Sarah Petit

Sigismund T A G Melissen

Moulay Loraine Duclaux

Tahar Sougrati

Tangui Le Bahers

Philippe Sautet

Damien Dambournet

Olaf Borkiewicz

Christel Laberty-Robert

Olivier Durupthy

Moulay T Sougrati

How Should Iron and Titanium be Combined in Oxides to Improve Photoelectrochemical Properties?

Keywords: pseudo-brookite, photoanode, water splitting, heterojunctions, "chimie-douce"

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

One of the grand challenges for the 21 st century is to secure the availability of energy on demand on the terawatt scale. Moreover, environmental concerns result in the need for renewable energy sources to satisfy this demand. The direct photoelectrochemical (PEC) conversion of solar energy into storable fuels, which is based on cheap and earth-abundant semiconductors and catalysts, has the potential to satisfy these requirements. [START_REF] Abdi | Efficient Solar Water Splitting by Enhanced Charge Separation in a Bismuth Vanadate-Silicon Tandem Photoelectrode[END_REF][START_REF] Joya | Water-Splitting Catalysis and Solar Fuel Devices: Artificial Leaves on the Move[END_REF][START_REF] Abdi | Efficient Bivo4 Thin Film Photoanodes Modified with Cobalt Phosphate Catalyst and W-Doping[END_REF][START_REF] Gratzel | Photoelectrochemical Cells[END_REF][START_REF] Mcevoy | Water-Splitting Chemistry of Photosystem Ii[END_REF][START_REF] Nocera | Personalized Energy: The Home as a Solar Power Station and Solar Gas Station[END_REF] Metal-oxide semiconductors are particularly appealing candidates for practical applications because of their low cost, nontoxicity, abundance, and stability toward corrosion.

For application in a water splitting device, a semiconductor must fulfill several requirements based on the microscopic steps involved in the working principle. First, the semiconductor must absorb sunlight efficiently to create an exciton. This property is governed by the bandgap (Eg) and the absorption coefficient (α). Then, the exciton is dissociated yielding a free electron and a free hole. Spontaneous dissociation is facile if the exciton binding energy (Eb) in the material is lower than the thermal energy at room temperature (25 meV). These free charges must diffuse toward the active sites for catalysis. The diffusion of charge carriers is characterized by the electron and hole mobilities (μ), which are themselves related to the effective masses of the charge carriers (m*). Then, the valence band (VB) and conduction band (CB) must be positioned such, that they straddle the redox potentials of the target reactions. [START_REF] Sivula | Metal Oxide Photoelectrodes for Solar Fuel Production, Surface Traps, and Catalysis[END_REF][START_REF] Hamann | Splitting Water with Rust: Hematite Photoelectrochemistry[END_REF] If the active sites are on another material, the band positions must be adapted to allow the charge transfer between the two systems thermodynamically. The efficiency of this final step is not only governed by bulk properties of the semiconductor, but also by the electronic structure of the interface. In addition to these requirements on the microscopic properties of the bulk or the interface, other more general constraints must be satisfied, such as the chemical stability in working conditions and the low cost of the materials.

Among the different materials already tested, [START_REF] Gan | Towards Highly Efficient Photoanodes: Boosting Sunlight-Driven Semiconductor Nanomaterials for Water Oxidation[END_REF] TiO2 and Fe2O3 have attracted a lot of attention in PEC water splitting applications, since they fulfill several of these requirements. [START_REF] Sivula | Solar Water Splitting: Progress Using Hematite (-Fe2o3) Photoelectrodes[END_REF][START_REF] Barroso | Charge Carrier Trapping, Recombination and Transfer in Hematite Water Splitting Photoanodes[END_REF][START_REF] Ginley | Photoelectrolysis of Water Using Iron Titanate Anodes[END_REF] TiO2 has been the first material tested for water oxidation. [START_REF] Fujishima | Electrochemical Photolysis of Water at a Semiconductor Electrode[END_REF][START_REF] Cowan | Water Splitting by Nanocrystalline Tio2 in a Complete Photoelectrochemical Cell Exhibits Efficiencies Limited by Charge Recombination[END_REF] The TiO2 absorption edge is 3.2 eV, allowing only 4% of the incident solar energy to be absorbed at best. [START_REF] Cowan | Water Splitting by Nanocrystalline Tio2 in a Complete Photoelectrochemical Cell Exhibits Efficiencies Limited by Charge Recombination[END_REF][START_REF] Dholam | Hydrogen Production by Photocatalytic Water-Splitting Using Cr-or Fe-Doped Tio2 Composite Thin Films Photocatalyst[END_REF][START_REF] Tang | Mechanism of Photocatalytic Water Splitting in Tio2. Reaction of Water with Photoholes, Importance of Charge Carrier Dynamics, and Evidence for Four-Hole Chemistry[END_REF] Its CB lies slightly above the hydrogen evolution redox potential, potentially decreasing the onset potential. The electron mobility of TiO2 in a nanoparticle (NP) film is about 0.01 cm 2 •V -1 •sec -1 , which is low compared to the mobility observed in monocrystals of 1 cm 2 •V -1 •sec -1 . [START_REF] Stamate | Dielectric Properties of Tio2 Thin Films Deposited by a Dc Magnetron Sputtering System[END_REF] Accordingly, TiO2 suffers from two major limitations which compromise high photon-to-hydrogen conversion efficiency: its wide bandgap and its fast electron-hole recombination. Lately, hematite has emerged as a more promising material due to its suitable bandgap of 2.1 eV, which corresponds to a maximum theoretical air mass 1.5 global (AM 1.5 G) photocurrent density of 12.3 mA•cm - 2 . [START_REF] Formal | Back Electron-Hole Recombination in Hematite Photoanodes for Water Splitting[END_REF] Hematite also has an excellent chemical stability in a broad pH range and is naturally abundant, nontoxic and cheap. The hematite conduction band minimum (CBM) and valence band maximum (VBM) are located at -0.3 V and 1.7 V vs. reversible hydrogen electrode (RHE) at pH = 0, respectively. [START_REF] Hamann | Splitting Water with Rust: Hematite Photoelectrochemistry[END_REF][START_REF] Sivula | Solar Water Splitting: Progress Using Hematite (-Fe2o3) Photoelectrodes[END_REF] However, the reported water oxidation efficiencies for hematite to date are notoriously lower than the predicted maximum value despite intense efforts. [START_REF] Formal | Back Electron-Hole Recombination in Hematite Photoanodes for Water Splitting[END_REF] This is caused by two principal limiting physical properties: the short lifetime of the photogenerated charge carriers (≤ 10 ps), and the low mobility of the minority carriers (0.2 cm 2 •V -1 •sec -1 ). [START_REF] Formal | Back Electron-Hole Recombination in Hematite Photoanodes for Water Splitting[END_REF] The combined effect is that the minority carrier (hole) diffusion length is only 2-4 nm, whereas full absorption of the incident light requires much thicker films due to the indirect nature of the bandgap (the penetration depth of light is ~118 nm at a wavelength of 550 nm). [START_REF] Formal | Back Electron-Hole Recombination in Hematite Photoanodes for Water Splitting[END_REF] This short literature overview underlines that no single material, whether Fe2O3 or TiO2, fulfilled all the criteria for efficient water oxidation. To overcome these intrinsic shortcomings, different approaches have been explored in the literature. For example, the modification of the TiO2 surface with a small bandgap semiconductor has been proven to enhance the photocatalytic activity. [START_REF] Yu | Preparation, Characterization and Photocatalytic Activity of in Situ Fe-Doped Tio2 Thin Films[END_REF][START_REF] Yu | Pore-Wall Chemistry and Photocatalytic Activity of Mesoporous Titania Molecular Sieve Films[END_REF] Another approach consists of tuning the optical and photoelectrochemical (PEC)

properties by element doping. [START_REF] Saremi-Yarahmadi | Nanostructured Ahem Electrodes for Solar Driven Water Splitting: Effect of Doping Agents on Preparation and Performance[END_REF] For example, ferric doping of TiO2 lowers the bandgap from 3.2 to 1.9 eV, depending on the Fe content and the synthesis procedure. [START_REF] Santos | Iron Insertion and Hematite Segregation on Fe-Doped Tio2 Nanoparticles Obtained from Sol-Gel and Hydrothermal Methods[END_REF][START_REF] Zhao | Direct Microwave Hydrothermal Synthesis of Fe-Doped Titania with Extended Visible-Light Response and Enhanced H2-Production Performance[END_REF] However, high doping levels often lead to heterogeneous materials. [START_REF] Zhang | Preparation of Photocatalytic Fe2o3-Tio2 Coatings in One Step by Metal Organic Chemical Vapor Deposition[END_REF][START_REF] Peng | Synthesis, Photoelectric Properties and Photocatalytic Activity of the Fe2o3/Tio2 Heterogeneous Photocatalysts[END_REF] Their functionality is then the result of both the bulk properties of the various components and their interface properties. We note that some reports in the literature highlight the benefit of designing heterojunctions to enhance the charge separation. [START_REF] Peng | Synthesis, Photoelectric Properties and Photocatalytic Activity of the Fe2o3/Tio2 Heterogeneous Photocatalysts[END_REF][START_REF] Courtin | New Fe2tio5-Based Nanoheterostructured Mesoporous Photoanodes with Improved Visible Light Photoresponses[END_REF][START_REF] Moniz | Fe2o3-Tio2 Nanocomposites for Enhanced Charge Separation and Photocatalytic Activity[END_REF][START_REF] Ahmed | A Facile Surface Passivation of Hematite Photoanodes with Tio2 Overlayers for Efficient Solar Water Splitting[END_REF] However, none of them studied together the impact of doping level, atoms ordering and heterostructures formation in the Ti-Fe-O system get high photo(electrochemical) efficiency.

All in all, good photoelectrode material must contain i) a bulk semi-conductor structure that efficiently harvests photons and allows their conversion into mobile charge carriers, titanium oxide structures being particularly efficient in that case; ii) appropriate surface catalytic sites, iron (III) being a priori better suited. The issue that is still to be answered is the elements mixing level coupled with the doping level that provide the best results. In other words, can Fe2TiO5 alone fulfill those conditions, is a low iron doping of titanium oxide enough or is it mandatory to combine metal oxide nanoparticles with completely different chemical compositions?

The goals of this article are to provide original syntheses of Ti-Fe-O nanomaterials with the iron and the titanium atoms combined in the same structure or into heterostructurated nanocomposites and to correlate the structural aspects of the materials used as photoanode with their efficiency for water oxidation. The huge versatility of aqueous sol-gel chemistry and more precisely that of iron and titanium metal ions is the perfect toolbox to prepare a wide range of photoelectrode materials with tailored Ti/Fe ratio and element vicinity. We have prepared single phase TiO2anatase (TiO2-A) doped with various amounts of ferric ions, Fe2TiO5 nanoparticles featuring two structural arrangements, i.e. a heterostructured system. We have also mixed phase pure NPs of TiO2, Fe2TiO5 and Fe2O3 hematite or maghemite to test their efficiency as photoelectrode material.

The PEC activities of the different pure phase and heterostructured systems were recorded and rationalized according to the known relative position of electronic bands and the charge dynamics in the different components. Additionally, an extensively tested theoretical approach 29- 33 was invoked to provide key data on the bulk properties of the different pure semi-conductors.

This study is the first complete study on the PEC activity of the whole Fe-Ti oxide composition range.

The manuscript is organized as follows. The methodology section is dedicated to the presentation of the experimental and computational details. In the first part of the results, the Density Functional Theory (DFT) calculations of the main TiO2, Fe2O3 and Fe2TiO5 phases will be presented in order to understand the ability of these materials in water splitting devices.

Subsequently, structural and morphological characterizations of the different synthesized systems are presented. The PEC activities are then determined for each material. Finally, in a discussion confronting both theoretical and experimental results we explain the relative efficiency of the different systems.

Methodology

Experimental setup

Nanoparticle syntheses

Preliminary studies have been performed with all the combinations of Fe(II)/(III) and Ti(III)/(IV) precursors in order to form mixed oxides. The best results were achieved with Fe (III) salts and Ti (IV) precursors.

The combination of Fe(III) and Ti(IV) oxides was created according to two different synthetic pathways. First, a crystallite structure involving both metal ions was obtained from the sol-gel reaction of the corresponding salts in aqueous solution. Alternatively, the composites were obtained through the grinding of a mixture of pure nanoparticles of TiO2, Fe2TiO5, α-Fe2O3 and γ-Fe2O3. The initial particles were synthesized through a microwave assisted process described below.

TiO2 anatase. TiO2 particles were obtained through the hydrothermal treatment of an aqueous TiCl4 solution with a pH set to 6 according to a previously described method. [START_REF] Pottier | Size Tailoring of Tio2 Anatase Nanoparticles in Aqueous Medium and Synthesis of Nanocomposites. Characterization by Raman Spectroscopy[END_REF] A stock solution with Ti(IV) ion concentration of 1.0 mol•L -1 was prepared by dilution of TiCl4 in HCl (3 mol•L - 1 ) solution. Then, 100 mL of the stock solution were introduced in 200 mL of Milli-Q water, the pH was set to 6 by addition of sodium hydroxide solution ([NaOH] = 3 mol•L -1 ) and the sample volume was increased to 500 mL with Milli-Q water to obtain a Ti(IV) ion concentration of 0.2 mol•L -1 in the reacting medium. A white sol immediately appeared; aliquots of this suspension were transferred in Teflon cups and heated in autoclaves for 1 h at 200 °C using a microwave oven (Synthos3000, Anton Paar). The resulting precipitate was collected by centrifugation, washed twice with water, then with nitric acid ([HNO3] = 3 mol•L -1 ) and at least one additional time with water. The obtained powders were dried under dry air flow overnight.

Fe doped TiO2 anatase. The former protocol was slightly modified to introduce various amounts of Fe (III) in the final product. The total metal ion concentration was maintained at 0.2 mol•L 1 and the targeted Fe(III) atomic ratio was varied between 5 and 66 %. The appropriate amount of the Ti(IV) stock solution and solid ferric nitrate nonahydrate (Fe(NO3)3•9H2O) were introduced in 200 mL of Milli-Q water to form a clear yellow solution.

Sodium hydroxide solution ([NaOH] = 3 mol•L -1 ) was then added to adjust the pH to 2-6 and the sample volume was increased to 500 mL with Milli-Q water. A brownish precipitate is obtained, hydrothermally treated in the microwave oven and washed the same way as the pure TiO2 sample. The samples obtained with a pH of the reacting medium of 6 are denoted by acronyms indicating the added iron content in at.%, for example TiO2-10%Fe. For reacting media with other pH values, this is indicated at the end of the sample name.

Fe2TiO5. Fe2TiO5 nanoparticles used for the mixing were obtained according to the former protocol with a well-defined set of parameters. The metal precursors were introduced with a relative Ti:Fe atomic ratio of 1:1 and the pH of the reactive medium was fixed at 6 with NaOH.

α-Fe2O3 Hematite. The preparation protocol is similar to that of iron doped titania particles, except that Fe(NO3)3 is the only metal ion source used and the pH of the reacting medium was not fixed with NaOH solution. The hydrolysis of the ferric precursor acidified the aqueous solution to a pH of around 2. A reddish precipitate was obtained.

γ-Fe2O3 Maghemite. The particles were synthesized according to an earlier published method using microwave assisted heating in order to reduce the aging time. [START_REF] Sreeja | Microwave-Hydrothermal Synthesis of -Fe2o3 Nanoparticles and Their Magnetic Properties[END_REF] First, 50 mL of a 0.05 mol•L -1 ferrous solution prepared from FeSO4•7H2O and 100 mL of a 0.05 mol•L -1 ferric solution prepared from FeCl3•6H2O were mixed together. Then, the pH of the reacting medium was maintained at 12 by adding small amounts of a NaOH solution. The obtained colloidal solution was heated in a microwave oven for 25 min at 150 °C. Finally, the dark brown precipitate was washed twice with water and dried under air flow.

Nanoparticle Characterizations

UV-visible diffuse reflectance spectra (DRS) of the samples were recorded on a Varian-Cary 5000 spectrometer with integrating sphere from 300 to 800 nm. Powder X-ray diffraction (XRD) measurements were performed with a Brücker D8 X-ray diffractometer operating in the Bragg-Brentano reflection mode using Cu K radiation. The data were collected in the 20 -80° 2 range with 0.02° steps and a counting time of 5 s per step. Transmission electron micrographs (TEMs) were obtained using a Tecnai spirit G2 apparatus operating at 120 kV. Samples were prepared by evaporating dilute suspensions in ethanol onto carbon-coated grids. Relative iron and titanium atomic composition of the samples was determined with a scanning electron microscope (Hitachi S-3400N) equipped with energy-dispersive X-ray spectroscopy (EDX, Oxford Instruments -X-max).

Total scattering data were collected at the 11-ID-B beamline at the Advanced Photon Source at Argonne National Laboratory, using high energy X-rays (λ = 0.2128 Å) to high values of momentum transfer Qmax = 22 Å -1 . [START_REF] Chupas | Applications of an Amorphous Silicon-Based Area Detector for High-Resolution, High-Sensitivity and Fast Time-Resolved Pair Distribution Function Measurements[END_REF][START_REF] Chupas | Rapid-Acquisition Pair Distribution Function (Ra-Pdf) Analysis[END_REF] One-dimensional diffraction data were obtained by integrating the raw 2D total scattering data in Fit2D. [START_REF] Hammersley | Two-Dimensional Detector Software: From Real Detector to Idealised Image or Two-Theta Scan[END_REF] Pair distribution functions (PDFs) G(r) were extracted from the background and Compton scattering corrected data following Fourier transform within PDFgetX2. [START_REF] Qiu | Pdfgetx2: A Gui-Driven Program to Obtain the Pair Distribution Function from X-Ray Powder Diffraction Data[END_REF] The PDFs were subsequently modeled using PDFgui. [START_REF] Farrow | Pdffit2 and Pdfgui: Computer Programs for Studying Nanostructure in Crystals[END_REF]

Film preparation

As photoanode, we synthesized 30 m mesoporous films (cf. Figure SI-1) by tape casting. 37 wt.% of the inorganic powders was dispersed in ethanol with 5 wt.% of ethylene glycol and 0.5 wt.% of Triton X. The ethanol suspensions were mixed in an ultrasonic bath for ½ h before deposition. The ethanol suspensions were then deposited directly onto a fluoride tin oxide (FTO) substrate to yield homogeneous thick films. The substrates used are TEC-15 FTO-coated glass (15/sq, Asahi, Co). After the deposition, the thick-films were annealed for 3 min. at 300 °C in air to decompose the organic material and to further improve the particle-particle contact.

Photo-electrochemical characterizations

PEC characterization was carried out in an aqueous solution 1 mol•L -1 NaOH (pH = 14). The solution was purged with nitrogen prior to and during the measurements to remove any dissolved oxygen. The working area of the electrodes exposed to the electrolyte was 28.5 mm 2 for all samples. The potential of the working electrode was controlled by potentiostat (Modulab, Ametek). In three-electrode measurements, a Pt wire and an Ag/AgCl electrode (Asashi, saturated AgCl solution) were used as the counter and reference electrodes, respectively. Cyclic voltammetry measurements were performed with a scan rate of 20 mV•s -1 . Light photocurrent measurements were performed using the visible range of a 300 W Xe-Lamp (Oriel and a UV filter with a 400 nm wavelength cutoff) providing a power density of 300 mW.cm -2 at the surface of the electrode. Electrical contact to the sample was made using a copper tape. For the cyclic voltammograms represented below, the potentials are referenced to RHE according to the following equation:

E (V/RHE) = E (V/Ag/AgCl) + E°Ag/AgCl + 0.059 pH.
(1)

Computational details

Density Functional Theory (DFT) Calculations

Geometry optimizations and frequency calculations were performed using the CRYSTAL14 code, due to its efficiency when doing calculations with hybrid functionals using Gaussian Type Orbitals. [START_REF] Dovesi | Crystal14: A Program for the Ab Initio Investigation of Crystalline Solids[END_REF][START_REF] Dovesi | Crystal14 User's Manual[END_REF] All optimized structures were characterized as minima on the potential energy surface by vibrational analysis. The all-electron 86-411G(2d) basis sets for Fe [START_REF] Catti | Theoretical Study of Electronic, Magnetic, and Structural Properties of Alpha -Fe2o3 (Hematite)[END_REF] and Ti [START_REF] Sophia | First-Principles Study of the Mechanisms of the Pressure-Induced Dielectric Anomalies in Ferroelectric Perovskites[END_REF] and an 8-41G(d) basis set for O were used. [START_REF] Sophia | First-Principles Study of the Mechanisms of the Pressure-Induced Dielectric Anomalies in Ferroelectric Perovskites[END_REF] All calculations, except for electronic structure calculations, were performed using the hybrid PBE0 functional, [START_REF] Adamo | Toward Reliable Density Functional Methods without Adjustable Parameters: The Pbe0 Model[END_REF] using an 884 K-point mesh [START_REF] Monkhorst | Special Points for Brillouin-Zone Integrations[END_REF] for Fe2TiO5 and a 121212 mesh for α-Fe2O3. Electronic structure calculations were performed with a 10106 mesh for Fe2TiO5 and a 242424 mesh for α-Fe2O3 and using the range-separated HSE06 functional [START_REF] Heyd | Hybrid Functionals Based on a Screened Coulomb Potential[END_REF][START_REF] Heyd | Erratum: Hybrid Functionals Based on a Screened Coulomb Potential[END_REF] with ω = 0.11 a0 -1 , [START_REF] Krukau | Influence of the Exchange Screening Parameter on the Performance of Screened Hybrid Functionals[END_REF] based on its positive comparison in terms of speed and accuracy with other hybrid functionals found here and reported earlier. [START_REF] Barone | Accurate Prediction of the Electronic Properties of Low-Dimensional Graphene Derivatives Using a Screened Hybrid Density Functional[END_REF][START_REF] Henderson | Accurate Treatment of Solids with the Hse Screened Hybrid[END_REF][START_REF] Bahers | Semiconductors Used in Photovoltaic and Photocatalytic Devices: Assessing Fundamental Properties from Dft[END_REF] (HSE06 reduces to PBE0 when setting ω to zero.) The dielectric constant εr is the sum of a vibrational contribution εvib and an optical contribution ε∞. The first term εvib was computed invoking the harmonic approximation to the lattice potential, with infrared intensities calculated using the Berry phase method. [START_REF] Noel | Polarization Properties of Zno and Beo: An Ab Initio Study through the Berry Phase and Wannier Functions Approaches[END_REF] The second term was computed using the Coupled-Perturbed Kohn-Sham (CPKS, although sometimes the acronym CPHF is used) approach. [START_REF] Ferrero | Coupled Perturbed Hartree-Fock for Periodic Systems: The Role of Symmetry and Related Computational Aspects[END_REF][START_REF] Ferrero | The Calculation of Static Polarizabilities of 1-3d Periodic Compounds. The Implementation in the Crystal Code[END_REF][START_REF] Ferrero | The Calculation of Static Polarizabilities of 1-3d Periodic Compounds. The Implementation in the Crystal Code[END_REF] The individual terms were calculated by taking the geometric mean of the diagonal elements of the (diagonalized) dielectric tensors. 57

Bulk properties calculations

The bulk properties computed to understand the semiconductors' capacities for photocatalytic reactions are the bandgap (noted Eg), the dielectric constant (noted εr), the effective masses (noted m*) and the exciton binding energy (noted Eb). The way to compute all these properties were presented in the following article [START_REF] Bahers | Semiconductors Used in Photovoltaic and Photocatalytic Devices: Assessing Fundamental Properties from Dft[END_REF] and used to compute properties of several families of semiconductors including oxides, sulfides, nitrides and halides. It can just be reminded that first the total dielectric constant (εr) is the sum of the electronic (ε∞) and the vibrational contribution (εvib) and then the exciton binding energy is computed in the framework of the Wannier model using the average values of m* and εr. This model is generally well adapted for delocalized excitons (large dielectric constant and low effective masses), which is usually the case for inorganic semiconductors.

Results and discussion

Bulk properties of the semiconductors

The materials used for the work presented in this manuscript fulfill several requirements of the bulk semiconductors properties presented in the introduction. This verification has been performed using the DFT calculations presented here.

Geometry optimizations. α-Fe2O3 (crystallographic cell in Figure 1a) was optimized in the 𝑅3 ̅ 𝑐 geometry as given by Adelstein et al., [START_REF] Adelstein | Density Functional Theory Based Calculation of Small-Polaron Mobility in Hematite[END_REF] using the conventional Anti-Ferromagnetic (AFM) spin configuration, reducing the symmetry to 𝑅3 ̅ . 59 TiO2 (crystallographic cell in Figure 1b) was optimized in the anatase structure at the same level of theory as in a previous publication. [START_REF] Bahers | Semiconductors Used in Photovoltaic and Photocatalytic Devices: Assessing Fundamental Properties from Dft[END_REF] For Fe2TiO5 (space group: Ccmm), the orthorhombic pseudo-brookite structure with ordered distribution of Fe was chosen and several spin isomers were investigated, since the exact spin state of the Fe(III) atoms is unknown. Figure 1c presents the structure of the most stable spin state found for Fe2TiO5. This is an anti-ferromagnetic spin state (AFM). The calculations revealed that several spin states (all AFM) are close in energy (see supporting information for more details). The ferromagnetic structure was significantly less stable (~1.1 eV/unit cell) than the AFM ones. The cell parameters of all the semiconductors are presented in Table 1. The larger discrepancy on a cell parameter between DFT and experiment is around 1.05%, which is the standard discrepancy at this level of theory. Electronic properties. The density of states (DOS) of the compounds are presented on Figure 2.

It appears that the valence bands is largely made up of the O atoms' orbitals while the conduction bands originate from Fe orbitals for α-Fe2O3 and Fe2TiO5 and from Ti atoms for TiO2. All relevant computed properties for photocatalysis are presented in Table 2. While the bandgap computed for TiO2 is relatively close to the experimental one, the Eg computed for Fe-containing semiconductors is overestimated. [START_REF] Pozun | Hybrid Density Functional Theory Band Structure Engineering in Hematite[END_REF] Unfortunately, this is the expected behavior when hybrid functionals are applied to materials having partially filled d-orbitals. A GGA+U approach [START_REF] Dudarev | Electron-Energy-Loss Spectra and the Structural Stability of Nickel Oxide: An Lsda+U Study[END_REF] could solve this problem, but GGA functionals are unreliable for the calculation of other properties explaining the choice for hybrid functionals as a compromise. About the effective masses, for α-Fe2O3, values for me of ~3.5 were found, whereas for mh values of ~1 were found, consistent with findings in the literature [START_REF] Huda | Electronic, Structural, and Magnetic Effects of 3d Transition Metals in Hematite[END_REF][START_REF] Tang | Titanium and Magnesium Co-Alloyed Hematite Thin Films for Photoelectrochemical Water Splitting[END_REF] that attribute the heavy electron mass to the strong localization of the VB on O, 10 confirming our approach. The anisotropy of the charge carrier mobilities in α-Fe2O3 can be explained by Hund's rule, [START_REF] Sivula | Solar Water Splitting: Progress Using Hematite (-Fe2o3) Photoelectrodes[END_REF] Table 2: DFT computed electronic properties α-Fe2O3, TiO2, and Fe2TiO5.

These calculations and the experimental data available clearly highlight their interest for this work. The strength of TiO2 is in its very good electronic properties, except for the bandgap.

Contrary to TiO2, hematite has a very adapted bandgap but is unadapted in terms of the other electronic properties. Finally, Fe2TiO5 has intermediate properties, inheriting the well-adapted bandgap from α-Fe2O3 and the good electronic properties from TiO2. For these reasons, it is interesting to study experimentally the PEC properties of each material individually as well as in combination with the others.

Experimental Results

Nanoparticle characterizations

Aqueous solution chemistry of Ti(IV) and Fe(III) metal ions is now rather well understood when studied separately. [START_REF] Cassaignon | Selective Synthesis of Brookite, Anatase and Rutile Nanoparticles: Thermolysis of Ticl4 in Aqueous Nitric Acid[END_REF][START_REF] Cassaignon | From Ticl3 to Tio2 Nanoparticles (Anatase, Brookite and Rutile): Thermohydrolysis and Oxidation in Aqueous Medium[END_REF][START_REF] Pottier | Size Tailoring of Tio2 Anatase Nanoparticles in Aqueous Medium and Synthesis of Nanocomposites. Characterization by Raman Spectroscopy[END_REF][START_REF] Jolivet | Iron Oxide Chemistry. From Molecular Clusters to Extended Solid Networks[END_REF][START_REF] Pottier | Synthesis of Brookite Tio2 Nanoparticles by Thermolysis of Ticl4 in Strongly Acidic Aqueous Media[END_REF][START_REF] Vayssieres | Size Tailoring of Magnetite Particles Formed by Aqueous Precipitation: An Example of Thermodynamic Stability of Nanometric Oxide Particles[END_REF][START_REF] Jolivet | Iron Oxides: From Molecular Clusters to Solid. A Nice Example of Chemical Versatility[END_REF][START_REF] Li | Synthesis, Properties, and Environmental Applications of Nanoscale Iron-Based Materials: A Review[END_REF] Much less is known about the ternary Ti-Fe-O diagram when dealing with nanoparticle synthesis in water. [START_REF] Adan | Structure and Activity of Nanosized Iron-Doped Anatase Tio2 Catalysts for Phenol Photocatalytic Degradation[END_REF][START_REF] Ambrus | Synthesis, Structure and Photocatalytic Properties of Fe(Iii)-Doped Tio(2) Prepared from Ticl(3)[END_REF][START_REF] Khaleel | Sol-Gel Synthesis, Characterization, and Catalytic Activity of Fe(Iii) Titanates. Colloids Surf[END_REF] In the 2-6 pH range, both metal ions first form an amorphous or very poorly organized hydroxide precipitate. Consequently, we used a subsequent microwave assisted hydrothermal treatment to enable the oxide crystallization. pH values strongly impact the nature of the stabilized phase. Particularly, low pH yields phase mixtures containing brookite and anatase polymorphs (cf. Table SI-1). In the following, we focused on samples prepared at pH 6 and monitored the structural dependence on the iron content. The obtained crystallized compounds were analyzed by XRD analysis. The evolution of the XRD patterns as a function of the Fe content is shown in Figure 3. X-ray diffraction analysis shows a strong structural dependence on the iron content as well as broad lines indicating the formation of nanosized particles. The TiO2 rutile structure was observed in none of the experiments, which is in good agreement with published resultsd. [START_REF] Dufour | Do Tio2 Nanoparticles Really Taste Better When Cooked in a Microwave Oven?[END_REF] For low iron contents, i.e. < 10%, only TiO2 anatase could be detected as crystalline phase. Increasing the Fe ratio renders the phase identification more complex owing to numerous phases occurring in the Ti-Fe-O system. Moreover, iron doping has shown to favor the brookite TiO2 polymorph at the expense of the anatase. [START_REF] Santos | Iron Insertion and Hematite Segregation on Fe-Doped Tio2 Nanoparticles Obtained from Sol-Gel and Hydrothermal Methods[END_REF][START_REF] Courtin | New Fe2tio5-Based Nanoheterostructured Mesoporous Photoanodes with Improved Visible Light Photoresponses[END_REF] For %Fe higher than 10, XRD patterns strongly evolve, indicating a phase transition which becomes more obvious in TiO2-33%Fe. A peak at around 18° (2) appears which can be assigned to orthorhombic Fe2TiO5 which has a pseudobrookite structure and eventually to a monoclinic polytype Fe2TiO5. [START_REF] Grey | Nonstoichiometric Li-Pseudobrookite(Ss) in the Li2o-Fe2o3-Tio2 System[END_REF][START_REF] Drofenik | A New Monoclinic Phase in the Fe2o3-Tio2 System .1. Structure Determination and Mossbauer-Spectroscopy[END_REF] Nevertheless, for the latter, the main line is absent from the XRD pattern. While increasing the Fe ratio to 50% seems to enable the preparation of a single phase of orthorhombic Fe2TiO5, the absence of the monoclinic variety cannot be completely ruled out. Moreover, pure Fe2TiO5 is expected in TiO2-66%Fe only. However, with that iron ratio, hematite is clearly detected by XRD.

Structural analyses were further conducted using synchrotron X-ray (λ = 0.2128 Å) scattering measurements from which we obtained the pair distribution function (PDF). PDFs are indeed particularly well adapted to study nanostructured materials as they provide a histogram of all the atom-atom (pair) distances within the sample independently of its crystalline state. [START_REF] Billinge | Beyond Crystallography: The Study of Disorder, Nanocrystallinity and Crystallographically Challenged Materials with Pair Distribution Functions[END_REF] The PDF of TiO2-50%Fe was fitted using different structural models of Fe2TiO5 (orthorhombic pseudo-brookite with random Fe distribution in the metal sites [START_REF] Guo | Crystal Structure and Cation Distributions in the Feti2o5-Fe2tio5 Solid Solution Series[END_REF] or ordered monoclinic structure [START_REF] Grey | Nonstoichiometric Li-Pseudobrookite(Ss) in the Li2o-Fe2o3-Tio2 System[END_REF] ). The fitting using solely the pseudo-brookite type structure largely failed in reproducing the PDF data. A good fit to the PDF data was only obtained using the two polymorphs of Fe2TiO5. The goodness of the fit could be further increased to Rw = 21.6 % [1-20 Å] by the addition of a third minor phase consisting of TiO2 anatase (Figure 4). The contributions of the orthorhombic and monoclinic phases of Fe2TiO5 to the PDF data (Figure 4) clearly indicate the co-existence of both phases. The structural parameters of Fe2TiO5 phases obtained by PDF refinement and their polyhedron representations are presented in Table 3 and Figure SI-2a respectively. The starting structural models of both Fe2TiO5 polytypes were based on structures resolved by Grey and Drofenik for the orthorhombic and monoclinic phases, respectively. [START_REF] Grey | Nonstoichiometric Li-Pseudobrookite(Ss) in the Li2o-Fe2o3-Tio2 System[END_REF][START_REF] Drofenik | A New Monoclinic Phase in the Fe2o3-Tio2 System .1. Structure Determination and Mossbauer-Spectroscopy[END_REF] The atomic occupancy was fixed to initial values due to the similar X-ray scattering intensity of iron and titanium atoms. The orthorhombic structure is composed of randomly distributed TiO6 and FeO6 octahedra featuring an edge-sharing configuration. In the monoclinic phase, iron and titanium occupy the 8f and 4a Wyckoff sites, respectively, forming a network with both edge and corner-sharing octahedra.

While unit cell parameters of the pseudo-brookite match those of literature data (within the estimated standard deviation), those of the monoclinic structure significantly deviate from the reference sample (cf. values in italics in Table 3) obtained by the high-temperature solution method as reported by Drofenik. [START_REF] Drofenik | A New Monoclinic Phase in the Fe2o3-Tio2 System .1. Structure Determination and Mossbauer-Spectroscopy[END_REF] The resulting changes in unit cell parameters explained the difficulty to detect this phase by conventional XRD since the most intense reflections were found to be superposed on the pseudo-brookite reflections. Moreover, the origin of the unit cell parameters variation might be a size effect and/or non-stoichiometry. Quantitative phase analysis deduced from PDF refinement indicates a total occurrence of 61-62% of Fe in contrary to the 50% used in the synthesis protocol, which indicates that (i) some titanium ions did not precipitate and that (ii) samples exhibit non-stoichiometric features. However, the chemical composition measurements determined with Inductively Coupled Plasma Mass Spectrometry (ICP-MS) and EDX on the different samples yielded values very close to the initial ratio with an error below 1 at.%. A recent study has shown that the pseudo-brookite phase may have more Ti than the exact composition Fe2TiO5, without significant conversion of ferric ions into ferrous ones. [START_REF] Seitz | Near the Ferric Pseudobrookite Composition (Fe2tio5)[END_REF] Indeed, the removal of Fe(III) that lower the 2 to 1 Fe/Ti stoichiometry can be compensated by oxygen defects. Mössbauer experiments reported in Figure SI-3 confirmed that only ferric ions are present in the material. More precisely, three sites are observed upon deconvolution of the spectrum that can be attributed to the 4c and 8f sites of the orthorhombic structure [START_REF] Courtin | New Fe2tio5-Based Nanoheterostructured Mesoporous Photoanodes with Improved Visible Light Photoresponses[END_REF][START_REF] Hirano | Direct Formation of Iron(Iii)-Doped Titanium Oxide (Anatase) by Thermal Hydrolysis and Its Structural Property[END_REF] and to the 8f site of the monoclinic one, in good agreement with PDF data. [START_REF] Drofenik | A New Monoclinic Phase in the Fe2o3-Tio2 System .1. Structure Determination and Mossbauer-Spectroscopy[END_REF] The diffractograms of phase pure TiO2 anatase, α-Fe2O3 (hematite) and γ-Fe2O3 (maghemite) are also reported in Figure 3 and all diffraction peaks could be indexed according to the corresponding reference cards (JCPDS 00-021-1272, 00-033-0664 and 01-039-1346 respectively). The γ-Fe2O3 maghemite diffractogram resembles that of the γ-Fe3O4 magnetite.

However, the brown color of the sample is in disagreement with a mixed iron oxide and all metal atoms are fully oxidized.

The presence of amorphous or very small crystalline nanoparticles was also studied through TEM. Selected micrographs corresponding to the phase pure samples and the TiO2-x%Fe series are reported in Figure 5. Micrographs of samples obtained at pH other than 6 are reported in

Figure SI-4. The first overview of the TEM images confirms that the particle size is about 20 nm except for hematite nanoparticle size that is closer to 100 nm. More precisely, the anatase particles with low iron content exhibit two slightly different morphologies: square bipyramids and small rods as already described in a previous article. [START_REF] Dufour | Morphological Control of Tio2 Anatase Nanoparticles: What Is the Good Surface Property to Obtain Efficient Photocatalysts?[END_REF] With increasing iron content, the TiO2 small rods population becomes predominant and the rods' aspect ratio also increases. HRTEM

and dark field images confirmed that there is almost no amorphous phase up to 10 at.% Fe. At 16.5 and 33 at.% Fe (figure 5d and5e), an amorphous or poorly crystalline phase is observed around crystalline anatase nanoparticles. In those two micrographs it is difficult to distinguish pseudo-brookite nanoparticles from anatase ones. The TiO2-50%Fe sample (figure 5f) seems to present two different nanoparticles morphologies with rods and spheres. However, lattice fringes analyses on HRTEM images reported in Figure SI-5 indicate that the two observed projections may correspond to the same 3D object: a thick disk with the small dimension being parallel to the (010) plane. The mean particle thickness is 10 nm and the mean diameter ~30 nm. Another explanation for the two morphologies may be the presence of two different Fe2TiO5 phases as observed in PDF analysis. No significant amorphous domains or small crystalline anatase nanoparticles could be detected on TEM micrographs. However, this does not imply that they are necessarily absent in the whole sample as they are detected in the PDF study. At even higher iron content (TiO2-66%Fe in Figure 5g) the pseudo-brookite nano-disks are thinner and larger. In addition, small quantities of large hematite rhombohedra with a length of about 300 nm can be observed (see Figure SI-6). The hematite nanoparticles synthesized as pure phase do not have that rhombohedral shape, but rather a round shape with sizes ranging from 50 to 300 nm. The maghemite nanoparticles also display a round shape without a clearly exposed family of facets and the particle size distribution is rather monodisperse at 10 nm in diameter.

Electrochemical characterization

Pure nanoparticles Fe-doped TiO2 anatase, Fe2TiO5 pseudo-brookite and a mechanical mixture of TiO2, and/or Fe2O3 and/or Fe2TiO5 were deposited onto FTO substrates to fabricate photoelectrodes. The synthesis procedure was described in section 2. increases the photocurrent mainly at low potential, as can be seen in Figure 6. This observation is consistent with the explanation that the photocurrent is limited by trap states present at the electrode/electrolyte interface. [START_REF] Formal | The Transient Photocurrent and Photovoltage Behavior of a Hematite Photoanode under Working Conditions and the Influence of Surface Treatments[END_REF] For unmodified TiO2, electrons in the CB may be injected into the surface states at low potentials while they are more strongly driven away from the surface at higher applied potentials, disabling the electrons from reaching the surface states. Doping TiO2 with iron narrows the band gap, which should favor the photocurrent density (see Figure SI-7a).

For 10% and more specifically 2% of Fe doping in TiO2 anatase structure it seems that additional discrete energy levels were added in the band gap. For higher concentration it is more like a combination of different contributions presenting different bandgap. All in all, the current density for TiO2-2%Fe is ~200 times higher than the one for TiO2, TiO2-10%Fe and TiO2-33%Fe at 1 V vs. RHE, where 1 V is close to a typical operating voltage when using this anode in conjunction with a single junction amorphous silicon solar cell. The improved JV characteristic is a sign that Fe doping TiO2 is beneficial since its presence enhances the oxidative power of the photoelectrodes. A more detailed look at the voltamogram at low potentials for TiO2-33%Fe reveals that the photocurrent increased starting at an onset potential at 0.75 V vs. RHE and that the increase of the photocurrent is more pronounced at higher potential, i.e. 1.6 V vs. RHE. This behavior can be linked to the fact that at low applied potentials (close to flat band potential), a dramatic electron/hole pair recombination is observed, i.e., the photogenerated holes have trouble to reach the semiconductor/electrolyte interface. In our photoelectrodes, sources of recombination are probably the numerous grain boundaries present in the nanostructure. At more positive potentials, the electric field created favors the electron/hole pairs separation, resulting in an increase of the photocurrent. behavior of the composite significantly differs from that of doped TiO2 for it looks like the superimposition of the anatase and hematite spectra. This means that both oxides absorb light depending on their bandgap without synergetic effect in the heterostructure while Fe doping seems to improve visible light absorbance of TiO2 anatase. The JV curves are completely different too as a significant photocurrent is observed on TiO2-2%Fe electrode at a potential of 1 V vs. RHE while its starts to be significant only at 1.7 V vs. RHE in the 98%TiO2-2% α-Fe2O3 electrode. With higher current density slope at higher voltage the photocurrent is still lower in the composite at 1.8 V vs. RHE with 220 C vs. 840 μA.cm -2 in Fe doped anatase. This definitively points out that, at that Fe:Ti ratio, the doping into a well crystallized structure gives better PEC properties than heterostructured system. Oppositely, the TiO2-50%Fe sample is almost completely made of well crystalline Fe2TiO5 structures and is not PEC active. Could it be possible to combine it with other oxides to recover PEC properties ?

Then, the light driven water oxidation by nanocomposite electrodes with Fe2TiO5 and single oxide, either TiO2 or Fe2O3, was monitored. To do so, nanocomposite electrodes with various compositions: x-Fe2TiO5/y-TiO2 or x-Fe2TiO5/y-Fe2O3 were fabricated; x and y represent the weight percentage (%) of Fe2TiO5 and pure oxides, respectively. For simplicity, Figure 7 summarizes the current density (expressed in μA.cm -2 ) measured at 1.8 V vs. RHE for the nanocomposite electrodes with various compositions. We monitored the photoactivity at this potential to ensure that all the photoanodes were photo(electro)chemically active. The JV curves of pure samples are reported in Figure SI-9. Several observations can be made: i) Fe2TiO5 needs to be combined with either TiO2 or Fe2O3 to achieve a high photocurrent; ii) the composite with Fe2TiO5 and Fe2O3 exhibits a high photocurrent for a wide range of compositions, iii) the best photocurrent for these nanocomposite electrodes is achieved for 25% Fe2TiO5 and 75% TiO2.

The poor electrochemical activity as well as the high onset potential (not shown here) of nanocomposite photoelectrodes with TiO2 and Fe2O3 is consistent with findings in the literature and is due to the poor separation of holes at the semi-conductor electrolyte interface that certainly increased back electron recombination at low bias voltage. 85 However, when the nanocomposite electrodes contain pseudo-brookite (Fe2TiO5/Fe2O3 or Fe2TiO5/TiO2) better charge separation is achieved as photocurrents measured are higher.

For Fe2TiO5/Fe2O3 nanocomposites, the heterojunction has a large effect on the charge separation process as better photocurrents are achieved for a wide range of compositions. Since both the valence and the conduction band extrema are higher in Fe2TiO5 than in Fe2O3 (as shown in figure 8), a straddling gap is created at the Fe2TiO5/Fe2O3 n-n heterojunction. However, the photocurrent onset is still low, indicating that the amounts of long-lived surface holes are not sufficient at low bias voltage. Electron/hole recombination should then occur, probably due to the existence of numerous grain boundaries in these thick electrodes acting as sources of recombination sites. Indeed, these grain boundaries contain a large amount of defects. photocurrent and ii) the lowest onset potential (not shown here). This is an indication that the pseudo-brookite/TiO2 heterojunction has a major impact on the charge separation process.

Compared to TiO2/Fe2O3 nanocomposite photoelectrodes, Fe2TiO5 favors the separation of holes at the semiconductor-electrolyte interface and greatly limits the e -/h + recombination at low bias voltages. This is supported by recent results on hematite-titania nanocomposite photoanodes where the charge dynamics of TiO2--Fe2O3 nanocomposite photoanodes were measured using transient absorption spectroscopy. [START_REF] Cowan | Activation Energies for the Rate-Limiting Step in Water Photooxidation by Nanostructured -Fe2o3 and Tio2[END_REF] DFT modeling results have shown that pseudo-brookite exhibits slightly more promising electronic properties than hematite, except for hole mobility, for which reason a good catalytic efficiency is expected for Fe2TiO5. Consequently, the gain is rather in the efficiency of the pseudo-brookite/TiO2 heterojunction for charge separation than the electrocatalytic activity of Fe2TiO5. Compared to Fe2O3/Fe2TiO5 nanocomposite photoelectrodes, TiO2 limits the fast charge carrier recombination due to both i) the high absorption coefficient of TiO2 compared to Fe2O3 and ii) the higher electron mobility in TiO2 compared to Fe2O3.

The impact of the structure of the iron oxide on the photocurrent has been explored. We performed equivalent studies as before, but hematite was replaced by maghemite. The interest of using maghemite instead of hematite is that maghemite contains cationic vacancies that can act as sites for electron trapping, limiting electron/hole recombination. The photocurrents measured on the various photoelectrodes are summarized and reported in Figure 7. We note that nanocomposite TiO2/Fe2O3 photoelectrodes exhibit higher photocurrents than TiO2/Fe2O3. This is probably linked to both better electron/hole separation in the photoelectrodes and holes with a longer life-time due to the presence of vacancies in Fe2O3. All of these facilitate the reaction with water.

In a more general comparison of the selected pure phase and the corresponding nanocomposites based on experimental and computational results, it is now possible to better explain their relative PEC activity. First pure TiO2, that has the best photon conversion efficiency suffers from a lack of catalytically active surface sites that can exploit trapped holes before their recombination. In accontrary, pure iron oxides α-and γ-Fe2O3 exhibit more active catalytic sites but also exhibit a lower photon conversion yield and charge mobility. The combination of TiO2 and Fe2O3 may be beneficial in nano-heterostructures, except if the solid-solid interface favors charge recombination. Moreover, the relative amount of each phase, their structure and size should be optimized for significant PEC activity.

The ordered orthorhombic Fe2TiO5 structure should exhibit fair PEC activity -between those of TiO2 and Fe2O3 -according to the calculation of its electronic and structural properties.

Unfortunately, our synthetic approach did not allow to produce phase pure orthorhombic Fe2TiO5, as a secondary monoclinic phase was detected with PDF analyses. Moreover, the iron occupancy in the pseudo-brookite structure is not fixed as proposed in the model but randomly distributed in the different metallic sites. Finally, the Fe:Ti chemical composition is slightly less than 2:1, implying oxygen defects in the structure that were not taken into account in the DFT calculations. These different points intuitively explain why the tested materials are inactive in PEC: with two phases and a random distribution of Fe atoms the Fe2TiO5 composite is much too disordered to allow good photon conversion and charge carrier mobility. The JV curve of Fe2TiO5 is compared in Figure SI-10 to that of a brookite sample prepared according to a procedure described in the literature. [START_REF] Magne | Effects of Tio2 Nanoparticle Polymorphism on Dye-Sensitized Solar Cell Photovoltaic Properties[END_REF] It confirms that the brookite structure of TiO2 presents weak photocurrents (20 μA.cm -2 at 1.8 V vs. RHE) and only at high bias voltage. However, it is still better that Fe2TiO5 sample which is still inactive at such bias voltage. In accontrary, its electrocatalytic activity may be enhanced due to the oxygen defects distorting the geometry of neighboring iron sites. Consequently the combination of small quantities of Fe2TiO5 with a good photoharvester such as TiO2 gave the best results among heterostructured systems.

Finally, this study demonstrates that tuning the electron/hole recombination processes in the bulk electrodes, but also at the electrode/electrolyte interfaces, favors the photoactivity. This was achieved by the design of the nanocomposite photoelectrodes made of materials with different properties. However, the photocurrent achieved as well as the onset potential are lower than that those for single Fe-doped photoelectrodes (onset potential is 0.8 V vs. RHE and at 1.2 V/RHE, the photocurrent is about 412 A•cm -2 and at 1.8 V/RHE is 735 A•cm -2 ). This is an indication that the nanocomposite electrodes probably contain a lot of defects on the grain boundaries that act as recombination traps.

Conclusion

In summary, we explored the photoelectrochemical performances of different Fe-Ti oxide electrodes for water oxidation. We proposed to use chimie douce coupled with micro-wave stimulus to achieve pure, mono-dispersed and crystalline nanoparticles with different levels of Fe. These nanoparticles have been characterized by various techniques including X-ray diffraction, PDF analysis, HRTEM and Mössbauer spectroscopy to monitor their structure, chemical composition and microstructure. Experimental conditions have been found and reported to achieve single phase TiO2 anatase doped with various amounts of Fe(III) up to 10 at.%, and pure α-Fe2O3 hematite or γ-Fe2O3 maghemite. Most notably, two polymorphs of Fe2TiO5 have been obtained in gentle synthesis conditions. The photoelectrochemical performances of pure Fe-doped TiO2 electrodes and nanocomposite electrodes made of Fe2O3 or/and Fe2TiO5 or/and TiO2 have been evaluated. We found that the electrocatalytic activity is strongly dependent on the nature of the hetero-junction and the best photocurrents are achieved for nanocomposite Fe2TiO5/TiO2 photoanodes with high TiO2 weight content (25/75). This increase in electrochemical performance is linked to the high absorption coefficient of TiO2 and its good electron mobility and to the higher catalytic efficiency of Fe2TiO5 compared to α-Fe2O3.

This highlights the interest of using heterostructure to cope with poor charge carrier dynamics.

However, the highest photocurrent is not achieved with heterostructured photoelectrodes but with Fe-doped TiO2 systems at a low doping content of Fe = 2 atom%. This demonstrates the necessity to engineer finely materials at the nanoscale to achieve both good charge carrier dynamics and high performance surface catalysis.

aFigure 1 :

 1 Figure 1: Representation of the conventional crystallographic cells of (a) α-Fe2O3, (b) TiO2, and

Figure 2 .

 2 Figure 2. Computed DOS projected on the atoms of (a) α-Fe2O3, (b) TiO2, and (c) Fe2TiO5. The

Figure 3 :

 3 Figure 3: X-Ray diffraction patterns of pure phase TiO2 anatase α-Fe2O3 and γ-Fe2O3, and TiO2-

Figure 4 .

 4 Figure 4. PDF refinement of TiO2-50%Fe (Rw = 21.6 %). Individual PDF contributions of the

Figure 5 :

 5 Figure 5: Transmission electron micrographs of a) pure TiO2, b) TiO2-5%Fe, c) TiO2-10%Fe, d)

1 . 3 .

 13 photoelectrodes. The synthesis procedure was described in section 2.1.3. and in Figure SI-1 An

Figure 6 .

 6 Figure 6. JV curve for TiO2-2%Fe (blue), TiO2-10%Fe (green) and TiO2-33%Fe (black and

  At this point we have compared our best monophasic photoanode TiO2-2%Fe with its equivalent where Ti and Fe have been precipitated as oxides separately (TiO2 anatase and Fe2O3 hematite) and combine afterwards into a nanocomposite electrode that present the same Fe:Ti atomic ratio of 2:98 (the electrode preparation is detailed in figure SI-1). The diffuse reflectance spectra of the 2% doped TiO2 and of the composite with the same metallic composition are reported in Figure SI-7b and the two JV curves are reported in Figure SI-8. The light absorption

Figure 7 .

 7 Figure 7. Photocurrents measured at 1.8 V vs. RHE and NaOH, pH = 14 for the nanocomposite photoelectrodes with various compositions. The photocurrents are given in μA.cm -2 and the area

Figure 8 .

 8 Figure 8. Charge generation, recombination, and transfer in the nanocomposite photoelectrodes

Figure SI- 3 :Figure SI- 7 :

 37 Figure SI-3: Mössbauer spectrum of TiO2-50%Fe and its deconvolution with increasing number of Fe 3+ sites; Figure SI-4: TEM images of selected TiO2-x%Fe-pHy samples; Figure SI-5:

  ABBREVIATIONS AFM Anti-Ferromagnetic

Table 3 .

 3 Structural parameters of Fe2TiO5 phases obtained by PDF analysis. Values in italics follows the nominal composition used during the synthesis. Slight deviations can be observed at high Fe content which, as mentioned above, can be due to non-stoichiometry.

	refer to reference literature data.
	PDF refinements conducted on the TiO2-10-to-66%Fe series indicate that both Fe2TiO5
	polymorphs can be detected even at low Fe content, i.e. 10%, -both phases are present in almost

equal proportions regardless of Fe content. Quantitative Fe content analysis (cf. Figure

SI-2c

) by PDF refinement

AUTHOR INFORMATION

Corresponding Author

Olivier Durupthy Telephone: +33144276306, Fax: +33144271504, E-mail: olivier.durupthy@upmc.fr Present Addresses †P.Sautet is now at Department of Chemical and Biomolecular engineering, University of California, Los Angeles, Los Angeles, CA 90095, United States.

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval


Vincent Chabot
email: vincent.chabot@meteo.fr

Maëlle Nodet
email: maelle.nodet@inria.fr

Arthur Vidard
email: arthur.vidard@inria.fr

Progressive assimilation of multiscale observations

The description of correlated observation error statistics is a challenge in data assimilation. Currently, the observation errors are assumed uncorrelated (the covariance matrix is diagonal) which is a severe approximation that leads to suboptimal results. It is possible to use multi-scale transformations to retain the diagonal matrix approximation while accounting for some correlation. However this approach can lead to some convergence problems due to scale interactions. In this paper we propose an online scale selection algorithm that improves the convergence properties in such case.

I. INTRODUCTION

Numerical weather prediction requires the determination of the initial state of the system. Indeed, the true state, at a given moment and in all points of space, is not accessible. In order to retrieve an optimal initial condition one uses the so called data assimilation methods that combine information from observations, model equations and their respective error statistics. Since the late 70s, satellites are a dominant source of information. Errors associated to such data are highly correlated in space, which can be detrimental if this is not properly accounted for. However their density in space allows for the efficient use of multi-scale transformation, which in turn permit a cheap but good approximation of said error statistics representation. For homogeneous spatially correlated Gaussian observation errors this approach is very efficient. For more complex errors, however, it can severely damage the convergence properties of the assimilation methods. In this paper, after a short introduction to the context (section II and III), we present, through a simple case mimicking a laboratory experiment (presented in section IV), an illustration of the above-mentioned problem and a possible solution using scale selection during the assimilation process (section V).

II. GENERAL FORMULATION OF VARIATIONAL DATA

ASSIMILATION

Let M be a dynamical model describing the evolution of the state variable X in space and time:

∂ t X(X 0 , x, t) + M(X(X 0 , x, t)) = 0 X(X 0 , x, t 0 ) = X 0 Let Y (t) be (partial) observations of this state variable.
The aim of data assimilation is to estimate an optimal initial condition X a 0 (often called analysed state) so that it is not far from the first guess X b 0 (in general coming from a previous forecast), and that the model trajectory X(X a 0 , x, t) is close to the observations Y (t). This is done by defining X a 0 as the minimum of the cost function:

J(X 0 ) = J b (X 0 ) + J o (X 0 ) = 1 2 X 0 -X b 0 2 V + 1 2 t f ti=t0 ||Y (t i ) -H(X(X 0 , x, t i ))|| 2 O
where V is the model state space, O the observation space and H : V → O the observation operator. Usually, in variational data assimilation, the minimisation is done using a gradient descent type algorithm and the gradient is computed using adjoint methods.

Typically in data assimilation one uses the Mahalanobis distance .

2 V = . 2 B and 2
O = . 2 R with X 2 K = X T K -1 X
where R and B are the observation and background error covariance matrices respectively.

III. ERROR COVARIANCE MATRIX REPRESENTATION

The choice of B and R is crucial in data assimilation, since it will drive the way information is spread and how redundancy of error is delt with. So far a strong research effort has been targeted to the background error statistics representation either using multi scale decomposition ( [START_REF] Berre | Filtering of Background Error Variances and Correlations by Local Spatial Averaging: A Review[END_REF], and references therein), recursive filters ( [START_REF] Purser | The filtering of meteorological fields[END_REF]) or general diffusion models ( [START_REF] Weaver | Correlation modelling on the sphere using a generalized diffusion equation[END_REF]). In this paper, we mostly focus on the observation error statistics representation, while B is constructed using the general diffusion approach.

The R matrix, however, has mostly been assumed diagonal. The main reason is that it simplifies greatly its management. Indeed the number of observations is in general non constant over time which prevents from using the same R matrix at each assimilation cycle. As a consequence, at each assimilation cycle, a new R matrix should be formed and inverted. Even without considering the formation of the R matrix, it size makes its storage and its inversion very difficult.

In [START_REF] Chabot | Accounting for observation errors in image data assimilation[END_REF] we proposed to use linear changes of variable A to allow for a simpler representation of the observation error covariances. Namely, assuming observation errors to be additive, unbiased and gaussian, i.e. Y = Y t + with ∼ N (0, R), Y t being the true signal. Then AY = AY t + β with β ∼ N (0, ARA T ).

By choosing A such that D A = diag(ARA T ) ARA T . one can retain a diagonal approximation, indeed after a bit of algebra, one gets

(Y -H(X)) T R -1 (Y -H(X)) (Y -H(X)) T A T D -1 A A(Y -H(X))
The existence of such A is no guaranteed, depending on the observation nature and its associated errors. For dense gridded observation with spatially correlated Gaussian errors, [START_REF] Chabot | Accounting for observation errors in image data assimilation[END_REF] used multi-scale transforms, such as Fourier, curvelet and wavelet, for designing A. Doing so, even with a diagonal covariance matrix, one can consider more, or less uncertainties for a given scale compare to the others, while without this change of variable one can only affect grid-point uncertainties. In this paper we will use orthonormal wavelet transform for the operator A (see [START_REF] Mallat | A Wavelet Tour of Signal Processing: The Sparse Way[END_REF]). The next sections present two cases study, one where such approach shows clear improvement compare to the classical one, and a second one where it renders the minimisation hieratic. In that case an algorithm based on online scale selection is proposed to circumvent this problem.

IV. EXPERIMENTAL SETTINGS

The experimental framework mimics the drift of a vortex on a turntable. The evolution of a vortex in the atmosphere is simulated at the CORIOLIS experimental turntable (Grenoble, France) which re-creates the effect of the Coriolis force on a thin layer of water. A complete rotation of the tank takes 60 seconds which corresponds to one Earth rotation. The vortex is created by stirring the water and made visible thanks to the addition of a passive tracer (fluorescein). The camera is placed above the turntable, and photographs of the vortex constitute the observed image sequence. For more details about these experiments, see [START_REF] Flór | The dynamics of monopolar vortices on a topographic beta-plane[END_REF].

A. Numerical configuration

In this configuration, the evolution of the fluid can be represented by the shallow-water equations involving the horizontal velocity w(x, t) = (u(x, t), v(x, t)), where u and v are the zonal and meridional components of the velocity, and the water elevation h(x, t). These unknown variables are defined on the spatial domain Ω x and the time interval [t 0 , t f ] t. Such a model reads:

∂ t u -(f + ζ)v + ∂ x B = -ru + κ∆u ∂ t v + (f + ζ)u + ∂ y B = -rv + κ∆v ∂ t h + ∂ x (hu) + ∂ y (hv) = 0.
The relative vorticity is denoted by

ζ = ∂ x v -∂ y u and the Bernouilli potential by B = g * h + u 2 + v 2 2
, where g * is the reduced gravity. The Coriolis parameter on the β-plane is given by f = f 0 +βy, κ is the diffusion coefficient and r the bottom friction coefficient. The following numerical values were used for the experiments: r = 9.10 -7 , κ = 0, f 0 = 0.25, g = 9.81 and β = 0.0406.

B. Observation operator

The vortex temporal evolution is shown through the fluorescein concentration evolution. This evolution is observed by an image sequence of the concentration of a passive tracer q transported by the velocity field.

Denoting w the velocity (computed by the model M) transporting the passive tracer and ν T the diffusion coefficient, we have

∂ t q + ∇q • w -ν T ∆q = 0 q(t 0 ) = q 0 . (1)
Assuming that the initial concentration of q is known at time t 0 , the dynamic of q on the time interval [t 0 ; t f ] is defined by the model [START_REF] Berre | Filtering of Background Error Variances and Correlations by Local Spatial Averaging: A Review[END_REF], where the diffusion coefficient is ν T = 10 -5 .

In the following experiments the considered observation sequence Y represents full maps of q. As a consequence, the observation operator reads :

H (X ti ) = q(t i ).
(

) 2
where q(t i ) comes from [START_REF] Berre | Filtering of Background Error Variances and Correlations by Local Spatial Averaging: A Review[END_REF].

In those experiments we assume that the initial concentration of the passive tracer is known. Therefor we do not control q 0 .

C. Twin Experiments Configuration

In order to focus on the methodological aspects we will use a so-called twin experiment framework. In this classical approach, synthetic observations are created thanks to a model simulation from a known "true state"; then an assimilation experiment is performed starting from another "background" state using the synthetic observations. The result of this analysis can be compared with the synthetic truth. Unless otherwise stated, the assimilation period will be of 144mn, with one snapshot of passive tracer concentration every 6mn (24 snapshot in total). A selection of such observations is shown in fig 1.

V. EXPERIMENTAL RESULTS

A. Effect of the nature of the errors

Two experiments are presented here, with two different kind of observation errors. The first one, with homogeneous Gaussian errors mimic what was done in [START_REF] Chabot | Accounting for observation errors in image data assimilation[END_REF]. The second one introduces more complex inhomogeneous observation errors. In both experiments, two minimization will be performed: a) the comparison between observation and its model equivalent is done grid point by grid point (hereafter Pixel) b) the same comparison is done wavelet coefficient per wavelet coefficient (hereafter Wavelet).

1) Homogeneous observation error:

The observation are obtained by adding a spatially correlated Gaussian noise to selected snapshot of the true trajectory

Y ti = Y t ti + with =∼ N (R, I)
In order to mimic the usual approach, only diagonal approximations of R are used: R pix = diag(R) for the Pixel minimisation and R wav = D A = diag(ARA T ) Figure 2 shows the residual error r = X t -X X t -X b along the iterations of minimisation for both Pixel and Wavelet. It shows that, event though D A is only an approximation of R, accounting for some part of the spatial correlation is clearly beneficial 2) Inhomogeneous observation error: In the second experiment, the observation error is still Gaussian and spatially correlated, but these correlations are now inhomogeneous in space. For the sake of clarity, this inhomogeneous noise is actually generated out of D A , through: Figure 3 shows the same residual error as for figure 2 for a spatially inhomogeneous observation error. Even though the observation error is exactly represented in the Wavelet case, its minimisation is struggling to converge toward a minimum similar or better than for the pixel case. The conditioning of the minimisation is probably affected, but the main reason comes from some sort of aliasing in the small scales, as illustrated by figure 4. It represents the norm of the difference between the background concentration and the subsequent observations Y ti -H(X b 0 ) 2 X for 0 ≤ i ≤ 240, with X = diag(R) for Pixel and X = ARA T for Wavelet. On the one hand, the blue line represents this norm for the Pixel case. It starts with a small value (the only difference comes from the noise) and, as time goes by, the vortex drifts and the difference with the initial concentration steadily increases. As one would expect the farther the vortex drift, the higher the difference with the initial concentration is, all the scales being given the same uncertainties. On the other hand, the wavelet-based norm (in green), shows a steep increase at the beginning, but then oscillate around a 'plateau'. This happens because, at this point, the norm is really dominated by the small scales. This is expected, since they are the least affected by the correlated noise, so their associated error variances are the smallest (i.e. one trusts more the small scales) and it is the inverse of the variances that is used as a weight in the norm. However it prevents to discriminate between two large scale signals, when the difference is too large (when the green curve stop being monotonic), so the minimisation problem becomes ill-posed. Red, black and purple curves show the same quantity as the green one, but removing the 1, 2 and 3 finest scales in the multi scale decomposition respectively. The problem appears later (i.e. for larger discrepancies) when removing the finest scales and even disappear for the purple one. Note that this problem arose as well for the homogeneous case, but after a longer time period (beyond the assimilation window) and in a less striking manner: the 'plateau' is less oscillating and still slightly monotonic.

Y ti = Y t ti + with = A T D 1/2 A β β ∼ N (0, I) So that D A is
A solution would be to first assimilate the large scales, in order to reduce the difference between H(X) and Y , and to be in the monotonic region of the green curve, and then progressively includes the smaller scales. The next section presents a way to progressively assimilate the smaller scales in order to circumvent the convergence problem.

B. Online scale selection

A simple way to sort out the above mentioned problem would be to remove the smaller scales altogether, but it would mean getting rid of an important part of the information. Instead we propose to gradually include the relevant scales. The contribution J s to the total cost from a given observed scale can be written

J s (X) = 1 2n s k (d Y s,k -d H(X) s,k ) 2 σ 2 s,k
where d s,k is a value of wavelet coefficient at scale s and σ 2 s,k the associated variance (i.e. the corresponding diagonal element in D A . One can consider that the information at a given scale is usable only if J s ≤ τ s so the modified cost function would replace J s , from each scale s by

J s,τs (X) = J s (X) if J s (X) ≤ τ s τ s otherwise
The green curve in figure 5 shows the evolution of residual error for such modification, with τ s = 4.5 for all s. This value has been chosen to retain Gaussianity in the retained scales. Indeed, for a Gaussian signal 99% of the considered population should lie within 3 std dev of the mean (here it is a square and divided by two, hence 4.5). Figure 6 shows the number of observed snapshot for which each scale is activated (i.e. J s,τs = τ s ). All the scales but the finest are gradually included for all snapshots after about 20 iterations. The finest scale is only activated for half of the snapshots after 200 iterations, showing that this information is very difficult to account for by the system. Figure 7 shows the contribution to the observation term of the cost function from each activated scale. The coarser scales are dominating at the very beginning of the minimisation and converge quite quickly (after 10 iterations), then scales 5 and 6 dominates and converge after 100 iterations. Scale 7 is still including snapshot and has not converged yet after 200 iterations.

VI. CONCLUSION

Using multi-scale transforms as changes of variables to represent observation error correlation in data assimilation is a cost effective and promising approach. However this can lead to convergence problems in some cases. A progressive assimilation of the finest scale can significantly improve the convergence, making it a more robust approach.

Fig. 1 .

 1 Fig. 1. "True" initial concentration of the passive tracer (top) and noisy observations at initial time, after 90mn, 150mn and 270mn (bottom)

Fig. 2 .

 2 Fig. 2. Ratio of residual errors along minimisation iterations for both pixel and wavelet based distances in presence of an homogeneous observation error

Fig. 3 .

 3 Fig. 3. Ratio of residual errors along minimisation iterations for both pixel and wavelet based distances in presence of an inhomogeneous observation error

Fig. 4 .

 4 Fig. 4. Discrepancy between the background initial concentration and the successive observations along time, as would be measured by the observation term of the cost function

Fig. 5 .

 5 Fig. 5.Ratio of residual errors along minimisation iterations for pixel and wavelet-based and progressive wavelet-based distances in presence of an inhomogeneous observation error.

Fig. 6 .

 6 Fig. 6.Number of snapshot for which a given scale is activated along iterations. Scale 1 being the soarse approximation and scale 7 the finest scale.

Fig. 7 .

 7 Fig. 7. Contribution to the observation term in the cost function represented by each activated scale along the minimization iteration

ACKNOWLEDGMENT

This work got support from the French national research agency (ANR) through contract ANR-09-COSI-005 (GeoFluids).