Romain Moury
email: romain.moury@univ-lemans.fr

Zbigniew Łodziana
email: zbigniew.lodziana@ifj.edu.pl

Arndt Remhof

Léo Duchêne

Elsa Roedern

Angelina Gigante

Hans Hagemann

Duchêne Léo

Elsa Da

Roedern

Study of the temperature and pressure dependent structural properties of alkali hydrido-closo-borates compounds

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from pressure dependent diffraction studies and DFT calculations; the shear moduli appear to be very low for all studied compounds revealing their high malleability (that can be beneficial for the manufacturing and stable cycling of all-solid-state batteries). The thermo-diffraction experiments also reveal a high coefficient of thermal expansion for these materials. We discover a pressure induced phase transition for K2B12H12 from Fm-3 to Pnnm symmetry around 2GPa. A temperature induced phase transition for Li2B10H10 was also observed for the first time by thermodiffraction and the crystal structure solved combining experimental data and DFT calculations. Interestingly, all phases of the studied compounds (including newly discovered high pressure and high temperature phases) may related via group-subgroup relationship, with the notable exception of the room temperature phase of Li2B10H10.

Introduction.

Since the discovery of superionic conductivity in the high temperature phase of Na2B12H12, [START_REF] Udovic | Sodium Superionic Conduction in Na2B12H12[END_REF] there has been an increasing interest in hydridocloso -(car)borates, and hydridonido -(car)borates and their solid solutions, [START_REF] Hansen | Metal Boranes: Progress and Applications[END_REF][START_REF] Duchêne | Status and Prospects of Hydroborate Electrolytes for All-Solid-State Batteries[END_REF][START_REF] Brighi | Closo-Hydroborate Sodium Salts as an Emerging Class of Room-Temperature Solid Electrolytes[END_REF][START_REF] Kim | A Complex Hydride Lithium Superionic Conductor for High-Energy-Density All-Solid-State Lithium Metal Batteries[END_REF][START_REF] Brighi | A Mixed Anion Hydroborate/Carba-Hydroborate as a Room Temperature Na-Ion Solid Electrolyte[END_REF][START_REF] Tang | Stabilizing Superionic-Conducting Structures via Mixed-Anion Solid Solutions of Monocarba-Closo -Borate Salts[END_REF][START_REF] Tang | Liquid-Like Ionic Conduction in Solid Lithium and Sodium Monocarba-Closo-Decaborates Near or at Room Temperature[END_REF][START_REF] Payandeh | Nido-Hydroborate-Based Electrolytes for All-Solid-State Lithium Batteries[END_REF][START_REF] Payandeh | Nido-Borate/ Closo-Borate Mixed-Anion Electrolytes for All-Solid-State Batteries[END_REF][11] as well as their halogenated derivatives. [START_REF] Jørgensen | Understanding Superionic Conductivity in Lithium and Sodium Salts of Weakly Coordinating Closo-Hexahalocarbaborate Anions[END_REF] Several of these compounds or solid solutions exhibit very high ionic conductivities above 1 mS/cm at room temperature and they are chemically and electrochemically very stable. 13,[START_REF] Asakura | Electrochemical Oxidative Stability of Hydroborate-Based Solid-State Electrolytes[END_REF] Recently, some prototype of all-solid-state batteries using hydridocloso -(car)borates as electrolytes have demonstrated very promising performances. [START_REF] Duchêne | A Stable 3 V All-Solid-State Sodium-Ion Battery Based on a Closo -Borate Electrolyte[END_REF][START_REF] Murgia | Room-Temperature-Operating Na Solid-State Battery with Complex Hydride as Electrolyte[END_REF] Furthermore novel cost effective methods have been developed to synthesis hydrido closoborates from solution, [START_REF] Moury | An Alternative Approach to the Synthesis of NaB3H8 and Na2B12H12 for Solid Electrolyte Applications[END_REF][START_REF] Gigante | Direct Solution-Based Synthesis of Na4(B12H12)(B10H10) Solid Electrolyte[END_REF] allowing electrodes impregnations to notably improve ionic contact between the electrode and electrolyte. [START_REF] Duchêne | Crystallization of Closo-Borate Electrolytes from Solution Enabling Infiltration into Slurry-Casted Porous Electrodes for All-Solid-State Batteries[END_REF] These results demonstrate that this family presents many excellent properties as solid ionic conductors for new generations of all-solid-state batteries.

Understanding the bulk mechanical properties of solid electrolyte in general and H-c-B in particular is crucial to develop manufacturing method for all-solid-state batteries as well as to improve their cycling stability. [START_REF] Brighi | Mechanical Behavior and Dendrite Resistance of Closo-Hydroborate Solid Electrolyte[END_REF] For example, charging and discharging cycles can induce volume changes of the electrodes materials, which should be accommodated by the solid electrolyte without mechanically disrupting the electrode-electrolyte interface. Some of the authors have demonstrated that a solid electrolyte based on a solid solution between H-c-B and hydrido-closo-carborate is stable for at least 800 charging/discharging cycles. [START_REF] Asakura | 4 V Room-Temperature All-Solid-State Sodium Battery Enabled by a Passivating Cathode/Hydroborate Solid Electrolyte Interface[END_REF] Mechanical properties such as the shear modulus of solid electrolyte is also considered as an important parameter in some models of dendrite formation in solid electrolytes. [START_REF] Frenck | Factors That Control the Formation of Dendrites and Other Morphologies on Lithium Metal Anodes[END_REF] Thermal expansion of the materials constituting the battery may as well induce mechanical stresses when the battery is subject to overheating. A colossal barocaloric effect has also been predicted for Li2B12H12, [START_REF] Sau | Colossal Barocaloric Effects in the Complex Hydride Li2B12H12[END_REF] hence the behavior of this family of materials upon pressure and temperature are important aspects in understanding these properties. Furthermore, while the fundamental of the crystal chemistry of hydridoclosoborates is well established, 24 some pieces are still missing. For instance, a temperature induced phase transition has been observed by DSC experiment for Li2B10H10 and (K,Cs)2B12H12 though no crystal structure was given. [START_REF] Wu | Structural Behavior of Li2B10H10[END_REF][START_REF] Verdal | Evidence of a Transition to Reorientational Disorder in the Cubic Alkali-Metal Dodecahydro-Closo-Dodecaborates[END_REF] In addition, only few reports on high pressure behavior for this family of compounds exist except for (Na,Cs)2B12H12. [START_REF] Moury | Pressure-Induced Phase Transitions in Na2B12H12, Structural Investigation on a Candidate for Solid-State Electrolyte[END_REF][START_REF] Benham | High-Pressure Diamond-Anvil-Cell Micro-Raman Spectra of Mercuric Cyanide, Hg(CN)2, and Cesium Dodecahydroborate, Cs2[B12H12[END_REF] In this context, insight in their structural behavior under external stimuli (pressure, temperature) is a critical aspect to extract some fundamental knowledge and physical properties, as the Coefficients of the Thermal Expansion (CTE) and the isothermal compressibility, of this class of compounds. In this work, we have investigated the temperature and pressure dependent X-ray diffraction of a series of Hydrido-closo-Borates (H-c-B) and of NaCB11H12 to provide experimental data of the thermal expansion and compressibility of constituents of solid state ionic conductors for sodium and lithium. In our investigations, we have discovered and solved two new polymorphs, namely the high temperature (ht, for temperature above room temperature) β-Li2B10H10, and the highpressure (hp) β-K2B10H10. These experimental results are completed by theoretical DFT calculations.

And a comprehensive analysis of the structural and vibrationnal behavior of these materials is given.

Background

Even though H-c-B exhibit a very rich crystal chemistry with numerous temperature and pressure induced phase transition, they share most of the time common aristotypes such as: cubic close packing (ccp), hexagonal close packing (hcp) and body center cubic (bcc) arrangement. [START_REF] Černý | The Crystal Chemistry of Inorganic Hydroborates[END_REF] Along the series of alkali dodeca H-c-B, at ambient pressure and temperature, the larger cations (K,Rb,Cs)2B12H12 crystallize in the cubic Fm-3 space group and Li2B12H12 in the cubic Pa-3 space group with undistorted ccp while Na2B12H12 adopts the monoclinic P21/c symmetry with the distorted ccp. With respect to the deca H-c-B, (Na,K,Rb)2B10H10 adopt the monoclinic P21/c space group with distorted ccp, hcp and hcp respectively whereas Li2B10H10 stand as an exception with the hexagonal space group P6422 without cubic or hexagonal compact underlying packing. Their temperature induced polymorphic phase transition have been studied and solved for, (Li,Na,Rb,Cs)2B12H12 and for Na2B10H10, [START_REF] Verdal | Evidence of a Transition to Reorientational Disorder in the Cubic Alkali-Metal Dodecahydro-Closo-Dodecaborates[END_REF][START_REF] Paskevicius | First-Order Phase Transition in the Li2B12H12 System[END_REF][START_REF] Verdal | Complex High-Temperature Phase Transitions in Li2B12H12 and Na2B12H12[END_REF] while pressure induced phase transitions have been investigated solely for Na2B12H12.

It is undergoing two phase transitions at relatively low pressure: P21/c → [0.3-0.8 GPa] Pbca → [5.7-8.1 GPa] Pnnm. [START_REF] Moury | Pressure-Induced Phase Transitions in Na2B12H12, Structural Investigation on a Candidate for Solid-State Electrolyte[END_REF] Fig. 1 gathers the different symmetries encountered for all the known H-c-B together with their underlying packing in a group-subgroup graph. It is worth noting that most of the H-c-B exhibit a direct group-subgroup relationship (red path on Fig. 1) that can come into play for the phase transitions for these compounds as discussed in detail for the pressure-induced transitions of Na2B12H12. [START_REF] Moury | Pressure-Induced Phase Transitions in Na2B12H12, Structural Investigation on a Candidate for Solid-State Electrolyte[END_REF] Furthermore, the preferred packing for this family is ccp with exception for ht polymorphs of Na2B12H12 adopting the bcc packing for the ht1-β-Na2B12H12 and ht2-γ-Na2B12H12.

Na2B12H12 possesses the richest phase diagram among all the H-c-B and it is the only one found in the bcc arrangement which is known to favor ionic conductivity.

Pressure dependence

High pressure X-ray diffraction

Six different samples (Li,Na,K)2B12H12, (Li,K)2B10H10 and NaCB11H12 were investigated at Swiss Norwegian Beam Line (SNBL) in order to study their behavior under pressure. Except for Na2B12H12, for which the hp phase transitions were already described, [START_REF] Moury | Pressure-Induced Phase Transitions in Na2B12H12, Structural Investigation on a Candidate for Solid-State Electrolyte[END_REF] K2B12H12 also undergoes to a reversible phase transition at pressure above 2 GPa toward a polymorph isostructural to hp2-ζ-Na2B12H12 with the orthorhombic Pnnm symmetry (Fig. 2 a). The cell parameters were first determined using Pawley refinement with hp2-ζ-Na2B12H12 as first input and manually increased to fit the diffraction pattern, once good approximation was found the refinement was carried out. The Rietveld refinement was subsequently achieved with the as-obtained cell parameters and hp2-ζ-Na2B12H12 atomic positions that enabled obtaining hp-β-K2B12H12 with a = 7.1670(13) Å, b = 9.212(6) Å and c = 7.560 (3) Å (Fig. 2 b).

Despite the low quality of the pattern, due to the strains and preferential orientations induced by the pressure, refinement successfully converged with the reliability factors of Rwp = 1.88, Rp = 1.16 and GoF = 6.5 (Fig. S1). S3) was performed prior to the comparison. The phase transition is diplacive combining a diffusionless (martensitic-like) transformation for the B12H12 2-units with the displacement of potassium cation similarly to Na2B12H12. [START_REF] Moury | Pressure-Induced Phase Transitions in Na2B12H12, Structural Investigation on a Candidate for Solid-State Electrolyte[END_REF] The martensitic-like transition is displayed in Fig. S4 during which the cubic lattice is transformed into the monoclinic one. The deformation leads to the following high values of the Lagrangian strain tensor with e11 = e33 = 0.1515, e22 = -0.2643 and e31 = e13 = 0.1272 hence must be taken into account to treat the phase transition using a finite strain approach. As consequence, the Landau free energy must be built with an order parameter-strain coupling. [START_REF] Schranz | Finite Strain Landau Theory of High Pressure Phase Transformations[END_REF] Using group-theory analysis with amplimode, [START_REF] Orobengoa | AMPLIMODES: Symmetry-Mode Analysis on the Bilbao Crystallographic Server[END_REF][START_REF] Perez-Mato | Mode Crystallography of Distorted Structures[END_REF] in Bilbao Crystallographic Server, one can identify that the symmetry reduction from Fm-3 to P21/c is driven by three one dimensional irreducible representations (irreps) Γ 4+ at wave vector k (0,0,0), X 1+ and X +2 and one three dimensional

X 2+ irrep at k (0,1,0).

Mechanical properties.

In the pressure range [0-6 GPa], Li2B12H12, (Li,K)2B10H10 and NaCB11H12 do not undergo pressureinduced phase transition. Together with hp2-ε-Na2B12H12 and hp-β-K2B12H12 their cell volumes were determined as a function of the pressure. When the data allowed, experimental bulk moduli were determined by fitting the Murnaghan equation of state (Eq. 1) with the experimental data (Fig. S5) and results are gathered in Table 1.

𝑉(𝑃) = 𝑉 0 (1 + 𝐾 0 ′ 𝑃 𝐾 0 ) -1/𝐵 0 ′ (1)
The experimental bulk moduli determined for the abovementioned compounds are in the range 16.3 to 25.5 GPa revealing very high compressibility in good agreement with our previous study about Na2B12H12. [START_REF] Moury | Pressure-Induced Phase Transitions in Na2B12H12, Structural Investigation on a Candidate for Solid-State Electrolyte[END_REF] Elastic properties of the ordered phases of alkali H-c-B were also determined by DFT calculation (Table 1). They are in good agreement with experimental values validating our calculation strategy.

One has to keep in mind that calculated values correspond to the adiabatic constants, while experimental data are for isothermal values. Evolution of the bulk (K) and the shear (G) moduli as a function of the volume per formula unit is represented in Fig. 3. For Li cation the bulk and the shear moduli are systematically larger than for other elements of the group. Along the series M2B10H10, the bulk modulus somewhat decreases for heavier cation while for the M2B12H12 family one can observe a slight increase with the mass of the cation. The shear modulus does not follow any obvious trend, however it is significantly smaller than for oxides or sulfides (>10 GPa) [START_REF] Wang | Elastic Properties of New Solid State Electrolyte Material Li10GeP2S12: A Study from First-Principles Calculations[END_REF] . This is an indication of malleability of these compounds, especially with dodeca H-c-B anions, for example following Pugh criterion for ductile materials is having ratio K/G > 1.75. 36 The relatively large compressibility and malleability of H-c-B is beneficial for solid-state battery manufacturing making it easier to densify the solid-electrolyte layer and to achieve intimate contact with the electrode. If a good contact between H-c-B and active material is established (via solution processing for example) [START_REF] Duchêne | Crystallization of Closo-Borate Electrolytes from Solution Enabling Infiltration into Slurry-Casted Porous Electrodes for All-Solid-State Batteries[END_REF] it can be maintained upon cycling thanks to their high deformability. It is worth mentioning that mechanical properties obtained from structural studies cannot always be translated to bulk properties in a battery where a solid electrolyte is typically a pressed polycrystalline powder. Nevertheless, H-c-B have proven to maintain stable interfaces in all-solid-state-batteries upon many cycles, including without applying significant external mechanical pressure. [START_REF] Asakura | 4 V Room-Temperature All-Solid-State Sodium Battery Enabled by a Passivating Cathode/Hydroborate Solid Electrolyte Interface[END_REF] The experimental and theoretical values appear to be characteristic for the entire series of compounds and can thus be extrapolated to predict the behavior of new ionic conductors based on H-c-B and derived compounds.

Additionally, the very low shear moduli is an indication that such materials easily adapt to the structural changes of the electrode.

Temperature dependence High temperature X-ray diffraction

Most of the crystal structures of deca and dodeca H-c-B of alkali metals are known for the low and high temperature polymorphs, [START_REF] Černý | The Crystal Chemistry of Inorganic Hydroborates[END_REF][START_REF] Verdal | Evidence of a Transition to Reorientational Disorder in the Cubic Alkali-Metal Dodecahydro-Closo-Dodecaborates[END_REF][START_REF] Verdal | Complex High-Temperature Phase Transitions in Li2B12H12 and Na2B12H12[END_REF] with exceptions concerning the high temperature phases of Li2B10H10 and (K,Rb)2B12H12 which were observed by DSC measurements but the structure has never been solved. [START_REF] Wu | Structural Behavior of Li2B10H10[END_REF][START_REF] Verdal | Evidence of a Transition to Reorientational Disorder in the Cubic Alkali-Metal Dodecahydro-Closo-Dodecaborates[END_REF] K2B12H12 undergoes to a phase transition at about 540°C which is around the transformation temperature of the glass capillary hence the transition was not recorded during our experiment. Nonetheless we did observe a phase transition for rt-α-Li2B10H10 starting to transform into ht-β-Li2B10H10 at 361°C. From this temperature both polymorphs coexist up to 384°C temperature at which rt-α-LiB10H10 totally transforms into ht-β-LiB10H10. At 390°C the diffraction peaks ht-β-LiB10H10 start to decrease with the appearance of an amorphous and a new, crystal phase. The possible nature of this new crystal structure will be discussed below. From 430°C only the new and amorphous phases are present up to 453°C, the temperature at which the compound becomes amorphous. Fig. 4 displays the diffraction patterns for the different steps described. These observations are in good agreement with the previous study on Li2B10H10 thermal behavior in which an entropically driven order-disorder phase transition was suggested. [START_REF] Wu | Structural Behavior of Li2B10H10[END_REF] However, the appearance of the unidentified crystal phase was never reported; due to the low quality of the diffraction pattern a direct structural determination was not possible during our experiments but the structure was likely determined with support of DFT calculations (see below).

DFT calculations for ht-β-Li2B10H10

These structures differ in the average orientation of B10H10 2-anions, as shown in the Fig. S6. For the structure with Pa-3 symmetry the B10H10 2-anions are oriented such that the longer anion axis is along one of the principal lattice directions (3 preferred orientations). This results an average quasioctahedral shape. For Fm-3m symmetry there are four orientations along the cubic unit cell diagonals preferred by B10H10 2-anions. They average to the effective cubic shape of anion, see Fig. S6. Since both geometrical figures, the cube and the octahedron have the same number of symmetry elements the distinction between two crystal structures of Li2B10H10 must be related to the positions of cations which is coupled to the anion orientation. In order to determine which of the two orientations are the preferred (has larger cohesive energy) we performed series of DFT calculations. Since the high temperature phase is disordered, calculations procedure was developed. The structures with random [START_REF] Wu | Structural Behavior of Li2B10H10[END_REF] . For the configurations that are the least stable the Li -H separation strongly differs from the low temperature one: Li -H spacing has broad distribution within the range from 1.9 Å to 2.7 Å. Such short interatomic distances indicate that Li is closely connected to anions. This can be seen in Fig. 5 where distribution of cations is shown for the structure with lowest energy, and cations confined to the tetrahedral interstitial voids. The least stable configuration cations are located at tetrahedral facets rather than in tetrahedral center. The projection of B10H10 2-anion orientations on (a,b) crystal plane indicate they are oriented with longer axis along (100), ( 010) or (001) lattice directions except for the least stable structure where the orientation is along unit cell diagonals, see Fig. 5. This point toward Pa-3 space group symmetry for the high temperature phase of Li2B10H10 and possible second phase transition to Fm-3m space group at higher temperatures prior the thermal decomposition of compound. Having this in mind one can suggest that the unidentified ht phase could be the cubic Fm-3m. A Rietveld refinement was then performed with a = 10.219(3) Å, however owing to the poor quality of the diffraction pattern the fit was not optimal but the solution cannot be excluded. Additional work would be necessary to demonstrate this last transition.

Coefficient of the Thermal Expansion

Coefficients of the Thermal Expansion (CTE) were determined for several alkali hydrido-closo-(car)borates studied here by fitting the evolution of the volume as a function of the temperature to a polynomial function (Eq. 2). The CTE α can be determined using Eq. 3. The results are gathered in Table 2.

𝑉 (Å 3 ) = 𝑉 0 + 𝐴 • 𝑇 + 𝐵 • 𝑇 2 + 𝐶 • 𝑇 3 (2) 𝛼(𝐾 -1 ) = 1 𝑉 ( 𝜕𝑉 𝜕𝑇 ) 𝑝 = 𝛼 0 + 𝐷 • 𝑇 + 𝐸 • 𝑇 2 (3)
The relatively high value of the CTE, compared to oxides which are usually two orders of magnitude lower than this family of compounds, point out a strong dilatation for these materials with temperature. This feature could induce mechanical stresses between the different component of the battery cathode (usually oxide-based), anode and electrolyte. These mechanical stressed would be detrimental in case of overheating. However the very low shear moduli for these compounds could overcome this issue, the material will flow and with proper construction hence shall not affect the interface. Low shear moduli can be seen as the behavior of the materials, upon external stimuli, approaching the one of the liquid. Additionally elastic constants (Table 1.) indicate malleability of this class of materials. The thermal expansion is a manifestation of the anharmonicity of the lattice vibrations and fitting the evolution of the volume as a function of the temperature with polynomial of order ≥ 2 indicate a strong anharmonicity, related to the orientational disorder of anions.

Structural and vibrational analysis

Symmetry analysis points out to relations between space group symmetry of all alkali metal deca-and dodeca H-c-B. The low temperature hexagonal phase of Li2B10H10 is somehow an exception from the family of structures originating from Pm-3m group.

In order to better understand the similarities between this family of compounds, below we report on structural and vibrational analysis. In Fig. 6 the distribution of the zone center modes for alkali metal deca and dodeca H-c-B together with the radial distribution function for metal-hydrogen separation is displayed. With regards to deca H-c-B a clear division into two groups is apparent: Li, Na show short separation between hydrogen and metal that can be correlated with broader extend of the lattice modes. Especially for lithium the lattice modes go beyond 250 cm -1 which is an indication of direct and strong Li -B10H10 2-interaction. More detailed information on internal B10H10 2-vibrations can be found elsewhere, [START_REF] Sethio | A Theoretical Study of the Spectroscopic Properties of B2H6 and of a Series of BxHy z-Species (x = 1-12, y = 3-14, z = 0-2): From BH3 to B12H12 2[END_REF][START_REF] Hagemann | Correlating Boron-Hydrogen Stretching Frequencies with Boron-Hydrogen Bond Lengths in Closoboranes: An Approach Using DFT Calculations[END_REF] however the splitting of B-H modes (>2400 cm -1 ) is related to the bond distortion and the internal closocage vibrations in the range 400 -1200 cm -1 are modified by small deformations of anion. The largest splitting of the highest frequency modes (B-H stretching modes) is observed for the heaviest cations, K, Rb, and Cs. This is related to the hcp packing of anions, [START_REF] Černý | The Crystal Chemistry of Inorganic Hydroborates[END_REF] rather than ccp that is observed for Li and Na, and symmetry of B10H10 2-molecule. This anion has point group symmetry D4d as for capped square antiprism, see Fig. 7. Distribution of cations is compatible with C4 and S8 symmetry elements of the molecule thus the differences can be ascribed to packing of anions.

For these modes similarities for K, Rb, and Cs are visible, which would be expected as these compounds have the same symmetry for the same anions. A large splitting of B-H modes for K, Rb, and Cs deca H-c-B shall be noticed, they are related to the distribution of these cations in the lattice which will be discussed below. For compounds with Li and Na B-H stretching have distinct splitting of frequencies. [START_REF] Hagemann | Correlating Boron-Hydrogen Stretching Frequencies with Boron-Hydrogen Bond Lengths in Closoboranes: An Approach Using DFT Calculations[END_REF] While the spectrum related to the internal vibrations of anions have similarities within each class of compounds the differences are related to different site symmetry of anion in Li and Na containing compounds. The largest differences between them are present in the upper range of lattice modes above 150 cm -1 . While for K, Rb, and Cs a clear gap between lattice and internal anion modes is present; this gap is smaller for Li and Na, especially for Li2B10H10 there are lattice vibrations present above 250 cm -1 , for Na they are less extended as well as for icosahedral dodeca anion they extend to lower frequencies.

In Fig. 6 c and d the pair distribution function calculated for metalhydrogen separation is presented.

For both classes of compounds clear differences are visible between Li/Na and heavier metals. While well-defined Li -H separation just above 2 Å is visible for Li2B10H10, these separations are still present in Na2B10H10 with some additional peaks below 3 Å. All heavier alkali metals are separated from nearest hydrogen by more than 2.5 Å, and the distribution of M -H distances is not well defined The short metalhydrogen distances for two lightest metals are correlated with broader range of their lattice modes and indicate direct M -H interaction. This is most apparent for Li2B10H10. From the Pauling rules for ionic compounds the coordination metals can be estimated from the ratio of ionic radii of anion and cations. [START_REF] Pauling | The Principles Determining the Structure of Complex Ionic Crystals[END_REF] This is particularly well observed in metal hydridoborates, where one can assume radii of B10H10 2-as 6.0 Å (5.8 Å for B12H12 2-). Size of alkali metal cations increases with atomic number and according to Shannon radii is: [START_REF] Shannon | Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides[END_REF] 1.2 Å for Li, 1.9 Å for Na, 2.66 Å for K, 2.96 Å for Rb, and 3.38 Å for Cs. The ionic size ratio for compounds with B10H10 2-anions is: 0.20 for Li (3), 0.32 for Na (4), 0.44 for K (6), 0.49 for Rb (6), and 0.56 for Cs (6); numbers in parentheses indicate coordination number for anion. For compounds with B12H12 2-anions formal coordination numbers are the same. In fact for all these compounds cations are located within coordination tetrahedra between nearest anions, as even for heavier alkali metals the ratio is close to 0.414 that is the limit of half of cations are located at octahedral voids. Exceptions are Li and Na, where each cation is surrounded by three anions, thus is located at the face of coordination tetrahedra. Relation between coordination number and lattice type is known to correlate with ionic conductivity, [START_REF] Wang | Design Principles for Solid-State Lithium Superionic Conductors[END_REF][START_REF] Sadikin | Modified Anion Packing of Na2B12H12 in Close to Room Temperature Superionic Conductors[END_REF] and bcc anion packing is ultimate for the best ion conductor.

In Fig. 7 we present coordination polyhedra for B10H10 anions by Li and Na. Such presentation reveals highly symmetric polyhedra for Li and Na. The positions of cations follow D4d symmetry of the anion, a deformed cubic coordination for Na2B10H10, and six Li cations surround the anion in Li2B10H10. Among the symmetry elements of D4d point group of capped square antiprism that is B10H10 only these related to rotations are accessible due to thermal excitations, see Fig. 2). As shown in Fig. 6 c for any configuration considered in the cubic phase the shortest distance between hydrogen and lithium do not increase above 2.1 Å which is equivalent to the fact that with increasing separation between anions the compound disintegrate into molecular entities consisting of cations and anions.

In the phase transition of Na2B10H10 from the low temperature P21/c structure to the cubic Fm-3m one is related to orientational disorder of anions, while eight Na + effectively forms regular cubic coordination around an anion (Fig. 7). This is related with disorder in the cationic sublattice but not with changes of the anion coordination number.

Conclusion

In this work a class of compounds, namely hydrido-closo-borates, has been subject to structural loaded in an argon filled glove box (MBraun O2<0.1 ppm, H2O<0.1ppm). The samples were loaded with ruby crystals, for pressure calibration, into a hole of 250 μm drilled in a stainless-steel gasket. No pressure transmitting media was used owing to the low bulk modulus of these families of materials. For high temperature experiments, the samples were loaded in 0.5 mm glass capillary in the glove box. The temperature was controlled using Cyberstar hot blower. For hp-β-K2B12H12 and ht-β-Li2B10H10 the structure were solved using the isostructural models hp3-ζ-Na2B12H12 and rt-α-Li2B12H12 manual adjustment of the cell parameters were done prior to their refinement using Pawley algorithm implemented in TOPAS, [START_REF] Coelho | Whole-Profile Structure Solution from Powder Diffraction Data Using Simulated Annealing[END_REF] this algorithm was used also for the refinement of the cell parameters as a function of temperature and pressure. The refinements of the structure were performed using the Rietveld method, [START_REF] Rietveld | A Profile Refinement Method for Nuclear and Magnetic Structures[END_REF] in the TOPAS program. [START_REF] Coelho | Whole-Profile Structure Solution from Powder Diffraction Data Using Simulated Annealing[END_REF] For the hp polymorph spherical harmonic approach was used to simulate the strong preferential orientation caused by the DAC. The cell parameters as a function of temperature and pressure were refined using Pawley algorithm.

DFT calculations

Calculations were performed within density functional theory (DFT) with periodic plane wave basis set as implements in Vienna ab initio Simulation Package. [START_REF] Kresse | Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set[END_REF][START_REF] Kresse | Efficiency of Ab-Initio Total Energy Calculations for Metals and Semiconductors Using a Plane-Wave Basis Set[END_REF] The calculations parameters were: cutoff energy for the basis set expansion 700 eV; the k-point sampling density k•a ≥ 20; the convergence criteria for the electronic degrees of freedom was 10 -6 eV/A; for the structural relaxations the conjugated gradient method with convergence 10 -2 eV/A was used; Projected Augmented Wave Potentials (PAW) [START_REF] Blöchl | Projector Augmented-Wave Method[END_REF][START_REF] Kresse | From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method[END_REF] were used for atoms with the electronic configuration (1s 1 ) for H, (2s 2 2p 1 ) for B, (1s 2 2s 1 ) for Li, (2p 6 3s 1 ) for Na , (3p 6 4s 1 ) for K , (4p 6 5s 1 ) for Rb , (5p 6 6s 1 ) for Cs. The gradient corrected (GGA) exchangecorrelation functional and the non-local corrections accounting for a weak dispersive interactions were applied. [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF][START_REF] Dion | Van Der Waals Density Functional for General Geometries[END_REF][START_REF] Maniadaki | Theoretical Description of Alkali Metal Closo-Boranestowards the Crystal Structure of MgB12H12[END_REF] The normal modes at the Γ-point were calculated in the real space with atomic displacements ±0.1 Å in all symmetry inequivalent directions, and visualized by placing Lorentzians with half-width 5 cm -1 for each mode. The normal mode frequencies were obtained by direct diagonalization of the dynamical matrix obtained from the forces calculated for displaced configurations. Elastic constants were calculated via deformation of the unit cell, ±1% in each relevant direction and angle. For normal mode and elastic properties fully optimized structures were used.

Figure 2 .

 2 Figure 2. Diffraction patterns of K2B12H12 at 0 GPa (black), 2 GPa (red) and back to ambient pressure (blue) depicting it reversible pressure induced phase transition (a). Representation of the hp-β-K2B12H12 crystal structure (b). Owing to the low quality of the diffraction pattern, DFT calculations were performed to further confirm the stability of the Pnnm symmetry. The diffraction peak at 8.2° cannot be explain by the refinement, which can be due either to the remaining 111 reflection from the lp-phase of another polymorph. The calculated pressure dependence of the free energy (F = E0 + pV) reveal the phase transition of K2B12H12 (Fm-3 → Pnnm) at the pressure 3.58 GPa hence further confirm the experimental data (Fig. S2). The phase transition is accompanied by ~7% specific volume change indicating first order transition. While a group-subgroup relationship exists between Fm-3 and Pnnm a

Figure 3 .

 3 Figure 3. Calculated bulk (squares) and shear (circles) moduli for alkali metal deca-and dodeca-H-c-B. The average (Xr+Xv)/2 values are presented.

Figure 4 .

 4 Figure 4. Temperature dependency of the diffraction patterns for Li2B10H10. At RT only rt-α-LiB10H10 is present, at 361°Cappearance of ht-β-LiB10H10 (arrow) and coexistence of both polymorphs up to 384°C, at 390°C appearance of the unidentified phase (arrow) and coexistence of both phases up to 430°C and at 453°C amorphization of the material.

Figure 5 .

 5 Figure 5. The energy distribution of Li2H10H10 in the cubic ccp structure with respect to the hcp (P6422) ground state. The inserts show radial distribution function for Li -H separation, red line is for the reference hexagonal structure. The green spheres are projections of anions on (a,b) plane for given energy range. The right side shows distribution of cations corresponding to the configurations with the lowest and highest energy, for simplicity only coordination tetrahedra of the fcc lattice are shown with light blue shade.

  for deca H-c-B (Fig.6 c), showing bond length distribution between 2.5 Å and 6 Å. The opposite is observed in dodeca H-c-B M-H spacing is well defined for Cs, Rb and K and it is decreasing with decreasing mass (radius) of metal cation (Fig 6. d). For Li and Na in dodeca H-c-B this spacing is smaller (~2 Å) and distance distribution similar to this in deca H-c-B is clear, also for sodium.

Figure 6 .

 6 Figure 6. Distribution of the phonon distribution at the Γ-point for alkali metal deca H-c-B (a) and dodeca H-c-B (b), and the radial distribution function for cationhydrogen separation for alkali metal deca H-c-B (c) and dodeca H-c-B (d). The calculated frequencies are broaden with 5 cm -1 Lorentzian.

Rysunek 1 .Rysunek 1 .Rysunek 1 .

 111 Distribution of the phonon distribution at the Γ-point for alkali metal deca-hydridoborates (left), and the radial distribution function for cation -hydrogen separation (right). The calculated frequencies are broaden with 5 Distribution of the phonon distribution at the Γ-point for alkali metal dodeca-hydridoborates (left), and the radial distribution function for cation -hydrogen separation (right). The calculated frequencies are broaden with 5 Distribution of the phonon distribution at the Γ-point for alkali metal dodeca-hydridoborates (left), and the radial distribution function for cation -hydrogen separation (right). The calculated frequencies are broaden with 5 The structure analysis indicates that in P21/c structure of (K,Rb,Cs)2B10H10

7 .

 7 Improperrotation by 45° (S8) without reflection is the less energy demanding process that preserves orientation of anions in the crystal opening additional sites for cations. This process will not change the hexagonal symmetry of the low temperature phase. Rotation around one of C2 axes by 90° changes anion orientation in the crystal lattice thus breaks a,b,a,b stacking of hcp lattice. This is in fact observed in the high temperature ccp structure of this compound, where the anions are still aligned along principal lattice directions with cations distributed in the tetrahedral void with similar Li -H separations as in the LT phase. The strong Li -H interaction in Li2B10H10 is also related to low thermal expansion of this compound within the H-c-B class (see Table

Figure 7 .

 7 Figure 7. Coordination polyhedra for B10H10 2-anions in alkali metal deca H-c-B. Small green/gray spheres are for boron/hydrogen; large green are Li, yellow -Na (a). The schematic view of the symmetry elements of D4d point group of B10H10 2-anion: C4/ S8 are rotations/improper rotations around four-fold and eight-fold axis, C2/C'2 are rotations around twoaxis, and σd stands for the mirror plane (b).

Rysunek 1 .

 1 investigation using in situ XRD methods under external pressure and temperature stimuli, combined with DFT calculations. Those materials have demonstrated very high compressibility and very low shear moduli revealing highly malleable materials which would allow fast structural reconstruction under mechanical stresses. Furthermore, this family of compounds has exhibited very high CTE, two orders of magnitude higher than oxides. Interestingly, our investigations discover two new crystal phases, the first one for K2B12H12 resulting from the pressure induced phase transition around 2 GPa toward Pnnm symmetry and the second phase for which the transition is induced by the temperature and transforms Li2B10H10 from P6422 to Pa-3 symmetry as suggested by DFT calculations. The present study allows completing knowledge about the crystal chemistry of this astonishing class of compounds and further confirms their trend for the ccp underlying anions packing.Experimental.PXRDSamples were purchased from Katchem company. The sample was measured at the Swiss-Norwegian Beam Lines BM01 of the European Synchrotron Radiation Facility in Grenoble, France. A 2D image plate detector Pilatus 2M positioned at 411 mm from the sample was used with a wavelength of 0.71414 Å. The 2D diffraction patterns were integrated with Bubble software.[START_REF] Dyadkin | A New Multipurpose Diffractometer PILATUS@SNBL[END_REF] The sample detector geometry was calibrated with a LaB6 NIST standard. For high pressure experiments the Diamond Anvil Cell (DAC), with flat culet of diameter 600 μm, was (a) Coordination polyhedra for B 10 H 10 2-anions in alkali metal decaborates. Small green/gray spheres are for boron/hydrogen; large green are Li, yellow -Na. (b) the schematic view of the symmetry elements of D4d point group of B 10 H 10 2-anion: C 4 / S 8 are rotations/improper rotations around four-fold and eightfold axis, C 2 /C' 2 are rotations around two-fold axis, and σ d stands for the mirror plane.

31 Figure 1. Group

  -subgroup relationship between the hydrido-closo-borates. t stands for translationengleich and k klassengleich subgroups. The asterisk * indicates the crystal structures determined in this work. rt and lt stand for room temperature and low temperature, here bellow 250 K, respectively.

	Fm-3m	k	Pm-3m	t	P4/mmm
	ht-Na 2 B 10 H 10 (ccp)
	Cs 2 B 6 H 6 (ccp)
	ht-Cs 2 B 12 H 12 (ccp) Cs 2 B 6 H 6 (ccp)		k			t
	t
	ht-β-Li 2 B 12 H 12 (ccp) Fm-3		ht2-γ-Na 2 B 12 H 12 (bcc) Im-3m		Cmmm
	Rb 2 B 12 H 12 (ccp) ht3-δ-Na 2 B 12 H 12 (ccp) K 2 B 12 H 12 (ccp) K 2 B 6 H 6 (ccp)		k	P6 4 22 rt-α-Li 2 B 10 H 10	t	t
	rt-Cs 2 B 12 H 12 (ccp) k		ht1-β-Na 2 B 12 H 12 (bcc) Pm-3n		C222
	Pa-3
	rt-α-Li 2 B 12 H 12 (ccp)
	ht-β-Li 2 B 10 H 10 * Pbca t		hp-β-K 2 B 12 H 12 * Pnnm hp2-ζ-Na 2 B 12 H 12 (ccp)
	hp1-ε-Na 2 B 12 H 12 (ccp)
	t	t
	P2 1 /c
	rt-α-Na 2 B 12 H 12 (ccp) rt-Na 2 B 10 H 10 (ccp) rt-α-K 2 B 10 H 10 (hcp)		P-1 lt-Rb 2 B 10 H 10 (hcp)
	rt-Rb 2 B 10 H 10 (hcp)

t Result and discussion.

Table 1 .

 1 Summary of the bulk (K) and shear (G) moduli obtained experimentally and by DFT of the alkali deca and dodeca H-c-B. The indexes r and v stands for Reuss and Voigt limit.

	Compound	Space group	Gr [GPa]	Gv [GPa]	Kr [GPa]	Kv [GPa]	Kexp
	Li2B10H10	P6422 (#181)	6.20	7.43	20.01	20.46	16.3(4.6)
	Li2B12H12	Pa -3 (#205)	11.31	8.96	20.84	20.84	21.3(1.4)
	Na2B10H10	P21/c (#14)	8.35	7.21	19.08	19.24	-
	Na2B12H12	P21/c (#14)	2.11	6.08	7.20	16.10	13.1(6)
	K2B10H10	P21/c (#14)	8.38	7.24	19.07	19.21	25.5(2.5)
	K2B12H12	Fm-3 (#202)	7.41	5.99	16.85	16.49	-
	Rb2B10H10	P21/c (#14)	5.25	4.45	17.10	17.56	-
	Rb2B12H12	Fm-3 (#202)	7.34	5.94	16.80	16.80	-
	Cs2B10H10	P21/c (#14)	2.33	3.14	10.25	15.27	-
	Cs2B12H12	Fm-3 (#202)	6.17	4.91	14.68	14.68	-

  anion orientations/cation distribution were used to calculate energy; details are presented in the Supporting Information. The energy distribution for the atomic configurations in the cubic phase of Li2B10H10 is presented in Fig.5. It consists of separated energy maxima starting with ΔE = 0.014 eV/atom above the P6422 ground state energy up to ΔE = 0.035 eV/atom for the least

	stable configurations. For four selected regions the radial distribution function (rdf) for Li -H
	separation were calculated, as shown in the inserts in Fig. 5. In general rdf for the most stable
	configurations resembles this of the low temperature phase, where two Li -H distances at 2.1 Å and
	2.3 Å are present (they are larger than 2.028 Å, 2.044 Å and 2.216 Å reported for experimental
	u.) structure of Li2B10D10)		
	(a.
	Intensity		 					390°C 384°C 361°C
							430°C
							453°C
	6	8	10	12	14	16	18	20
				2()	°RT

Table 2 .

 2 Summary of the coefficients (V0, A, B, C) of the polynomials used to fit the evolution of the volume of the cells as a function of the temperature together with the coefficients (α0,D,E) of the equations of the CTE as a function of the temperature and the average of the CTE (α(avg)) calculated on the indicated temperature range.

	Compound	Space group	V0 (Å 3 )	A	B	C	α (avg) (K -1 )	α0 (K -1 )	D (K -2 )	E (K -3 )	T (K)
	Li2B10H10	P6422	600.9(3)	0.020(1)	3.9(1) 10 -5	-	0.95 10 -4	3.333(3) 10 -5	1.342(1) 10 -7	-1.67(1) 10 -11	320-620
	Na2B10H10	P21/c	870.8(3)	0.123(1)	-	-	1.35 10 -4	1.410(1) 10 -4	-	-	300-380
	Na2B10H10	Fm-3m	891.2(4)	0.1376(7)	-	-	1.4 10 -4	1.533(7) 10 -4	-	-	400-795
	Na4(B10H10)(B12H12)	Fm-3m	900.5(6)	0.468(3)	-5.50(7) 10 -4	3.19(4) 10 -7	1.8 10 -4	4.69(1) 10 -4	-1.098(3) 10 -6	9.32(2) 10 -10	300-796
	K2B12H12	Fm-3	1141(1)	0.1838(2)	-	-	1.4 10 -4	1.90(3) 10 -5	-	-	305-724
	NaCB9H10	Pna21	406.1(2)	0.1129(4)	-	-	2.5 10 -4	2.751(1) 10 -4	-	-	325-500
	Na2B12H12	P21/c	492	0.19	-3.8 10 -4	3.67 10 -7	1.3 10 -4	3.902 10 -4	7.723 10 -7	7.459 10 -10	273-540
	Na2B12H12	Pm-3n	457.9	0.12	-	-	2.3 10 -4	2.621 10 -4	-	-	540-580
	Na2B12H12	Im-3m	471.4	0.24	-	-	1.8 10 -4	5.091 10 -4	-	-	580-700
	Na2B12H12	Fm-3m	1051.1	0.14	-1.88 10 -5	-	1.1 10 -4	1.109(1) 10 -4	-1.075(6) 10 -11 -	540-750

Acknowledgements.

We thank V. Dmitriev, I. Dovgaliuk and D. Chernyshov of the SNBL at ESRF for their support with the high-pressure experiments. We thank the financial support from the SNSF Sinergia project CRSII2_160749 and the SNSF project 200021_169033/1. Z.L. acknowledges NCN support through 2019/01/Y/ST5/00046 project and CPU allocation at PL-Grid infrastructure.

Associated content.

Supporting Information available: Calculation procedure for ht-Li2B10H10, the result of the Rietveld refinement, of the free energy as a function of the pressure and the transformation of the structures from Pnnm to P21/c for hp-K2B12H12 (Fig. S1-4). The evolution of the cell volume as a function of the pressure (Fig. S5) and the orientation of B10H10 2-anions of for ht-Li2B10H10 with Fm-3m and Pa-3

Table of Content

Synopsis. Herein we studied the pressure and temperature dependancies of alkali hydrido-closo-borates to extract mechanical propertie of this class of compound which have promising future as solid electrolyte. In our search, we have discovered and have solved two new high pressure and high temperature crystal structures.