α-quartz, and a specific volume of 0.63 cm 3 /g. It is thus a good candidate for the synthesis of a pure silica zeolite with large negative linear compressibility. Other candidates are ATT, ABW, APC, MSO, MER, PHI, UEI, AHT, AWO and LAU.

Finally, looking at the frameworks that exhibit NLC (the three frameworks with the largest NLC are depicted in Fig. 8) immediately reveals that they share a common feature: they all have a wine-rack topology. Thus, while NLC in zeolites has not been reported before, its mechanism is quite similar to that observed in dense inorganic materials, molecular frameworks and porous metal-organic frameworks.

Conclusion

We have performed the first systematic study of the elastic properties of pure silica zeolites. By means of density functional theory calculations, we have calculated the secondorder elastic constants of 121 SiO 2 frameworks, including members that have already been synthesized experimentally as well as hypothetical structures. We showed that DFT calculations confirm the correlation established by earlier work between framework density and energetic stability. In order to shed light into the little-known mechanical behavior of pure silica zeolites, we analyzed their Young's and shear moduli, as well as their linear compressibility. We found that most of the frameworks studied have high anisotropy of their elastic properties. However, the frameworks that have been experimentally synthesized tend to be among the most mechanically stable ones, with relative low anisotropy. Thus, we propose to extend the "feasibility" criterion proposed in earlier literature for pure silica zeolites to include elastic anisotropy, allowing to further reduce the number of possible targets for synthesis of SiO 2 frameworks. Finally, we predict that a small number of hypothetical pure silica zeolites present large negative linear compressibility (NLC) due the wine-rack motif of their framework. If they were synthesized, such inorganic nanoporous materials with NLC could have applications as nano-actuators, pressure or gas sensors.

Fig. 2

 2 Fig.2Energy of the pure silica zeolites relative to α-quartz versus specific volume. Left: all material studied; right: zoom on the low energy part. Red symbols: materials experimentally synthesized; green symbols: other topologies; blue: α-quartz, the reference material.

Fig. 3

 3 Fig.3Histogram of framework energy (relative to α-quartz) as a function of dimensionality of the channel system.

Fig. 4

 4 Fig.4Young's modulus and shear modulus of pure silica zeolites: each point represent a single framework, red points correspond to synthesized pure silica zeolites.

Fig. 5

 5 Fig.5Left: correlation between stiffness (measured by smallest eigenvalue of the elastic matrix, λ 1 ) and specific volume. Right: correlation between stiffness and lattice energy relative to α-quartz; different colors correspond to the different lattice systems.

Fig. 6

 6 Fig.6Anisotropy of the Young's modulus (E, left) and shear modulus (G, right) of pure silica zeolites, by plotting their minimal value against their maximal value. The black lines are the y = x axes, and the grayed areas indicates anisotropy lower than 20%.

Fig. 7

 7 Fig.7Plot of the elastic anisotropy of pure silica zeolites vs. lattice energy relative to α-quartz; red points correspond to synthesized pure silica zeolites. The gray area corresponds to the feasibility criterion proposed in the text.

Fig. 8

 8 Fig. 8 Representations of the GIS, SOS and ATT zeolite frameworks (from left to right). Edges correspond to Si atoms, vertices to Si-O-Si links.

Table 1

 1 Distribution of the pure silica zeolites with respect to the dimensionality of their framework.

  List of the 206 zeolite frameworks of the IZA database. In red: pure silica zeolites that have been synthesized to date; with blue background: energy optimization and elastic constants calculated in this work; green background: energy optimization performed in this work.

	ABW	ACO	AEI	AEL	AEN	AET	AFG	AFI	AFN	AFO	AFR	AFS
	AFT	AFX	AFY	AHT	ANA	APC	APD	AST	ASV	ATN	ATO	ATS
	ATT	ATV	AWO	AWW	BCT	*BEA	BEC	BIK	BOF	BOG	BOZ	BPH
	BRE	BSV	CAN	CAS	CDO	CFI	CGF	CGS	CHA	-CHI	-CLO	CON
	CZP	DAC	DDR	DFO	DFT	DOH	DON	EAB	EDI	EMT	EON	EPI
	ERI	ESV	ETR	EUO	EZT	FAR	FAU	FER	FRA	GIS	GIU	GME
	GON	GOO	HEU	IFR	IHW	IMF	IRR	ISV	ITE	ITH	ITR	-ITV
	ITW	IWR	IWS	IWV	IWW	JBW	JOZ	JRY	JSN	JST	JSW	KFI
	LAU	LEV	LIO	-LIT	LOS	LOV	LTA	LTF	LTJ	LTL	LTN	MAR
	MAZ	MEI	MEL	MEP	MER	MFI	MFS	MON	MOR	MOZ	*MRE	MSE
	MSO	MTF	MTN	MTT	MTW	MVY	MWW	NAB	NAT	NES	NON	NPO
	NPT	NSI	OBW	OFF	OSI	OSO	OWE	-PAR	PAU	PCR	PHI	PON
	PUN	RHO	-RON	RRO	RSN	RTE	RTH	RUT	RWR	RWY	SAF	SAO
	SAS	SAT	SAV	SBE	SBN	SBS	SBT	SFE	SFF	SFG	SFH	SFN
	SFO	SFS	*SFV	SGT	SIV	SOD	SOF	SOS	SSF	SSY	STF	STI
	*STO	STT	STW	-SVR	SZR	TER	THO	TOL	TON	TSC	TUN	UEI
	UFI	UOS	UOZ	USI	UTL	UWY	VET	VFI	VNI	VSV	WEI	-WEN
	YUG	ZON
	Fig. 1

Acknowledgement

The author thanks German Sastre for a helpful discussion. This work was performed using HPC resources from GENCI-IDRIS (project 087069).