Marie Jeffroy

Carlos Nieto Draghi

Anne Boutin

Carlos Nieto-Draghi

A new molecular simulation method to determine both aluminum and cation location in cationic zeolites

come

New molecular simulation method to determine both aluminum and cation location in cationic zeolites

INTRODUCTION

Zeolites are nanoporous crystalline materials that find wide applications in industrial processes, as catalyst, adsorbent or ion exchanger. Over the past 100 years a huge number of zeolites have been synthesized with various pore size and geometry. [START_REF] Baerlocher | Atlas of zeolite framework types[END_REF] The adsorption and diffusion properties of molecules inside the zeolitic porous volume play a crucial role in the various processes using zeolites. 2 Zeolitic materials are aluminosilicate porous solids. Starting from a purely siliceous material, the substitution of a tetravalent silicon atom by a trivalent aluminum atom on a T-site introduces a charge defect that is compensated by a non-framework cation. Adsorption properties of aluminosilicate zeolites are closely related to the location of nonframework cations [START_REF] Fang | Identification of High-CO2-Capacity Cationic Zeolites by Accurate Computational Screening[END_REF] . Location of aluminum atoms can strongly influence the cation location. [START_REF] Mellot-Draznieks | Placement of cations in NaX faujasite-type zeolite using (N,V,T) Monte Carlo simulations[END_REF][START_REF] García-Pérez | A computational method to characterize framework aluminum in aluminosilicates[END_REF][START_REF] Liu | Understanding aluminum location and non-framework ions effects on alkane adsorption in aluminosilicates: A molecular simulation study[END_REF] It is therefore important to get some insights on the aluminum repartition inside the unit cell of zeolite. Due to their large range of applications, many experimental studies have been devoted to zeolites. Structure determinations by RX or neutron experiments have been done for several materials allowing the creation of a database of zeolites structures. [START_REF] Baerlocher | Atlas of zeolite framework types[END_REF] Although the determination of the mean structure and the chemical composition of a zeolite are well established, experimental aluminum distribution measure remains a challenge. Aluminum and silicon atoms are almost impossible to discriminate by the conventional RX methods. Recent experimental developments have however allowed to determinate the aluminum repartition in few cases (see below). Since it is very hard to get a direct determination of aluminum siting, this repartition remains a subject of much debate with large number of dedicated research studies. The so-called Lowenstein rule, which states that two aluminum atoms cannot be neighbors of a common oxygen atom, is the only commonly admitted rule in this subject, [START_REF] Lowenstein | The distribution of aluminum in the tetrahedra of silicates and aluminates[END_REF] even if some exceptions have been observed. [START_REF] Tarling | The structure and Si,Al distribution of the ultramarines[END_REF] For materials having a Si:Al ratio very close to one, the Lowenstein rule imposes an alternated repartition of the silicon and aluminum atoms in the Tsites of the material. For the other Si:Al ratio, very few is known about the aluminum repartition on the zeolitic framework. Even if direct determination is limited to very few cases, other technics such as [START_REF] Oumi | Structural and physicochemical properties of high-silica mordenite[END_REF] Si or [START_REF] Alberti | The determination of the Al-content in the tetrahedra of framework silicates[END_REF] Al NMR technics as well as the analysis of the TO distances from the average structure measured by RX give some indications that in many cases the distribution may be nonrandom. [START_REF] Massiani | 29 Si and 27 Al MAS NMR study of the distribution of Si and Al atoms in various forms of synthetic zeolite omega[END_REF][START_REF] Newsam | Aluminum partitioning in zeolite-L[END_REF][START_REF] Alberti | The determination of (Si. Al) distribution in zeolites. In Guidelines for Mastering the Properties of Molecular Sieves: Relationship Between the Physicochemical Properties of Zeolitic Systems and Their Low Dimensionality[END_REF][START_REF] Han | Direct evidence for the nonrandom nature of Al substitution in zeolite ZSM-5: an investigation by 27 Al MAS and MQ MAS NMR[END_REF] Silicon or aluminum NMR techniques have been applied to various zeolites. [START_REF] Massiani | 29 Si and 27 Al MAS NMR study of the distribution of Si and Al atoms in various forms of synthetic zeolite omega[END_REF][START_REF] Newsam | Aluminum partitioning in zeolite-L[END_REF][START_REF] Han | Direct evidence for the nonrandom nature of Al substitution in zeolite ZSM-5: an investigation by 27 Al MAS and MQ MAS NMR[END_REF][START_REF] Fyfe | Detailed interpretation of the 29 Si and 27 Al high-field MAS NMR spectra of zeolites offretite and omega[END_REF][START_REF] Sanz | Distribution of Si and Al in clintonites: A combined NMR and Monte Carlo study[END_REF][START_REF] Kato | Characteristics of MOR framework zeolites synthesized in Fluoride-Containing media and related ordered distribution of Al atoms in the framework[END_REF][START_REF] Sklenak | Aluminum siting in silicon-rich zeolite frameworks: A combined high-resolution 27 Al-NMR spectroscopy and quantum mechanics/molecular mechanics study of ZSM-5[END_REF][START_REF] Dedecek | Effect of Al-Si-Al and Al-Si-Si-Al pairs in the ZSM-5 zeolite framework on the 27 Al NMR spectra. A combined highresolution 27 Al NMR and DFT/MM study[END_REF][START_REF] Sklenak | Aluminum siting in the ZSM-5 framework by combination of high resolution 27 Al NMR and DFT/MM calculations[END_REF] They allow identifying different environments for aluminum atoms. In materials with a small number of different Tsites, the attribution of the different picks can be done but the determination of the aluminum repartition of the sites remains subject to interpretation. [START_REF] Massiani | 29 Si and 27 Al MAS NMR study of the distribution of Si and Al atoms in various forms of synthetic zeolite omega[END_REF][START_REF] Fyfe | Detailed interpretation of the 29 Si and 27 Al high-field MAS NMR spectra of zeolites offretite and omega[END_REF] Recently, the aluminum repartition has been determined for high silica materials. [START_REF] Newsam | Aluminum partitioning in zeolite-L[END_REF][START_REF] Sklenak | Aluminum siting in silicon-rich zeolite frameworks: A combined high-resolution 27 Al-NMR spectroscopy and quantum mechanics/molecular mechanics study of ZSM-5[END_REF]19 Sklenak et al. observed that the aluminum repartition depends on the synthesis mode of the material. [START_REF] Sklenak | Aluminum siting in silicon-rich zeolite frameworks: A combined high-resolution 27 Al-NMR spectroscopy and quantum mechanics/molecular mechanics study of ZSM-5[END_REF] Extended X-ray adsorption fine structure (EXAFS) experiments can bring some information about the direct environment of aluminum atoms such as distances, number and nature of neighbors atoms, though this information cannot be directly interpreted to deduce the aluminum repartition. [START_REF] Bokhoven | Effect of aluminum on the local structure of silicon in zeolites as studied by Si K edge Xray absorption Near-Edge fine structure: Spectra simulation with a Non-Muffin tin atomic background[END_REF][START_REF] Bugaev | Effect of aluminum on the local structure of silicon in zeolites as studied by Si K edge Xray absorption Near-Edge fine structure: Spectra simulation with a Non-Muffin tin atomic background[END_REF][START_REF] Van Bokhoven | Three-coordinate aluminum in zeolites observed with in situ Xray absorption Near-Edge spectroscopy at the al K-Edge: flexibility of aluminum coordinations in zeolites[END_REF] Using X-ray standing waves, van Bokhoven et al. succeeded in determining unambiguously the aluminum repartition in Scolecite. [START_REF] Van Bokhoven | Determining the aluminium occupancy on the active T-sites in zeolites using X-ray standing waves[END_REF] Very recently, a quantitative analysis was reported that uses a combination of EXAFS and [START_REF] Alberti | The determination of the Al-content in the tetrahedra of framework silicates[END_REF] Al MAS NMR spectroscopy [START_REF] Vjunov | Quantitatively Probing the Al Distribution in Zeolites[END_REF] . Average T-O distance and TOT angle for each T-site of the average structure can give some indirect indication of the aluminum location on the zeolite framework. Some information about aluminum occupancy of the different T-sites are usually extracted from analysis of the mean T-O distance of a particular T-site, longer T-O distance been indicative of a higher Al occupancy. [START_REF] Jones | Al-O and Si-O tetrahedral distances in aluminosilicate framework structures[END_REF][START_REF] Ribbe | Statistical analysis and discussion of mean Al/Si-O bond distances and the aluminum content of tetrahedra in feldspars[END_REF] Those results have to be taken with particular caution since extraframework cations location also has an influence on this mean average distance. [START_REF] Alberti | The determination of the Al-content in the tetrahedra of framework silicates[END_REF] The cation location can also be incorporated in models, to take this effect into account. [START_REF] Alberti | The determination of (Si. Al) distribution in zeolites. In Guidelines for Mastering the Properties of Molecular Sieves: Relationship Between the Physicochemical Properties of Zeolitic Systems and Their Low Dimensionality[END_REF] Simulation studies have been used to help in the determination of aluminum repartition in the framework. Several studies have been done to determine the energy of replacing an aluminum atom by a silicon one, using quantum mechanics (QM) or empirical potentials. 28,29, ,30 However, these calculations do not allow to predict the aluminum repartition for low silica materials, since long-range electrostatic repulsion play an important role in those cases. [START_REF] Ruiz-Salvador | Aluminum distribution in low Si/Al zeolites : Dehydrated Naclinoptilolite[END_REF][START_REF] Channon | A computational investigation into the effect of extra framework cations on the structural composition of heulandite-type zeolites[END_REF] Lewis and co-worker have employed energy minimization using an empirical forcefield to study the aluminum repartition of different natural materials. Ruiz Salvador et al. showed that the most stable aluminum repartition inside the framework for a small Si:Al ratio cannot be predicted by calculating the substitution energy of one silicon by aluminum atom. [START_REF] Ruiz-Salvador | Aluminum distribution in low Si/Al zeolites : Dehydrated Naclinoptilolite[END_REF] They studied the different possible configurations of aluminum (the number has been reduced by taking into account the Lowenstein rule as well as symmetry constraints) to find the most stable one. This methodology was successfully applied to Heulandite, and Clinoptilolite zeolites. [START_REF] Ruiz-Salvador | Aluminum distribution in low Si/Al zeolites : Dehydrated Naclinoptilolite[END_REF][START_REF] Ruiz-Salvador | Silicon-aluminium distribution in dehydrated calcium heulandite[END_REF][START_REF] Ruiz-Salvador | Clinoptilolite-heulandite polymorphism : structural features from computer simulation[END_REF] However, in the case of Goosecreekite the method fails to predict the experimental aluminum repartition. [START_REF] Ruiz-Salvador | Interplay of water, extra-framework cations and framework atoms in the structure of low-silica zeolites : the case of the natural zeolite goosecreekite as studied by computer simulation[END_REF] French et al. have demonstrated that Monte Carlo and energy minimization methods can successfully be used to explore aluminum repartition in zeolites using a large distributed computing approach. [START_REF] French | Probing the structure of complex solids using a distributed computing approach-Applications in zeolite science[END_REF] A different approach was introduced by Soukoulis using Monte Carlo simulations to generate a large number of Si:Al configurations in a diamond rigid lattice. [START_REF] Soukoulis | Monte Carlo simulations of zeolites[END_REF] The model only contained interactions between first and second neighbors, and the parameters where adjusted to reproduce the experimental Si NMR spectra of Faujasite. Similar methods were applied by Herrero et al. to Faujasite, LTA, and Chabasite using the experi-mental crystallographic structures. In a first approach, Herrero et al. include only short-range interactions. [START_REF] Herrero | Short-range order of the silicon/aluminum distribution on the faujasite framework[END_REF] Later version of the model includes long-range interactions [START_REF] Herrero | Long-versus shortrange Si, Al ordering in zeolites X and Y[END_REF] between the aluminum and silicon atoms, as well as the polarization energy of the oxygen atoms. 14 , 40 The methodology is restricted to zeolites with a single T-site. Soukoulis model was extended to materials having different T-sites by adjusting different parameters for the different sites, [START_REF] Li | A simulated annealing study of Si,Al distribution in the omega framework[END_REF] or by adding in the calculation the relative energy of the different T-sites. [START_REF] Hong | In situ disorder-order transformation in synthetic gallosilicate zeolites with the NAT topology[END_REF] In this family of methods, extraframework species are never taken into account. Adsorption and diffusion properties of the zeolitic materials have been extensively studied by molecular simulation. 43 , 44 , 45 , 46 , 47 Forcefields to describe interactions between adsorbate and adsorbent, but also to model the flexibility of the framework have been developed with success. [START_REF] Auerbach | Behavior of benzene in Na-X and Na-Y zeolites : Comparative study by H2 NMR and molecular mechanics[END_REF][START_REF] Jaramillo | New force field for Na cations in faujasite-type zeolites[END_REF][START_REF] Di Lella | Molecular simulation studies of water physisorption in zeolites[END_REF][START_REF] Calero | Understanding the role of sodium during adsorption: A force field for alkanes in sodiumexchanged faujasites[END_REF][START_REF] Demontis | Molecular dynamics studies on zeolites. 3. Dehydrated zeolite A[END_REF][START_REF] Kramer | Interatomic force fields for silicas, aluminophosphates, and zeolites: Derivation based on ab initio calculations[END_REF][START_REF] Smirnov | A molecular dynamics computer study of window fluctuations in zeolite A[END_REF][START_REF] Demontis | Structure and dynamics of zeolites investigated by molecular dynamics[END_REF][START_REF] Ramsahye | Cation mobility and the sorption of chloroform in zeolite NaY: molecular dynamics study[END_REF][START_REF] Leroy | Self-diffusion of nalkanes in silicalite using molecular dynamics simulation: A comparison between rigid and flexible frameworks[END_REF][START_REF] Nicholas | Molecular modeling of zeolite structure. 2. Structure and dynamics of silica sodalite and silicate force field[END_REF][START_REF] Jeffroy | Molecular simulation of zeolite flexibility[END_REF] To overcome the general problem of the lack of information about aluminum location, two general approaches have been developed. The first one models explicitly the aluminum and silicon atoms in the framework and considers one or several random aluminum distribution (taking into account the Lowenstein rule). [START_REF] Calero | Understanding the role of sodium during adsorption: A force field for alkanes in sodiumexchanged faujasites[END_REF][START_REF] Demontis | Molecular dynamics studies on zeolites. 3. Dehydrated zeolite A[END_REF][START_REF] Kramer | Interatomic force fields for silicas, aluminophosphates, and zeolites: Derivation based on ab initio calculations[END_REF][START_REF] Ramsahye | Cation mobility and the sorption of chloroform in zeolite NaY: molecular dynamics study[END_REF][START_REF] Maurin | Combining the Monte Carlo technique with 29 Si NMR spectroscopy: Simulations of cation locations in zeolites with various Si/Al ratios[END_REF] The results are then usually averaged over different repartitions of aluminum atoms. However, in these methods, the number of considered repartitions remains low to be statistically fully representative. The second approach, usually called T-atom model, considers that aluminum and silicon can be modeled as an average atom, whose physical properties are intermediate between the aluminum and the silicon ones. [START_REF] Di Lella | Molecular simulation studies of water physisorption in zeolites[END_REF][START_REF] Smirnov | A molecular dynamics computer study of window fluctuations in zeolite A[END_REF]61 Both approaches can lead to satisfactory results. Different studies analyzed the influence of the explicit representation of aluminum atoms on extraframework cation location and adsorption properties. Mellot-Draznieks et al. compared the results given by two different models, one with explicit Si:Al and the other using average T-atoms on the extraframework cation distribution of the NaX zeolite. [START_REF] Mellot-Draznieks | Placement of cations in NaX faujasite-type zeolite using (N,V,T) Monte Carlo simulations[END_REF] Calero et al. studied the influence of the aluminum repartition on the adsorption properties of alkanes on eight zeolitic materials. [START_REF] García-Pérez | A computational method to characterize framework aluminum in aluminosilicates[END_REF][START_REF] Liu | Understanding aluminum location and non-framework ions effects on alkane adsorption in aluminosilicates: A molecular simulation study[END_REF] They divided the materials in two groups: some in which the adsorption properties are sensitive to the chosen aluminum repartition, and the other which are not. By comparing the calculated adsorption properties with the experiments for the first group of materials, they can indirectly deduce the aluminum repartition on the framework of these materials. However, since the adsorption properties have to be calculated for each possible aluminum repartition of the material, this method is restricted to simple cases, where only a small number of distributions are possible. Therefore, a need exists to design a new methodology to take into account a large number of aluminum atoms configurations, corresponding to the experimental repartition over the different unit cells. In the present paper, we propose a new methodology to predict aluminum atoms repartition in a given zeolitic structure, as well as the extraframework cation location in various thermodynamic conditions, such as hydra-tion. This methodology allows generating a huge number of aluminum atoms configurations along the simulation. To our knowledge this is the first time that the properties of the materials are averaged over all the different configurations in the thermodynamic limit (and not only over the few lowest-energy configurations). This method differs from and complements energy minimization approaches on three main points: 1. it does not include any local relaxation of the framework, thus does not look for the most stable structure of the complete material. 2. It focus on the determination of aluminum atoms and cations distribution in a given average zeolite framework. It thus allows predicting a distribution for different experimental structures that can be obtained for the same materials but various synthesis ways. 3. It is fully compatible with standard MC adsorption simulation and allows computing thermodynamic properties averaged over equilibrium distribution. The method has been applied to tens of zeolitic materials. The paper is organized as follow. The proposed methodology will be detailed in the next section. Later, the aluminum repartition obtained for various zeolites will be presented, and compared with experimental data. Finally, the question of the extraframework cation distribution will be addressed in the presence of adsorbed molecules (water) followed by the conclusion and perspectives.

METHODOLOGY 2.1 Simulation method:

This method aims at determining the aluminum repartition over the different T-sites of zeolites. The proposed method introduces in standard Monte Carlo simulations a new move to exchange aluminum and silicon atoms on the different Tsites of the material. The new configuration is accepted on the basis of an energy criterion, like for standard Monte Carlo simulations. In this way, the calculated properties of the material on the simulation are averaged on many aluminum and silicon atoms configurations in the thermodynamic limit (statistics are usually performed on ~1.10 5 configurations). It is worth noticing that the generation of the configuration takes into account not only the interactions between framework atoms (NAl, NSi), but also interactions with extraframework cations (Ncations) and potential adsorbed species (Nads). Simulations are done at constant (NAl,NSi,Ncations,V,T), where the extra-framework cations can explore together the whole space of configurations whereas the silicon and aluminum atom coordinates are restricted within the T-sites of the structure. One can then write the partition function of the system as follows:

ܳ N Al ,N Si ,N cations ,V,T ∝ ∑  ݀r N expቀ-ߚܷሺr N ሻቁ r cations N r Al N ,r Si N (1)
The new MC move concerns the exchange of randomly selected Si and Al atoms and is accepted with the following probability,

ܲ = షഁ∆ೆ ೕ ொ N Al ,N Si ,N cations ,V,T (2)
The energy (∆Uij) term is calculated using an analytical forcefield (see section 2.4). This MC move can be easily implemented in Grand Canonical ensemble when simulating adsorption phenomena (fix µads). Moreover, the framework structures are considered fixed along the simulation, the aluminum and cation locations thus correspond to the expected thermodynamic equilibrium for a given set of T and O coordinates. Additional technical details and a critical analysis of the method are available in the SI.

2.2 Zeolite model: the zeolitic structure is taken from RX experimental structure determination. The zeolitic structure is considered rigid to limit the cost of the simulation. In principle, the method could be applied for flexible structures (using a realistic force field for the framework) with however a significant increase of the computation time. Since the Si:Al ratio simulated corresponds to the experimental one; the footprint of the experimental aluminum repartition is contained in the average structure. Thus without taking into account explicitly the flexibility, our model can still be sensible to differences on the T-atoms environment.

Zeolite structures and composition:

The structures of zeolites were taken from experimental diffraction studies characterizations: Analcime (ANA), [START_REF] Haüy | Traité de minéralogie[END_REF] Faujasite (FAU), [START_REF] Mortier | International Zeolite Association. Structure Commission. Compilation of extra frameword sites in zeolites[END_REF] Ferrierite (FER), [START_REF] Vaughan | Crystal structure of zeolite ferrierite[END_REF] Goosecreekite (GOO), [START_REF] Rouse | Crystal structure of the zeolite mineral goosecreekite, CaAl2Si6O16. 5H2O[END_REF] Linde Type A (LTA), [START_REF] Fyfe | Investigation of the dealumination of high silica zeolite A (ZK-4) by 29 Si magic angle-spinning NMR spectroscopy[END_REF] Linde Type L (LTL), [START_REF] Artioli | Synchrotron X-Ray rietveld study of perlialite, the natural counterpart of synthetic zeolite-L[END_REF] Mordenite (MOR), [START_REF] Klinowski | A re-examination of Si,Al ordering in zeolites NaX and NaY[END_REF] Natrolite (NAT), [START_REF] Sulikowski | Dealumination of zeolites with silicon tetrachloride vapour. Part 6. Zeolites[END_REF] Theta-1 (TON), [START_REF] Barri | Structure of theta-1, the first unidimensional medium-pore high-silica zeolite[END_REF] Mobil FIve (MFI), [START_REF] Olson | Crystal-structure and structure-related properties of ZSM-5[END_REF] and Mobil ELeven (MEL). [START_REF] Kokotailo | Synthesis and structure of synthetic zeolite ZSM-11[END_REF] Si:Al ratio of the simulated material was chosen as close as possible to the experimental one. Cations and water molecules are explicitly represented in the simulation box. In this work all extraframework cations in the simulation are sodium. For experimental hydrated structures, the experimental number of water molecules has been incorporated in the simulation box.

The exact composition of the simulated zeolites can be found in supplementary information (SI).

Forcefield:

Within the framework, the silicon, aluminum and oxygen atoms interact via electrostatic interactions. Parameters describing the interaction between adsorbent and extraframework cation were taken from previous studies. [START_REF] Di Lella | Molecular simulation studies of water physisorption in zeolites[END_REF] The total energy of the system was computed using an electrostatic and a repulsive dispersion terms. We used the Kiselev approximation for the dispersion-repulsion term between adsorbate or extraframework cations and adsorbent. [START_REF] Fuchs | Adsorption of guest molecules in zeolitic materials: Computational aspects[END_REF] Interaction energy between sodium and oxygen was modeled with a Buckingham potential adjusted by Jaramillo et al. [START_REF] Jaramillo | New force field for Na cations in faujasite-type zeolites[END_REF] The Lennard-Jones potential parameters for oxygen atoms (O-O) were adjusted in a previous work. 74 These parameters were successfully used to reproduce alkanes, alkenes and water adsorption in various types of zeolites. [START_REF] Di Lella | Molecular simulation studies of water physisorption in zeolites[END_REF][START_REF] Pascual | Adsorption of hydrocarbons in zeolites from molecular simulations. The alkane-ferrierite system revisited[END_REF]76,[START_REF] Trzpit | The Effect of local silanol defects on water adsorption in Silicalite-1 zeolite. A joint experimental and molecular simulation study[END_REF][START_REF] Coudert | Water nanodroplets confined in zeolite pores[END_REF][START_REF] Cailliez | Thermodynamics of water intrusion in nanoporous hydrophobic solids[END_REF] Water molecules were modeled with the well-known model TIP4P. 80 This model has already successfully been used to simulate the adsorption properties of various siliceous and cationic zeolites. [START_REF] Di Lella | Molecular simulation studies of water physisorption in zeolites[END_REF][START_REF] Coudert | Water nanodroplets confined in zeolite pores[END_REF] The sodium cation-water LJ interactions parameters are taken from the work of Dang et al. [START_REF] Dang | Mechanism and thermodynamics of ion selectivity in aqueous-solutions of 18-crown-6 ether. A molecular-dynamics study[END_REF] All cross parameters have been obtained using the Lorentz-Berthelot mixing rules. All interaction parameters are provided in SI.

2.5 Simulation details: All simulations were done using the canonical ensemble (N,V,T). The method is easily extendable to the Grand Canonical ensemble to calculate adsorption isotherms. The usual translation and rotation moves were used for the extraframework species. In order to accelerate convergence a bias "translational jump move" was used. This move consists in a deletion followed by an insertion for a particular molecule in order to do a long distance translation inside the simulation box. More detail about this move can be found in ref. 50. The runs lasted for some 200 million steps. For our simulations, we used a structure from Fälth and coll. [START_REF] Fälth | Structure of scolecite from Poona, India[END_REF] of a natural Scolecite. We used a Si:Al ratio of 1.55 which is slightly higher than in ref. 23. The reason is that with a Si:Al ratio of 1.5 we rapidly reach a blocked configuration since all Si-Al exchanges performed lead to neighboring aluminum atoms, which is forbidden due to the Lowenstein rule. To avoid this problem, we could replace the total interdiction of having two neighboring aluminum atoms by an energetic penalty for a pair of aluminum neighbors. However, the value of this penalty would have to be adjusted, probably to a different value for the different zeolites, and we choose not to introduce an additional parameter.

Our simulation results show that aluminum atoms are located in sites T1a and T1b. The occupancy of the different T sites is represented in Figure 2. These results are in perfect agreement with the experimental results of Van Bokhoven et al. [START_REF] Van Bokhoven | Determining the aluminium occupancy on the active T-sites in zeolites using X-ray standing waves[END_REF] These results indicate the capacity of our model to capture the physics of the studied systems.

The same method has been applied to different materials in order to find some insight on the parameters governing the aluminum distribution on these systems. This method leads to better agreement between the estimated Si:Al ratio and the experimental one, though some discrepancy still exists. [START_REF] Alberti | The determination of (Si. Al) distribution in zeolites. In Guidelines for Mastering the Properties of Molecular Sieves: Relationship Between the Physicochemical Properties of Zeolitic Systems and Their Low Dimensionality[END_REF][START_REF] Alberti | The determination of the Al-content in the tetrahedra of framework silicates[END_REF][START_REF] Alberti | On the influence of (Si-Al) disorder on the TO distances measured in zeolites[END_REF] To compare our simulation results to the estimations obtained by those methods, we studied a material belonging to the MOR family completely exchanged with sodium whose structure has been published by Schlenker et al. [START_REF] Schlenker | Positions of cations and molecules in zeolites with the mordenite-type framework. VIII. Dehydrated sodium-exchanged mordenite[END_REF] The simulated Si:Al ratio was of 5, leading to 8 sodium cations/uc. Our simulation results are represented in Figure 3 whereas the percentage of aluminum occupancy obtained in our simulations are compared with experiments and the results obtained by Jones and Alberti methods in Figure 4. [START_REF] Dedecek | Effect of Al-Si-Al and Al-Si-Si-Al pairs in the ZSM-5 zeolite framework on the 27 Al NMR spectra. A combined highresolution 27 Al NMR and DFT/MM study[END_REF][START_REF] Sklenak | Aluminum siting in the ZSM-5 framework by combination of high resolution 27 Al NMR and DFT/MM calculations[END_REF] The order of occupancy of the sites obtained by all the methods is the same: T3 > T4 > T1 > T2. These results are also in perfect agreement with other experimental studies. [START_REF] Shiokawa | Crystal-structure of synthetic mordenites[END_REF] [START_REF] Oumi | Structural and physicochemical properties of high-silica mordenite[END_REF] Si MAS NMR, one group is synthesized in the presence of fluoride ion and the other following the conventional way. [START_REF] Kato | Characteristics of MOR framework zeolites synthesized in Fluoride-Containing media and related ordered distribution of Al atoms in the framework[END_REF] Their results suggest that aluminums are located in T1, T2 and T4 in the fluored samples whereas they occupy all the different T-sites in conventional structures. This suggests that the aluminum atoms location depends on the synthesis mode. This question will be further discussed in the following section. Calero et al. studied by molecular simulation the adsorption of alkanes in sodium Mordenite. [START_REF] Calero | Understanding the role of sodium during adsorption: A force field for alkanes in sodiumexchanged faujasites[END_REF] They calculated the adsorption isotherm and Henry constants for the different possible aluminum repartitions in this material. By comparison with the experimental adsorption properties, they concluded that the aluminum atoms are located in the T3 and T4 sites of the Mordenite framework. Moreover French et al. by minimizing a large amount of randomly distributed aluminum have obtained a larger occupancy of T3 over T4 for lowest energy structures. [START_REF] French | Probing the structure of complex solids using a distributed computing approach-Applications in zeolite science[END_REF] Our simulation results fairly agree with all these findings. The comparison between our simulation results and the prediction by the Alberti's method agrees well. This is not only the case for Mordenite materials, but also for several other materials we studied by simulation belonging to the GOO and ANA families. All the results are reported in SI. It is important to note that our simulation model is sensible to the geometric parameters distances of the framework, but also to the angles and cation location.

Comparison with other methods

3.4

The case of Analcimes: influence of the synthesis way: It has been suggested that the aluminum location may depend on the synthetic way. Sklenak et al. reported different aluminum repartition on MFI zeolites [START_REF] Sklenak | Aluminum siting in silicon-rich zeolite frameworks: A combined high-resolution 27 Al-NMR spectroscopy and quantum mechanics/molecular mechanics study of ZSM-5[END_REF] and Kato et al. for different MOR zeolites. [START_REF] Kato | Characteristics of MOR framework zeolites synthesized in Fluoride-Containing media and related ordered distribution of Al atoms in the framework[END_REF] Mazzi and Galli published a series of natural Analcime structures, whose samples come from different origins. [START_REF] Mazzi | Is each analcime different?[END_REF] The samples almost have the same composition and Si:Al ratio. We calculated the aluminum distribution of the different samples. We chose to use a Si:Al ratio of 2.2, instead of 2 to avoid having blocked configurations due to the Lowenstein rule. The detailed composition of the simulated samples is reported in SI.

In this case we simulated a fully hydrated sample. Simulation results are reported for two different samples ANA1 and ANA6 in Figure 5 (for the description of the samples see ref 87). Their structure contains two different T-sites T1 and T2. The repartition obtained by Mazzi et al. [START_REF] Mazzi | Is each analcime different?[END_REF] and by Jones equation and the Alberti methods [START_REF] Alberti | The determination of (Si. Al) distribution in zeolites. In Guidelines for Mastering the Properties of Molecular Sieves: Relationship Between the Physicochemical Properties of Zeolitic Systems and Their Low Dimensionality[END_REF] are also reported. The results show that in ANA1 the most occupied site is T1 whereas in ANA6 the most occupied site is T2. This inversion for two materials having the same composition, and belonging to the same zeolitic group shows that the aluminum location is strongly dependent of the synthesis conditions. It must be noticed that again our method is in agreement with the prediction of Alberti It is worth noticing that the synthesized structures are metastable and the zeolite formation may not be thermodynamically driven. This limits the validity of the energy minimization approaches and may explain the discrepancy with experimental results obtained for some materials. [START_REF] Ruiz-Salvador | Interplay of water, extra-framework cations and framework atoms in the structure of low-silica zeolites : the case of the natural zeolite goosecreekite as studied by computer simulation[END_REF] While not fully predictive, our method allows taking into account the variability in the framework structures depending on the synthetic routes. The proposed method is thus a useful alternative for aluminum location determination in complement to diffraction experiments.

Correlation with energy of sites:

Aluminum distribution is very often predicted by calculating the energy of one aluminum atom located in the different T-sites of the material. This approach considers the energy of aluminum atoms embedded in a particular configuration, thus the repartition of aluminum atoms predicted this way does not take into account the interactions between distant aluminum atoms. We calculated with our method the energy minimum of each configuration of a system containing one aluminum atom located in one of the different T-sites of Mordenite and one cation free to move. The results are reported in Table 1. The aluminum distribution can then be estimated using Boltzmann probabilities (see SI for technical details). The predicted distribution is represented in Figure 6 and compared to the simulation results. T3 is the most favorable site using both approaches. However the occupancy of the site obtained from energy calculation is overestimated compared to the simulation results.

Consequences of aluminum position on extraframework cation location:

Since extraframework cations are free to move inside the simulation box, we can also obtain some information about their location. The cation distribution obtained corresponds to an average of the different cations distribution over different aluminum configurations. The position of extraframework cation influences the material adsorption properties. [START_REF] Di Lella | Molecular simulation studies of water physisorption in zeolites[END_REF][START_REF] Bellat | Unusual hysteresis loop in the Adsorption-Desorption of water in NaY zeolite at very low pressure[END_REF] Thus, it is interesting to know the influence of the aluminum distribution on the extraframework cation location. This question is the object of the following sections.

Cation location-Comparison with experimental data:

Extraframework cation location have been calculated for several zeolitic materials and compared to the experimental data when available. In this section, we will describe only the results for the materials for which we found some experimental data on cation location. All the additional results can be found in SI.

In Faujasite type zeolites, we observe in our simulations that extraframework cations location varies with the number of cations (Si:Al ratio). In Na48Y, sodium cations occupy sites I, II. When the Si:Al ratio increases, the number of occupied site I decreases in favor of sites I' until all sites I' are occupied. Then sites III/III' start to be filled. These results are in agreement with experimental data (see Table 2). [START_REF] Di Lella | Molecular simulation studies of water physisorption in zeolites[END_REF] Schlenker et al. have studied the extraframework cation location in Na-exchange Mordenite by RX diffraction experiment. [START_REF] Schlenker | Positions of cations and molecules in zeolites with the mordenite-type framework. VIII. Dehydrated sodium-exchanged mordenite[END_REF] They published the zeolitic structure as well as the extraframework cation location inside the porous volume. A more recent work, by Maurin et al. studied the cation location in dehydrated Mordenite by combined [START_REF] Oumi | Structural and physicochemical properties of high-silica mordenite[END_REF] Si NMR and molecular simulation. [START_REF] Maurin | Combining the Monte Carlo technique with 29 Si NMR spectroscopy: Simulations of cation locations in zeolites with various Si/Al ratios[END_REF] These results are reported with our simulation results in Table 2. In Mordenite cations occupy sites I, IV and VI in agreement with experimental results.

Zeolite-L has been the subject of different experimental studies and the extraframework cation distribution has been characterized for various samples. We calculated the cationic distribution of sodium-exchanged LTL containing 9 sodium per unit cell. The structure is taken from a study of Baerlocher et al. on a hydrated sample. [START_REF] Baerlocher | The structure of the synthetic zeolite (K, Ba)-G[END_REF] The cation repartition has been calculated and is reported in Table 2 together with experimental data. We observe a good agreement when compared with the experiments. In all cases, sodium cations fully occupy sites B whereas sites C and D are partially occupied. 3.6.2 Influence of the aluminum location on the cationic distribution: Due to the atomic charge difference, the repulsion of sodium is expected to be smaller with an aluminum than with silicon atoms. Extraframework cations are thus expected to be located close to aluminum atoms.

In order to quantify the potential influence of an aluminum atom on the location of an extraframework cation we calculate the number of aluminum atoms close to an extraframework cation. We choose a threshold of 4 Å, which in most of the structures corresponds to the first minimum in the Na-Al radial distribution function. However, this value is dependent on the Si:Al ratio, i.e. the higher the Si:Al ratio is, the smaller this value is expected to be. If the local Si:Al ratio around a cation is lower than the global Si:Al ratio, the cation is sensible to the presence of aluminum atom, and is located close to them. By analogy with selectivity coefficient in adsorption, we define an aluminum order parameter α NaAl for extraframework cation:

 α NaAl = n Al,loc n Si,loc n Si,glob n Al,glob = (Si : Al) glob (Si : Al) loc (3)
where nAl,loc is the average number of aluminum atom close to an extraframework cation, nSi,loc is the average number of silicon atoms close to an extraframework cation, nAl,glob the number of aluminum atoms per unit cell, and nSi,glob the number of silicon atoms per unit cell. (Si:Al)glob is the global Si:Al ratio, and (Si:Al)loc represents the average local Si:Al ratio around an extraframework cation. If αNaAl > 1 it indicates that extraframework cations are located close to the aluminum atom. An αNaAl < 1 would indicate that extraframework cations are located close to silicon atoms. And αNaAl = 1 indicates that there is no influence of the aluminum atom location on the cation location. This order parameter has been calculated for various materials, having different Si:Al ratio and different zeolitic structure, for each cationic site. The data obtained is reported in SI. Except for one case, which will be discussed in further detail later, the order parameter is higher than one. The value of this order parameter highly differs from one cationic site to another. It goes from 1 to 95. We did not found any general factor to rationalize the value of α. We concentrate our attention to FAU structure, for which all the T-sites are geometrically equivalent, thus having a homogenous aluminum distribution. We studied four different Si:Al ratio: Si:Al = 3, Si:Al = 2.69, Si:Al = 2 and Si:Al = 1.4. As seen before, in these samples, cations occupy sites I, I', II, III, and III', where cationic site were labeled using Smith classification. [START_REF] Smith | Chapter 15 : Faujasite-type structures : Aluminosilicate framework: Positions of cations and molecules : Nomenclature[END_REF] Values of the αNaAl calculated for faujasite are reported in Table 3. We can see that for cationic sites that are fully occupied, αNaAl is equal to one. This is the case of sites II for all Si:Al ratio and sites I for Na48Y and of site I' in Na64Y and Na80X. This means that Si:Alloc is equal to Si:Alglob. This is linked to the symmetry of Faujasite. We will concentrate on sites II to develop this point. In fact, as said before, all T-sites are equivalent in Faujasite materials, which means that they are all neighbors of site II. Since all site II are occupied, all T sites are neighbors of an occupied site II. When we average the Si:Al ratio on all occupied sites II we thus average on all the different T-sites. Thus, we find that Si:Alloc is equal to Si:Alglob. In Na52Y we calculated a value of αNaAl that is higher than one for site I'. This means that the presence of an aluminum atom close to a site I' stabilizes the cation in this site. Surprisingly, in this material the ratio for site I is smaller than one. Since cations are more likely to go closer to aluminum atoms, we did not expect a ratio smaller than one. This unexpected behavior of aluminum atoms close to site I is also due to symmetry reasons. All T-sites are neighbors of a cation in site I or in site I'. If there is an excess of aluminum atoms around site I', this induce a local defect of aluminum atoms in sites I. In Na80Y we can see that for sites III and III', αNaAl is higher than one, which means that the presence of an aluminum atom stabilizes a cation in those sites.

We can see that many factors may affect the relation between aluminum atoms and extraframework cations.

The question is then to know how the aluminum distribution affects the average cationic repartition.

To answer this questions, we have compared the cationic location on two materials belonging to the same zeolitic group, having the same chemical composition but two different structures and thus different aluminum atom distributions. This is the case for Analcimes. We can then compare the cation distribution in different Analcimes samples. We will detail the observed extraframework cation location in ANA1 and ANA6 structures. The results concerning those two structures are reported in Table 4. Experimental data from Mazzi and Galli is also reported. [START_REF] Mazzi | Is each analcime different?[END_REF] Structural information as well as additional results for the other Analcime materials can be found in SI. In ANA1 cations mainly occupy site A whereas in ANA6 cations mainly occupy sites B. In those two materials, the predicted aluminum atom distribution is different. In ANA1 aluminum are mainly located in site T1 whereas that in ANA6 aluminum atoms mainly occupy site T2. Cationic site A is located closer to site T1, whereas site B is closer to site T2. We can clearly see in this case the influence of the aluminum atoms distribution in the extraframework cation position among the unit cell.

Comparison between full Si:Al representation vs. T-atom approximation:

In order to see the influence of the aluminum atom distribution on the extraframework cation location of zeolites, we can also compare the results of simulation with the new method and the results obtained using the T-atom approach. For different zeolitic materials we calculated the extraframework cation location obtained with both methods. Part of the results is reported in Table 5. Other results can be found in SI. From those results we classified the zeolites in three different groups.

The first group corresponds to zeolites whose cationic distribution is not sensitive to aluminum atom position. This is the case of LTA and FAU having Si:Al ratio higher than 2. The second group corresponds to zeolites whose cationic distribution is only slightly modified by the aluminum atom distribution. We thus do not expect a strong influence of the aluminum atoms distribution on the adsorption properties of those materials. This group contains TON, and MFI, MOR and FAU with Si:Al higher than 2. The different distributions of extraframework cations on MFI zeolite between T-atom model and Si/Al explicit representation are represented on Figure 7. In MFI, cations are mainly located in intersections but their position in this site slightly differs between the two models. Results obtained for Faujasite zeolites are represented in Table 5. In Faujasite zeolites we can observe a displacement from site III to site III'. This effect has already been noticed by Mellot et al. In the third group we find zeolites were the aluminum distribution induces strong rearrangement of the extraframework cations. This group contains MEL, FER and LTL. Figure 8 represent a projection of the cation distribution obtained with the two models for MEL. We can clearly see the formation of a new cationic site when the aluminum atoms are represented explicitly in the simulation.

CONCLUSION AND PERSPECTIVES

We have presented a new simulation method that allows predicting the aluminum atom distribution in zeolitic framework. The method based on a Monte Carlo algorithm allows predicting both aluminum location and cationic distribution in the material which can be hydrated or not. The only required input of this method is the average experimental structure of the framework, which can be obtained by standard diffraction experiments. This method has been tested on various zeolites with success. The prediction of aluminum atom repartition is in agreement with the experimental and model data available in the literature. The method using experimental average structure allows to indirectly taking into account the kinetic effects existing in the synthesis pathway. Interactions between aluminum atoms (electrostatics) as well as the local environment (T-O-T angles, T-O distances) are determinant to predict the aluminum distribution.

The cationic location was also predicted with success in various materials. The influence of the aluminum distribution varies from one material to another. This method can be used for other zeolites, exchanged with different cations and to calculate properties of the material, such as adsorption isotherms.

3. RESULTS AND DISCUSSION 3 . 1

 31 Aluminum atoms location: convergence test:The average aluminum atoms repartition has been calculated for several zeolitic topologies (containing between 1 and 12 different crystallographic T-sites): MOR, TON, MFI, NAT, FER, LTL, ANA, FAU, LTA, GOO, MEL. Among these structures, in the LTA and FAU framework all the T-sites are crystallographic equivalent. NAT and GOO are usually classified in the group of ordered materials, meaning that the aluminum atoms repartition is considered non-random. The average occupancy of the T-sites has been calculated from the configurations obtained during the simulations using the average experimental structure of the framework as an input information. The example of Goosecreekite is presented in Figure1. We observed that crystallographically equivalent T-sites have the same aluminum occupancy within statistical uncertainties. This indicated that the convergence is reached in our simulations.

Figure 1 . 23 Figure 2 .

 1232 Figure 1. Aluminum repartition in Goosecreekite material. The percentage of occupancy for the different T-site is represented. Site T1 and T2 are almost fully occupied. The only other significantly occupied site is site T8.

  based on average distances and angles: Different methods exist for the determination of the aluminum repartition of the different T-sites using the average experimental structure as an input information. The first one extracts the aluminum occupancy of the different T-sites direct-from average T-O distance of each T-site. Jones 25 and later Ribbe et al. 26 established two different linear relations between the T-O distance and the aluminum atom occupancy. Those relations works well for completely disordered aluminum atoms repartitions, but always underestimate the total number of aluminum atoms of the partially ordered materials. Alberti et al. proposed a corrected version of this method including angular constraints and the presence of extra framework cations.

Figure 3 .

 3 Figure 3. Aluminum repartition calculated by simulation of the different T sites (T1, T2, T3 and T4) of Mordenite. The sphere diameter is proportional to the aluminum population in the corresponding site.

Figure 4 .

 4 Figure 4. Comparison of aluminum occupancy of the different T-sites of Mordenite calculated with different methods: our results (grey), using Jones' 25 model (red), using Alberti's 11 model (purple) and experiments from refs 17 and 18.

Figure 5 .

 5 Figure 5. Aluminum repartition of (a) ANA1 and (b) ANA6.

  Simulation results reveal a partial occupancy of T1 and T2 whereas energies calculation predicts almost zero population. This example illustrates the influence of aluminum-aluminum interaction on the aluminum distribution. Ruiz-Salvador et al. reached similar conclusions in the case of Clinoptilolite 31 and ZSM-5.[START_REF] Ruiz-Salvador | Aluminium distribution in ZSM-5 revisited: The role of Al-Al interactions[END_REF]

Table 1 .Figure 6 .

 16 Figure 6. Aluminum repartition of MOR according to site energies (see text for definition).

Figure 7 .

 7 Figure 7. Comparison of Na + cation distribution on MFI zeolite using full Si:Al representation (top) and T-atom approximation (bottom). Two orientations are shown (The MFI model contains 4 aluminums). Small points represent the projection of the center of mass of cations. Size of green spheres (top) represents the percentage of aluminum occupancy on different T sites.

4

 4

Figure 8 .

 8 Figure 8. Comparison of Na + cation distribution on MEL zeolite using full Si:Al representation (left) and T-atom approximation (right). Small green points represent the center of mass projection of cations. Green spheres (left) represent aluminum occupancy on different T sites. Sphere diameters are proportional to the percentage of occupancy of the sites. Small blue points represent a new site for Na + cations when the full Si:Al representation is used.

Table 2 .

 2 Extraframework cation location in FAU, MOR and LTL zeolites. Simulation results are compared to the available experimental data. The simulated LTL material is hydrated and contains 44 water molecules per unit cell. To our knowledge there is no experimental data concerning cation location in a completely exchange zeolite L.

		FAU Sample	I	I'	II	III	III'	non-localized	ref
		Na48Y	15.5	0.5	32.0	0	0	0
	Simulation	Na52Y Na64Y	11.9 0.7	8.1 31.3	32.0 32.0	0 0	0 0	0 0
		Na80Y	1.0	31.0	32.0	12	4	0
	Experiments	Na54Y	7.04	13.76	29.44	0	0	3.76	91
		Na58Y	9.34	16.67	31.04	0	0	0.94	92
		Na81X	3.84	32.0	30.78	7.87	0	6.50	92
		MOR Sample	I	III	IV	VI		non-localized	Ref
	Simulation	Na8	3.0	0	3.75	1.23		0.02
	Experiments	Na8	3.1	0	2.6	1.5		0.8	84
		Na8.1	4.0	0	4.1				60
		LTL Sample	A	B	C	D	Total num-ized cation ber of local-		Ref
	Simulation	Na9	0	2.00	1.22	5.59	8.81
	Experiments	Na3K6	1.4	2.0	2.7	3.6	9.6		93
		NaK	0	2.0	2.9	5.3	10.2		94
		NaK	0.0	2.0	3.0	5.1	10.1		95
			0	2.0	3.0	4.0	9.0		96

Table 3 .

 3 Order parameter αNaAl for the different sites of Faujasite materials (see the text for the definition of αNaAl).

	Zeolite	Na48Y	Na52Y	Na64Y Na80X
	Cationic
	site
	I	1.03	0.85
	I'		1.58	1.00	1.01
	II	1.00	1.00	1.00	1.00
	III				1.32
	III'				1.16

Table 4 .

 4 Cation location observed in two different analcimes: ANA1 and ANA6 (structures are labeled according to Mazzi and Galli[START_REF] Mazzi | Is each analcime different?[END_REF] ). The cation distribution as well as the aluminum occupancy of the different T sites of analcime is reported. Experimental data about cation location are shown for comparison.[START_REF] Mazzi | Is each analcime different?[END_REF]

		Aluminum
		repartition	Sodium cation location
		(simulation)
	Site	T1	T2	Sim	A	Exp	Sim	B	Exp
	ANA1	44.81	4.13	11.99		13.12	3.01		1.84
	ANA6	17.77	58.20	7.22		9.76	7.77		6.16

Table 5 .

 5 Comparison between cation locations calculated with two different models: the T-atoms model and our method with Si/Al differentiation (labeled Si/Al) for Faujasite

		Model	I	I'	II	III	III'
	Na48Y	Si/Al T-atoms	15.5 15.9	0.5 0.1	32.0 32.0	0 0	0 0
	Na52Y	Si/Al T-atoms	11.9 12.0	8.1 8.0	32.0 32.0	0 0	0 0
	Na64Y	Si/Al T-atoms	0.7 0.5	31.3 31.5	32.0 32.0	0 0	0 0

ACS Paragon Plus EnvironmentChemistry of Materials

ACKNOWLEDGMENT

The authors thank Francesco di Renzo for fruitful discussion on this topic.

ASSOCIATED CONTENT

AUTHOR INFORMATION Corresponding Author

* E-mail: anne.boutin@ens.fr Present Addresses † Marie Jeffroy is now at Saint-Gobain Glass, 39 quai Lucien Lefranc, 93303 Aubervillliers. France

Notes

The authors declare no competing financial interest.


Antonin Grenier

Ana Gabriela Porras Gutierrez

Henri Groult

Damien Dambournet
email: damien.dambournet@upmc.fr

Ana Gabriela

Porras Gutierrez

Modified coin cells to evaluate the electrochemical properties of solid-state fluoride-ion batteries at 150°C

Keywords: Electrode, metal fluoride, conversion reaction, fluoride shuttle, reversible capacity, hermetical sealing

Introduction

Owing to the high electronegativity of fluorine, the electrochemical reaction of a metal M with fluoride ions F -to form a metal fluoride MFx can lead to a large change in Gibbs free energy, inducing high theoretical electromotive forces (emf) in electrochemical cells. Although the study of systems relying on fluoride shuttle started in the mid 70's [START_REF] Gutierrez | Electrochemical secondary cells which contains only solid materials[END_REF][START_REF] Kennedy | Thin-Film Galvanic Cell Pb/PbF2/PbF2,CuF2/Cu[END_REF][START_REF] Schoonman | A Solid-State Galvanic Cell with Fluoride-Conducting Electrolytes[END_REF][START_REF] Borger | Galvanic cell with solid fluoride ion-conductive electrolyte[END_REF][START_REF] Lucat | Etude des propriétés de conductivité ionique de quelques fluorures de structure fluorine[END_REF], Anji Reddy and Fichtner were recently the first to valid the concept and the feasibility of a rechargeable Fluoride-Ion Battery (FIB) [START_REF] Reddy | Batteries based on fluoride shuttle[END_REF]. Upon discharge, the MFx cathode is reduced down to its metallic state M, while the metal anode M' is simultaneously oxidized to give M'Fx, according to the following reactions: At the cathode:

𝑀𝐹 𝑥 + 𝑥𝑒 -→ 𝑀 + 𝑥𝐹 - 1.
At the anode: 𝑀′ + 𝑥𝐹 -→ 𝑀′𝐹 𝑥 + 𝑥𝑒 -

2.

Overall: 𝑀′ + 𝑀𝐹 𝑥 → 𝑀′𝐹 𝑥 + 𝑀

3.

These reactions, which may be classified as conversion reactions, are multi-electronic processes. Associated with large emf, some combinations of fluorides could deliver theoretical energy densities as high as 5000 Wh.L -1 [START_REF] Rongeat | Development of new anode composite materials for fluoride ion batteries[END_REF], values much larger than those obtained with conventional lithium-ion batteries.

However, considering the relatively poor conductivity of solid fluoride electrolytes [START_REF] Chable | Fluoride solid electrolytes: investigation of the tysonite-type solid solutions La1-xBaxF3-x (x < 0.15)[END_REF][START_REF] Rongeat | Solid Electrolytes for Fluoride Ion Batteries: Ionic Conductivity in Polycrystalline Tysonite-Type Fluorides[END_REF][START_REF] Patro | Fast fluoride ion conducting materials in solid state ionics: An overview[END_REF][START_REF] Rongeat | Nanostructured Fluorite-Type Fluorides As Electrolytes for Fluoride Ion Batteries[END_REF][START_REF] Trnovcová | Fluoride solid electrolytes[END_REF][START_REF] Sorokin | Nonstoichiometric fluorides-Solid electrolytes for electrochemical devices: A review[END_REF], it is required to evaluate solid-state FIBs at higher temperatures, typically 150 °C. Indeed, performing electrochemical testing at 150 °C decreases the cell internal resistance and enables the use of higher current densities (10 µA/cm² [START_REF] Rongeat | Development of new anode composite materials for fluoride ion batteries[END_REF][START_REF] Gschwind | Fluoride ion batteries: Theoretical performance, safety, toxicity, and a combinatorial screening of new electrodes[END_REF]). These current densities are in the order of magnitude of those typically employed in solid state lithium batteries operating at room temperature (≥ 64 µA/cm² [START_REF] Hakari | All-solid-state lithium batteries with Li3PS4 glass as active material[END_REF][START_REF] Luntz | Interfacial Challenges in Solid-State Li Ion Batteries[END_REF][START_REF] Boulineau | Electrochemical properties of allsolid-state lithium secondary batteries using Li-argyrodite Li6PS5Cl as solid electrolyte[END_REF][START_REF] Delaizir | The stone age revisited: Building a monolithic inorganic lithium-ion battery[END_REF][START_REF] Sakuda | Interfacial Observation between LiCoO 2 Electrode and Li 2 S-P 2 S 5 Solid Electrolytes of All-Solid-State Lithium Secondary Batteries Using Transmission Electron Microscopy †[END_REF][START_REF] Ohta | Enhancement of the highrate capability of solid-state lithium batteries by nanoscale interfacial modification[END_REF]).

Typical battery testing systems rely on coin cells or Swagelok type cells. Coin cells generally use polyethylene (PE) or polypropylene (PP) sealing O-rings that soften and age quickly at high temperatures, leading to the loss of sealing properties of the cell, and therefore, to the oxidation of the battery materials. Similar observations can be made with Swagelok type cells. The sealing of Swagelok type cells is generally ensured by polytetrafluoroethylene (PTFE) rings. While PTFE presents a higher fusion temperature than PE or PP, it is also subject to softening and to significant creeping at elevated temperature, due to its glass transition temperature of 115 °C.

The creeping of PTFE eventually leads to a loss of the sealing properties of the Swagelok type cell. Early experiments carried out with a conventional Swagelok type cell placed in an oven at 150 °C confirmed that the sealing properties were lost after a few days. Moreover, the PTFE sealing rings were permanently damaged as they were significantly deformed. Other commercial systems are available, but they all rely on polymeric seals which are not adapted for high temperature operation. The development of a high-temperature (300 °C) Swagelok type cell has already been reported elsewhere [START_REF] Muñoz-Rojas | Development and implementation of a high temperature electrochemical cell for lithium batteries[END_REF], but its development requires significant modifications and machining work.

Here, we report a simple and rapid method to prepare modified coin cells particularly adapted to the study of FIBs at 150 °C, over long periods of time.

Results and discussion

Cell modification

CR2032 coin cells (MTI Corp) were chosen as the starting setup. In order to obtain a coin cell featuring high temperature resistance, the cell was covered with an epoxy resin (Loctite EA 9492) selected for its room temperature curing and high temperature resistance properties over long periods of time (i.e. no degradation of the strength retention after 3000 h at 150 °C, according to specifications). The use of an epoxy resin on the surface of the coin cell constitutes a simple and rapid approach to obtain electrochemical cells that can achieve high-temperature (150 °C) hermetical sealing.

A schematic representation of the modified CR2032 coin-cell is presented on Figure 1. All parts are made from 304 stainless steel. The case sealing O-ring is made of PP, and is already mounted on the negative side of the case. Some modifications are necessary to adapt the CR2032 coin cell dimensions (⌀int 16.4 mm, 2.8 mm thickness when crimped empty) to the FIB's dimensions assembled for this study (⌀int 10 mm). A flat PTFE washer is used to maintain the FIB at the center of the coin-cell, and a flat wave spring (⌀14.4 mm, 1.5 mm thickness when fully expanded) and spacers (⌀15.4 mm, 0.5 mm thickness) are used to ensure electrical contact and to adjust the thickness of the FIB to the internal space available in the coin cell. After assembling and crimping the coin cell within the glove box, the epoxy resin is deposited under air atmosphere. A steel pellet, used as a support and for electrical contact, is attached to the negative side of the coin-cell case with silver paint. Both the coin-cell and the contact steel pellet are thoroughly degreased with acetone prior to resin application. Additionally, the positive side of the coin-cell is masked with tape to avoid resin covering. The modified coincells obtained after curing (24 h) of the epoxy resin (Figure 2 The coin cells are tightened between a contact rod (bottom) and a screw (top) and connected to a potentiostat. The pressure screw maintains the modified coin-cell in place and keeps a constant pressure on the FIB placed within the coin-cell to maintain electrical contact.

To confirm the proper sealing provided by the epoxy resin, electrochemical testing is performed on pristine coin cells and coin cells covered by epoxy resin.

Electrochemical testing

A symmetrical cell relying on the Bi 3+ /Bi redox couple was selected to confirm the sealing since its reversibility was demonstrated by Anji Reddy and Fichtner [START_REF] Reddy | Batteries based on fluoride shuttle[END_REF]. The electrolyte selected for our electrochemical investigation is Ba-doped LaF3. Ball-milling LaF3 and BaF2 is an easy method to produce nanocrystalline La1-xBaxF3-x (0 ≤ x ≤ 0.15) solid solutions of tysonite-type structure (trigonal, P-3c1) which offer a fluoride-ion conductivity of about 10 -5 S•cm -1 at 150 °C [START_REF] Rongeat | Solid Electrolytes for Fluoride Ion Batteries: Ionic Conductivity in Polycrystalline Tysonite-Type Fluorides[END_REF][START_REF] Chable | Fluoride solid electrolytes: from microcrystalline to nanostructured tysonite-type La0.95Ba0.05F2.95 solid solution[END_REF]. For our study, La0.95Ba0.05F2.95 (denoted LBF hereafter) was used.

LBF (60 wt %) and carbon black (C, 10 wt %) were hand-milled with Bi and BiF3 (15 and 15 wt %) to produce the Bi-BiF3-LBF-C composite electrode. Two Bi-BiF3-LBF-C|LBF|Bi-BiF3-LBF-C stacks were pressed between two graphite disks, acting as current collectors, to form a pellet. A cross section of the stack obtained is shown on Figure 3.

Figure 3. Optical micrograph of the cross section of the graphite|Bi-BiF3-LBF-C|LBF|Bi-BiF3-LBF-C|graphite stack.

The stacks obtained were placed in a coin cell as described above. One coin cell was covered with epoxy resin and the other was not. The two cells were placed on the electrochemical test bench at 150 °C and cycled between 1 and -1 V at 4.5 µA/cm² for 3 cycles (Figure 4.)

Figure 4. Charge-discharge curves obtained for the Bi-BiF3-LBF-C|LBF|Bi-BiF3-LBF-C symmetrical cells with epoxy resin sealing (top) or without (bottom). The cells were cycled with a current of 4.5 µA/cm² between +1 V and -1 V. The numbers indicate the order in which

each charge/discharge was recorded.

Both cells show similar behavior in the first cycle with a charge capacity lower than the discharge capacity, which can be explained by the fact that the BiF3:Bi ratio (1:1 initially) is modified after the 1 st charge, creating an excess of Bi at the cathode and BiF3 at the anode. The Bi and BiF3 converted during the 1 st charge can then react during the 1 st discharge giving rise to a larger capacity compared to the 1 st charge step.

The specific capacity observed during the 1 st charge is largely inferior to the theoretical capacity expected for the cells (302.3 mAh/g based on the theoretical capacity of BiF3). This may be explained by the presence of isolated particles of active material (i.e. Bi and/or BiF3 not in contact with LBF and carbon), which cannot contribute to the electrochemical reactions. It is anticipated that an optimization of the electrode composition, associated with a more thorough mixing (e.g. ball-milling), would lead to higher capacities in the 1 st charge.

The capacities obtained for both cells upon cycling are reported on The discharge capacity of the sealed cell remains stable at around 190 mAh/g for the first two cycles and slightly fades in the 3 rd cycle (182 mAh/g).

For the non-sealed cell, the discharge capacity quickly fades from 145, to 96, to 53 mAh/g from the 1 st , to the 2 nd , and 3 rd cycle, respectively. The capacity fading observed in the non-sealed cell is significantly higher than in the coin-cell covered with epoxy resin.

Capacity fading is a typical phenomenon observed in conversion electrodes for lithium-ion batteries. [START_REF] Teng | Investigation of the conversion mechanism of nanosized CoF2[END_REF][START_REF] Badway | Carbon Metal Fluoride Nanocomposites[END_REF][START_REF] Bervas | Reversible Conversion Reactions with Lithium in Bismuth Oxyfluoride Nanocomposites[END_REF] It is generally attributable to the volumetric variations of the active material upon cycling (lithium input and output), leading to a partial loss of contact with the conductive additive, creating isolated particles of active materials which cannot contribute to the electrochemical reactions. Similar considerations can be made for FIBs. Reduction of BiF3 (Orthorhombic, Pnma space group) to Bi (trigonal, R-3mH space group) leads to a volumetric variation of -57 %, and oxidation of Bi to BiF3 leads to a volumetric variation of +36 %. For the sealed cell, the slight capacity fading observed after 3 cycles may be explained by these volumetric variations. As for the increase of capacity between the 1 st and 2 nd discharge, a reorganization of the electrode structure may be responsible. The low melting point of Bi (271 °C) may be responsible for the grain growth of the Bi phase [START_REF] Rongeat | Development of new anode composite materials for fluoride ion batteries[END_REF]. While this phenomenon could be detrimental to the capacity due loss of reactive interfaces, it could also be beneficial depending on the starting microstructure of the composite. Indeed, previously isolated Bi particles may merge with other reactive Bi particles, increasing the net amount of Bi available for electrochemical reaction.

As for the non-sealed cell, the larger decrease of capacity upon cycling may be explained by a loss of the sealing properties of the coin-cell after prolonged air exposure at 150 °C (55 h at the end of the 1 st charge, more than 2 weeks at the end of the 3 rd discharge). Slow diffusion of air within the coin-cell would lead to a progressive oxidation of the Bi particles upon cycling. The passivated Bi particles would be therefore rendered electrochemically inactive, inducing a progressive loss of capacity upon cycling.

Figure 6 shows the Nyquist plot of impedance spectra recorded for the two cells, after the 1 st and the 3 rd charge-discharge cycles. The spectra are all composed of a single visible semi-circle.

The low frequency zone of the Nyquist plots (Figure 6, insets) shows the end of another contribution which is attributed to the ionic resistance of LBF layer. It is expected that the electrolyte layer yields a semi-circle at high frequencies, but the frequency used here is too low (i.e. 300 kHz) for the entire semi-circle to be visible. A value of approximately Re = 750 Ω for both cells is found, which is concomitant with the resistance given by a layer of LBF in our experimental conditions (σi(150 °C) ≈ 9.4×10 -5 S.cm -1 for La0.95Ba0.05F2.95) [START_REF] Rongeat | Solid Electrolytes for Fluoride Ion Batteries: Ionic Conductivity in Polycrystalline Tysonite-Type Fluorides[END_REF].

The visible semi-circle is linked to electrode phenomena, such as charge transfer resistance. In the case of the sealed coin-cell, the resistance remains relatively similar between the 1 st and 3 rd cycles. For the coin-cell not sealed with resin, the resistance dramatically increases between the 1 st and 3 rd cycles.

Figure 6. Nyquist plots of the impedance spectra recorded on the Bi-BiF3-LBF-C|LBF|Bi-BiF3-LBF-C cells with epoxy resin sealing (left) or without (right) after the 1 st and 3 rd cycles. Insets: zoom in on the low frequency regions showing the end of the 1 st semi-circle corresponding to the resistance of the LBF layer. Re denotes the resistance of the electrolyte.

The impedance measurements support the cycling performance observed for the two cells. Indeed, the relatively stable electrodes' resistance yielded by the sealed cell between the 1 st and 3 rd cycles supports the capacity retention observed upon cycling.

In the case of the coin cell not sealed with epoxy resin, impedance measurements support the large capacity fading observed. Indeed, the loss of sealing properties of the coin-cell upon prolonged air exposure at 150 °C leading to a passivation of Bi within the composite electrode would lead to an increased resistance of the electrodes.

Figure 7 shows photographs on the cells before and after the cycling experiments carried out at 150 °C. The total length of the experiment was approximately 1 month.

After such exposure, the outside surface of the epoxy resin is browned due to oxidation. The resin was broken down to check the coin cell O-ring. The resin in contact with the coin-cell was not browned like on the outside surface and adhered relatively well with the metal surface of the coin-cell case. The PP O-ring of the coin cell protected with resin was swollen but did not present any cracks or any sign of significant aging. However, the O-ring of the coin-cell not protected with resin was significantly aged and presented multiple cracks. The aged O-ring was brittle and could be easily broken down.

The observations made on the coin-cells suggest that the epoxy resin protected the PP O-ring from significant degradation, which allowed the cell to remain hermetical.

For the non-covered cell, the brittle nature and cracks present on the O-ring suggest that the sealing of the coin-cell was not achieved anymore, which must have allowed air to diffuse within the coin-cell.

These observations are in agreement with the electrochemical results obtained on the corresponding cells. Indeed, the diffusion of air within the non-sealed coin-cell may be responsible for the progressive oxidation of the Bi-BiF3 composite, eventually leading to a passivation of the Bi particles, rendering them electrochemically inactive. The passivation of the Bi particles would explain the increase of the impedance of the cell as well as the progressive decrease of the capacity upon cycling. Covering the coin cell with a high temperature epoxy resin provides protection of the O-ring, preventing air to diffuse within the cell and allowing to maintain the inert atmosphere throughout the electrochemical investigation.

Conclusions

Electrochemical investigations performed at 150 °C on a symmetrical cell assembled using composite electrodes composed of a mixture of Bi, BiF3, electrolyte (La0.95Ba0.05F2.95) and carbon black, show that an epoxy-sealed coin cell can be cycled with little capacity fading (about 190 mAh/g of Bi after 3 cycles), while a non-sealed cell shows significant capacity loss (about 60 mAh/g of Bi after 3 cycles). The cycling results are in agreement with the significant increase of the internal resistance of the non-sealed cell, as observed by impedance spectroscopy. Post mortem observations of the coin cells after experiment corroborate the electrochemical results as the epoxy sealed coin cell O-ring shows no sign of significant degradation, while the non-sealed coin cell O-ring was burnt.

The covering of a coin cell with a high temperature epoxy resin is particularly adapted to study cells based on fluoride shuttle as it enables electrochemical investigation under inert atmosphere, outside a glovebox, at 150 °C and over long periods of time. Moreover, the setup described allows to maintain the cell under constant pressure, which may be beneficial to the electrochemical performance of such system.

Experimental

Unless specified otherwise, all the manipulation of the materials and assembly of the cells were carried out in a dry Ar glove box (H2O < 2 ppm).

Materials preparation

La0.95Ba0.05F2.95 (LBF) electrolyte was prepared by direct mechanosynthesis from commercial fluorides LaF3 (Alfa Aesar, 99.99 % anhydrous REO) and BaF2 (Strem 99.99 +% Puratrem) in purified N2 atmosphere. A Fritsch pulverisette 7 premium line planetary mill with ZrO2 jars and milling balls (10 balls, 10 mm diameter) was used. To produce 4 g of LBF, the mixture was ball-milled for 12 hours at 400 rpm. A comprehensive study of the influence of the mechanosynthesis parameters on the structural and conduction properties of LBF can be found in the paper of Chable et al. [START_REF] Chable | Fluoride solid electrolytes: from microcrystalline to nanostructured tysonite-type La0.95Ba0.05F2.95 solid solution[END_REF] To prepare the Bi-BiF3-LBF-C composite electrode, β-BiF3 (Alfa Aesar, 99.999%) and Bi (Alfa Aesar, -200 mesh, 99.999%) commercial powders were mixed with LBF and carbon black (Pure Black, Superior Graphite) in 15:15:60:10 % wt ratio. The mixture was hand-milled for about 30 minutes in a dry Ar glove box with an agate mortar and pestle.

Cell assembly

Symmetrical cells were assembled, in a dry Ar glove box, using approximately 16 mg of the Bi-BiF3 composite, and 200 mg of LBF. The powdered stack was pressed in a ⌀10 mm stainless steel mold with an uniaxial press (Specac Atlas), at a load of 5 tons. Two graphite disks (Cg)

were pressed with the powdered stack to form a Cg|Bi-BiF3-LBF-C|LBF|Bi-BiF3-LBF-C|Cg cell in a single step. The graphite disks were directly pressed with the powders to provide a good contact between the composite and the current collector and to ensure the easy handling of the pellet stack.

Graphite disks (Cg) were cut from high purity graphite rods (Mersen). Graphite rods were previously machined to reduce their diameter so that they would fit the pressing mold. The discs were then polished with grit P400 SiC abrasive paper, degreased in acetone then sonicated three times in ultra-pure water for ca. three minutes. Finally, Cg disks were outgassed at 150 °C for 12 h under vacuum, and entered in the glovebox without any contact with air. Typical thickness of a graphite disc was 0.5 -0.7 mm.

Finally, the Cg|Bi-BiF3-LBF-C|LBF|Bi-BiF3-LBF-C|Cg stack was placed in a modified coin cell as described in the discussion.

Flat PTFE washers (⌀ext 16.0 mm, ⌀int 9.95 mm, 1.5 mm thickness) were machined from a PTFE rod. For the electrochemical investigation, one flat wave spring and one spacer were used to maintain electrical contact within the coin cell.

Electrochemical measurements

Electrochemical measurements were performed with an AMETEK PARSTAT PMC 500 potentiostat connected to the cell in a two electrode setup, i.e. the reference and counter electrodes were plugged together. The cells were placed in the oven at 150 °C on the custommade electrochemical test bench described in the discussion.

Galvanostatic (I = 3.5 µA, i ≈ 4.5 µA/cm²) charge-discharge experiments were carried out between 1 V and -1 V. For impedance measurements, the cells were left to rest for 2 hours prior to recording. The spectra were recorded between 300 kHz and 10 mHz with a sinusoidal excitation amplitude of 30 mV RMS.

Figure 1 .

 1 Figure 1. Schematic representation of the cross-section of the modified coin-cell placed on the test bench.

  .a) are placed on the custom made electrochemical test bench (Figure 2.b) placed in an oven, heated to 150 °C for electrochemical evaluation.

Figure 2 .

 2 Figure 2. Photograph of (a) the modified coin-cells and (b) the electrochemical test bench placed in an oven.

Figure 5 .

 5 Charge and discharge capacities remain almost similar for the sealed cell, while the cell not sealed with resin shows a progressive loss of capacity upon subsequent charge or discharge.

Figure 5 .

 5 Figure 5. Comparative of the cycling behavior of the two Bi-BiF3 symmetrical cells with or without epoxy resin.

Figure 7 .

 7 Figure 7. Photographs of coin-cells before and after exposure at 150 °C for about 1 month, without or with the use of epoxy resin. The browned resin was broken down to let the O-ring visible.

Acknowledgements

We gratefully acknowledge J. Chable and A. Gil-Martin for the preparation of the solid fluoride electrolyte, F. Chrétien for the preparation of the composite electrode, the French National Research Agency (Project FLUOBAT-ANR-12-PRGE-0009-01) for financial support, and S.

Leclerc for technical support.


Roope Vehkalahti

Laura Luzzi
email: laura.luzzi@ensea.fr

Jean-Claude Belfiore
email: belfiore@enst.fr

Shifted inverse determinant sums and new bounds for the DMT of space-time lattice codes

This paper considers shifted inverse determinant sums arising from the union bound of the pairwise error probability for space-time codes in multiple-antenna fading channels. Previous work by Vehkalahti et al. focused on the approximation of these sums for low multiplexing gains, providing a complete classification of the inverse determinant sums as a function of constellation size for the most well-known algebraic spacetime codes. This work aims at building a general framework for the study of the shifted sums for all multiplexing gains. New bounds obtained using dyadic summing techniques suggest that the behavior of the shifted sums does characterize many properties of a lattice code such as the diversity-multiplexing gain trade-off, both under maximum-likelihood decoding and infinite lattice naive decoding. Moreover, these bounds allow to characterize the signal-to-noise ratio thresholds corresponding to different diversity gains.

I. INTRODUCTION

Shifted inverse determinant sums appear naturally when analyzing the union bound for the pairwise error probability (PEP) of space-time codes over MIMO channels [START_REF] Tarokh | Space-Time Codes for High Data Rate Wireless Communications: Performance Criterion and Code Construction[END_REF]. The high-SNR approximation of these sums was analyzed in [START_REF] Vehkalahti | Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra[END_REF], providing general bounds on the performance of algebraic space-time codes from division algebras and number fields. In particular, it was shown that the approximate sums are enough to characterize the diversity-multiplexing gain tradeoff (DMT) [START_REF] Zheng | Diversity and Multiplexing: A Fundamental Tradeoff in Multiple-Antenna Channels[END_REF] of these codes in the multiplexing gain range r ∈ [0, 1]. However, in order to study the DMT for higher multiplexing gains r, it becomes necessary to consider the original shifted determinant sums. In this work we provide a general framework to analyze shifted sums, which are able to predict the correct DMT curve for r > 1 in some cases. We also discuss the characterization of the "high SNR" threshold as a function of constellation size. Moreover, we show that while their high-SNR approximations never converge, shifted sums always converge if the number of receive antennas is large enough; this provides new bounds on the DMT performance of naive lattice decoding.

Inverse determinant sums in the sense we are discussing were considered by Tavildar and Viswanath in [START_REF] Tavildar | Approximately Universal Codes Over Slow-Fading Channels[END_REF], where the authors analyzed the DMT of several simple space-time codes. The most recent appearance of these sums is in the work of Belfiore and Oggier concerning the eavesdropper's error probability in the MIMO wiretap channel [START_REF] Belfiore | An Error Probability Approach to MIMO Wiretap Channels[END_REF].

Our take on the subject follows the general setting of [START_REF] Vehkalahti | Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra[END_REF], but we replace the approximation of PEP by the more accurate version. The idea to consider symmetric polynomials and their relations to analyze PEP was given in [START_REF] Vehkalahti | Reducing complexity with less than minimum delay space-time lattice codes[END_REF].

A. Matrix Lattices and spherically shaped coding schemes

Before we can introduce inverse determinant sums, we need a few definitions.

Definition 1.1:

A matrix lattice L ⊆ M n×T (C) has the form L = ZB 1 ⊕ ZB 2 ⊕ • • • ⊕ ZB k ,
where the matrices B 1 , . . . , B k are linearly independent over R, i.e., form a lattice basis, and k is called the rank or the dimension of the lattice. Definition 1.2: If the minimum determinant of the lattice L ⊆ M n×T (C) is non-zero, i.e. inf 0 =X∈L |det(XX * )| > 0, we say that the lattice satisfies the non-vanishing determinant (NVD) property.

We now consider a spherical shaping scheme based on a kdimensional lattice L in M n×T (C). Given M > 0 we define

L(M ) = {a ∈ L : a F ≤ M, a = 0}.
Here • F refers to the Frobenius norm.

B. Motivation and problem statement

Let us suppose that we are considering the complex Gaussian channel and a finite code L(M ) ∈ C n . If the codewords are sent equiprobably, we can upper bound the average error probability by the sum

P e ≤ x∈L, 0< x E ≤2M e -x 2 ,
where the term 2M follows from the fact that we have to consider differences of codewords. The right-hand-side is then a well known truncated exponential sum taking values on lattice points of L. Let us now describe the analogous bound in the fading channel. Suppose that we have a lattice L ⊂ M n×T (C) and that we have chosen a finite code L(M ) and a constant θ such that θL(M ) has average energy 1.

Consider the Rayleigh block fading MIMO channel with n = n t transmit and n r receive antennas. The channel is assumed to be fixed for a block of T channel uses, but to vary in an independent and identically distributed (i.i.d.) fashion from one block to another. Thus, the channel input-output relation can be written as

Y = ρ n HθX + N, (1)
where H ∈ M nr×n (C) is the channel matrix and N ∈ M nr×T (C) is the noise matrix. The entries of H and N are assumed to be i.i.d. zero-mean complex circular symmetric Gaussian random variables with variance 1. The matrix X ∈ L(M ) is the transmitted codeword, and the term ρ denotes the signal-to-noise ratio (SNR).

Following [START_REF] Tarokh | Space-Time Codes for High Data Rate Wireless Communications: Performance Criterion and Code Construction[END_REF], we can upper bound the pairwise error probability between two codewords X = X , when transmitting with SNR ρ, as follows:

P (ρ, X → X ) ≤ 1 (det(I + ρθ 2 4n (X -X )(X -X ) * )) nr
, where * denotes complex conjugate transpose. The scaling factor 4n for the SNR ρ is irrelevant for our asymptotic analysis so we will omit it in the sequel. We can upperbound the average probability, when transmitting a codeword from L(M ), as

P e ≤ X∈L, 0<||X|| F ≤2M 1 (det(I + ρθ 2 XX * )) nr .
This discussion leads us to consider sums of the type

X∈L, 0<||X|| F ≤M 1 (det(I + cXX * )) m , ( 2
)
where c is considered a variable. Remark 1.1: We remark that when c is very large, the terms in (2) are well-approximated by 1/ det(cXX * ) m . In the case T = n, we can consider sums of the type

S m L (M ) := X∈L(M ) 1 | det(X)| m . (3)
The asymptotic behavior of these sums, and its relation to the diversity-multiplexing trade-off of space-time codes, were analyzed in [START_REF] Vehkalahti | Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra[END_REF].

In this paper, we will address some additional aspects of MIMO space-time code optimization that are not captured by the approximate sums (3), but instead require to study the original sums [START_REF] Vehkalahti | Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra[END_REF]. In particular, we will consider the following problems:

-Find upperbounds of the type

X∈L, 0<||X|| F ≤M 1 (det(I + cXX * )) m ≤ c -k f (M )
for some function f and positive constant k.

-How large should m be for the sum (2) to converge? -What is the highest power k such that

X∈L,||X|| F ≤M 1 (det(I + cXX * )) m ≤ c -k G
for some constant G and for every M ? In the following we will give some general answers to these questions and build a framework for using these sums to analyze codes.

II. DYADIC SUMMING AND UPPER BOUNDS FOR SHIFTED INVERSE DETERMINANT SUMS

Let's start by considering the decomposition of the shifted determinant. Let λ 1 , λ 2 , . . . , λ n be the eigenvalues of XX * . Then

det(I + cXX * ) = (1 + λ 1 c)(1 + λ 2 c) • • • (1 + λ n c) = = 1 + n 1 p 1 c + n 2 p 2 c 2 + • • • + p n c n ,
where n i p i is the i-th symmetric polynomial of variables λ 1 , . . . , λ n . One should note that

p 1 = Tr(XX * ) = ||X|| 2 F and p n = det(XX * ).
The following inequalities will be useful in the sequel: Proposition 2.1 (McLaurin's and Newton's inequalities):

The coefficients p i satisfy

p 1 ≥ √ p 2 ≥ 3 √ p 3 ≥ • • • ≥ n √ p n , (4) p 2
i ≥ p i-1 p i+1 . Corollary 2.2: Let us suppose that det(XX * ) ≥ 1. With the previous notation we have that

p k ≥ 2 k-1 √ p 1 , for all n -1 ≥ k.
Proof: We have that

p i ≥ √ p i+1 p i-1 .
Due to the condition p n ≥ 1 we have that

p k ≥ 1 ∀k. Therefore p i ≥ √ p i-1 .
Induction now gives us the result.

In the following we are interested in asymptotics and convergence and therefore we can forget the binomial terms and concentrate on the terms p i . The following inequalities formalize this approach:

(det(I + cXX * )) m = = 1 + n 1 p 1 c + n 2 p 2 c 2 + • • • + p n c n m ≥ ≥ (1 + p 1 c + p 2 c 2 + • • • + p n c n ) m ≥ ≥ (c||X|| 2 F + c n | det(XX * )|) m = = m i=0 m i c i+n(m-i) X 2i F |det(XX * )| m-i
In particular we have

X∈L(M ) 1 (det(I + cXX * )) m ≤ ≤ X∈L(M ) 1 c i+n(m-i) X 2i F |det(XX * )| m-i (5)
for every 0 ≤ i ≤ m.

The following two Lemmas are useful to provide bounds for the sum in equation ( 5).

Lemma 2.3 (Dyadic Summing): Let f : R → R be a positive valued function, and I ⊂ R be a discrete set. Suppose that there exist positive constants K and s such that ∀M ≥ 1,

x∈I, 1≤x≤M f (x) ≤ KM s .
We then have that

x∈I, 1≤x≤M f (x) x t < K 1 if t > s, x∈I, 1≤x≤M f (x) x t < K 2 log(M ) if t = s, x∈I, 1≤x≤M f (x) x t < K 3 M s-t if t < s,
for some constants K 1 , K 2 , K 3 (depending on s and t).

Proof: By partitioning the interval [1, M ] into subinter- vals of the form [2 i-1 , 2 i ], we get x∈I, 1≤x≤M f (x) x t ≤ log 2 (M ) i=1 x∈I, 2 i-1 ≤x≤2 i f (x) x t ≤ ≤ log 2 (M ) i=1 x∈I, 2 i-1 ≤x≤2 i f (x) 2 (i-1)t ≤ log 2 (M ) i=1 K2 is 2 (i-1)t = = 2 t K log 2 (M ) i=1 (2 (s-t)i ).
Lemma 2.4: Let L ⊂ M n×T (C) be a lattice such that X F ≥ 1 for all the non-zero points X ∈ L. Let g be a positive valued function defined in all the non-zero points of the lattice. If

X∈L(M ) g(X) ≤ KM s
for some fixed positive constants K and s, then

X∈L(M ) g(X) ||X|| t F < K 1 if t > s, X∈L(M ) g(X) ||X|| t F < K 2 log(M ) if t = s, X∈L(M ) g(X) ||X|| t F < K 3 M s-t if t < s, for some constants K 1 , K 2 , K 3 .
Proof: This is simply the previous proposition applied to the function f (x) = X∈L, ||X|| F =x g(X). Note that the hypothesis X F ≥ 1 in Lemma 2.4 doesn't incur any loss of generality since we can just rescale the lattice. However, in that case the constants K 1 , K 2 , K 3 will depend on the scaling factor.

We can now obtain a set of upper bounds for shifted inverse determinant sums: Proposition 2.5: Let us suppose that L is a k-dimensional lattice in M n×T (C), such that X F ≥ 1 for all the non-zero points X ∈ L, and that we have a bound

X∈L(M ) 1 | det(XX * )| l ≤ KM s(l) .
We then have that

X∈L(M ) 1 (det(I + cXX * )) m ≤ min 0≤i≤m {W i (M )},
where for i ∈ {0, . . . , m -1} we have

W i (M ) = G i c i+n(m-i) , if s(m -i) < 2i, W i (M ) = G i c i+n(m-i) log(M ), if s(m -i) = 2i, W i (M ) = G i c i+n(m-i) M s(m-i)-2i if s(m -i) > 2i, where G i are some constants. When i = m, we have W m (M ) = G m c -m if k < 2m, W m (M ) = G m c -m log M if k = 2m and W m (M ) = G m c -m M k-2m M if k > 2m.
Proof: The conclusion follows from equation ( 5) and from Lemma 2.4 with g(X) = 1/ det(XX * ) m-i . For the special case i = m, observe that the number of lattice points in L(M ) is proportional to the volume of the ball of radius M in R k .

As a consequence of Proposition 2.5 in the case i = m, the shifted determinant sum will converge for m > k/2: Proposition 2.6 (Convergence): Let us suppose that L is a k-dimensional lattice in M n×T (C) such that X F ≥ 1 for all the non-zero points X ∈ L. We then have that

X∈L(M ) 1 (det(I + cXX * )) k/2+ ≤ G c -k/2 ,
where is any positive number and G a constant independent of M , but dependent on .

We can conclude that while

X∈L(M ) 1 det(XX * ) m
does not usually converge for any m [START_REF] Vehkalahti | Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra[END_REF], quite the opposite is true for the sum

X∈L(M ) 1 (det(I + cXX * )) m .
As long as the power m is large enough this sum will always converge.

III. SHIFTED INVERSE DETERMINANT SUMS AND SNR LEVEL ANALYSIS

Let us introduce one more use for shifted inverse determinant sums. If we have a finite space-time NVD lattice code in M n (C), then in the high SNR regime the diversity order is nn r = n t n r . However this regime is rarely visible on error performance curves. We will now see how shifted inverse determinant sums can explain this behavior, and provide an estimate of the SNR threshold beyond which higher diversity kicks in.

Proposition 3.1: Suppose that we have a k-dimensional lattice L ∈ M n×T (C) such that a determinant sum upper bound

X∈L(M ) 1 (det(I + cXX * )) m ≤ KM t c -d ,
holds for some constants d,K and t. Then the average error probability is upper bounded as

P e (ρ) ≤ KM d+t ρ -d , (6)
when transmitting with signal-to-noise ratio ρ. Proof: The average energy of the code 1 √ M L(M ) is less than 1. Let us now suppose that θ > 1 is such a constant that θ 1 √ M L(M ) has average energy 1. We then have that

P e (ρ) ≤ X∈L(M ) 1 (det(I + ρθ 2 XX * )) m ≤ KM t+d (ρθ 2 ) -d = KM t+d ρ -d , (7)
which concludes the proof. This result has several implications. The first is that we can estimate the SNR threshold beyond which we can see diversity order d. We can see from equation ( 7) that when SN R = ρ ≥ K M (t+d)/d the diversity d will appear; before this point we don't have guaranteed diversity d.

Another implication is easier to explain through an example. Example 3.1: Let us suppose that we have an 8-dimensional lattice code L in M 2 (C) and bounds

X∈L(M ) 1 (det(I + cXX * )) 4 ≤ min{K 1 M 4 c -8 , K 2 c -4 }.
In order to have guaranteed diversity 8 we must have ρ ≥ K 1 M 3/2 . For guaranteed diversity 4 the SNR condition is ρ ≥ K 2 M 1 . This shows that when the code grows we need eventually considerably more energy to have guaranteed diversity 8, independently of the size of the constants K 1 and K 2 .

IV. SHIFTED INVERSE DETERMINANT SUMS AND DMT

ANALYSIS

It was proven in [START_REF] Vehkalahti | Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra[END_REF] that the growth of the inverse determinant sums of a lattice code L ∈ M n (C) describes the diversity-multiplexing gain trade-off (DMT) [START_REF] Zheng | Diversity and Multiplexing: A Fundamental Tradeoff in Multiple-Antenna Channels[END_REF] of the code L for multiplexing gains r ∈ [0, 1]. We will now show that the shifted determinant sum bounds are useful to analyze the DMT of a code for higher multiplexing gains. We will also show how we can use these sums to evaluate the DMT of a lattice code under naive lattice decoding.

A. Lower bounds for the DMT under ML decoding Definition 4.1: Given the lattice L ⊂ M n×T (C), a spacetime lattice coding scheme associated with L is a collection of STBCs where each member is given by

C L (ρ) = ρ -rT k L ρ rT k (8)
for the desired multiplexing gain r and for each ρ level. Proposition 4.1: Let L be a k-dimensional lattice in M n×T (C) and suppose that the determinant sum upper bound X∈L(M ) 1 (det(I + cXX * )) m ≤ Kc -a f (M ), holds for some positive constants K and a. We then have that for SNR ρ the average error probability of the code C L (ρ) has an upperbound

ACKNOWLEDGEMENT

The authors are grateful for Eeva Suvitie for suggesting the use of dyadic summing. The research of R. Vehkalahti is supported by the Academy of Finland grant #252457.

Proof: The average energy for the code C L (ρ)is less than 1. Now for transmission with SNR ρ, each of the codewords gets multiplied with ρ 1/2 as in [START_REF] Tarokh | Space-Time Codes for High Data Rate Wireless Communications: Performance Criterion and Code Construction[END_REF]. We now have

where c = ρ 1-2rT /k . The final result is then simply gotten by substitution.

The following DMT bound is a direct corollary of the previous result: Corollary 4.2: Let us suppose that we have an upperbound

We then have that the DMT of the code L is lowerbounded by the following line:

We also have the following curiosity, which shows that any full dimensional lattice achieves the full multiplexing gain, when we have enough receive antennas:

Corollary 4.3: Let L be a 2nT -dimensional lattice code in M n×T (C). If n r > nT +1, the code C L (ρ) has an upperbound for the average error probability P e ≤ ρ (1-r/n)nr , when transmitting with SNR ρ.

Proof: This follows from Proposition 2.5 with m = n r > k/2, yielding a = n r and b = 0 in Corollary 4.2.

B. Lower bound for the DMT under naive lattice decoding

Let us consider naive lattice decoding as defined in [START_REF] Taherzadeh | On The Limitations of The Naive Lattice Decoding[END_REF], which consists in minimizing the Euclidean metric with respect to the received signal over all the lattice points X ∈ L, regardless of whether they belong to the finite code.

It is clear that the average probability of error of naive lattice decoding can be upper bounded by a determinant sum over the whole lattice. This bound is relevant only when the sum is converging. We thus state the following Proposition, which can be proven in the same way as Corollary 4.2:

Proposition 4.4: Let L ∈ M n×T (C) be a k-dimensional lattice, and suppose that a determinant sum upper bound

holds for some positive constants K and a. Then the DMT of the code C L (ρ), under naive lattice decoding, is lower bounded by [r, (a -2rT a/k) + ].

V. EXAMPLES A. Analyzing the Golden code

Let us consider the Golden code L ⊂ M 2 (C), which is an 8-dimensional lattice. According to [START_REF] Luzzi | A new design criterion for spherically-shaped division algebra-based space-time codes[END_REF], if n r > 1 we have

where K is a positive constant.

With the previous notation we have that We can see that even with naive lattice decoding the Golden code does achieve the optimal multiplexing gain. However, the maximal diversity is only 4.

Remark 5.1: Here one should note that for multiplexing gains r ∈ [0, 1] the sum

does provide the best upper bound. However, when r ∈ [START_REF] Tarokh | Space-Time Codes for High Data Rate Wireless Communications: Performance Criterion and Code Construction[END_REF][START_REF] Vehkalahti | Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra[END_REF] the sum

, gives a tighter upper bound.

B. Analyzing diagonal number field codes

Let us now consider a complex diagonal number field code. Such a code is 2n-dimensional NVD lattice in M n (C). As proved in [START_REF] Vehkalahti | Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra[END_REF] we have that for m ≥ 1 we have

for some constant K. Proposition 5.2: Let L be a diagonal number field code in M n (C) such that det(XX * ) ≥ 1 for all X ∈ L, X = 0 and let m > 1. We then have that

where K is some constant independent of M .

Proof: We begin with