Erwan Peigney

Angéline Poulon-Quintin

Marjorie Cavarroc

Guillaume Aubert

Mathieu Duttine

Christine Labrugère Sarroste

Cyril Aymonier
email: cyril.aymonier@icmcb.cnrs.fr

Investigation of the interaction between adsorbed water and various morphologies of boehmite nanoparticles prepared by continuous supercritical hydrothermal synthesis

Keywords: Synthetic boehmite nanoparticles, Water and nitrate adsorption, pH-dependent morphology, Continuous supercritical hydrothermal synthesis, Supercritical water

Boehmite applications are often subject to specific characteristics in terms of morphologies and surface properties. The versatile continuous supercritical hydrothermal method allows synthesizing various boehmite morphologies from rhombic to hexagonal platelet-like nanoparticles depending on the preferential adsorption of part of the precursor after transformation species on the (020) surface resulting in a preferential growth along [001] and mostly [100] directions. This adsorption on surface free sites plays the role of morphology stabilizer and results in the reduction of adsorbed water.

The interaction between adsorbed water and the boehmite morphology depends on various parameters; 1) the structure order of boehmite as exhibited by XPS analysis associated with 1 H Solid-State NMR spectroscopy and, 2) the number of available free sites, which tend to be filled by the adsorption of nitrates as observed by Raman spectroscopy.

Introduction

An explosion of research works on nanotechnologies has marked the last decade. In particular, inorganic nanoparticles are of interest in broad domains ranging from electronics to pharmaceuticals. These fields require high performance crystallized materials with controlled sizes and morphologies. Additionally, industrial scale applications need low-price, reproducibility and environmentally friendly conditions to meet society challenges [START_REF] Aymonier | Review of supercritical fluids in inorganic materials science[END_REF]. This can be achieved through continuous supercritical hydrothermal synthesis [START_REF] Aymonier | Playing with chemistry in supercritical solvents and the associated technologies for advanced materials by design[END_REF].

Boehmite, an aluminium oxyhydroxide (γ-AlOOH), is widely used as catalyst [START_REF] Takagaki | Solid Lewis acidity of boehmite γ-AlO(OH) and its catalytic activity for transformation of sugars in water[END_REF], adsorbent [START_REF] Granados-Correa | Chromium (VI) adsorption on boehmite[END_REF][START_REF] Li | The vanadate adsorption on a mesoporous boehmite and its cleaner production application of chromate[END_REF][START_REF] Zhang | In Situ Synthesis of γ-AlOOH and Synchronous Adsorption Separation of V(V) from Highly Concentrated Cr(VI) Multiplex Complex solutions[END_REF], coating [START_REF] Wang | In situ growth of hierarchical boehmite on 2024 aluminum alloy surface as superhydrophobic materials[END_REF], membrane [START_REF] Vatanpour | Boehmite nanoparticles as a new nanofiller for preparation of antifouling mixed matrix membranes[END_REF] and fire-retardant [START_REF] Zhang | Thermal stability and flame-retardancy mechanism of poly(ethylene terephthalate)/boehmite nanocomposites[END_REF]. Furthermore, it is an important precursor for the preparation of aluminium oxides such as ϒ-, θ-, χ-, κand α-Al 2 O 3 [START_REF] Boumaza | Transition alumina phases induced by heat treatment of boehmite: An X-ray diffraction and infrared spectroscopy study[END_REF].

Boehmite has an orthorhombic crystal structure (space group Cmcm), shown on Figure 1, characterized by double sheets of distorted edge-sharing octahedra with oxygen atoms at the top and aluminium ones at the centre. These double layers are linked together by hydrogen bonds forming parallel rows of chains in zig-zag perpendicular to the b-axis.

Because of the weak interaction, the crystal surface ends in the interface between two double layers. This produces surfaces full of hydroxyl groups [START_REF] Bokhimi | Dependence of Boehmite Thermal Evolution on Its Atom Bond Lengths and Crystallite Size[END_REF][START_REF] Bokhimi | Relationship between Crystallite Size and Bond Lengths in Boehmite[END_REF]. Consequently, boehmite nanoparticles tend to have a large excess of adsorbed water up to 30 wt.%, leading to the name of "pseudo-boehmite" [START_REF] Baker | Water content of pseudoboehmite: A new model for its structure[END_REF]. At first, the literature hypothesized that this adsorbed water was incorporated in the sheet interlayer along the b-axis [START_REF] Lippens | Structure and texture of aluminas[END_REF]. The lattice deformation along this specific axis appears to be insufficient to insert water molecules which are considered to be only adsorbed on the boehmite surfaces [START_REF] Guzmán-Castillo | Effect of Boehmite Crystallite Size and Steaming on Alumina Properties[END_REF].

The structure tends to show platelet-like particles with a dominant (010) basal surface and less stable (100), ( 101) and (001) edge facets. A Density Functional Theory (DFT) study of boehmite surface properties exhibited the basal (010) surface forms about 50 % of the total surface [START_REF] Raybaud | Morphology and Surface Properties of Boehmite (γ-AlOOH): A Density Functional Theory Study[END_REF]. This surface is characterized by the smallest interfacial energy and a weak chemical reactivity with water. That is why water is mainly physisorbed on it.

The relative surfaces of each facet can be modified through the synthesis route, which may influence the adsorbed water content. The morphology of boehmite particles can be easily tuned thanks to the versatility of the continuous supercritical hydrothermal synthesis method.

Adschiri et al. have reported for the first time the use of this technique for the preparation of boehmite [START_REF] Adschiri | Rapid and Continuous Hydrothermal Synthesis of Boehmite Particles in Subcritical and Supercritical Water[END_REF] which, was traditionally synthesized using sol-gel precipitation or hydrothermal synthesis [START_REF] Hsu | Effect of salts on the formation of bayerite versus pseudo-boehmite[END_REF][START_REF] Tettenhorst | Crystal Chemistry of Boehmite[END_REF]. They showed the influence of various parameters such as temperature, pressure, pH and precursor concentration on the particle characteristics. Hakuta et al. were able, with a hydrothermal process, to modify the shape of boehmite particles from hexagonal platelet-like to rhombic shaped by increasing the temperature from 350 °C to 400 °C [START_REF] Hakuta | Effects of hydrothermal synthetic conditions on the particle size of γ-AlO(OH) in sub and supercritical water using a flow reaction system[END_REF]. More recently, Fujii et al. demonstrated the ability of alkylcarboxylic acids to change the aspect ratio of boehmite particles up to 5-fold [START_REF] Fujii | High-Speed Morphology Control of Boehmite Nanoparticles by Supercritical Hydrothermal Treatment with Carboxylic Acids[END_REF]. They suggested that carboxylic acid inhibits crystal growth along the [001] and [010] axes relative to the [100] direction through surface-capping, resulting in the formation of rodlike particles. The variation of the particle morphology necessarily generates an evolution of the active surfaces, the surface properties and thus of the potential quantity of adsorbed water. All of which play a crucial role in the final material properties and thus performance for applications where boehmite is used and need to be investigated. This paper reports the ability of supercritical boehmite γ-AlOOH surfaces to interact with water. Supercritical boehmite particles were synthesized using the continuous supercritical hydrothermal process. Microstructures were determined by Rietveld analysis of Powder X-Ray Diffraction (PXRD) data and Transmission Electron Microscopy (TEM). Adsorbed water was investigated using X-Ray Photoelectron Spectroscopy (XPS), 1 H Solid-State Nuclear Magnetic Resonance (NMR) and Raman Spectroscopy.

Experimental

Materials

Boehmite particles were prepared using the continuous supercritical hydrothermal synthesis.

Aluminium nitrate nonahydrate Al(NO 3 ) 3 •9H 2 O (> 98 % purity) was selected as precursor.

The metal salt was dissolved into distilled water (10-18 MΩ.cm). Sodium hydroxide NaOH (> 98 % purity) was used to increase the pH of the precursor solution from 3.9 up to 6.7. All the chemicals were purchased from Sigma-Aldrich and used as received.

Hydrothermal flow synthesis apparatus and procedure

Continuous flow synthesis was carried out in a homemade reactor custom-designed to minimize clogging problems. It is made of a 0.74 m long stainless steel pipe with an internal diameter of 4.78 mm. The set-up consists of a HPLC pump, two ceramic band heaters and a TESCOM back pressure regulator. This equipment has been thoroughly described elsewhere [START_REF] Musumeci | Suband supercritical hydrothermal route for the synthesis of xonotlite nanofibers for application to green concrete materials[END_REF].

The reactor is first pressurized to 25 MPa at ambient temperature using distilled water and then set to the process temperature of 400 °C. Precursor solution is fed at a flow rate of 3.5 or 6.6 mL.min -1 corresponding to residence times of 60 s and 20 s, respectively. At the outlet of the reactor, the chemical reaction is stopped by ice bath quenching. A stainless steel fritted filter is placed inside the quenching system to collect the produced nanoparticles. The product is recovered as a gel, which is washed several times with distilled water. Produced powders were then dried in an oven at 60 °C overnight.

Material characterization

Powder X-Ray Diffraction (PXRD)

Powder X-Ray Diffraction (PXRD) patterns were collected at room temperature on a PANalitycal X'pert PRO MPD diffractometer with the Bragg-Brentano θ-2θ geometry and over a 2θ angular range from 10 ° to 130 ° (minimum step of 0.008 °). The diffractometer was equipped with a Ge (111) monochromator, which generated a perfectly Cu Kα 1 radiation (λ = 1.540598 Å) at 45 kV and 40 mA. A multichannel X'Celerator detector was used to collect the XRD patterns. Prior to measurement, samples were finely ground to break up the aggregates and passed through a 50 μm mesh sieve. Afterward, powder was placed and randomly distributed on a silicone sample holder.

Rietveld refinement

Data were analysed by Rietveld structural model refinements using the FullProf Suite software package [START_REF] Rodriguez-Carvajal | FullProf Suite[END_REF]. The Thompson-Cox-Hasting formulation of the pseudo-Voigt function was used to fit the Bragg reflection. Refined parameters were the 8-coefficient polynomial functions of background, instrumental zero, scale factor, unit cell parameters, Gaussian and Lorentzian contributions and microstructural strain parameters. The model included only the Al and O atom positions and isotropic thermal parameters. Crystallite shape anisotropy, specific to boehmite particles, was taken into account through spherical harmonic functions. Up to six harmonics were used to describe size broadening for the orthorhombic structure [START_REF] Rodríguez-Carvajal | Study of Micro-Structural Effects by Powder Diffraction Using the Program FULLPROF[END_REF]. Simulated X-ray diffractogram quality was measured with the weighted-profile R-factor (R wp ).

Transmission Electron Microscopy (TEM)

The boehmite particle morphology was investigated using Transmission Electron Microscopy (TEM) with a ThermoFisher Talos F200S G2 microscope. Micrographs were captured with an electron beam voltage of 200 keV allowing an accessible range of magnification up to x1,050,000 and a resolution of 0.1 Å. Prior to the observation, less than 2 mg of powder were put in ethanol (≈ 1 mL) and dispersed with ultrasound during 5 min. Then, TEM thin foils were prepared by drop casting the solution made beforehand onto a holey carbon-coated grid and dried at room temperature.

X-Ray Photoelectron Spectroscopy (XPS)

XPS spectra of boehmite were collected with a ThermoFisher Scientific K-Alpha spectrometer using a monochromatic Al Kα X-radiation (1486.6 eV) and a hemispherical electron energy analyser. Powder was pressed onto indium foil fixed on a metallic support. This technique was used to characterize extreme surfaces of a set of grains (i.e. about 3 to 5 nm in depth) with a 400 µm diameter X-Ray spot size. The binding energies reported were referenced with the C 1s peak at 284.9 eV. Survey and the high-resolution spectra Al 2p , O 1s , and C 1s are reported. Elemental composition was calculated from areas of high-resolution spectra acquired with a pass energy of 40 eV. Experimental data were curve fitted into components with Gaussian-Lorentzian peaks using the ThermoFisher Scientific Avantage software.

Raman Spectroscopy

The vibrational properties of boehmite were characterized by a Raman spectrometer Horiba LaBRAM HR Evolution. Spectra were acquired with a 532 nm laser wavelength and a spot size of 1 μm. Boehmite particles dispersed in ethanol were drop-casted onto a glass slide and air dried during one day. Before each analysis, the spectrometer was calibrated using the 520.5 cm -1 peak of Si acquired on a silicon wafer. The measurements were performed with a long working distance x100 objective lens, a 600 lines/mm grating and a CCD detector.

Acquisitions were done for an exposure time of 20 s. Each spectrum was accumulated 40 times to optimize the signal-to-noise ratio on the 200 -1200 and 2900 -3800 cm -1 ranges to visualize the major bands of boehmite with the hydroxyl deformation modes located at 360, 495 and 674 cm -1 , the hydroxyl stretching bands at 3085 and 3220 cm -1 and any remaining water beyond [START_REF] Ruan | Comparison of Raman spectra in characterizing gibbsite, bayerite, diaspore and boehmite[END_REF]. The aluminium nitrate band is mainly located at 1049 cm -1 [START_REF] Kloprogge | Raman microscopy study of basic aluminium nitrate[END_REF].

Solid-State Nuclear Magnetic Resonance Spectroscopy

Results

Influence of the process operating parameters on the grain morphologies

Influence of different operating parameters on the hydrothermal flow synthesis of boehmite was widely reported in the literature [START_REF] Adschiri | Rapid and Continuous Hydrothermal Synthesis of Boehmite Particles in Subcritical and Supercritical Water[END_REF][START_REF] Hakuta | Effects of hydrothermal synthetic conditions on the particle size of γ-AlO(OH) in sub and supercritical water using a flow reaction system[END_REF][START_REF] Lock | Continuous flow supercritical water synthesis and crystallographic characterization of anisotropic boehmite nanoparticles[END_REF]. pH is one of the major parameters to modify the boehmite particle morphology without adding additives. Using aluminium nitrate as precursor, pH variation can be achieved by: 1) adjusting the precursor concentration and, 2) adding NaOH.

The syntheses were performed at 25 MPa, 400 °C and for different pH values 2.9, 3.5, 3.9 by varying the nitrate concentration and at pH = 6.7 by adding NaOH. In this article, these samples are labelled S2.9, S3.5, S3.9 and S6.7, respectively. Table 1 summarizes process operating parameter variations we used. The residence time was fixed at 20 s except for S3.5 with 60 s as residence time.

Sample name

S2.9 S3.5 S3.9 S6.7

Residence time (s) 20 60 20 20

Precursor concentration (mol.L

-2 pH 2.9 3.5 3.9 6.7

Morphology of the as-synthesized particles was investigated using TEM analysis. As shown in Figure 2, boehmite particles are anisotropic. S2.9 and S3.5 powders exhibit strongly aggregated hexagonal particles of about a few hundred nanometers in length (Figure 2a) and b)). The particles are very thin, less than 15 nm as observed on TEM images of S3.5 and S3.9

(Figure 2b) and c)). The increase of the residence time from 20 to 60 s does not seem to favour the particle growth. This has already been observed by Hakuta et al. who assumed the Table 1: Process operating parameters for the boehmite syntheses performed at 25 MPa and 400 °C.

particles grow only at the very early stage of the reaction (< 2 s) at 25 MPa and 400 °C in acidic conditions [START_REF] Hakuta | Effects of hydrothermal synthetic conditions on the particle size of γ-AlO(OH) in sub and supercritical water using a flow reaction system[END_REF]. with the electron diffraction pattern of a hexagonal particle of S3.9 on e) and a scheme of a hexagonal platelet grain on f).

A further increase of the pH value results in the formation of rhombic particles. S3.9 powder shows both aggregated hexagonal and rhombic particles. Moving to a near neutral pH allows the synthesis of perfectly rhombic boehmite particles. Alkaline pH of 9.3 and 12.0 were also tested, but are not presented here. Slightly basic conditions gave a mix of boehmite and bayerite, an aluminium hydroxide α-Al(OH) 3 whereas strong basic ones did not produce any particles. This is in agreement with Hakuta et al. observations [START_REF] Hakuta | Effects of hydrothermal synthetic conditions on the particle size of γ-AlO(OH) in sub and supercritical water using a flow reaction system[END_REF]. They explained the particle disappearance by a redissolution of very small particles during cooling. However, recent research brings new hypotheses with the discovery of Hydrothermal Molten Salts (HyMOS) [START_REF] Voisin | A new solvent system: Hydrothermal molten salt[END_REF]. The precipitation at a higher temperature than their melting point i.e. between 318 and 323 °C for NaOH is instantaneously followed by the fusion of the salts. It produces a unique system composed of supercritical water at low density with a fraction of dissolved salt and a molten salt phase with a fraction of water [START_REF] Duverger-Nédellec | Unveiling the complexity of salt(s) in water under transcritical conditions[END_REF]. The high polarity and density of molten salts results in dissolving a large amount of inorganic particles that could be the case of boehmite. Voisin et al. [START_REF] Voisin | A new solvent system: Hydrothermal molten salt[END_REF] demonstrated the formation of NaOH molten salts in supercritical water at 400 °C and 25 MPa. One of the conditions for observing molten salt is the prior NaOH precipitation (before melting) meaning that a salt concentration above its solubility (133 ppm at 400 °C and 25 MPa) is necessary. In this study, this condition is achieved with a NaOH concentration of 400 ppm approximately at a pH of 12. Although molten salts have never been observed under these specific conditions, all conditions required to obtain them are met and could also explain the redissolution of boehmite nanoparticles. Zhang et al.

discussed the effect of pH on the growth of boehmite using traditional hydrothermal synthesis based on sol-gel process. In their study, a pH higher than 12 is required to obtain rhombic grains [START_REF] Zhang | Boehmite and Gibbsite Nanoplates for the Synthesis of Advanced Alumina Products[END_REF]. The [010] zone axis electron diffraction pattern in Figure 2e) reveals diffraction spots of (100), ( 001) and (101) planes. Such a result is consistent with a hexagonal grain with a dominant (020) surface and (100), ( 001) and ( 101) edge surfaces as represented on the scheme shown in Figure 2f).

The aggregation makes particle size determination complex. To overcome this issue, A deep study of the crystal structure is proposed by simulating the experimental X-ray diffraction patterns thanks to the Rietveld refinement method. An example for the S2.9

sample is displayed in Figure 4. The accuracy of the Rietveld refinement presented is relatively good with a weighted-profile R-factor of 9%. Refined crystal parameters for all the studied powders are reported in Table 2.

Refined lattice parameters, and especially along the b-axis, do not show significant discrepancies showing that no atom or water molecule could be inserted within the structure.

Guzmán-Castillo et al. have also made this observation [START_REF] Guzmán-Castillo | Effect of Boehmite Crystallite Size and Steaming on Alumina Properties[END_REF]. As expected for anisotropic particles, peak broadening varies from peak to peak. In agreement with the scheme of a hexagonal grain and the electron diffraction pattern (Figure 2e) and f)), boehmite particles can be characterized by the (200), (020), ( 002) and ( 202 S6.7 powder presents relatively large crystallites compared to the three others (at least twice the size along all crystallographic directions). For instance, S2.9 has l 200 , l 020 and l 002 of 28 ± 5 nm, 10 ± 2 nm and 23 ± 6 nm while those of S6.7 are 47 ± 4 nm, 39 ± 4 nm and 47 ± 4 nm, respectively. The ratios between these particular distances report on specific growth rate.

From acidic to near neutral conditions at a constant residence time, the ratio l 200 / l 020 is reduced from 2.8 to 1.2, l 002 / l 020 from 2. 2), of S3.5 powder with l 200 , l 020 and l 002 of 19 ± 4 nm, 12 ± 2 nm and 18 ± 4 nm, respectively, agreed with a non-significant impact of the residence time on particle growth.

Acidic conditions have led to hexagonal platelets whereas rhombic grains are produced in near neutral solutions. Morphology modification is correlated to pH and so is the variation of nitrates in the solution.

Adsorption of nitrates

Modification of boehmite morphology can be obtained through anisotropic adsorption of chemical species, nitrates for instance. He et al. characterized sulfates adsorption on boehmite particles using Raman spectroscopy [START_REF] He | Hydrothermal Preparation of Boehmite Nanorods by Selective Adsorption of Sulfate[END_REF]. In the case of a sulfate H-bonded complex, a sulfate would not form a direct chemical bond with the surface and the vibrational spectrum is very similar to the spectra of sulfate in solution. Raman spectroscopy appears to be a suitable technique to analyse chemical bonds of adsorbed species thanks to its high sensitivity to molecule vibration. The low wavenumber region Raman spectra of the different boehmite samples is reported in Figure 5. The 400 -800 cm -1 region consists of the hydroxyl translation modes. In particular, the most intense peak at 359 cm -1 corresponds to the A g fully symmetric mode due to the movement of aluminium and oxygen atoms parallel to the [010] direction [START_REF] Doss | Raman studies of sol-gel alumina: Finite-size effects in nanocrystalline AlO(OH)[END_REF]. The AlO 6 octahedron structure causes two peaks at 494 and 674 cm -1 [START_REF] Kiss | Raman and i.r. spectra and structure of boehmite (γ-AlOOH). Evidence for the recently discarded D172h space group[END_REF][START_REF] Yang | Synthesis and Characterization of Boehmite Nanofibers[END_REF].

The small and sharp peak at 1049 cm -1 which is slightly intense for S2.9 and S3.9 and almost extinct for S6.7 is attributed to NO 3 -vibrations [START_REF] Kloprogge | Raman microscopy study of basic aluminium nitrate[END_REF][START_REF] Doss | Raman studies of sol-gel alumina: Finite-size effects in nanocrystalline AlO(OH)[END_REF]. Although Raman spectroscopy gives qualitative information, this peak confirmed the presence of nitrates at the boehmite surfaces, especially on S2.9 and S3.9.

A further study at high wavenumber was performed. Results are presented in Figure 6. This region exhibited up to four bands at 3078, 3219, 3340, 3515 cm -1 and eventually a shoulder at 3561 cm -1 . The two first peaks characterized the hydroxyl stretching vibrations within boehmite structure and are observable for all the samples [START_REF] Ruan | Comparison of Raman spectra in characterizing gibbsite, bayerite, diaspore and boehmite[END_REF][START_REF] Kiss | Raman and i.r. spectra and structure of boehmite (γ-AlOOH). Evidence for the recently discarded D172h space group[END_REF][START_REF] Yang | Synthesis and Characterization of Boehmite Nanofibers[END_REF].

The assignment above 3300 cm -1 is extremely complex because of the intricate hydrogen bonding network [START_REF] Casella | Water O-H Stretching Raman Signature for Strong Acid Monitoring via Multivariate Analysis[END_REF]. This region is associated with the stretching vibrations of surface OH groups or residual water on the surfaces [START_REF] Ruan | Comparison of Raman spectra in characterizing gibbsite, bayerite, diaspore and boehmite[END_REF][START_REF] Tunega | Theoretical study of structural, mechanical and spectroscopic properties of boehmite (γ-AlOOH)[END_REF]. Bands at 3340, 3515 and 3561 cm -1 are displayed for S2.9 and more particularly for S3.5. On the top of Figure 6, a spectrum of a boehmite sample synthesized in a batch reactor using gibbsite Al(OH) 3 as a precursor and so without nitrates is depicted. This spectrum shows the structural hydroxyl stretching vibrations between 3000 and 3300 cm -1 but does not display any bands at higher frequency.

Consequently, bands above 3300 cm -1 may have a link with surface hydroxynitrates.

Observations done by Herzog-Cance et al. who studied the nitric acid autoprotolysis using Raman spectroscopy confirmed the substantial correlation between these bands and nitrates [START_REF] Herzog-Cance | Étude, par spectroscopie de vibration, de l'autoprotolyse de l'acide nitrique absolu. Nouvelle interprétation[END_REF]. Thus, the band at 3340 cm -1 may correspond to the stretching contribution of weakly bonded HNO 3 also named as "free" nitric acid. The greatest band at 3515 cm -1 could be attributed to the stretching vibration of water associated with nitric acid molecules through the Figure 6: Raman spectra of the high wavenumber region from 3000 to 3650 cm -1 of; on the top, a boehmite sample synthesized without nitrates; on the bottom, S2.9, S3.5, S3.9 and S6.7. Each part of the spectra is described with its corresponding assignment.

molecular association H 2 O … HONO 2 produced from the dissociation of the HNO 3 molecule [START_REF] Herzog-Cance | Étude, par spectroscopie de vibration, de l'autoprotolyse de l'acide nitrique absolu. Nouvelle interprétation[END_REF]. The shoulder at 3561 cm -1 may assimilate to differently bonded water molecules.

Assignments of the Raman spectra indicated a strong adsorption of nitrate species, especially in acidic conditions. Bands correlated to nitrate signatures almost disappear for S3.9 and are inexistent for S6.7. Raman spectroscopy of S3.5 at a longer residence time has demonstrated many signatures of nitrates under various forms.

Nitrate adsorption occurred more under acidic conditions as shown in Raman spectroscopy results, leading to the formation of hexagonal particles. The next part will deal with the interaction between these specific morphologies and the adsorption of water molecules.

Adsorbed water determination

Nguefack et al. developed a model to investigate adsorbed water content on boehmite crystallites based on free sites available on each facet [START_REF] Nguefack | Preparation of alumina through a sol-gel process. Synthesis, characterization, thermal evolution and model of intermediate boehmite[END_REF]. They obtained the maximum adsorption capability by expanding the coordination number of surface aluminium atoms to six and surface oxygen atoms to four. Stoichiometric water content adsorption x is calculated by dividing the number of free sites by the total number of atoms in the crystal. To estimate the free site number, the model uses the A.C.S. and lattice parameters refined from the Rietveld method. The corresponding results are summarized in Table 3. % for S2.9 and S6.7, respectively. However, the basal surface decreases when the pH Table 3: Determination of the relative facet areas of the three mains faces (100), ( 010) and ( 001) and of the overall adsorbed water calculated using the model of Nguefack et al. [START_REF] Herzog-Cance | Étude, par spectroscopie de vibration, de l'autoprotolyse de l'acide nitrique absolu. Nouvelle interprétation[END_REF].

Sample name

becomes more neutral to the benefit of the (100) and ( 001) facets. This is in agreement with the observation of the Rietveld refinement with the inhibition of (100) and (001) facets in acidic conditions. Stoichiometric water content increased from 0.16 to 0.21 following the inverse of the (010) relative facet area. S3.5 sample follows the same tendency and does not show a significant effect of residence time. Bigger crystallites obtained for S6.7 have less adsorption sites in proportion to the total number of atoms in the crystal and so less water is adsorbed following this model.

To sum up, a raise of the pH from 2.9 to 3.9 is followed by a slight increase of water content whereas a further increase of the pH to 6.7 generated a drastic reduction of water content.

As explained in the introduction, two possibilities have been suggested for the location of water. The first assumption was inside the structure along the b-axis. However, the Rietveld refinement revealed a restricted variation of the crystal dimension along the b-axis, insufficient to insert any atoms within the structure. Consequently, the excess of water is probably adsorbed on the crystal surfaces.

Investigation of surface species and their chemical environment

XPS survey scans, shown in Figure 7, exhibit very clean samples with mostly Al and O atoms and the adventitious C between 4.2 and 9.7 at. %. This is confirmed by the Al/O ratio equal to 0.5. In addition, no significant shift of the Al 2p binding energies is observed with a position at approximately 74.0 and 74.1 eV meaning that the oxidation state of Al atoms is stable for all the samples. eV for S2.9 and S6.7, respectively. S3.5 sample exhibits the more ordered structure with the lowest peak broadening for aluminium. This means that an increase in the residence time to 60 s does not promote the particle growth but improves the local structure order. The contribution of ads. H 2 O revealed some variations of the adsorbed water on the boehmite surfaces. Indeed, S3.9 has the highest adsorbed water contribution with 20.4 at.%. In comparison, the lowest is contained on S3.5 surfaces with only 6.1 at.%. The study of the ratios between these contributions makes it possible to get rid of the impurity effects such as carbon. The adsorbed water content is quite similar for both S2.9 and S3.9 with a ratio close to 1 and is reduced down to 0.7 for S6.7 at high pH.

Sample name

However, an increase in the residence time with S3.5 proposed to reduce significantly this ratio and so the adsorbed water content to 0.2. These observations are almost in agreement with the adsorption site calculation (Table 3) except for S3.5. The ratio is expected to be nearly 1 according the AlOOH chemical formula [START_REF] Kloprogge | XPS study of the major minerals in bauxite: Gibbsite, bayerite and (pseudo-)boehmite[END_REF]. This ratio is about 1.2 for S3.5, S3.9 and S6.7 and raised to 1.5 for S2.9, highlighting an excess of surface hydroxyls in these conditions. The depth probed by XPS (≈ 3 to 5 nm) coupled with crystallite thickness near 10 nm for S2.9, S3.5 and S3.9 likely might overestimate the adsorbed water and surface hydroxyl compared to thicker crystallites of S6.7.

Even if the assignment of 1 H resonance lines is very challenging, 1 H Solid-State NMR spectra of the S2.9 and S3.5 samples confirmed the observations done using XPS and summarized in Figure 8.

In Figure 9, the 1 H MAS NMR spectrum of S3.5 can be described as the sum of four main resonance lines centered at 1.2, 3.5, 7.1 and 10.4 ppm. Peaks at 7.1 and 10.4 ppm are tentatively assigned to structural octahedral Al 2 OH sites which involve different strength of bonds while the one located at 3.5 ppm is attributed to octahedral AlOH sites [START_REF] Fitzgerald | Dehydration Studies of a High-Surface-Area Alumina (Pseudo-boehmite) Using Solid-State $^1H$ and $^27Al[END_REF]. The narrow resonance line at 1.2 ppm is often referred as another contribution of octahedral AlOH sites. The slight variation in 1 H NMR shifts between different authors is likely due to difference in morphology of boehmite particles [START_REF] Paluch | Application of 1H and 27Al magic angle spinning solid state NMR at 60kHz for studies of Au and Au-Ni catalysts supported on boehmite/alumina[END_REF]. In comparison, an additional peak at 4.7

ppm is exhibited for the sample S2.9. This contribution counting for 25 % of the overall resonance area coincides undoubtedly to the protons of "physisorbed" water on boehmite particles [START_REF] Fitzgerald | Dehydration Studies of a High-Surface-Area Alumina (Pseudo-boehmite) Using Solid-State $^1H$ and $^27Al[END_REF][START_REF] Paluch | Application of 1H and 27Al magic angle spinning solid state NMR at 60kHz for studies of Au and Au-Ni catalysts supported on boehmite/alumina[END_REF].

As stated previously, A.C.S. plays an important role but are insufficient to explain all the involved surface reactions. Especially, the local environment order would appear to have a major impact.

Discussion

Firstly, the discussion is focused on the effect of pH considering the samples S2.9, S3.9 and S6.7. pH has a great impact on the fluctuation of the boehmite particle morphology synthesized using the continuous supercritical hydrothermal process. Indeed, a highly acidic pH or a neutral one will give anisotropic hexagonal-plate and rhombic particles, respectively.

Boehmite synthesis, described by the chemical reaction below, produces also HNO 3 [START_REF] Adschiri | Hydrothermal Synthesis of Metal Oxide Nanoparticles at Supercritical Conditions[END_REF].

These HNO 3 molecules may adsorb on boehmite particle surfaces. Raman spectroscopy underlined the strong presence of nitrates on the hexagonal-plate boehmite powders (Figure 5 and Figure 6). In acidic conditions, the anion adsorption is favored due to the positive charges at the boehmite surface determined with a zeta potential study considering an isoelectric point of boehmite between 7 and 9.2 depending on the morphology and the particle size [START_REF] He | Different nanostructures of boehmite fabricated by hydrothermal process: effects of pH and anions[END_REF][START_REF] Pandey | Development of Commercial Trace Moisture Sensor: A Detailed Comparative Study on Microstructural and Impedance Measurements of Two Phases of Alumina[END_REF].

On the contrary, at near neutral pH, there is no adsorbed anions leading to no or a little preferential growth and so to large rhombic particles. The less acidic the solution is, the less anion adsorption will occur. Furthermore, the anion adsorption may be surface-dependent.

Nitrates seem to adsorb preferentially on (020) surface resulting in a preferential growth along the [001] and mostly [100] directions. Consequently, the relative (020) surface is increased compared to the edge surfaces (Table 3).

According to the model developed by Nguefack et al., the less prominent the (020) relative surface is, the less water is adsorbed on the boehmite surfaces. The XPS results tend to this conclusion too with a strong reduction of the adsorbed water content for pH from 2.9 to 6.7.

This is directly linked to the crystal structure of boehmite. As the octahedral double layer interactions are weak in the boehmite structure, the crystal is easily cleaved between these layers causing (010) surfaces to be covered by hydroxyl groups [START_REF] Bokhimi | Dependence of Boehmite Thermal Evolution on Its Atom Bond Lengths and Crystallite Size[END_REF][START_REF] Bokhimi | Relationship between Crystallite Size and Bond Lengths in Boehmite[END_REF]. However, the difference is not significant between pH values equal to 2.9 and 3.9, which could be explained by a reduction of the available sites for water adsorption. Even if the sample S2.9 has more adsorption sites, they are probably mostly filled with nitrate species. In addition, a very low pH seems to promote a dissociative adsorption of water resulting in more surface hydroxyl groups.

Secondly, the residence time does not significantly influence the morphology. In acidic conditions, hexagonal-plate particles with the same range of size were synthesized at longer residence time (60 s instead of 20 s). Surprisingly, this sample exhibits a very small adsorbed water content, lower than the one of sample done at pH 6.7, which is totally at odds with the model of Table 3. The increase of the residence time implies a real improvement of the local structure environment (Table 4, see FWHM). In this case, oxygen with a free orbital will form a ligand between an aluminium atom and the non-reacted nitrates in solution. Thus, nitrate hydroxyls such as Al 2 O 2 (OH)NO 3 can be formed at the surfaces preventing further adsorption of water [START_REF] Ross | Desorption of Nitric Acid From Boehmite and Gibbsite[END_REF]. These species may have the time to be stabilized and so to reduce the local disorder.

Surfaces may be fully adsorbed by H-bonded liquid HNO 3 . During the synthesis, produced nitric acid in the fluid will react with the surface of the particles and especially at low pH. The strong peak at 3515 cm -1 in Figure 6 may prove that the molecular association H 2 O … HONO 2 is weakly bonded at the surface of boehmite. The intensity of this peak differs from the small content of adsorbed water observed using XPS and 1 H MAS NMR spectroscopy. As a result, the residence time increase may promote the adsorption of hydroxynitrates combined with the one of liquid HNO 3 leading to the full filling of the free adsorption sites of boehmite.

The scheme displayed in Figure 10 summarizes the discussion.

From the pH variation, two factors are pointed out on the adsorbed water modification. A weakly alkaline pH will give rhombic grains with a little relative (020) surface. This will lead to a reduced adsorbed water content validated by XPS results (Figure 8).

In the case of an acidic environment, it is more complex. The decrease of the pH will allow the production of hexagonal-plate grains associated with an increase of the relative (020) surface area (Table 3). Nevertheless, the more acid the environment is, the more nitrates will be adsorbed (Figure 5). These molecules may block the further adsorption of water at the surface of boehmite particles.

Decrease of the available adsorption sites at high residence time is due to the improvement of the intricate structure with the hydroxynitrates and of the HNO 3 content at the surface.

Finally, the prediction of the adsorbed water quantity is very tricky. Indeed, it involves complex interdependent phenomena linking the growth along specific directions, anisotropic surface reactions of chemical species such as anions and available adsorption free sites.

Conclusion

This study investigated the adsorption of water on various morphologies of boehmite particles synthesized using the continuous supercritical hydrothermal synthesis. The use of this process allowed to synthesize pure and crystalline submicrometer boehmite particles. pH modification from 2.9 to 6.7 offered a way to tune the particle morphologies from hexagonal platelets to rhombic grains and the sizes of the particles. In fact, the anisotropic nature of boehmite results in a surface-dependent anion adsorption. Nitrates were adsorbed preferentially on the (020)

surface leading to a preferential growth along [001] and mostly [100] directions.

Our results showed that the adsorption of water cannot be characterized only by a morphological analysis. It depends substantially on two phenomena. First, the more ordered the crystal structure is, the less the adsorbed water will be. XPS analysis combined with Then interaction between the boehmite morphology and the adsorption of water is very difficult to be predicted because it is correlated with different parameters such as crystallite size, structural order and other adsorbed species such as nitrates.

Figure 1 :

 1 Figure 1: Schematic representation of two unit cells of orthorhombic crystal structure of

1 H

 1 Solid-State Nuclear Magnetic Resonance (NMR) analyses were carried out on a Bruker Avance III 300 WB spectrometer (B0 = 7.05 T). Experiments were performed with a 2.5 mm solid probe under Magic Angle Spinning (MAS) conditions (30 kHz) with a rotorsynchronized Hahn-echo pulse sequence (90 ° -180 °) with a 90 ° pulse length of 2 μs and an optimized recycle delay of 5 s. The chemical shift scale was calibrated using H 2 O milliQ (δ( 1 H) = + 4.7 ppm) as reference. Desummation of the 1 H MAS NMR spectra was performed with Dmfit software [27].

Figure 2 :

 2 Figure 2: TEM images of boehmite particles: a) S2.9, b) S3.5, c) S3.9 and d) S6.7 associated

  boehmite particles are then defined by their Average Crystallite Size (A.C.S). X-ray diffraction patterns of the as-synthesized powder are shown in Figure 3. They confirmed the synthesis of well-crystallized boehmite powders independently of the pH value. Using the PDF # 00-021-1307 file for an orthorhombic boehmite (space group Cmcm), plan identification of boehmite is assigned in Figure 3. Crystallized boehmite is described by a most intense (020) peak (2θ = 14.5 °) followed by (021) (2θ = 28.2 °) and (130) (2θ = 38.4 °) peaks of half the intensity.

Figure 3 :

 3 Figure 3: X-ray diffraction patterns of the four studied samples. In blue, boehmite peak

  ) diffraction peaks observed at 49.3°, 14.5°, 65.0° and 85.7°, respectively. The assumption of a parallelepipedic grain is done by taking into account the crystallite dimensions only along the [100], [010] and [001] directions to facilitate the study. Table 2

Figure 4 :

 4 Figure 4: Rietveld refinement plot performed for the S2.9 X-ray diffraction pattern.

Figure 5 :

 5 Figure 5: Raman spectra of the low wavenumber region of S2.9, S3.5, S3.9 and S6.7; on the left, from 200 to 1200 cm -1 ; on the right, a focus on the 1020 -1080 cm -1 region. Each part of the

Figure 7 :

 7 Figure 7: XPS survey spectrum of the boehmite S2.9 sample.

Figure 8

 8 Figure 8 exhibits the high-resolution O 1s spectra of the four samples fitted into three

Figure 8 :

 8 Figure 8: Fitted XPS O 1s spectra of S2.9, S3.5, S3.9 and S6.7 into three contributions: Al-O-Al

Figure 9 : 1 H

 91 Figure 9: 1 H MAS NMR spectra of boehmite samples S2.9 and S3.5. Fitted curves with the

Figure 10 :

 10 Figure 10: Schematic representation of the adsorption phenomena on different

1 H

 1 Solid-State NMR experiments have effectively highlighted the very small amount of adsorbed water on well-ordered boehmite particles. Secondly, in acidic conditions, anions such NO 3 are attracted to the boehmite surfaces leading to the diminution of the adsorption sites as shown by Raman spectroscopy.

Table 2 :

 2 3 to 1.2 whereas l 200 / l 002 shows a slight decrease from 1.2 to 1.0. Refined parameters and residuals from the Rietveld refinement using spherical harmonic peak shape functions. Calculation error values are mentioned in brackets. Considering only a decrease of the pH, a preferential growth may be underlined along the [100] and [001] directions compared to the [010] one. In addition, crystallite dimensions, measured on TEM images (figure

	Sample name	S2.9	S3.5	S3.9	S6.7
	a (nm)	0.2868 (1)	0.2869 (1)	0.2868(1)	0.2868 (1)
	b (nm)	1.2225 (1)	1.2221 (2)	1.2222 (1)	1.2219 (1)
	c (nm)	0.3693 (1)	0.3694 (1)	0.3693 (1)	0.3694 (1)
	R wp (%)	9	8	10	11
	l 200 (nm)	28 (5)	19 (4)	17 (3)	47 (4)
	l 020 (nm)	10 (2)	12 (2)	10 (2)	39 (4)
	l 002 (nm)	23 (6)	18 (4)	17 (3)	47 (4)
	l 200 / l 020	2.8	1.6	1.8	1.2

  As stated above, the (010) facet is dominant with a relative surface area between 56 % and 38

		S2.9	S3.5	S3.9	S6.7
	S (100) (%)	20	26	30	31
	S (010) (%)	56	45	41	38
	S (001) (%)	24	28	29	31
	AlOOH.xH 2 O	0.16	0.18	0.21	0.07

Table 4 :

 4 Summary of the C 1s , Al 2p and O 1s spectra description with the full width at half maximum (FWHM), the atomic proportions of each environment and the ratio between contributions.

	Al 2p		31.3	33.3	31.3	29.9
		Al-O-Al	18.3	25.7	19.8	20.5
	O 1s	Al-OH	26.6	30.7	22.1	24.8
		Ads. H 2 O	17.7	6.1	20.4	15.1
	Ratio
	Al 2p /O 1s		0.5	0.5	0.5	0.5
	O 1s (Ads. H 2 O/Al-O-Al)	1.0	0.2	1.0	0.7
	O 1s (Al-OH/Al-O-Al)	1.5	1.2	1.1	1.2

Acknowledgement

The authors would like to thank the "Agence de l'Innovation de Défense (AID)" and the Safran company for their financial support. They would also like to thank E. Lebraud and B.

Dusolle both from ICMCB laboratory for their respective contributions for the XRD analyses and for the graphical abstract.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Luc Le Magoarou

Taha Yassine

Stéphane Paquelet

Matthieu Crussière

Channel charting based beamforming

Keywords: channel charting, cell-less network, dimensionality reduction, MIMO signal processing, machine learning

Channel charting (CC) is an unsupervised learning method allowing to locate users relative to each other without reference. From a broader perspective, it can be viewed as a way to discover a low-dimensional latent space charting the channel manifold. In this paper, this latent modeling vision is leveraged together with a recently proposed location-based beamforming (LBB) method to show that channel charting can be used for mapping channels in space or frequency. Combining CC and LBB yields a neural network resembling an autoencoder. The proposed method is empirically assessed on a channel mapping task whose objective is to predict downlink channels from uplink channels.

I. INTRODUCTION

M ACHINE learning techniques have been applied with success to several wireless communication problems in recent years. One can cite for example detection [START_REF] Samuel | Learning to detect[END_REF], [START_REF] He | Modeldriven deep learning for mimo detection[END_REF], channel estimation [START_REF] He | Deep learning-based channel estimation for beamspace mmwave massive mimo systems[END_REF], [START_REF] Balevi | Massive MIMO channel estimation with an untrained deep neural network[END_REF], [START_REF] Yassine | mpNet: variable depth unfolded neural network for massive MIMO channel estimation[END_REF], [START_REF] Chatelier | Efficient deep unfolding for siso-ofdm channel estimation[END_REF] and even end-to-end system learning [START_REF] Timothy | An introduction to deep learning for the physical layer[END_REF], [START_REF] Aoudia | Model-free training of endto-end communication systems[END_REF] (see [START_REF] Zhang | Deep learning in mobile and wireless networking: A survey[END_REF], [START_REF] Wang | Deep learning for wireless physical layer: Opportunities and challenges[END_REF] for exhaustive surveys on this topic). However, most of these techniques fall within the supervised learning paradigm and thus require labeled data to operate. The acquisition of such data may be impractical or complex within existing communication systems.

For instance, location-based beamforming (LBB) [START_REF] Kela | Location based beamforming in 5G ultra-dense networks[END_REF] is a method allowing to choose precoders based on users' locations, thus suppressing the need for base stations to perform channel estimation or beam sweeping. LBB has recently been tackled successfully with supervised machine learning techniques [START_REF] Le Magoarou | Deep learning for location based beamforming with NLOS channels[END_REF]. However, the main drawback of LBB is that it requires having access to the precise users locations, which are difficult to acquire within current systems.

On the other hand, channel charting (CC) [START_REF] Studer | Channel charting: Locating users within the radio environment using channel state information[END_REF] recently appeared as an unsupervised method allowing to locate users relative to each other, based solely on channel knowledge. Predicting this way the relative locations of users from channel measurements has many potential applications, ranging from SNR prediction [START_REF] Kazemi | Snr prediction in cellular systems based on channel charting[END_REF] and pilot reuse [START_REF] Ribeiro | Channel charting for pilot reuse in mmtc with spatially correlated mimo channels[END_REF], to user grouping, proactive handover management or beam-finding (see [START_REF] Studer | Channel charting: Locating users within the radio environment using channel state information[END_REF] for more details on potential applications). CC is an active area of research [START_REF] Le | Efficient channel charting via phase-insensitive distance computation[END_REF] [17] [START_REF] Rappaport | Improving channel charting using a split triplet loss and an inertial regularizer[END_REF] [19] and rapid progress is expected in this direction. The channel chart can actually be seen as a pseudo-location of the users. In this regard, channel charting could be used to relax the constraints of LBB, by replacing the needed location (difficult to estimate) by a pseudo-location obtained by channel charting.

A. Contributions

In this paper, channel charting is taken as input for location based beamforming. This allows a base station to choose appropriate precoders based on chart locations instead of spatial locations. This alleviate the need for a precise estimation of user locations and opens the way to several applications such as channel mapping in space and frequency [START_REF] Alrabeiah | Deep learning for tdd and fdd massive mimo: Mapping channels in space and frequency[END_REF]. The proposed method is empirically assessed on realistic synthetic channels, clearly showing the potential of the approach.

B. Related work

To the best of the authors' knowledge, this paper is the first to propose a combination between CC and LBB. The approach relies on previous work done in both areas. The CC method chosen here is inspired by the ones of [START_REF] Le | Efficient channel charting via phase-insensitive distance computation[END_REF], [START_REF] Yassine | Leveraging triplet loss and nonlinear dimensionality reduction for on-the-fly channel charting[END_REF]. The used LBB method is very close to the one of [START_REF] Le Magoarou | Deep learning for location based beamforming with NLOS channels[END_REF], despite the fact that its input is different (chart locations instead of spatial locations). On the application side, one contribution pursues an objective quite similar to this paper [START_REF] Ponnada | Best beam prediction in non-standalone mm wave systems[END_REF]. Indeed, it is proposed in [START_REF] Ponnada | Best beam prediction in non-standalone mm wave systems[END_REF] to learn a mapping from chart locations to beam indices. However there are major differences between this approach and the present paper. First of all, the beam is taken from a discrete set of candidate beams in [START_REF] Ponnada | Best beam prediction in non-standalone mm wave systems[END_REF] (classification problem) whereas a continuous beamforming is sought here (infinite number of beams, regression problem). Second, features of much higher dimension than the channel vectors and requiring temporal averaging (second order moments) are used in [START_REF] Ponnada | Best beam prediction in non-standalone mm wave systems[END_REF], whereas an especially designed distance measure that operates directly on instantaneous channel vectors (thus much less complex) is used here. Finally, the neural network used for the mapping is generic (multilayer perceptron) in [START_REF] Ponnada | Best beam prediction in non-standalone mm wave systems[END_REF], whereas a specialized architecture based on random Fourier features is used here.

II. PROBLEM FORMULATION

Let us formulate the problem by describing the studied system and stating clearly the pursued objective.

A. Setting

A cell-less multi-user massive MIMO network is considered, in which B base stations (BS) each comprising A antennas communicate over S subcarriers with U user equipments (UE) each having a single antenna. The location of the jth user equipment is denoted l j , and the vectorized uplink (respectively downlink) channel between it and the ith base station is denoted h i,j ∈ C AS (respectively g i,j ∈ C AS ). Note that the uplink and downlink channel are not necessarily over the same band. The considered setting is shown on Fig. 1.

B. Objective

The goal of the proposed contribution is to map channels in space or frequency, which amounts to determine a downlink precoder w k,j that should be highly correlated with the downlink (target) channel g k,j , based on the knowledge of the uplink (origin) channel h i,j . The origin and target channels are not necessarily on the same band nor even on the same base station (k and i may be different). Performance measure. In order to evaluate precoders, the normalized correlation between the precoder w k,j and the target channel g k,j is used. It is expressed as

η k,j |w H k,j g k,j | 2 g k,j 2 2
.

(

It is between zero and one (for a perfect precoder), and is tightly linked to the downlink channel capacity (considering a single user), whose expression is

log(1 + η k,j .SNR opt ),
for received signal of the form y = √ P w H k,j g k,j s + n where n ∼ N (0, σ 2 ) is additive noise, s is the sent symbol and P the transmit power. In that setting, SNR opt

P g k,j 2 2 σ 2
is the highest achievable signal to noise ratio (SNR) [START_REF] Björnson | Massive mimo networks: Spectral, energy, and hardware efficiency[END_REF]. In summary, the correlation is a single number between zero and one allowing to determine the maximum achievable downlink spectral efficiency for any transmit power and noise variance considering a given precoder.

III. PROPOSED METHOD

In this section, the inference and training phases of the proposed channel mapping method are described in details.

A. Inference

The proposed method comprises three steps: 1) At first, D latent variables corresponding to the vector z i,j ∈ R D are computed from the origin channel h i,j using channel charting methods [START_REF] Studer | Channel charting: Locating users within the radio environment using channel state information[END_REF], [START_REF] Le | Efficient channel charting via phase-insensitive distance computation[END_REF] (with D SA) at the ith BS. In particular, the efficient channel charting method of [START_REF] Le | Efficient channel charting via phase-insensitive distance computation[END_REF] is used in this paper. 2) Then, the latent variables z i,j , that can be seen as a compressed version of the channel are sent from the ith BS to the kth BS. 3) Finally, the kth BS performs location based beamforming using the latent variables as inputs, so as to output a precoder w k,j .

These three steps are depicted on Fig. 2. It is important to notice that the dimension is greatly reduced when performing channel charting, which limits the overhead induced by transmitting the latent variables from the ith to the kth BS.

B. Training

In order to calibrate the CC and LBB steps, a training phase is required.

The CC step is calibrated using the method proposed in [START_REF] Le | Efficient channel charting via phase-insensitive distance computation[END_REF]. A phase-insensitive distance between training channels

{h i,n } N n=1
is first computed and then used within the Isomap nonlinear dimensionality reduction method to obtain chart locations

{z i,n } N n=1 .
The resulting matrices D (h i,1 , . . . , h i,N ) ∈ C AS×N and Z (z i,1 , . . . , z i,N ) ∈ R D×N are then used to form the feedforward neural network shown on Fig. 3 used at inference time to rapidly compute chart location (as first proposed in [START_REF] Yassine | Leveraging triplet loss and nonlinear dimensionality reduction for on-the-fly channel charting[END_REF]).

The LBB step is calibrated using the method proposed in [START_REF] Le Magoarou | Deep learning for location based beamforming with NLOS channels[END_REF], except that the inputs are here chart locations instead of true spatial locations. To do so, a database containing chart locations and associated target channels has to be built (classical downlink channel estimation methods have to be used here): Then, a neural network comprising a random Fourier features (RFF) layer (see detailed architecture in [START_REF] Le Magoarou | Deep learning for location based beamforming with NLOS channels[END_REF]) is trained with a cost function expressed as

{z i,n , g k,n } N n=1 .
CF k 1 - 1 N N n=1 |w H k,n g k,n | 2 g k,n 2 2
.

(

) 2
This cost measures the misalignment of the predicted precoders with respect to the training target channels and is between 0 (if the precoders are perfectly aligned with the target channels) and 1 (if the precoders are orthogonal to the target channels).

C. Potential applications

The proposed method can be applied to several tasks, depending on the source and target channels. For example, if CC and LBB are done in the same base station (i = k) but the frequency bands are different for the uplink (source) and downlink (target) channels, then channel mapping in frequency is carried out. In the more general case where CC and LBB are done in two distinct base stations (i = k), the proposed method allows to determine a precoder at one BS relying on a channel estimated at another BS (potentially in another frequency band), so that no channel estimation is required at the precoding BS. This is particularly interesting if the CC done at one BS is used to perform LBB at several other BSs (say B -1), since in that case channel estimation would be needed at only one BS to determine precoders at B BSs. The overhead due to channel estimation would then be reduced by a factor of B. Moreover, the backhaul requirements are kept reasonable, since only D real numbers (dimension of the channel chart which is much smaller than the dimension of the channel) have to be shared for each user.

IV. EXPERIMENTS

To evaluate the proposed approach, a set of experiments is conducted on synthetic data and the results are analysed in this section.

A. Simulation settings

Multipath channel vectors are generated using the Deep-MIMO dataset [START_REF] Alkhateeb | DeepMIMO: A generic deep learning dataset for millimeter wave and massive MIMO applications[END_REF]. It consists of channels obtained using a ray-tracing software in an urban outdoor scenario ('O1') (c.f. [23, Figure 2]). Two base stations (i.e. B = 2) on opposite sides of a street communicate with U = 10000 UEs spread across the map. The first BS corresponds to BS 16 in the original scenario and the second one corresponds to BS 15. Both BSs are equipped with A = 8 × 8 UPAs. The first one operates at 3.5 GHz while the second one operates at 28 GHz and both transmit over sixteen subcarriers (i.e. S = 16) spanning a bandwidth of 20 MHz. Note that for simplicity, the precoding results reported in this paper are for a single subcarrier (the central one). All in all, the obtained dataset allows to have access to a triplet {h 1,j , g 2,j , l j } for each UE (uplink channel to the first BS, downlink channel from the second BS, user location).

B. Implementation details

The PyTorch library [START_REF] Paszke | Pytorch: An imperative style, high-performance deep learning library[END_REF] The on-the-fly setting has the obvious advantage of being independent from training once performed, but is generally less robust than its one-shot counterpart. LBB. The input is of size D = 5 (i.e. dimension of the learned chart), unless otherwise indicated. The number of RFF is set to 600. The MLP consists of 4 layers. Each layer but the last one contains 300 neurons followed by a ReLU nonlinearity. The model is trained for 30 epochs using the Adam optimizer [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF] with a batch size of 100. Finally, the model outputs a vector w ∈ R 2A . For simplicity and to limit the complexity of the model in these initial experiments, the task is restricted to the prediction of the precoder corresponding to the central subcarrier only. Extension to the prediction of the full precoder is trivial but likely requires increasing the number of parameters of the NN.

C. Compared variants

Five variants of the model are considered for comparison.

• LBB is performed at the first BS using the real UEs' locations. Performance is evaluated using the uplink channels. This serves as a baseline. • CC (one-shot) and LBB are both performed at the 1 st BS and chained together as described in III. The uplink channels are fed as input for training and used to evaluate the performance. • CC (one-shot) is performed first at the 1 st BS on the uplink channels. The learned chart is then transmitted to the 2 nd BS for the LBB step on the downlink channels. This variant corresponds to the channel mapping setting. • This variant is similar to the second one, except this time the learned chart is of dimension D = 3. The chart is computed in a one-shot. • This variant is similar to the third one, except this time CC is computed on the fly. This last variant corresponds to the most realistic setting that is described in section III-B.

D. Results

The five variants of the model are evaluated on the test set. Fig. 5 shows the cumulative distribution function (CDF) of the correlations η-the closer to a Dirac at x = 1 the better. All methods perform relatively well and achieve a satisfactory level of performance. Interestingly, the CC based LBB variant performs better than the classical one, meaning that the 5 dimensions of the learned chart allow to convey more information than the 3-dimensional spatial locations. This hypothesis is consolidated when comparing classical LBB to the CC based one when keeping only 3 dimensions to the chart where we observe a small decline in performance. Furthermore, the figure shows that in the channel mapping setting, the model performs slightly worse but is on par with the classical one. Finally, the CC based LBB model where the chart is computed on the fly performs the worst, which is expected and highlights the influence of the quality of the learned chart on the subsequent tasks. It worth noting that in this last case, an end-to-end training of the network (i.e., training both the CC NN and the LBB NN) would certainly improve the performance. Fig. 7 shows the spatial distribution of the correlations for the last approach described in IV-C. The model succeeds in learning an adequate precoder for most of the spatial locations for both LoS and NLoS channels. Some areas show a high concentration in blue dots (top left and bottom left of the map) corresponding to precoders of low quality and could be explained by locations where propagation conditions with respect to the two base stations are very different (very weak channel between the user and one of the BSs).

V. CONCLUSION AND OUTLOOK

In this paper, a way to directly predict precoders from channel chart locations was proposed. It amounts to combining wireless channel charting and location-based beamforming.

It has the potential of greatly reducing the overhead due to channel estimation in cell-less systems, since CC can be carried out in a base station, and LBB in another one (or several others) that does not need to perform channel estimation nor beam sweeping. The amount of data to transfer between the two BS is reduced since the channel chart is of very low dimension. The proposed method was empirically assessed on realistic synthetic urban channels, showing the great potential of the approach, since it was found that the chart information (in dimension five) was as good as the real location information.

In future work, it would be interesting to train the whole system end-to-end (CC and LBB), which raises the question of how to handle the transmission of gradients and parameters between base stations (decentralized federated learning). Moreover, a theoretical motivation based on the channel manifold in order to set hyperparameters in a principled way would be valuable. Finally, a similar approach could be envisioned for beam prediction within a finite set (as studied in [START_REF] Ponnada | Best beam prediction in non-standalone mm wave systems[END_REF]), since the problem can then be seen as a discrete version of the problem studied here.

Fig. 1 .

 1 Fig. 1. Considered setting with B = 2 and U = 4.

Fig. 2 .

 2 Fig. 2. Schematic view of the proposed method (the real dimension being shown below each variable).

Fig. 3 .

 3 Fig. 3. Proposed similarity-based neural network for the CC step.

Fig. 4 .

 4 Fig. 4. Proposed RFF-based neural network for the LBB step.

  is used to implement and conduct the experiments, and implicitly handles the intricacies of neural network training (gradient computation, backpropagation, etc.). Complex vectors are converted to their real representation by stacking the real and imaginary parts resulting in vectors of twice the original size. The dataset is split into a training set and a test set. The training set size is set at N = %70 × U = 7000. CC. The channel charting step could be performed in one of two manners: in a one-shot (to form a baseline) or on the fly (as proposed). The one-shot variant is obtained by applying Isomap to the distance matrix of all of the U channels. Only then is the chart split into the training and test sets. On the other hand, the on the fly variant is obtained by first computing matrices D and Z, restricted to the training set of size N , that serve to initialize the NN. The test set is then obtained by computing the chart for the rest of the dataset using the latter. The neighborhood for Isomap is of size 5 for both variants.

Fig. 5 .

 5 Fig. 5. CDF of the correlations. Blue curve corresponds to the original LBB at BS1. Orange curve corresponds to CC (one-shot) and LBB both at BS1 at the same frequency. Green curve corresponds to CC (one-shot) at BS1 and LBB at BS2 at different frequencies. Red curve corresponds to CC and LBB both at BS1 at the same frequency with a chart of dimensionality 3. Purple curve corresponds to CC (on the fly) at BS1 and LBB at BS2 at different frequencies.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Spatial distribution of the UEs in the test set (left) and their corresponding learned chart obtained on the fly (right). Color is added to distinguish local neighborhoods.

ACKNOWLEDGEMENT

This work has been partly funded by the European Commission through the H2020 project Hexa-X (Grant Agreement no. 101015).


Percam Observations

L Mandon
email: lucia.mandon@obspm.fr

C Quantin-Nataf

C Royer

P Beck

T Fouchet

J R Johnson

O Forni

F Montmessin

C Pilorget

F Poulet

S Le Mouélic

E Dehouck

O Beyssac

A Brown

J Tarnas

S Maurice

R C Wiens

team . The Supercam

INFRARED REFLECTANCE OF ROCKS AND REGOLITH AT JEZERO CRATER: ONE YEAR OF SU

Introduction:

The Mars 2020 Perseverance rover landed in Jezero crater, Mars on February 18th, 2021. The main science objectives of the mission are the characterization of past habitable environments, the search for preserved biosignatures and the collection of samples to be returned to Earth by the MSR (Mars Sample Return, NASA/ESA) program [1]. The payload includes the SuperCam instrument, which plays a critical role in the exploration strategy, as it combines various remotesensing techniques to investigate the elemental and mineralogical composition of rocks and soils [2,3]. In particular, the SuperCam reflectance spectrometer covers the visible and near-infrared range, a novelty on the Martian surface, while this range is primarily used from orbit to detect and map minerals. As of Sol 300 (December 2021), the rover has driven onto two geological units of the crater floor: the Máaz formation (mapped from orbit as the Crater Floor fractured rough, or the Cf-fr), a darktoned, crater retaining unit that shows some signature of pyroxene from orbit, and the Séítah formation (in a wider geological unit mapped as the Crater Floor fractured 1, or Cf-f-1), a lighter-toned unit, less resistant to erosion and that exhibits a signature of olivine from orbit, similarly to the olivine-unit that spans the Nili Fossae region where Jezero crater is located [4,5]. Here we present the initial results of the SuperCam near-infrared spectrometer, with calibrated spectral data collected in the first year of operation of the instrument, with a focus on the rocks and soils present in the vicinity of the rover [6].

Data and methods:

The SuperCam VISIR spectrometer measures the reflected light from 385 to 475 nm, 535 to 855 nm and 1.3 to 2.6 µm [2,3,7]. In this contribution, we focus on the 1.3-2.6 µm spectral range, which enables the identification of a wide variety of minerals, especially mafic minerals and secondary phases such as phyllosilicates, sulfates, or carbonates. VIS data are presented in [8]. The calibration of the IR spectra [9] includes smoothing of the dark current spectra, I/F conversion, removal of anomalously high-values "spikes", and preliminary atmospheric correction. At wavelengths longer than ∼2.5 µm, the calibration is still uncertain (owing to temperature sensitivities in the IRS instrument) and data are not yet usable. For targets a few meters from the rover mast, the analytical footprints of individual locations within a raster are a few mm, hence encompassing several mineral phases and combining the spectra of different components. To unmix these components, several data types are presented: single-point spectra, ratios, and Independent Components Analysis (ICA) components. In particular, the ICA computational approach aims at finding independent components whose relative variations could explain a set of multivariate signalsin our case the different minerals affecting the measured spectra.

Rocks of the Máaz formation: Most of the spectra measured on rocks of the Máaz formation exhibit an absorption near 1.9 µm (indicating the presence of water), a band that is usually not observed as strongly in the soils. Some spectra of poorly dusty rocks make the exception, with relatively featureless spectra that indicate a low mafic minerals content, or that these minerals have a grain size too small to be detected. In addition to the common absorption at ∼1.9 µm, subtle absorptions near 1.4, 2.28 and 2.39 µm show up locally, suggesting the presence of an Fe-rich phyllosilicate (e.g., nontronite or hisingerite), potentially mixed with an oxyhydroxide (e.g., ferrihydrite) as some spectra with weak or nonexistent 2.28 and 2.39 µm bands are observed, along with a shift of the 1.9 µm band position from 1.92 to 1.94 µm (Fig. 1c). Additionally, some spectra ratioed by the median spectrum of the raster exhibit an absorption at 1.93 µm, a drop at 2.4 µm, often paired with an absorption at 2.14 µm (Fig. 1b); such spectral features are consistent with a relative enrichment in mono and poly-hydrated Mg-sulfates. Also, an absorption at ∼2.2 µm is commonly observed and can either be explained by gypsum, an Al-phyllosilicate or hydrated silica (Fig. 1c). Finally, results of the ICA analysis highlighted a component with a large absorption centered at 2.3 µm and consistent with high-Ca pyroxene (Fig. 1a).

Rocks of the Séítah formation:

The VISIR spectra of Séítah rocks are significantly different from those measured in the Máaz formation, with a distinct downturn toward shorter wavelengths typical of olivine. A different secondary mineral assemblage from that of the Máaz formation is inferred: possible Fe-phyllosilicate (e.g., nontronite or hisingerite) and Mg-phyllosilicate(s), as expressed by a narrow 2.32 µm band -the consistent association with the absorptions at 2.28 and 2.39 µm might be indicative of talc. In some spectra, a significant enlargement of the 2.32 µm absorption is observed (Fig. 1c), consistent with the presence of the Fe/Mg-carbonates

1631.pdf 53rd Lunar and Planetary Science Conference (2022)

that were also detected in some locations by the LIBS and Raman techniques of SuperCam (although the 2.5 µm band of carbonates is not unambiguously observed owing to calibration uncertainties described above). By comparison with laboratory measurements, we estimate that the carbonates are present in the rock at a few wt.%. Additionally, the ICA components of abraded rocks are consistent with unaltered high-Ca pyroxene, and olivine with carbonate and/or Fe/Mg-phyllosilicates (Fig. 1a); this suggests that the alteration products might be preferentially spatially associated with olivine than with pyroxene (e.g., in cracks or on the rims of the olivine crystals).

Máaz formation soils: ICA analysis of the Máaz formation regolith spectra highlighted the three following components (in no preferential order): a primary component consistent with intermediate-Ca pyroxene or a mixture of various pyroxene types of distinct composition, a second component consistent with olivine mixed with carbonates and/or Fe/Mg-phyllosilicates (similarly to Séítah rocks), and a hydrated phase of uncertain nature (Fig. 1a). Individual point spectra show that the olivine:pyroxene ratio varies with soil type: fine soils (µm-scale particles) exhibit a weak signature of pyroxene mixed with olivine, while coarser soils showing additional dark-toned mm-sized have spectra with deeper absorption bands and the distinct features of olivine, carbonates and/or Fe/Mg-phyllosilicates.

Discussion: One of the implications regarding the history of Jezero is that hydration pervasively affected its crater floor, with hydrated minerals present at least to some degree in most of the rocks analyzed. The mineralogical assemblage shows a diversity of secondary phases, with both Fe 3+ and Mg-phyllosilicates, oxyhydroxides, carbonates and sulfates. On the other hand, Al-phyllosilicates do not appear to be abundant. In Séítah, the aqueous alteration was likely limited, or the rock burial was fast, in order to preserve a significant amount of olivine crystals, which alter rapidly at geological timescales when interacting with water [10]. Nevertheless, the carbonates detected by SuperCam are likely products of olivine alteration, and the rocks in Séítah might have been marginally altered through the carbonation process, an alteration environment that could have been favorable for life. Additionally, the Mg-sulfates observed by SuperCam might record some late aqueous environment corresponding to the evaporation of the lake.

As SuperCam measures the first near-IR spectra on the surface of Mars, it provides some insights into the complex effects of spectral mixing and scaling between orbital and in situ measurements. In particular, the Máaz formation was interpreted as a lava flow given its pyroxene signature from orbit and its dark-toned, craterretaining morphology. SuperCam data suggest that py- roxene is spectrally more dominant in the soils than in the rocks of the Máaz formation, whose spectra are more strongly affected by hydrated minerals. Hence, the soils might be significant contributor to the signature measured from orbit, at least at the Máaz formation locations explored by the rover so far. Also, this regolith has a variable composition, and shows similar features to the rocks in Séítah, which likely provide local inputs of material through eolian processes [11].

Figure 1 :

 1 Figure 1: Spectral endmembers identified in the rocks and soils of Jezero crater floor: (a) selected ICA components (n: number of ICA components in input), (b) point spectra ratioed by the raster median, (c) point spectra. Target names along with point number in the raster are indicated on the right side. The white mask corresponds to CO2 absorptions domain.

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