Guillaume Gouget

Mathieu Duttine

Etienne Durand

Antoine Villesuzanne

Vincent Rodriguez

Frédéric Adamietz

Thierry Le Mercier

Marc-David Braida

Alain Demourgues

Isolating the two room-temperature polymorphs of NaNbO 3 : structural features, optical band gap and reactivity

Keywords: sodium niobates, perovskites, acentric structure, structural distortions, competing bonds, hydrothermal synthesis, second harmonic generation, reducibility

The two room-temperature NaNbO3 polymorphs crystallizing with orthorhombic symmetry have been successfully isolated at using a new preparative method. The pure polar phase, annealed at 600 °C under air after hydrothermal treatment at 200 °C, adopts the P21ma space group, whereas the well-known and thermodynamically stable form with the Pbma structure is obtained at higher temperatures under air (T = 950 °C). Thanks to the combination of powder-XRD Rietveld analysis, 23 Na solid-state NMR spectroscopy and second harmonic generation studies, structural features of both these polymorphs reveal clear structural differences. The stability of each atom site is investigated by mean of bond distances, Madelung potentials and DFT calculations. The key role of Na atomic positions is highlighted, especially how they influence the [NbO6] octahedron distortion. In the polar P21ma-phase, the higher distortion both along the apical axis and the equatorial plane is consecutive to the relaxation of the overall network with the stabilization of Na atoms in two sites sharing the same symmetry. Focusing on oxygen mobility, both polymorphs show distinct reactivities toward reductive heat treatments: as characterized by thermogravimetric analysis and ESR measurements, the Pbma framework is relatively insensitive, while the P21ma one yields a non-stoichiometric oxide with a Nb 4+ content of 18 % corresponding to NaNbO2.91 chemical composition. The fine control in phasic purity together with advances on the structural features of room-temperature phases should benefit both non-linear optical applications and photocatalytic performances of sodium niobates.

INTRODUCTION

The lead-free alternatives to the piezoelectric material PbZr1-xTixO3 (PZT) have been widely investigated these last 15 years; [START_REF] Saito | Lead-Free Piezoceramics[END_REF][START_REF] Cross | Lead-Free at Last[END_REF][START_REF] Rödel | Perspective on the Development of Lead-Free Piezoceramics[END_REF][START_REF] Rödel | Transferring Lead-Free Piezoelectric Ceramics into Application[END_REF] among them, KxNa1-xNbO3 (KNN) is the most promising solid solution, whereas Li1-xNa1-xNbO3 (LNN) phases also display interesting ferroelectric properties at low Li contents (x < 0.15). [START_REF] Ringgaard | Lead-Free Piezoceramics Based on Alkali Niobates[END_REF][START_REF] Peel | Unusual Phase Behavior in the Piezoelectric Perovskite System, LixNa1-xNbO3[END_REF] The composition and temperature dependences of KNN and LNN oxides have been largely explored, with complex structural features associated to various tilting modes of [NbO6] octahedra in these derived perovskite networks. [START_REF] Peel | Unusual Phase Behavior in the Piezoelectric Perovskite System, LixNa1-xNbO3[END_REF][START_REF] Ahtee | Lattice Parameters and Tilted Octahedra in Sodium-Potassium Niobate Solid Solutions[END_REF][START_REF] Handoko | Hydrothermal Synthesis of Sodium Potassium Niobate Solid Solutions at 200 °C[END_REF] The common elements Na and Nb in these series combined to the occurrence of various polymorphs at room temperature naturally leads to focus specifically on the end member NaNbO3. [START_REF] Cai | Electronic Characterization of Polar Nanoregions in Relaxor-Type Ferroelectric NaNbO3 Films[END_REF][START_REF] Gu | Piezoelectric Active Humidity Sensors Based on Lead-Free NaNbO3 Piezoelectric Nanofibers[END_REF][START_REF] Koruza | Grain-Size-Induced Ferroelectricity in NaNbO3[END_REF][START_REF] Singh | Electrically Tuned Photoelectrochemical Properties of Ferroelectric Nanostructure NaNbO3 Films[END_REF][START_REF] Chao | Macroscopic Ferroelectricity and Piezoelectricity in Nanostructured NaNbO3 Ceramics[END_REF] This composition has attracted interest more recently as a photocatalyst for several reactions, namely hydrogen evolution, [START_REF] Li | The Effects of Crystal Structure and Electronic Structure on Photocatalytic H2 Evolution and CO2 Reduction over Two Phases of Perovskite-Structured NaNbO3[END_REF][START_REF] Li | Surface-Coordination-Induced Selective Synthesis of Cubic and Orthorhombic NaNbO3 and Their Photocatalytic Properties[END_REF][START_REF] Gu | Modified Solvothermal Strategy for Straightforward Synthesis of Cubic NaNbO3 Nanowires with Enhanced Photocatalytic H2 Evolution[END_REF][START_REF] Opoku | Tuning the Electronic Structures, Work Functions, Optical Property and Stability of Bifunctional Hybrid Graphene Oxide/V-doped NaNbO3 type-II Heterostructures: A Promising Photocatalyst for H2 production[END_REF] CO2 reduction [START_REF] Li | The Effects of Crystal Structure and Electronic Structure on Photocatalytic H2 Evolution and CO2 Reduction over Two Phases of Perovskite-Structured NaNbO3[END_REF][START_REF] Li | Surface-Coordination-Induced Selective Synthesis of Cubic and Orthorhombic NaNbO3 and Their Photocatalytic Properties[END_REF][START_REF] Zeng | A Review on Photocatalytic CO2 reduction Using Perovskite Oxide Nanomaterials[END_REF][START_REF] Lais | Semiconducting Oxide Photocatalysts for Reduction of CO2 to Methanol[END_REF] and degradation of various organic molecules. [START_REF] Yu | Preparation and Photocatalytic Activity of Triangular Pyramid NaNbO3[END_REF][START_REF] Wang | Hybrid Density Functional Study on Mono-and Codoped NaNbO3 for Visible-Light Photocatalysis[END_REF][START_REF] Xu | BiOCl Decorated NaNbO3 Nanocubes: A Novel p-n Heterojunction Photocatalyst With Improved Activity for Ofloxacin Degradation[END_REF][START_REF] Chen | Surface Interaction between Cubic Phase NaNbO3 nanoflowers and Ru Nanoparticles for Enhancing Visible-Light Driven Photosensitized Photocatalysis[END_REF] In the final catalyst consisting of the niobate alone or combined to other species, the structural variety of NaNbO3 has again been demonstrated to have a strong impact on the catalyst efficiency, especially between the centrosymmetric orthorhombic structure and the cubic one. [START_REF] Li | The Effects of Crystal Structure and Electronic Structure on Photocatalytic H2 Evolution and CO2 Reduction over Two Phases of Perovskite-Structured NaNbO3[END_REF][START_REF] Li | Surface-Coordination-Induced Selective Synthesis of Cubic and Orthorhombic NaNbO3 and Their Photocatalytic Properties[END_REF] Interestingly, all the catalyzed reactions involve oxygen-based small species as intermediates, underlying the role of oxygen deficiency and mobility in sodium niobates. [START_REF] Chen | Surface Interaction between Cubic Phase NaNbO3 nanoflowers and Ru Nanoparticles for Enhancing Visible-Light Driven Photosensitized Photocatalysis[END_REF] However, to our knowledge, some polymorphs of NaNbO3 are still to be investigated, e.g. the acentric room-temperature polymorph crystallizing in an orthorhombic space group.

Several authors mention the occurrence of at least seven allotropic forms of NaNbO3 as a function of temperature from 930 to 15 K, corresponding to cubic, tetragonal, orthorhombic and rhombohedral space groups related to the perovskite network. [START_REF] Glazer | Studies of the Lattice Parameters and Domains in the Phase Transitions of NaNbO3[END_REF][START_REF] Mishra | Phase Stability and Structural Temperature Dependence in Sodium Niobate: A High-Resolution Powder Neutron Diffraction Study[END_REF][START_REF] Peel | New Twists on the Perovskite Theme: Crystal Structures of the Elusive Phases R and S of NaNbO3[END_REF][START_REF] Bondarev | Thermal and Physical Properties of Sodium Niobate Ceramics over a Wide Temperature Range[END_REF] The highand room-temperature phase transitions are mainly consisting in the tilting of the [NbO6] octahedra, whereas the low-temperature transitions are associated to the off-centered displacements of niobium atoms as well as the distortion of the [NbO6] octahedra. It must be pointed out that similar Madelung energies or free energies of all these varieties account for competing structural instabilities, giving rise to all these allotropic forms. [START_REF] Shigemi | Enthalpy of Formation of Various Phases and Formation Energy of Point Defects in Perovskite-Type NaNbO3 by First-Principles Calculation[END_REF][START_REF] Fritsch | Electronic and Optical Properties of Sodium Niobate: A Density Functional Theory Study[END_REF] However, as far as the accurate structure determination is concerned, the main difficulties arise from the slight differences between diffraction patterns especially for the phases which stabilize around room temperature. [START_REF] Peel | New Twists on the Perovskite Theme: Crystal Structures of the Elusive Phases R and S of NaNbO3[END_REF][START_REF] Johnston | The Polar Phase of NaNbO3: A Combined Study by Powder Diffraction, Solid-State NMR , and First-Principles Calculations[END_REF] In order to find the best model for structural analysis of NaNbO3 at room temperature, complementary experimental techniques are necessary. The combination of structural refinements from X-Ray powder Diffraction (XRD) and [START_REF] Chen | Surface Interaction between Cubic Phase NaNbO3 nanoflowers and Ru Nanoparticles for Enhancing Visible-Light Driven Photosensitized Photocatalysis[END_REF] Na solid-state Nuclear Magnetic Resonance (NMR) measurements to probe the local environments of sodium ions allows solving these ambiguities. In addition, the specific technique of optical Second Harmonic Generation (SHG) is useful in the case of compounds that do not possess inversion symmetry. Moreover, Madelung potentials and Density Functional Theory (DFT) calculations allow extracting the stability and bonding of each atomic site. All these slight differences of NaNbO3 allotropic forms must be critically described in regard of their reactivity and their optical band gap, which give valuable information about the chemical bonding and the consequences of the K + or Li + substitution for Na + in the case of KNN and LNN piezoelectric and ferroelectric phases, but also on surface activity towards several catalyzed reactions.

Depending on the synthesis route, solid-state approach and sol-gel techniques for instance, two varieties are observed at room temperature. [START_REF] Johnston | The Polar Phase of NaNbO3: A Combined Study by Powder Diffraction, Solid-State NMR , and First-Principles Calculations[END_REF] Both structures are distorted perovskites network. Accordingly, the Goldschmidt tolerance factor, describing the deviation to the ideal cubic structure using a geometrical argument, is 0.97 for NaNbO3, taking into account ionic radii of from Shannon et al. with r(Na + [XII]) = 1.39 Å, r(Nb 5+ [VI]) = 0.64 Å and r(O 2-[VI]) = 1.40 Å. [START_REF] Goldschmidt | Crystal Structure and Chemical Constitution[END_REF][START_REF] Shannon | Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides[END_REF] The first form crystallizing in the orthorhombic system and Pbma space group is the most cited in the literature with a = ap2, b = 4ap, c = ap2, where ap  3.9 Å represents the lattice parameter of the idealized cubic perovskite. [START_REF] Sakowski-Cowley | The Structure of Sodium Niobate at Room Temperature, and the Problem of Reliability in Pseudosymmetric Structures[END_REF] Three independent [NbO6] octahedra tilts occur in this supercell that also contains two crystallographic distinct Na sites. By applying an electrical field, an acentric ferroelectric phase is detected. [START_REF] Shuvaeva | Crystal Structure of the Electric-Fieldinduced Ferroelectric Phase of NaNbO3[END_REF] This polar phase crystallizes also in the orthorhombic symmetry in the P21ma space group, but the smaller unit cell with a twice smaller b parameter (a  ap2, b  2ap, c  ap2) is associated with less tilting modes for [NbO6] octahedra. In addition, several authors mention that the number of phase transitions in NaNbO3 vary as a function of particle size and that NaNbO3 with submicrometer particle sizes favors the polar P21ma space group at room temperature. [START_REF] Shiratori | Polymorphism in Micro-, Submicro-, and Nanocrystalline NaNbO3[END_REF][START_REF] Shiratori | Temperature-Induced Phase Transitions in Micro-, Submicro-, and Nanocrystalline NaNbO3[END_REF] By sol-gel technique, starting with Na and Nb alkoxides followed by annealing under air at T = 950 °C, a probably monoclinic phase strongly related to the orthorhombic space group P21ma can be stabilized. [START_REF] Johnston | The Polar Phase of NaNbO3: A Combined Study by Powder Diffraction, Solid-State NMR , and First-Principles Calculations[END_REF][START_REF] Darlington | On the Lattice Parameters of Sodium Niobate at Room Temperature and Above[END_REF] Indeed, high resolution X-ray and neutron powder diffraction patterns reveal slight deviations from P21ma space group. However, [START_REF] Chen | Surface Interaction between Cubic Phase NaNbO3 nanoflowers and Ru Nanoparticles for Enhancing Visible-Light Driven Photosensitized Photocatalysis[END_REF] Na solid-state NMR and second harmonic generation measurements confirm the occurrence of a polar phase. Furthermore, a mixture of the two orthorhombic varieties, Pbma and P21ma, is systematically obtained at room temperature when starting with a classical solid-state route after several grinding and annealing under air up to T = 950°C. [START_REF] Johnston | The Polar Phase of NaNbO3: A Combined Study by Powder Diffraction, Solid-State NMR , and First-Principles Calculations[END_REF] Then, a question remains unanswered: which synthesis route would stabilize separately the two orthorhombic forms at room temperature, in order to perfectly characterize their structural features, their reactivity and their optical properties?

In this paper, we implement another synthesis route to stabilize the two orthorhombic varieties as pure phases at room temperature. A comparative study then highlights their relative stability as well as their distinct reactivity and optical properties, which are discussed consequently to their structural differences. On the basis of powder XRD Rietveld analysis and [START_REF] Chen | Surface Interaction between Cubic Phase NaNbO3 nanoflowers and Ru Nanoparticles for Enhancing Visible-Light Driven Photosensitized Photocatalysis[END_REF] Na solidstate NMR investigation, the local distortion of [NbO6] octahedra, their tilting as well as the Na + local environments are described. The chemical bonding and the stability of all atomic sites are evaluated using various models: (i) Madelung potentials are calculated based on an ionic model and (ii) partial and total densities of states are obtained from DFT. Diffuse reflectance spectra of both oxides allows to compare their optical band gap. As the dominant SHG interaction mechanism exists only in materials that lack inversion symmetry, the SHG intensity versus the incident power is measured for both oxides in the Pbma (centrosymmetric) and P21ma (acentric) space groups. Finally, Thermogravimetric Analysis (TGA) under Ar:H2 (95:5 %vol.) and Electron Spin Resonance (ESR) allow quantifying the Nb 5+ reduction into Nb 4+ related to the stability of the Nb site in both polymorphs.

RESULTS AND DISCUSSION

Phase selectivity among room-temperature polymorphs. In a first step, a Rietveld refinement was carried out on powder obtained after hydrothermal route in autoclave at 200 °C before annealing under air. Considering separately the two structural hypotheses, Pbma and P21ma (Supporting Information, Figure S1), the fit remains unsatisfactory. The differences appear clearly on XRD patterns by analyzing the key low intensity peaks between 35 ° < 2 < 45 °. Finally, a phase mixture with 2/3 of Pbma and 1/3 of P21ma structural hypotheses leads to the best fit with remarkable accordance in the 35 ° < 2 < 45 ° region. The sample consisting of a mixture of two phases was then annealed under air at either T = 600 °C during 6 h or at T = 950 °C during 24 h, yielding a white powder in both cases. Rietveld refinements of XRD data revealed an excellent fit to the P21ma and Pbma space groups, respectively, as observed on Figure 1. The key inset in the 35 ° < 2 < 45 ° region clearly shows the good agreement of the refinements in both cases considered separately, but also the difference between the two space groups. Indeed, both structures are very similar and thus, their XRD pattern are similar as well.

The main difference consists in the cell parameter b being twice as large in the Pbma case. Consequently, inter-reticular planes (hkl) with k-odd Miller indices in Pbma space group have no equivalent in the P21ma space group, and thus no signal is detected at the 2 angles corresponding to their Bragg distance dhkl. Accordingly, Table 1 displays good reliability factors of the Rietveld refinements with similar cell parameters and atomic positions in both cases, after the doubling of the b cell parameter has been taken into account. The higher crystallinity of the Pbma phase synthesized at T = 950°C for 1 day (average coherent domain size around 170 nm) is qualitatively observed on the XRD patterns with narrower peaks compared to the oxide crystallizing in the P21ma space group (average coherent domain size around 120 nm) and obtained at lower temperature (T = 600 °C) during a shorter time (6 h). As mentioned in the literature, NaNbO3 undergoes a particle-size-induced phase transition from Pbma to P21ma space group. [START_REF] Shiratori | Polymorphism in Micro-, Submicro-, and Nanocrystalline NaNbO3[END_REF][START_REF] Shiratori | Temperature-Induced Phase Transitions in Micro-, Submicro-, and Nanocrystalline NaNbO3[END_REF] Limited temperature and time applied during the annealing are clearly key aspects to control crystal size expansion and thus monitoring the phase purity. More importantly, coupling a hydrothermal route and an annealing treatment under air at various temperatures happens to be a very simple method to access both pure NaNbO3 polymorphs at room temperature, and especially in the P21ma space group.

Table 1. Refined unit-cell parameters and atomic positions of NaNbO3 structures in the Pbma and P21ma space groups.

There is a clear structural relationship between these frameworks, taking into account the doubling of the b cell parameter for the Pbma hypothesis. The atomic networks are further compared from the point of view of distortions by means of Nb-O bond lengths (Figure 2). Along the b direction, the O2-Nb-O1 bonding is more distorted in the P21ma space group, with a long Nb-O2 distance (2.05 Å) compared to Nb-O1 in the same structure, but also compared to both Nb-O1 and Nb-O2 in Pbma space group (1.97 and 1.98 Å). In both allotropes, [NbO6] exhibits a second distortion perpendicular to the b axis which involves infinite O4-Nb-O3-Nb-O4 bond sequences in the equatorial plane with alternating long (ca. 2.1 Å) and short (ca. 1.9 Å) bonds (red distances in Figure 2a). However, this distortion is more pronounced for the P21ma polymorph, as the absolute deviations from average distance value are systematically higher than for Pbma (red values in Figure 2b). The second order Jahn-Teller (JT) effect of Nb 5+ ions arises from mixing vacant 4dcharacter conduction bands and filled 2p-character valence bands. It occurs when the d levels are sufficiently lowered as the cation becomes smaller and more highly charged. It is induced in structures with relatively low bandgap energy such as sodium niobiates with perovskite frameworks. This aspect is further discussed in the forthcoming paragraph related to the optical band gap and associated figures. Then a spontaneous second-order JT distortion can be observed in the P21ma and Pbma polymorphs. It is important to analyze the reason for the [NbO6] octahedron to be more distorted in the P21ma network, although it is prepared at lower temperatures. Looking closely at Na sites, both phases display two structurally distinct coordination sites Na1 and Na2. Average bond distances <Na1-O> and <Na2-O> are almost identical in the P21ma structure but rather different in the Pbma network (Table S1). The largest average bond distance is identified for <Na1-O> in the Pbma space group corresponding to the Na1 coordination site with a rotation axe, a symmetry operator that is exclusive to this Na site. The local environments of sodium atoms were further investigated using [START_REF] Chen | Surface Interaction between Cubic Phase NaNbO3 nanoflowers and Ru Nanoparticles for Enhancing Visible-Light Driven Photosensitized Photocatalysis[END_REF] Na solid-state NMR. Sodium nuclei, with a nuclear spin quantum number I = 3/2, carry an electric quadrupole moment Q = 10.4 fm 2 . As a consequence, they are highly sensitive to Electric Field Gradient (EFG) arising from a non-spherical distribution of the electric charges around the nuclei. Then, [START_REF] Chen | Surface Interaction between Cubic Phase NaNbO3 nanoflowers and Ru Nanoparticles for Enhancing Visible-Light Driven Photosensitized Photocatalysis[END_REF] Na NMR spectra, including central and satellite transition lines, may be of complex shape. This is reflected by the strength of the quadrupolar interaction and the anisotropy of the EFG. Nonetheless, spectra can be simulated considering the effects of first and/or second order quadrupolar interactions. Such a simulation depends of three main parameters: iso the isotropic chemical shift, CQ the quadrupolar coupling constant (CQ = 0 for spherically symmetric EFG) and Q the asymmetry parameter of the EFG (0 ≤ Q ≤ 1; Q = 0 for axially symmetric EFG). [START_REF] Chen | Surface Interaction between Cubic Phase NaNbO3 nanoflowers and Ru Nanoparticles for Enhancing Visible-Light Driven Photosensitized Photocatalysis[END_REF] Na MAS NMR spectra of both NaNbO3 polymorphs are presented on Figure 3 (black lines). Each spectrum can be described as the sum of two components (green subspectra) characterized by different sets of iso, CQ and Q parameters (Table 2) and associated with two distinct Na site environments. The structural features clearly show that Na2 sites in Pbma and P21ma space groups (4d Wyckoff position in Pbma and 2b Wyckoff position in P21ma) share the same mirror symmetry and the absence of rotation axis of symmetry, while Na1 coordination sites are characterized by a different rotational symmetry with a C2 axis in Pbma space group (4c Wyckoff position in Pbma and 2a Wyckoff position in P21ma). Thus, the NMR subspectra with CQ = 1 MHz and ηQ = 0.8 (deep green components in Figure 3) can be associated with Na2 sites while the other subspectra (light green) are unambiguously attributed to Na1 sites where the sodium nuclei are influenced by more or less asymmetric electric field gradients. The asymmetry parameter of the EFG tensor ηQ is a measure of deviation from cylindrical symmetry. Accordingly, it is close to 0 for Na1 sites in Pbma polymorph and close to 1 for Na1 sites in P21ma (Table 2). Moreover, the values reported in Table 2 are in good agreement with theoretical and experimental data from the literature predicting the same site assignement. [START_REF] Johnston | The Polar Phase of NaNbO3: A Combined Study by Powder Diffraction, Solid-State NMR , and First-Principles Calculations[END_REF] Comparing experimental 1D [START_REF] Chen | Surface Interaction between Cubic Phase NaNbO3 nanoflowers and Ru Nanoparticles for Enhancing Visible-Light Driven Photosensitized Photocatalysis[END_REF] The two Na sites exhibit different symmetries in the Pbma space group, whereas they share the same mirror symmetry in the case of P21ma. The absence of rotational symmetry for the two Na coordination sites in the polar phase, P21ma, creates less constraints in the network and, consequently, the [NbO6] octahedron distortion is higher. On the basis of competitive bonds along Na-O-Nb or inductive effects around O 2-ligands in the perovskite network, the stabilization of a larger (Na1) and a smaller (Na2) coordination sites with two various symmetries in Pbma structure contributes to decrease the [NbO6] second order Jahn-Teller distortion. The driving force of second Jahn-Teller distortion of the [NbO6] octahedron is then the degree of liberty and site symmetry of Na local environments. The symmetry of NaNbO3 weakens from Pbma to P21ma space group as well as the crystallinity (annealing under lower temperature and shorter time), whereas the [NbO6] octahedron distortion in the metastable P21ma phase increases thanks to the flexibility of the network and especially the Na coordination sites. Importantly, [START_REF] Chen | Surface Interaction between Cubic Phase NaNbO3 nanoflowers and Ru Nanoparticles for Enhancing Visible-Light Driven Photosensitized Photocatalysis[END_REF] Na NMR experiments confirm the XRD results that evidenced the phase selectivity of the chosen synthesis routes.

Phase selectivity can also be assessed by measurement of optical non-linear properties of the structure in the P21ma space group. Optical Second Harmonic Generation (SHG) is a process that combines two photons of an incident wave at frequency , into a single photon of frequency 2. The dominant electricdipole SHG interaction involves three photons, two input photons of frequency , both taken from the same laser source, and one output photon at frequency 2, which can occur only in materials that do not possess inversion symmetry. SHG measurements of the intensities backscattered can be recorded as a function of the incident power to determine the quadratic coefficient of the non-linear optical response and perform relative quantitative measurements of the susceptibility of a material. Such non-linear optical quantitative 3D µ-imaging measurements in a spincrossover micro-crystallite have been successfully applied in the past. [START_REF] Lacroix | From Magnetic to Nonlinear Optical Switches in Spin-Crossover Complexes[END_REF] A generalization of this previous semi-quantitative approach can also be applied to a statistic assembly of micro-crystallites with an average granulometry:

𝐼 ∝ 𝑁. 𝜒 𝐼 , ( 1
)
where N is the number density of micro-crystallites. It allows to quantify the slope of the quadratic dependence of the SHG intensity with the incident power, slope which is proportional to the square of the average effective non-linear optical susceptibility 𝜒 . Note that this experimental procedure is quite different but complementary to the well-known method developed by Kurtz and Perry in the late 1960's. [START_REF] Kurtz | A Powder Technique for the Evaluation of Nonlinear Optical Materials[END_REF] SHG quantitative measurements were performed on both oxides crystallizing with the Pbma and P21ma space groups and are reported on Figure 4. The intensity of back-scattered photons at 2 is plotted as a function of the incident source power at  and fit nicely the quadratic function given by Equation 1. The phase annealed at T = 600 °C, which adopts the acentric orthorhombic P21ma network, exhibits a very intense SHG active signal, with a slope 100 times higher than the other oxide annealed at higher temperatures. The signal detected in this case cannot be attributed to the centrosymmetric structure in Pbma space group. It could be due to P21ma contamination, which then, following Equation 1, would be ca. 1 %vol., since the density of the two forms are very similar. Another possibility is the local surface polarization of small molecules, such as water, which would provide acentric species to generate an induced second harmonic signal. The surface charge density and the associated potential are important characteristics of interfaces. This interfacial contribution is a Kerr contribution called Electric Field Induced Second Harmonic (EFISH) which involves the third-order susceptibility χ [START_REF] Rödel | Perspective on the Development of Lead-Free Piezoceramics[END_REF] of the material (which is strong for such perovskites materials) and the interfacial potential. Despite the fact that the origin of this residual signal remains unclear, SHG is in agreement with powder XRD Rietveld and [START_REF] Chen | Surface Interaction between Cubic Phase NaNbO3 nanoflowers and Ru Nanoparticles for Enhancing Visible-Light Driven Photosensitized Photocatalysis[END_REF] Na NMR data, confirming unambiguously the stabilization of two pure polymorphs of NaNbO3 at room temperature. They share a lot of structural similarities, however some differences must be pointed out, such as the distortion of Nb site and symmetries of Na sites, in addition to the well-known absence of inversion center in the case of a P21ma space group. In the following section, the stability of all atomic sites in the polymorphs are compared using energy calculations, and the occurrence of the metastable acentric phase is discussed. Energy considerations. In order to investigate site energies and chemical bonding in P21ma and Pbma frameworks, two approaches were considered. The first one, based on a simple ionic model, [START_REF] Ewald | Die Berechnung Optischer Und Elektrostatischer Gitterpotentiale[END_REF] allows calculating the Madelung potential of each crystallographic site, whereas the second one consists in determining the partial and total density of states (DOS), giving access to the center of gravity of the partial DOS of each nonequivalent atom. As reported in Table 3 for niobium and oxygen, the relative stability of atoms or atomic sites follows the same trend in both approaches. Table 4 shows that whatever the space group, the Na1 site is the most unstable from the point of view of Madelung potential, and the differences in Madelung potentials and average energy between Na1 and Na2 crystallographic site are larger in the Pbma structure. The lowest absolute potential and energy are obtained in this space group, for the Na1 site involving a rotational symmetry. Furthermore, the Madelung potential and average energy of O1 species in the vicinity of this Na1 site are the highest among the four oxygen positions. Lastly in the Pbma space group, the same effect is observed with the stabilization of Na2 site and the destabilization of O2 atomic position in the vicinity of Na2. Concerning the P21ma space group, the Madelung potentials and average energies of Na1 and Na2 sites are closer. The stability of O1 and O2 apical sites are also closer but in average lower than in Pbma space group. Besides, the higher the absolute values of potentials and energies of Na1 (Na2) site, the lower the values of O1 (O2) environment. In both structures, the two Na1 and Na2 atomic positions thus strongly influence O1 and O2, i.e. the apical oxygen atoms in the [NbO6] octahedron. Table 3 Regarding oxygen sites in the equatorial planes, the O3 is more stable in the P21ma network, while O4 is more stabilized in the Pbma space group. However, by considering average Madelung potentials and energies between O3 and O4, it appears that the oxygen atoms in the equatorial planes are more stable in the P21ma framework. Furthermore, the average potentials and energies between O1 and O2 sites are higher in the Pbma than in the P21ma networks. O2 and O3 are the most unstable in the Pbma structure, whereas only the O2 atomic position is clearly the unstable anionic species in the P21ma network. Finally, a stabilization of the apical oxygen atoms in the phase of Pbma space group (higher potentials and energies of O1 and O2) is accompanied by a destabilization of equatorial oxygen atoms (lower potentials and energies of O3 and O4). This effect of apical oxygen destabilization in the P21ma space group contributes, by competing interactions, to stabilize the O3 and O4 oxygen atoms in the equatorial planes (average values).

. Madelung potentials (VM) and average valence energies (<E>) with respect to the Fermi level, for Nb and O atomic sites in

The total DOS of NaNbO3 in the Pbma and P21ma structures are depicted on Figure 5, as well as partial DOS for non-equivalent sodium and oxygen atoms. The partial DOS for Na atoms (Figure 5a) shows clear differences between P21ma and Pbma structures in the low energy region of 3s-character bands, which especially reveals again the clear differentiation of Na1 and Na2 in the Pbma space group. For oxygen sites (Figure 5b), the partial DOS contribution around the Fermi level in the Pbma framework shows the higher instability of O3 compared to the other ones, in agreement with the previous analysis. In the P21ma network, the O2 atomic position in the vicinity of Na2 is the most unstable, as mentioned in the previous paragraph. The total DOS in the Pbma and P21ma frameworks (Figure 5c) displays very close band gaps around 3.5 eV, with a more pronounced band edge in the case of Pbma. A slightly higher band gap is expected for the P21ma structure associated to a slightly lower crystal field splitting of Nb 5+ (4d). Accordingly, diffuse reflectance spectra of the two polymorphs, represented on Figure 6, confirm band gaps both at ca. 3.4 eV, though slightly higher in the P21ma network. All in all, taking into account the structural study and relative stability considerations, the relaxation of equatorial Nb-O bond distances in the P21ma framework is associated to similar Na1 and Na2 sites (stability and symmetry), leading to the destabilization of O1 and O2 apical oxygen atoms, especially O2 in the vicinity of Na2. These effects, which are not found in the Pbma structure, are key parameters to understand the stabilization of the lower symmetry structure in the P21ma space group. Calculations confirm the strong influence of Na chemical bonding which contribute to perturb the Nb 5+ second-order Jahn-Teller distortion in NaNbO3. From these results, it is pertinent to question the mobility of oxygen in two overall closely related perovskite-derived anion networks, with one of them exhibiting adjacent oxygen sites that are more similar, although more destabilized. Niobium being prone to stabilize the +IV oxidation state (4d 1 ), the reducibility of sodium niobates vs. their crystal structures is investigated in the following part.

Reducibility of sodium niobates. Reduction processes under a gas mixture of argon and dihydrogen (Ar:H2 = 95:5 %vol.) were followed by thermogravimetric analysis during a heat treatment consisting of a 3 °Cꞏmin -1 ramp, to a temperature of 1000 °C that was maintained 1 h. The curves on Figure 7 show striking differences in term of weight loss upon such treatment: no significant weight loss is detected in NaNbO3-Pbma during the whole experiment, whereas NaNbO3-P21ma experiences a continuous weight loss up to 500 °C; then, the weight stabilizes before strongly decreasing from 800 °C. ESR spectra of the two resulting samples are reported on Figure 8. The reduction treatment of NaNbO3-Pbma at high temperature results in two superimposed signals associated with a low content of paramagnetic centers. The signal complexity is the consequence of anisotropic g-values and spectral overlap of hyperfine (h.f.) and superhyperfine (s.h.f.) structures arising from h.f. interaction with 93 Nb nuclear spin (I = 9/2) and s.h.f. additional interaction with [START_REF] Chen | Surface Interaction between Cubic Phase NaNbO3 nanoflowers and Ru Nanoparticles for Enhancing Visible-Light Driven Photosensitized Photocatalysis[END_REF] Na nuclear spins (I = 3/2). Nonetheless, the average value of g-factor (≤ 2), the values of hyperfine coupling constants (≤ 10 mT), the observation of [START_REF] Chen | Surface Interaction between Cubic Phase NaNbO3 nanoflowers and Ru Nanoparticles for Enhancing Visible-Light Driven Photosensitized Photocatalysis[END_REF] Na superhyperfine structure and the linewidth (≤ 1 mT) suggest that they are due to paramagnetic centers such as surface defects involving Nb 4+ , unpaired electrons or holes located on the oxygen atoms and/or oxygen vacancies. [START_REF] Kim | ESR Studies of Radiation-Induced Niobium Centers in Nb2O5-Na2O-SiO2 Glasses[END_REF][START_REF] Sweeney | Oxygen vacancies in lithium niobate[END_REF] Based on ESR data, the concentration in paramagnetic centers was roughly estimated at 100±20 ppm. The expected weight loss associated to oxygen removal during such a reduction is of the order of 0.0001 %, which is not significant with respect to the sensitivity of the TGA apparatus. Regarding NaNbO3-P21ma treated in the same reductive conditions, a broader signal in the same magnetic field region is detected by ESR. It is attributed to a Nb 4+ concentration several orders of magnitude higher in this polymorph, high enough to generate strong dipolar interactions and magnetic coupling at the origin of the signal broadening. Accordingly, the color of the Ar:H2 treated sample vary from light blue-grey for NaNbO3 originally in the Pbma space group to dark grey for P21ma. For the latter, no quantification was possible from ESR. Based on the weight loss from 800 °C (Figure 7), the deduced composition after reduction is NaNbO2.91, i.e. 18% of Nb 4+ . This nonstoichiometric oxide with Nb mixed valence crystallizes with the Pbma space group (Figure S2). Such a phase transition is not surprising, given the temperature achieved during the reductive treatment. The weight loss at lower temperatures during the heat treatment of NaNbO3-P21ma is attributed to adsorbed moieties at the surface, such as water and hydroxyl groups. [START_REF] Xu | A New Series of Oxygen-Deficient Perovskites in the NaTixNb1-xO3-0.5x System: Synthesis, Crystal Chemistry, and Energetics[END_REF] No weight loss is observed starting from Pbma-NaNbO3, even below 500 °C. It could be due to smaller specific surface area for this sample, which corresponds to larger average crystallite size (170 nm instead of 120 nm for P21ma) and higher annealing temperature (950 °C instead of 600 °C). Indeed, a more pronounced aggregation of larger crystals accounts for significantly less surface area where adsorption occurs. This could then explain the difference in weight loss between both samples, below 500 °C. This last investigation demonstrates the higher reducibility of NaNbO3-P21ma. It is probably related to the distortion of NbO6 octahedra in this framework, where Na sites acts indirectly by competing bonds on Nb-O chemical bonding. The Nb 4+ rate stabilized in the P21ma network remains unusually high in this perovskite-related network and might be even increased if the structure in P21ma space group is maintained after the treatment, for instance by optimizing the conditions of the reductive treatment. The reducibility of Nb in NaNbO3 is also necessarily connected to oxygen mobility. O3 and O4 sites are always closer to the Nb site than O1 and O2. On the basis of energy considerations and previous calculations, it is hypothesized that O2 sites are the most prone to participate to oxygen diffusion in P21ma network The reduction treatment of Pbma phase in the same conditions show this composition is stable without the formation of oxygen vacancies (only 100 ppm of Nb 4+ determined by ESR measurements leading to consider the quasi-absence of oxygen vacancies). Then, the oxygen and sodium sites stabilities, strongly connected to the creation of oxygen vacancies, affect the Madelung energy of P21ma polymorph which is lower than of Pbma form. In NaNbO2.91, it was not possible to localize O-vacancies based on XRD data (Figure S2). Intense XRD light (synchrotron) and neutron sources will be required to further investigate structural features of mixed valence sodium niobates, as well as possible occurrence of phase transitions vs. temperature. Moreover, the control of mixed valence ratios and oxygen mobility in room-temperature polymorphs of NaNbO3 rises questions concerning the photocatalytic properties that recently attracted interest for NaNbO3, especially in hydrogen evolution, CO2 reduction and degradation of organic molecules. [START_REF] Li | The Effects of Crystal Structure and Electronic Structure on Photocatalytic H2 Evolution and CO2 Reduction over Two Phases of Perovskite-Structured NaNbO3[END_REF][START_REF] Li | Surface-Coordination-Induced Selective Synthesis of Cubic and Orthorhombic NaNbO3 and Their Photocatalytic Properties[END_REF][START_REF] Gu | Modified Solvothermal Strategy for Straightforward Synthesis of Cubic NaNbO3 Nanowires with Enhanced Photocatalytic H2 Evolution[END_REF][START_REF] Opoku | Tuning the Electronic Structures, Work Functions, Optical Property and Stability of Bifunctional Hybrid Graphene Oxide/V-doped NaNbO3 type-II Heterostructures: A Promising Photocatalyst for H2 production[END_REF][START_REF] Zeng | A Review on Photocatalytic CO2 reduction Using Perovskite Oxide Nanomaterials[END_REF][START_REF] Lais | Semiconducting Oxide Photocatalysts for Reduction of CO2 to Methanol[END_REF][START_REF] Yu | Preparation and Photocatalytic Activity of Triangular Pyramid NaNbO3[END_REF][START_REF] Wang | Hybrid Density Functional Study on Mono-and Codoped NaNbO3 for Visible-Light Photocatalysis[END_REF][START_REF] Xu | BiOCl Decorated NaNbO3 Nanocubes: A Novel p-n Heterojunction Photocatalyst With Improved Activity for Ofloxacin Degradation[END_REF][START_REF] Chen | Surface Interaction between Cubic Phase NaNbO3 nanoflowers and Ru Nanoparticles for Enhancing Visible-Light Driven Photosensitized Photocatalysis[END_REF] Future works focusing on phasic purity should enable to better understand catalytic properties of sodium niobates and improve their performance.

CONCLUSION In summary, a new selective synthesis route has been presented towards the two polymorphs of NaNbO3 at room temperature. A thorough and detailed investigation of both phases by powder XRD, [START_REF] Chen | Surface Interaction between Cubic Phase NaNbO3 nanoflowers and Ru Nanoparticles for Enhancing Visible-Light Driven Photosensitized Photocatalysis[END_REF] Na solid-state NMR and SHG complementary analyses revealed without ambiguities the formation of a pure polar phase which crystallizes with the P21ma space group at relatively low temperature (600 °C). This oxide transforms irreversibly at higher temperature (below 950 °C) into the thermodynamically stabilized compound which adopts the centrosymmetric Pbma space group. Despite the similarities of XRD patterns of both these forms, the [NbO6] octahedron distortion and complex octahedral tilt schemes have been described and discussed. Significant structural changes were pointed out. The second order Jahn-Teller distortion of Nb 5+ (4d 0 ) is exalted in P21ma network where the two Na atom sites display the same mirror symmetry. In the Pbma space group, the Na atoms are located in two sites with distinct symmetry. Both Na sites contribute by competing bonds to constrain [NbO6] octahedron to be less distorted but more rigid in the Pbma framework. On the contrary, for the polar phase, the large difference of the two apical Nb-O bond lengths and the alternation of longer and shorter Nb-O bond distances in the equatorial plane underline the relaxation of the P21ma network, though more distorted. This is due to the key role of Na atomic positions which influence indirectly the second order Jahn-Teller distortion of [NbO6]. Madelung potentials and exploration of partial density of states by DFT allowed identifying the site stability and the differences in chemical bonding. Both approaches clearly show the effect of sodium on the oxygen atoms and their consequence on the [NbO6] tilt and distortion. The optical band gap of both these polymorphs are very similar but the energy is slightly higher for the polar phase, in good agreement with the total density of states around the Fermi level calculated by DFT. The structural features of NaNbO3 polymorphs, and especially the relaxation of the network and the distortion of NbO6 octahedron, are discussed in the light of drastically distinct reactivities upon reduction treatments. The thermodynamically stabilized form (Pbma space group) of NaNbO3 is not affected by heat treatments under Ar:H2 (95:5 %vol.) up to 1000 °C, whereas a large content of Nb 4+ (18 %), detected by TGA and ESR, is stabilized in the centrosymmetric perovskite network with a composition NaNbO2.91, starting from NaNbO3-P21ma. These preparative methods and consecutive characterizations shed new light on several existing fields of research regarding sodium niobates, especially (i) in the stabilization at low temperature of LNN and KNN solid solution, with the influence of polarizing (Li + ) and polarizable (K + ) species on Na sites, leading to ferroelectric and piezoelectric properties and (ii) concerning photocatalytic processes already demonstrated in various reactions of interest.

EXPERIMENTAL SECTION Synthesis. 0.810 g of Nb2O5 (99.9 %, Sigma-Aldrich) was dispersed in 20 mL of NaOH 6 M and heated in an autoclave equipped with 40 mL Teflon vials. The sample was heated at 200 °C during 24 h with heating and cooling ramps of 2 °Cꞏmin -1 . The white sediment and supernatant were both transferred in a 200 mL centrifuge tube (Nalgene) and centrifuged 15 min at 7 krpm. The supernatant was removed, the pellet was dispersed in ca. 100 mL of deionized water and the dispersion was centrifuged 15 min at 7 krpm. The separation, dispersion and centrifugation steps were repeated once. Finally, the supernatant was removed and the pellet was dried at 80 °C overnight, yielding a white powder. Annealing treatments were performed under air in tubular ovens. NaNbO3 structures crystallizing with Pbma and P21ma space groups were obtained after 24 h at 950 °C and 6 h at 600 °C, respectively, with systematic 2 °Cꞏmin -1 heating and cooling ramps.

Powder X-ray diffraction was performed on a PANalytical X'pert PRO MPD apparatus equipped with an X'Celerator multichannel detector. CuKα1/Kα2 radiations were generated at 45 kV and 40 mA. Data were acquired on the reflection Bragg-Brentano θ-θ mode, 8 ° ≤ 2θ ≤ 80 ° with 0.017 ° steps and acquisition time of 6 h 37 min. The resolution of XRD diffraction lines with this apparatus is equal to 0.08°. Samples were prepared from soften powder (particle size < 45 µm) flattened with the edge of a razor blade to avoid any preferential orientation. Refinements were performed using the JANA2006 software. [START_REF] Petrícek | Crystallographic Computing System JANA2006: General Features[END_REF] Rietveld refinements were performed with fixed βiso value of 0.5 for Nb, 1 for Na and 1 for O.

23 Na solid-state nuclear magnetic resonance spectra were acquired at a 7.05 T static magnetic field (Larmor frequency: 79.4 MHz) with a single short pulse (π/8) excitation of 1.5 μs (radio frequency field strength: ca. 166 kHz) and a recycling delay of 30 s. Magic Angle Spinning (MAS) experiments were performed at a spinning frequency of 10 kHz with a 4 mm Bruker probe. The resolution of NMR peaks is 0.06 ppm. Chemical shifts were calibrated according to a molar solution of NaCl (δ = 0.0 ppm). Magnitude and anisotropy of electric field gradient of each contribution were refined from experimental data using dmfit software. [START_REF] Massiot | Modelling One-and Two-Dimensional Solid-State NMR Spectra[END_REF] Second harmonic generation. Samples were pressed between two glass slides and analyzed with a home-built scanning SHG microscope. The excitation source is a femtosecond laser emitting at 1030 nm (Mikan, Amplitude system), which delivers 250 fs pulses at a repetition rate of 50 MHz. The laser light is focused into the sample with a microscope objective (10X Nikon Plan Fluor, NA = 0.3) and the resultant SHG at the wavelength 1030/2 nm is collected in the backward direction using a photomultiplier tube. Typical large SHG images of 500 x 500 µm [START_REF] Cross | Lead-Free at Last[END_REF] where recorded with a spatial resolution of ca. 10 µm and with a total acquisition time less than 2 s. SHG images were acquired by incrementing the average incident power between 1 and 10 mW to achieve a semi-quantitative quadratic power law dependence of the powder samples by averaging the SHG response for some selected areas.

Madelung potentials and hybrid DFT calculations. On the basis of the crystal structure determination, the Madelung potentials of each atomic sites were calculated by the Ewald method. [START_REF] Ewald | Die Berechnung Optischer Und Elektrostatischer Gitterpotentiale[END_REF] In this pure ionic model, atoms are considered as point (formal) charges located at all relevant crystallographic sites. Density functional theory (DFT) calculations were performed using the projector augmented-wave (PAW) method as implemented in the VASP code. [START_REF] Kresse | Ab Initio Molecular Dynamics for Liquid Metals[END_REF][START_REF] Kresse | Efficiency of Ab-Initio Total Energy Calculations for Metals and Semiconductors Using a Plane-Wave Basis Set[END_REF][START_REF] Blöchl | Projector Augmented-Wave Method[END_REF] PAW datasets were used to describe valence electronionic core interactions, with valence Na 2s, 3p, Nb 4p, 4d, 5s and O 2s, 2p electrons treated explicitly. The exchange and correlation energies were treated using the generalized gradient approximation (GGA) according to the Perdew-Burke-Ernzerhof formulation, [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] further corrected by the Heyd-Scuseria-Ernzerhof (HSE06) screened hybrid scheme, where 25 % of GGA exchange is replaced by exact Hartree-Fock (HF) exchange. [START_REF] Heyd | Hybrid Functionals Based on a Screened Coulomb Potential[END_REF][START_REF] Krukau | Influence of the Exchange Screening Parameter on the Performance of Screened Hybrid Functionals[END_REF] The Brillouin zone was sampled using -centered grids of 626 and 646 kpoints for the Pbma and P21ma polymorphs, respectively. Calculations were carried out on the two structural models leading to determine the total and the partial density of states (DOS) for each atom. In order to extract an information on the stability and bonding of each atomic site from partial densities of states, and for comparison with Madelung potentials, the center of gravity of the partial DOS (with respect to the Fermi level) can be calculated for each non-equivalent atom, according to:

𝐸 . . . ( 2 )
where ni(E) is the partial DOS for the atom i. For Nb and O atoms, the integrals run over the valence (2pcharacter) bands. For Na atoms, the partial DOS in O 2p-character bands is very small, because the Na-O bonding is mostly ionic. Therefore, it is more relevant to examine the Na partial DOS values and center of gravity in the Na 3s-character (empty) bands.

Diffuse reflectance spectra were recorded on powders in the 200-800 nm range using a Carry-Varian 5000 spectrometer equipped with an integrating sphere. MgO powder was used as a white reference.

Thermogravimetric analyses were carried out on a SETARAM Setsys Evolution apparatus under an Ar:H2 (95:5 %vol.) flux (50 mLꞏmin -1 ) with a heating rate of 3 °Cꞏmin -1 up to 1000 °C. The sample was then maintained at 1000 °C for 1 h. Mass fluctuations of a blank sample (empty Pt nacelle) due to the heat treatment were subtracted from the raw signal.

Electron spin resonance experiments were conducted with a Bruker EMX spectrometer operating at X-Band (9.45 GHz) and equipped with a helium flow cryostat. The presented powder spectra were recorded at low temperature (4 K) and normalized to the sample mass (100 mg). The frequency and amplitude of the magnetic field modulation were set to 100 kHz and 0.5 mT, respectively. The magnetic field scale was calibrated using a DPPH standard sample (g = 2.0036) and the Bruker software suite (WinEPR and WinSimfonia) was used to analyze the observed ESR signals.

Figure 1 .

 1 Figure 1. Powder XRD Rietveld refinements of NaNbO3 structures crystallizing in the Pbma and P21ma space groups, obtained after hydrothermal synthesis at T = 200°C during 24 h followed by heat treatment under air at 600 °C during 6 h and 950 °C during 24 h, respectively. Insets: zoom of the powder XRD patterns in the region 35.5 ≤ 2θ ≤ 44.5 °.

Figure 2 .

 2 Figure 2. [NbO6] octahedra distortion of NaNbO3 structures for Pbma and P21ma space groups. a) Each structure displays two adjacent octahedra with Nb atoms at the center (grey) and Nb-Oi distances in angströms for i = 1, 2 (black) and i = 3, 4 (red); b) table of Nb-Oi distances and deviation to average for each structure, red values highlight higher absolute deviations for the phase in P21ma space group.

  Na MAS-NMR spectra, both phases are distinguished in the chemical shift region -30 <  < -20 ppm, where Na1 component is exclusively detected.

Figure 3 .

 3 Figure 3. 23 Na solid-state NMR of NaNbO3 structures in the Pbma and P21ma space groups and Na local environments calculated from powder XRD Rietveld refinements. 23 Na MAS (10 kHz) NMR experimental spectra (black) and simulated signals (deep and light green) associated with the two Na sites. Structures: sodium (deep and light green) surrounded by oxygen with Na-O

Figure 4 .

 4 Figure 4. a) Bright field pictures of the powders with selected areas (white rectangles) for SHG analysis. b) SHG of NaNbO3 structures in the Pbma (centrosymmetric) and P21ma (acentric) space groups.

  NaNbO3, in the Pbma and P21ma space groups. <E> values were calculated from DFT. The experimental crystal structures were used as input for all calculations. Table 4. Madelung potentials (VM)) and average energies (<E>) of Na 3s-character bands, in the Pbma and P21ma space groups. <E > values were calculated from DFT.

Figure 5 .

 5 Figure 5. Partial and total density of states of NaNbO3 in the Pbma and P21ma space groups. a) Na1 and Na2, b) O1, O2, O3 and O4, and c) total density of states.

Figure 6 .

 6 Figure 6. Diffuse reflectance of NaNbO3 after hydrothermal synthesis followed by annealing 6 h at 600 °C (P21ma) and 24 h at 950 °C (Pbma).

Figure 7 .

 7 Figure 7. TGA of NaNbO3 structures in the Pbma and P21ma space groups under a Ar:H2 mix (95:5).

Figure 8 .

 8 Figure 8. Low-temperature (4 K) X-band ESR spectra of reduced sodium niobates from the Pbma and P21ma structures after exposure to a Ar:H2 mix (95:5 %vol.) at 1000 °C during 1 h. *unidentified paramagnetic signal.

ACKNOWLEDGMENT

The authors are thankful to Dany Carlier for the discussions on 23 Na NMR.

ASSOCIATED CONTENT

Supporting Information. The following file are available free of charge (Word document): Figure S1. Rietveld refinements of NaNbO3 powder obtained after hydrothermal synthesis (no annealing); Table S1. Na-Oi distances in NaNbO3 structures in the Pbma and P21ma space groups; Figure S2. XRD pattern of NaNbO3-x after exposure of P21ma-NaNbO3 to Ar:H2 (95:5 %vol.) at 1000 °C during 1 h, and refinement using profile matching method with Pbma structural hypothesis. Two .cif files (NaNb_Pbma.cif and NaNb_P21ma.cif) have been uploaded as supporting information.

AUTHOR INFORMATION

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.


Investigating the Recognition of Local Shapes Using Mid-air Ultrasound Haptics

Thomas Howard 1 , Gerard Gallagher 3 , Anatole Lécuyer 2 , Claudio Pacchierotti 1 , and Maud Marchal 3

Abstract-Mid-air haptics technologies are able to convey haptic sensations without any direct contact between the user and the haptic interface. One representative example of this technology is ultrasound haptics, which uses ultrasonic phased arrays to deliver haptic sensations. Research on ultrasound haptics is only in its beginnings, and the literature still lacks principled perception studies in this domain. This paper presents a series of human subject experiments investigating important perceptual aspects related to the rendering of 2D shapes by an ultrasound haptic interface (the Ultrahaptics STRATOS platform). We carried out four user studies aiming at evaluating (i) the absolute detection threshold for a static focal point rendered via amplitude modulation, (ii) the absolute detection and identification thresholds for line patterns rendered via spatiotemporal modulation, (iii) the ability to discriminate different line orientations, and (iv) the ability to perceive virtual bumps and holes. These results shed light on the rendering capabilities and limitations of this novel technology for 2D shapes.

I. INTRODUCTION

Mid-air haptics is an emerging group of technologies for providing haptic sensations at a distance, without any direct physical contact with the interface creating the stimuli. Several physical principles can be used to provide midair haptic stimuli: magnetism [START_REF] Adel | Rendering 3d virtual objects in mid-air using controlled magnetic fields[END_REF], acoustics [START_REF] Iwamoto | Airborne ultrasound tactile display[END_REF], electric arcs [START_REF] Spelmezan | Sparkle: Towards haptic hover-feedback with electric arcs[END_REF], optics [START_REF] Jun | Laser-induced thermoelastic effects can evoke tactile sensations[END_REF], and aerodynamics [START_REF] Gupta | Airwave: Non-contact haptic feedback using air vortex rings[END_REF]. Among these technologies, the currently most mature one uses focused airborne ultrasound. Arrays of ultrasonic transducers produce phase-shifted acoustic waves which constructively interfere at points in space called focal points and destructively interfere elsewhere, conveying sensations by varying acoustic radiation pressure on the skin [START_REF] Iwamoto | Airborne ultrasound tactile display[END_REF], [START_REF] Carter | Ultrahaptics: multi-point mid-air haptic feedback for touch surfaces[END_REF]. The device used in this study is commercialized by Ultrahaptics and capable of creating multiple focal points whose intensity and position are updated at up to 40 kHz [START_REF] Carter | Ultrahaptics: multi-point mid-air haptic feedback for touch surfaces[END_REF]. Ultrahaptics technology is used in a large set of application domains, from entertainment and Virtual Reality (VR) to 3D modeling and automotive [START_REF] Magazine | Automotive infotainment cruises on high at ces[END_REF]. For example, Long et al. [START_REF] Long | Rendering volumetric haptic shapes in mid-air using ultrasound[END_REF] presented an algorithm for rendering three-dimensional shapes. Users were shown to be able to differentiate five different shapes: a sphere, a pyramid, a horizontal prism, a vertical prism and a cube. Harrington et al. [START_REF] Harrington | Exploring the use of mid-air ultrasonic feedback to enhance automotive user interfaces[END_REF] and Shakeri et al. [START_REF] Shakeri | May the force be with you: Ultrasound haptic feedback for mid-air gesture interaction in cars[END_REF] demonstrated the use of ultrasound haptics in vehicular applications, showing significant reductions in visual demand, interaction times, and improved accuracy. Finally, Georgiou et al. [START_REF] Georgiou | Touchless haptic feedback for VR rhythm games[END_REF] used ultrasound haptics for a VR mid-air haptic rhythm game, and Martinez et al. [START_REF] Martinez | Touchless haptic feedback for supernatural vr experiences[END_REF] for an immersive fantasy narrative.

This research has received funding from the EU's H2020 research and innovation programme (grant agreement No 801413, project "H-Reality"). From a perceptual point of view, there are several studies investigating the effect of providing ultrasound haptic stimulation. Gavrilov et al. [START_REF] Gavrilov | A study of reception with the use of focused ultrasound. i. effects on the skin and deep receptor structures in man[END_REF] perfomed pioneering work on tactile sensations evoked by ultrasound, determining detection thresholds for ultrasound stimuli from a single transducer at various frequencies. Dalecki et al. [START_REF] Dalecki | Tactile perception of ultrasound[END_REF] expanded on this work, showing the main phenonmenon contributing to elicited tactile sensations was acoustic radiation pressure. However, to the best of our knowledge, no perception studies have been carried out using focused ultrasound phased arrays like the Ultrahaptics device.

Toward this objective, we present a series of human subjects experiments investigating the perceptual aspects associated with the rendering of 1D and 2D shapes, such as points, lines, bumps, and holes. We expect these results to be useful for all researchers aiming at designing new rendering techniques and immersive experiences using this technology.

II. EXPERIMENTAL OBJECTIVE AND SETUP

The objective of this work is to evaluate to what extent we can render 1D and 2D shapes using the Ultrahaptics technology. The following experiments measure and discuss (i) the absolute detection threshold for a static focal point rendered via amplitude modulation (Sec. III), (ii) the absolute detection and identification thresholds for line patterns rendered via spatiotemporal modulation (Sec. IV), (iii) the ability to discriminate different line orientations (Sec. V), and (iv) the ability to perceive virtual bumps and holes (Sec. VI).

Experiment #1: detecting focal points. The first experiment evaluated the absolute detection threshold for a single static focal point. For this point to be felt, the acoustic radiation pressure must vary at a given frequency, i.e., 200 Hz in this experiment [START_REF] Hasegawa | Aerial vibrotactile display based on multiunit ultrasound phased array[END_REF]. The focal point was rendered using Amplitude Modulation (AM). When using this modality, the device generates output from every transducer such that constructive and destructive interferences at and around the focal point result in its acoustic radiation pressure oscillating at the desired frequency. The focal point intensity can then be varied by changing the output intensity, affecting the peak of the resulting acoustic radiation pressure. For the Ultrahaptics STRATOS platform, changes in intensity ranging between 0% and 100% correspond to peak acoustic radiation pressure at the focal point between 0-1125 Pa.

Experiment #2: detecting and identifying lines. Our second experiment evaluated the detection threshold for a linear pattern of a given length. For this experiment, AM is not a suitable modality, because the update rate for point positions and intensities is constrained by the chosen focal point frequency. In fact, a certain number of acoustic radiation pressure oscillation periods is required at a given position and intensity for the user to perceive the haptic stimulus. Spatiotemporal modulation (STM) was introduced by Kappus and Long [START_REF] Kappus | Spatiotemporal modulation for mid-air haptic feedback from an ultrasonic phased array[END_REF] to address this limitation. In STM, focal points are generated with a fixed frequency (usually the maximum achievable by the device, i.e., 40kHz). Since this frequency is very high, it poses significantly fewer constraints on the temporal evolution of the peak intensity and focal point position. Thus, by moving the focal point through space and/or varying the peak intensity, it is possible to achieve complex modulations of the resulting acoustic radiation pressure along 3D paths. Frier et al. [START_REF] Frier | Using spatiotemporal modulation to draw tactile patterns in mid-air[END_REF] have also investigated the trade-off between pattern repetition rate in STM and perceived intensity. For these reasons, we used STM to display lines above the board at the optimal focal point movement speed with respect to their length (7 m/s in our case) while varying the intensity of the moving focal point. The experiment was performed in two parts: the first one investigated the detection threshold regardless of the perceived pattern, while the second one investigated the intensity required for users to reliably feel the displayed pattern as a line.

Experiment #3: discriminating line orientations. Our third experiment aimed at evaluating how well users can discriminate lines oriented in four different ways. Again, we rendered line patterns on the subject's palm using STM. However, here we investigated the effect of stimulus intensity on line orientation discrimination as well as potential effects of hand movements for the same metric (passive vs. active touch).

Experiment #4: recognizing bumps and holes. Finally, our fourth and last experiment aimed at evaluating the device's capability for displaying shapes along the z-axis, i.e., modulating the perceived local height profile. While in planes parallel to the array surface, the Ultrahaptics device can achieve a spatial resolution below 1 cm, this is not true in planes perpendicular to the array surface. In fact, focal points tend to "stretch" along the vertical direction, with the acoustic energy spreading out over several centimeters. This is especially an issue when rendering small-scale shapes along the z-axis. To overcome the above rendering issue, we propose to modulate the focal point intensities in a plane parallel to the device, so as to introduce the illusion of a variation of height.

Setup. The experimental setup, shown in Fig. 1, is composed of an armrest, an Ultrahaptics STRATOS platform, an LCD screen, and a pair of noise-canceling headphones. The armrest was situated 20 cm above the Ultrahaptics device, and was designed to keep subjects comfortable and rested throughout the duration of the experiments. Subjects were asked to sit in front of the screen, place their arm on the armrest, with their palm facing the ultrasound array and provide responses to various stimuli, depending on the experimental protocols. Subjects used a mouse to report their answers to the questions displayed on the screen. Subjects wore headphones playing pink noise throughout all experiments in order to avoid potential effects due to auditory cues arising from the device operation. Starting conditions as well as starting trial blocks were randomized between subjects so as to rule out any possible learning and fatigue effect. All subjects provided informed consent.

III. DETECTION OF SINGLE FOCAL POINTS

We aim at evaluating the 50%-detection threshold for a single focal point generated using AM, both in a passive touch (static hand) and an active touch (moving hand) condition. Subjects were presented with a static 200 Hz focal point located 20 cm above the center of the board (see Fig. 2) for a duration of 2 s at different intensity levels and were asked a binary question regarding whether they felt the stimulus or not. In the passive touch condition, subjects were asked to keep their hand still while the focal point was rendered on the palm. In the active touch condition, subjects were asked to swipe their hand back and forth while following an on-screen display of a dot moving at 0.5 m/s.

A. Materials and Methods

We used a 1-up, 1-down staircase method to determine subjects' 50-%detection thresholds. Trial blocks were structured so as to randomly interleave one upward staircase trial run (starting from 0% intensity) and one downward staircase trial run (starting from 100% intensity). Intensities were adjusted in 4% increments, and subject responses were recorded for each level. Each subject performed three trial blocks per condition (passive touch and active touch). Trial blocks continued until a total of 9 reversals were recorded for each of the interleaved upward and downward trial runs. The first reversal was ignored in subsequent analyses and the remaining 8 were averaged to obtain an estimate of the detection threshold, yielding two measures per trial block.

15 healthy subjects (8 male, 7 female; ages 22-32 (M=25); 2 left-, 13 right-handed) participated in the experiment.

B. Results and discussion

In the passive condition, mean subject thresholds ranged from 32% to 77% (interquartile 41%-52%) with a mean of 49.5% (median 46%). In the active condition, thresholds are slightly lower, ranging from 28% to 72% (interquartile 41%-51%) with a mean of 48.7% (median 45.3%). This difference between conditions is not significant (t-test). Therefore, we cannot conclude any effect of hand movement on the detection threshold for a focal point generated through AM.

IV. DETECTION AND IDENTIFICATION OF LINE PATTERNS

We aim to investigate the 50%-detection and 50%identification thresholds for lines generated using STM. Subjects were presented with 15-cm-long lines aligned with the device x-or y-axis at a height of 20 cm. We selected a focal point movement speed of 7 m/s, which is close to the best perceived intensity [START_REF] Frier | Using spatiotemporal modulation to draw tactile patterns in mid-air[END_REF]. In both variations of the experiment, the intensities of the focal point were varied, and subjects were asked to respond to binary questions regarding either the detection or identification of the stimuli. Subjects were left free to explore the plane in which the stimuli were presented (z=20 cm) as they saw fit.

A. Materials and methods

We used 1-up, 1-down staircase methods to determine both thresholds, as we did in Sec. III. Each subject performed two trial blocks, one in a α = 0 • and another in a α = 90 • orientation. Subjects responded to the binary questions "Did you feel a stimulus?" and "Did you feel a line?" for the detection and identification experiments, respectively. 7 healthy subjects (5 male, 2 female, ages 22-32 (mean: 27.2), all right-handed) participated in the detection thresholds experiment, and 12 different healthy subjects (9 male, 3 female, ages 22-47 (mean: 28.3), 11 right handed, 1 left handed) participated in the identification thresholds experiment.

B. Results and discussion

For the detection thresholds experiment, 6 out of 7 subjects showed a lower median threshold in the 0 • condition when compared to the 90 • condition, and only 1 showed a higher median threshold in the 0 • condition (none of these differences were found to be significant, Wilcoxon rank-sum tests). Looking at mean subject thresholds, they ranged from 21.3% to 33.3% in the 0 • orientation, with a mean of 28.4% (median 29.6%), as shown in Fig. 4. In the 90 • orientation, subject mean thresholds ranged from 21.8% to 36.3%, with a mean (and median) of 31%. The observed differences in mean thresholds between orientation were not found to be significant (Wilcoxon rank-sum test).

For the identification thresholds experiment, subject mean thresholds ranged from 23.1% to 56.8% in the 0 • orientation, with a mean of 42.8% (median 46%), as shown in Fig. 4. In the 90 • orientation, subject mean thresholds ranged from 29.8% to 79.6%, with a mean of 48.9% (median 47.8%). These mean identification thresholds tended toward a lower median value in the 0 • condition, but the difference was not found to be significant (Wilcoxon rank-sum test). For both the 0 • and 90 • orientations, the measured identification thresholds are well above the detection thresholds (median 46% for identification vs. median 29.6% for detection in 0 • , median 47.8% for identification vs. median 31% for detection in 90 • ). Pairwise comparisons between these thresholds in the 0 • and 90 • orientations show these differences to be significant with p<0.05 and p<0.01 respectively (Wilcoxon rank-sum tests).

Both detection and identification thresholds for line patterns generated through STM were found to be lower than detection thresholds for a single point generated through AM (which is consistent with prior results from Takahashi et al. [START_REF] Takahashi | Lateral modulation of midair ultrasound focus for intensified vibrotactile stimuli[END_REF]). This result can be attributed to a number of factors, such as the change in modulation and possible constructive interference between successive focal points in STM, the stimulation of a larger surface area (which is consistent with previous findings on contact vibrotactile displays [START_REF] Verrillo | Effect of contactor area on the vibrotactile threshold[END_REF]), and even the fact that lines aligned with the y-axis allowed subjects to feel out the pattern with more sensitive parts of the hand than the palm (e.g., the fingertips).

V. DISCRIMINATION OF LINE ORIENTATIONS

During the previous experiment, subjects frequently reported a certain degree of confusion regarding the orientation of the line patterns, especially at lower intensities. Therefore, we decided to carry out a preliminary investigation into the capacity for discrimination of line pattern orientations at different intensity levels.

A. Materials and methods

Subjects were presented a randomized sequence of lines with 4 different orientations, α = {0 • , 45 • , 90 • , 135 • }, at 3 different intensity levels, a low-intensity level (54%), an intermediary intensity level (77%), and a high intensity level (100%). The low intensity level was chosen to be above the 50%-identification threshold of 90% of the sample population (see Sec. IV), ensuring that subjects almost certainly felt the pattern as a line. The intermediary intensity was chosen as the midway point between the low and maximum intensity. Subjects performed two trial blocks with 120 trials each (10 for each pair of intensity and orientation levels), one block in a passive touch condition, one in an active touch condition.

10 healthy subjects (8 male, 2 female, ages 22-44 (M=27.2), all right handed) participated in the study. Confusion matrices were obtained for each intensity level and averaged (see Fig. 5). Overall, the majority of orientations were correctly identified. 0 • lines were occasionally confused with 45 • or 135 • , but never with 90 • lines (both in active and passive conditions). 90 • lines are similarly never confused with 0 • lines in the passive condition, but occasionally in the active condition. The most confused orientation is 135 • with 90 • .

B. Results and discussion

In the passive condition, 0 • lines are best identified, with no differences observed regarding the intensity level of the display. The 45 • and 135 • orientations seem to be the hardest to identify, showing the lowest means and largest variability in subject responses. Correct identification rates increase in mean for the 135 • orientation, while they tend to decrease for both the 45 • and 90 • orientations as intensity increases. Analyzing the effect on hit rates of orientation and intensity, a 2-way ANOVA shows that orientation has a significant effect on hit rates (p<0.05), while intensity and interactions between both do not.

In the active condition, correct identification rates do not differ much from one orientation to another. There seems to be a slight tendency toward greater variability and lower mean correct identification rates with increasing intensity, but a 2-way ANOVA shows neither orientation, nor intensity, nor interactions between both significantly affect hit rates.

When evaluating the effect of hand movement, it seems that subjects do worse on identifying 0 • lines in the active condition. However, overall, they show slightly higher mean correct identification rates and less variability than in the passive condition. Analysing the effect of intensity and condition (passive vs. active) on hit rates for the 0 • orientation, a 2-way ANOVA shows no effect of intensity, but a significant effect of condition (p<0.01) and no effect of interactions. 2-way ANOVAs analyzing the effects of intensity and condition on hit rates in other orientations show no significant effects.

2-way ANOVAs analyzing the effects of intensity and condition on confusion rates between pairs of orientations show no effects of intensity, condition or interactions in most cases. The confusion rate between orientations 0 • and 90 • is an exception, with a significant effect of the condition on the confusion rate (p<0.05). The confusion rate between orientations 90 • and 135 • is significantly affected by the interaction between intensity and condition (p<0.1).

VI. DISPLAYING BUMPS AND HOLES ALONG THE Z-AXIS

We aim to investigate the identification rates of bumps and holes rendered along the device z-axis, together with effects of orientation on shape recognition.

A. Materials and methods

Subjects were presented with a random selection of five haptic patterns (one bump, one hole, and three lines of different intensities) and asked to identify the stimulus as a line, a bump, or a hole. Each pattern was presented 30 times, i.e., the subjects felt a total of 30 bumps, 30 holes, and 10 lines for each of the 3 intensity levels. For this initial investigation, we chose to use the largest possible difference between baseline and peak intensity for both patterns, in order to maximise chances of correct identification. However, because of this, the baseline intensities for the bump and hole patterns are necessarily different. We chose to divide the line patterns into three intensity levels so as to act as potential confounders, in the case where subjects would not actually identify bumps and holes but rather differences in baseline, peak or mean intensities. Lines were designed in Fig. 6: Stimuli are presented in two possible orientations, 0 • and 90 • (top), with 5 different intensity profiles (bottom). Focal point intensity as a function of position is show in blue for a hole pattern; in yellow, orange and red respectively for the minimum, intermediary and maximum intensity lines; and in green for the bump pattern.

the same way as presented in Secs. IV and V, with the three intensity levels selected so as to match the baseline intensities of, the hole pattern (100%), the bump pattern (54%) and an intermediary value (77%), respectively. To maximize chances of identification, bumps and holes were designed with the intensity profile following a step function, with the central 1/3rd of the line at peak intensities of 100% (bump) and 54% (hole), respectively (see Fig. 6).

In addition to the recognition rate, we also investigated the effects of orientation on pattern discrimination. Subjects performed the experiment in two blocks, one with patterns aligned with the device x axis (α =90 • ) and one with patterns aligned with the device y axis (α =0 • ).

12 healthy subjects (8 male, 4 female; ages 23-30 (M = 25.3); 10 right-handed, 2 left-handed) participated in the experiment. A questionnaire (7-point Likert scale) was filled out at the end of the experiment, rating subject confidence, fatigue, ease of the task and difficulties in identifying the different shapes. Correct identification rates are shown in Fig. 7. A 2way ANOVA analyzing the effects of orientation and shape on correct identification rates showed that none of these variables and their interactions have a significant effect on the identification rates.

B. Results and Discussion

In the 0 • condition, there is no difference between identification rates for lines of different intensities (69.7% mean), and barely any difference between identification rates of bumps and holes (52.2% and 53.9% means respectively). Bumps and holes are identified correctly much less of the time than lines. None of the observed differences are significant (1-way ANOVA).

In the 90 • condition, minimum intensity lines are identified correctly less of the time (60% mean) than intermediary or maximum intensity lines (73.3% mean). Bumps are identified (62.8% mean) less than holes (75.3% mean) or intermediate and maximum lines (mean identification rate of all lines was 68.9%), but still twice above the random chance (33.3%). Holes are identified as much as intermediary or maximum intensity lines and above lines on average. A 1-way ANOVA shows none of the observed differences are significant. These tendencies show that it is slightly easier to correctly identify high-intensity lines, and that holes tend to be easier to identify than bumps, mirroring the subjective reports of subjects who generally expressed a greater degree of certainty when feeling holes than when feeling bumps.

In the 90 • condition, bumps and lines are more often confused (24.2% of the time) than holes and lines (11%) or bumps and holes (9.7%) (p<0.01, t-test). Confusion between bumps and lines is primarily due to bumps being misidentified as lines, indicating no discrimination of shape on the part of subjects (difference in medians significant with p<0.1, Wilcoxon rank-sum test). Confusion between holes and lines seems equally due to holes being misidentified as lines, indicating no discrimination of shape, as to lines being misidentified as holes, indicating an erroneous perception of a dip in intensity along the line. Confusion between bumps and holes is mainly due to holes being misidentified as bumps, i.e., it appears that a lower intensity in the centre of a high-intensity pattern is easier to miss than a high intensity in the centre of a low intensity pattern (difference is significant with p<0.05, t-test). Holes seem to be more often misinterpreted as lines than as bumps, confirming the idea that subjects probably "missed" the low intensity section in the centre. In the 0 • condition, lines and bumps are confused 25.3% of the time on average, mostly from bumps being misidentified as lines (difference in median misidentification rate significant with p<0.01, Wilcoxon rank-sum test). Again, confusion between lines and holes (15.1% of the time) is more or less equally distributed among lines being misidentified as holes and holes being misidentified as lines. Bumps are confused with holes 21.3% of the time, mostly from holes being misidentified as bumps. Again, this result indicates that subjects may perceive that there is an evolution in the intensity profile but do not interpret it correctly.

A detailed look at the influence of line intensity on misidentification of lines in the 90 • condition shows that, for minimum and intermediate intensity lines, misidentification rates are similar for bumps and holes. Maximum intensity lines are more often misidentified as bumps than holes (p<0.05, t-test). This means that intensity at the center of the pattern is important for shape identification. In the 0 • condition, minimum intensity lines are mostly misidentified as bumps (p<0.1, t-test), but more of them are identified as holes than in the 90 • condition. Minimum intensity lines are more often misidentified as holes than as bumps (p<0.1, t-test), while intermediary intensity lines are equally misidentified as bumps and holes. This again tends to indicate that intensity in the centre of the pattern is determining in identification, more than intensity on the edge.

Comparisons between orientations reveal no significant differences. Subjects tended to perform similarly for both (contrary to their subjective perception indicating major preference for the 90 • condition). Only the identification rates for bumps and holes appear to be slightly lower, and the confusion rate between bumps and holes higher in the 0 • condition, which does not match subjective assessment by our subjects.

Concerning the subjective questionnaire, participants reported being quite confident (mean M = 4, standard deviation SD = 1.6). They found the task quite easy (M = 3, SD = 3.45) and were almost not fatigued (M = 2, SD = 2.45). Concerning the difficulty to perceive the different shapes, they found it moderately difficult to feel bumps (M = 3.1, SD = 1), and slightly more difficult to feel lines (M = 4.36, SD = 1.6) and holes (M = 4.73, SD = 1.7). 50% of the subjects found it easier to identify bumps in the 90 • condition, against 25% for the 0 • condition and 25% which expressed no preference. Concerning identification of lines, most subjects (83%) did not prefer a specific orientation, reflecting the almost identical identification rates for lines in both orientations. Concerning the identification of holes, 42% of subjects showed no preference while another 42% preferred the 90 • condition. Overall, it seems that subjects expressed greater ease of identification in the 90 • condition, which reflects the tendencies shown in both identification rates for bumps and holes as well as in confusion rates for all patterns. When asked about why they believed that the 90 • condition allowed for better identification, most subjects responded that exploring a shape along a line using the palm made things easier thanks to (i) the less differences in tactile sensitivities between the various parts of the skin being stimulated and (ii) the stimulation of a contiguous patch of skin (e.g., against fingers, that are separated by gaps).

VII. GENERAL DISCUSSION AND CONCLUSION

This paper presented a series of user studies investigating perceptual aspects associated with the rendering of 1D and 2D shapes, such as points, lines, bumps, and holes. We determined the 50%-detection threshold for a single focal point generated with AM, finding it to be largely independent of hand movement and on average between 48.7% and 49.5% intensity, i.e., approx. 560 Pa peak acoustic radiation pressure. Then, we proceeded to investigate the effects of changing the displayed pattern and modulation to gain insights about displaying lines. Using STM, we found it was possible to reduce the detection threshold to around 30% intensity, i.e., approx. 338 Pa. However, at such low intensities, subjects did not readily identify the patterns as lines, with the threshold for a proper identification of lines being closer to 45% intensity. An extension of the current protocol to other 2D shapes with various parameters could yield further insights into the intensity required to properly feel and discriminate between them. We also performed a preliminary investigation into line orientation discrimination at various intensities, finding no significant effect of intensity or hand movement on line orientation discrimination capabilities. However, a comprehensive evaluation of angular JNDs for linear patterns and possible effects of intensity could provide insights into the best intensity setting when displaying directions. Finally, we proposed a novel solution for displaying bumps and holes by modulating the intensity profile along a linear path, and we investigated the corresponding subject discrimination abilities. While our method was found to be effective to a certain extent, an evaluation of the JND for the intensity of line patterns would be interesting both for insights into rendering of 2D shapes and for possibly enhancing the resolution obtained in our rendering method for displaying bumps and holes along the z axis.

In the future, we plan to study to what extent it is possible to render other 2D (and 3D) shapes with this technology, while extending the perceptual tests to other parts of the body. Combining ultrasound haptics with visual stimuli will let us study how immersive virtual environments can facilitate sensations and possibly lower thresholds. Finally, we plan to combine ultrasound haptics with other types of haptic stimuli (e.g., using wearables [START_REF] Pacchierotti | Wearable haptic systems for the fingertip and the hand: Taxonomy, review, and perspectives[END_REF]), to extend the range of sensations provided and the range of features we can render.

Fig. 1 :

 1 Fig. 1: Experimental setup: Participants place their dominant arm on the armrest, receiving haptic stimuli on their palm.

Fig. 2 :

 2 Fig. 2: A static focal point (light blue) is rendered at 20 cm above the centre of the board, so that the user can feel it on the palm. Only the focal point intensity is varied throughout the experiment.

Fig. 3 :

 3 Fig. 3: Left: line pattern aligned with the device y-axis (α = 0 • ). Right: line pattern aligned with the device x-axis (α = 90 • ). Using STM, a high-frequency focal point (red dot) is rapidly moved along a path (light blue segment). The resulting sensation is a continuous pattern in the shape of the focal point path. The perceived intensity of the pattern can be modulated by changing the focal point intensity.

Fig. 4 :

 4 Fig. 4: Mean subject detection (yellow) and identification (blue) thresholds for lines aligned with the x-axis (right) and y-axis (left). Medians are dark blue, means are in red.

Fig. 5 :

 5 Fig. 5: Mean confusion matrices for each condition. Hit rates are highlighted in green. The most prevalent confusions are highlighted in orange, the second most prevalent in yellow.

Fig. 7 :

 7 Fig. 7: Correct identification rates grouped by shape, in the 90 • (blue) and 0 • (green) conditions. Medians are shown in blue and means in red.


D Gaudrie

R Le Riche

V Picheny

B Enaux

V Herbert

Targeting Well-Balanced Solutions in Multi-Objective Bayesian Optimization under a Restricted Budget

Keywords: Gaussian Processes, Parsimonious Optimization, Computer Experiments, Preference-Based Optimization

Multi-objective optimization aims at finding trade-off solutions to conflicting objectives. These constitute the Pareto optimal set. In the context of expensive-to-evaluate functions, it is impossible and often non-informative to look for the entire set. As an end-user would typically prefer solutions with equilibrated trade-offs between the objectives, we define a Pareto front center. We then modify the Bayesian multi-objective optimization algorithm which uses Gaussian Processes to maximize the expected hypervolume improvement, to restrict the search to the Pareto front center. The cumulated effects of the Gaussian Processes and the center targeting strategy lead to a particularly efficient convergence to a critical part of the Pareto set.

Introduction

Multi-objective optimization aims at minimizing m objectives simultaneously: min

x∈X⊂R d
(f 1 (x), .., f m (x)). As these objectives are generally competing, optimal trade-off solutions x * 1 , .., x * q known as the Pareto optimal set P X are sought. These solutions are non-dominated (ND): it is not possible to improve one objective without worsening another (∀i = 1, .., q, z x * i ). The image of P X in the objective space is called the Pareto front, P Y = {f (x), x ∈ P X }. The Ideal and the Nadir points bound the Pareto front and are defined respectively as I = ( min Multi-objective optimization algorithms aim at constructing the best approximation to P Y , called the empirical Pareto front P Y which is made of nondominated observations. At the end of the search, P Y is delivered to a Decision Maker (DM) who will choose the solution he/she prefers.

However, when dealing with expensive computer codes, only a few designs x can be evaluated. In Bayesian optimization, a surrogate is built for each objective upon all past evaluations using Gaussian Process (GP) regression [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF].

Information given by these metamodels is used in order to sequentially evaluate new promising inputs with the aim of reaching the Pareto front. As the latter encompasses a large number of solutions when many objectives are considered, it may be impossible to compute an accurate approximation within the restricted computational budget. That approximation being used afterwards by a DM, it may anyway be irrelevant to provide the whole Pareto front because it will contain many uninteresting solutions from the DM's point of view.

Without additional knowledge about the preferences of the DM, we argue that "well-balanced" solutions, in a sense that will be defined hereafter as the central part of the Pareto front, are the most interesting ones. Therefore, we show how to estimate the center of the Pareto front and how classical infill criteria used in Bayesian optimization can be tailored to intensify the search towards it.

Center of the Pareto Front: definition and estimation

Definition. We define the center, C, of the Pareto front as the projection (intersection in case of a continuous front) of the closest ND point on the Ideal-Nadir line (in the Euclidean objective space). An example of Pareto front center can be seen in Fig. 1. This center corresponds visually to an equilibrium among all objectives. Alternative definitions involving e.g. the barycenter of the Pareto front, are likely to be harder to calculate in high-dimensional spaces. Furthermore, this center has the property of being insensitive to a linear scaling of the objectives in a bi-objective case Estimation. As the Ideal and the Nadir of the empirical Pareto front will sometimes be weak substitutes for the real ones (leading to a biased estimated center), those two points have to be truly estimated for the purpose of computing the center. The probabilistic nature of the metamodels (GPs) allows to simulate possible responses of the objective functions. Conditional GP simulations are thus performed to create possible Pareto fronts, each of which defines a sample for I and N. The estimated Ideal and Nadir are the medians of the samples. The intersection between the line L joining those points and the empirical Pareto front (or the projection if there is no intersection) is the estimated center C.

Targeting Infill Criteria for Bayesian Optimization

Articulating preferences has already been addressed in evolutionary multi-objective optimization [START_REF] Bechikh | Chap. 4: Preference incorporation in evolutionary multiobjective optimization: A survey of the state-of-the-art[END_REF]. In Bayesian multi-objective optimization, fitted to costly objectives, new points are sequentially added using an infill criterion whose purpose is to guide the search towards the Pareto front. After having been evaluated, that point is used to update the metamodel. The Expected Hypervolume Improvement (EHI, [START_REF] Emmerich | Hypervolume-based expected improvement: Monotonicity properties and exact computation[END_REF]) is a commonly used multi-objective infill criterion. EHI chooses the input x that maximizes the expected growth of the hypervolume dominated by P Y up to a reference point R. Classically, R is taken beyond the observed Nadir, e.g. [START_REF] Ponweiser | Multiobjective optimization on a limited budget of evaluations using model-assisted s-metric selection[END_REF], in order to cover the entire front. WHI [START_REF] Brockhoff | Directed multiobjective optimization based on the weighted hypervolume indicator[END_REF] is a variant of EHI for targeting particular regions of the objective space through a user-defined weighting function.

Our approach targets the central part of the Pareto front with EHI by solely controlling the reference point R. Indeed, the choice of R is instrumental in deciding the combination of objectives for which Improvement occurs: I R = {y ∈ Y : y R}. Positioning R at the (estimated) center of the Pareto front C will favour EHI's search at objective vectors belonging to I C .

When the objectives are modeled by independent GPs and the used reference point R P Y , one has EHI(•; R) = mEI(•; R), where mEI stands for the product of the famous mono-objective Expected Improvement (EI) [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF] considering R as the observed minimum in each objective, mEI(

x; R) = m i=1 EI i (x; R i ).
This observation is particularly appealing from a computational point of view: EHI requires the computation of m-dimensional hypervolumes involving expensive Monte-Carlo estimations in a many-objectives case. For this reason, in our algorithm where R ≡ C is not dominated, EHI is replaced by mEI.

Targeting a particular (here, central) part of the Pareto front leads to a fast local convergence. Once R is on the real Pareto front, the algorithm will try to improve non-improvable values (see left of Fig. 2). To avoid wasting costly evaluations, the convergence has to be checked. To this aim, we estimate p(y), the probability of dominating y ∈ Y , simulating Pareto fronts through conditional GPs. p(y)(1 -p(y)) is a measure of domination uncertainty, which tends to 0 as p(y) tends to 0 or 1. We assume local convergence when the line-uncertainty, L p(y)(1 -p(y))dy, is small enough. The complete algorithm, called C-EHI, is summarized below. We apply the proposed methodology to a benchmark built from real-world airfoil aerodynamic data. Fig. 2 shows that, compared with standard techniques, the proposed methodology leads to a faster and a more precise convergence to the central part of the Pareto front at the cost of a narrower covering of the front.

The results are shown at the iteration which triggers the convergence criterion: only marginal gains would indeed be obtained continuing targeting the same region. Further work will consider the continuation of the search to a broader but central part of the Pareto front. That newly targeted region, controlled again through R, will be as large as possible while being still attainable within the remaining budget.

y∈P Y y 1

 1 , .., min y∈P Y y m ) and N = ( max y∈P Y y 1 , .., max y∈P Y y m ).

  4 . C is also very little sensitive to perturbations of the Ideal or the Nadir point: under mild regularity conditions on the Pareto front, | ∂Ci ∂Ij | and | ∂Ci ∂Nj | < 1, i, j = 1, . . . , m.

Fig. 1 Fig. 1 .

 11 Fig. 1. Reference points R successively used for directing the search. They lie close to the dashed Ideal-Nadir line (IN) and lead the algorithm to the center of the Pareto front (C).

Fig. 2 .

 2 Fig. 2. Two objectives optimization with C-EHI (left). The initial approximation (black) has mainly been improved around the center. Compared with a standard EHI (right), the proposed methodology achieves convergence to the central part of the front. EHI considers more compromises between objectives, but cannot converge within the given budget (46 evaluations).

Non-sensitivity to a linear scaling of the objectives is true when the Pareto front intersects the Ideal-Nadir line. Without intersection, exceptions may occur for m ≥ 3.