Daria Ryaboshapka

Thomas Len

Pascal Bargiela

Mimoun Aouine

Christophe Geantet

Valerie Briois

Laurent Piccolo

Pavel Afanasiev

Genesis of active phase and support effect in ultradispersed Mo sulfide catalysts

Keywords: Ultradispersed metals, operando XAS, molybdenum sulfide, support effect

Molybdenum sulfide -based catalysts are widely used for hydrotreating, hydrogen evolution and many other reactions. Recently, we demonstrated that not only the edges of MoS2 slabs, but few-atom ultradispersed MoSx clusters possess high intrinsic activity. However, the structure and the genesis of such ultradispersed species remain unknown. Herein we present a comparative study of MoSx catalysts ultradispersed on different supports (carbons, SiO2, Al2O3, TiO2). Evolution of the Mo species during sulfidation and HDS reaction was studied by means of operando Quick-X-ray absorption spectroscopy (QXAS) at the Mo K edge, assisted by chemometric analysis (MCR-ALS). Significant differences of the structure of Mo species and their temperature evolution as a function of support were observed. The sulfidation pathway involves formation of oxysulfide and sulfur-rich MoS3-like intermediates, which are further transformed into the final MoSx clusters. As compared with MoS2 nanoslabs, the coordination numbers of Mo in the ultradispersed clusters are decreased and the interatomic Mo-S and Mo-Mo distances are shortened. Other characterizations, in particular STEM-ADF, confirm that few-atom clusters and single-atom (SA) species are predominant in all the catalysts. The materials show high activity per Mo atom in hydrodesulfurization (HDS) of thiophene, varying in step with the MoSx dispersion as determined from XAS, in the sequence: Mo/carbons > Mo/TiO2 > Mo/Al2O3 ≈ Mo/SiO2.

Introduction

Lamellar molybdenum sulfide MoS2 is a well-known catalyst for hydrodesulfurization (HDS) reaction, used to remove sulfur from petroleum fractions. [START_REF] Topsøe | Hydrotreating Catalysis Science and Technology[END_REF] For a long time it was considered that only the edge sites of hexagonal and triangular MoS2 slabs are catalytically active. [START_REF] Šarić | Modeling the Adsorption of Sulfur Containing Molecules and Their Hydrodesulfurization Intermediates on the Co-Promoted MoS2 Catalyst by DFT[END_REF][START_REF] Lauritsen | Atomic-Scale Structure of Co-Mo-S Nanoclusters in Hydrotreating Catalysts[END_REF] Theoretical models predict the specific activity to depend on the amounts of exposed crystallographically distinct M-and S-edges. [START_REF] Krebs | A DFT Study of the Origin of the HDS/HydO Selectivity on Co(Ni)MoS Active Phases[END_REF][START_REF] Raybaud | Structural and Electronic Properties of the MoS2(1010) Edge-Surface[END_REF] Recent studies describe tuning the slabs shape, aiming at the control of M to S edge ratios. [START_REF] Chen | Effect of high pressure sulfidation on the morphology and reactivity of MoS2 slabs on MoS2/Al2O3 catalyst prepared with citric acid[END_REF] However, some experimental results require to go beyond the conventional 2D slabs model. Thus in the studies of model unsupported MoS2 and supported MoS2/Al2O3 catalysts, [START_REF] Afanasiev | The influence of reducing and sulfiding conditions on the properties of unsupported MoS2-based catalysts[END_REF][START_REF] Baubet | The Influence of MoS2 Slab 2D Morphology and Edge State on the Properties of Alumina-Supported Molybdenum Sulfide Catalysts[END_REF] the specific HDS activity did not evolve in step with the amount of exposed edges as estimated from TEM. Moreover, as the STEM-HAADF technique became widely used for studies of supported sulfides, the features smaller than 1 nm or even single atoms were systematically observed alongside the MoS2 slabs. [START_REF] Baubet | Quantitative Two-Dimensional (2D) Morphology-Selectivity Relationship of CoMoS Nanolayers: A Combined High-Resolution High-Angle Annular Dark Field Scanning Transmission Electron Microscopy (HR HAADF-STEM) and Density Functional Theory (DFT) Study[END_REF][START_REF] Zavala-Sanchez | Formation and Stability of CoMoS Nanoclusters by the Addition of Citric Acid: A Study by High Resolution STEM-HAADF Microscopy[END_REF][START_REF] Girleanu | HAADF-STEM High-Resolution Study of Nanometric MoS2 inside Mesoporous SBA-15[END_REF] These subnanometric or single atomic features persisted even after the thiophene HDS test at 350 °C. [START_REF] Zavala-Sanchez | Formation and Stability of CoMoS Nanoclusters by the Addition of Citric Acid: A Study by High Resolution STEM-HAADF Microscopy[END_REF] However, the catalytic activity of such species was never accessed. Recently, combining a high -surface activated carbon support, low Mo loading and a simple adsorption preparation technique, we prepared MoSx/C catalysts that contained no TEM-detectable slabs, but mostly SAs and subnanometric clusters. [START_REF] Ryaboshapka | Ultradispersed (Co)Mo Catalysts with High Hydrodesulfurization Activity[END_REF] These catalysts demonstrated high specific HDS activity and high degree of promotion with cobalt. However, the structure of ultradispersed MoSx species in the conditions of HDS reaction remained unclear, because merely ex-situ XAS measurements were carried out in our previous work. [START_REF] Ryaboshapka | Ultradispersed (Co)Mo Catalysts with High Hydrodesulfurization Activity[END_REF] The support effect in the conventional sulfide HDS catalysts has been widely studied for a long time. [START_REF] Breysse | Overview of Support Effects in Hydrotreating Catalysts[END_REF][START_REF] Hensen | The Relation between Morphology and Hydrotreating Activity for Supported MoS2 Particles[END_REF][START_REF] Pratt | Morphology and activity of MoS2 on various supports: Genesis of the active phase[END_REF][START_REF] Gutiérrez | Effects of the Support on the Performance and Promotion of (Ni)MoS2 Catalysts for Simultaneous Hydrodenitrogenation and Hydrodesulfurization[END_REF] The nature of the support strongly influences the MoS2 slabs size, stacking, orientation as well as the coordination of Mo atoms. [START_REF] Breysse | Support Effects on Hydrotreating Catalysts[END_REF][START_REF] Payen | Morphology Study of MoS2-and WS2-Based Hydrotreating Catalysts by High-Resolution Electron Microscopy[END_REF][START_REF] Kibsgaard | Scanning Tunneling Microscopy Studies of TiO2-Supported Hydrotreating Catalysts: Anisotropic Particle Shapes by Edge-Specific MoS2-Support Bonding[END_REF][START_REF] Ninh | Support Effect for CoMoS and CoNiMoS Hydrodesulfurization Catalysts Prepared by Controlled Method[END_REF] Several (not mutually excluding) models of the support effect were proposed, focusing on different properties, such as the support acidity, [START_REF] Nguyen | Use of Kinetic Modeling for Investigating Support Acidity Effects of NiMo Sulfide Catalysts on Quinoline Hydrodenitrogenation[END_REF][START_REF] Blanchard | Acidic Zeolites and Al-SBA-15 as Supports for Sulfide Phases: Application to Hydrotreating Reactions[END_REF] its impact on the MoS2 slabs dispersion via interactions in the oxidic form, [START_REF] Dominguez Garcia | New Insight into the Support Effect on HDS Catalysts: Evidence for the Role of Mo-Support Interaction on the MoS2 Slab Morphology[END_REF] its impact on the reducibility of slabs edges, [START_REF] Kaluža | Effect of Support Type on the Magnitude of Synergism and Promotion in CoMo Sulphide Hydrodesulphurisation Catalyst[END_REF] different properties of exposed crystallographic facets of support particles, [START_REF] Garcia De Castro | Surface-Dependent Activity of Model CoMoS Hydrotreating Catalysts[END_REF] or the electronic effects. [START_REF] Galhenage | MoS2 Nanoclusters Grown on TiO2: Evidence for New Adsorption Sites at Edges and Sulfur Vacancies[END_REF][START_REF] Lee | Direct Evidence of Electronic Interaction at the Atomic-Layer-Deposited MoS2 Monolayer/SiO2 Interface[END_REF] In all these models nanoparticulate monolayer or stacked MoS2 particles were considered, in which only a minority of Mo atoms is potentially involved in the interaction with a support. To observe the support effect by physical techniques, the greater possible part of the active phase (MoS2) should interact with the support, i.e. the greater is the dispersion of the active phase, the higher should be the observed effect. Therefore in the ultradispersed MoSx catalysts, a strong influence of support on the properties of the supported sulfide species could be expected.

Since the active species of sulfide catalysts are air-sensitive, in-situ and operando techniques are the most appropriate to characterize them. X-ray absorption spectroscopy (XAS), due to its element-specificity and relatively high penetration of hard X-rays through the catalyst, is one of the most valuable techniques to study sulfide catalysts upon activation and under working conditions. [START_REF] Grunwaldt | X-ray absorption spectroscopy under reaction conditions: suitability of different reaction cells for combined catalyst characterization and time-resolved studies[END_REF] Dynamic conditions of temperature and gas atmosphere during sulfidation and reaction cause chemical transformations, from the oxide precursor to the final sulfide active phase, stable under reaction conditions. In earlier works, in-situ XAS was applied to study genesis of active phase in the unpromoted and Co-promoted Mo-based hydrotreating catalyst. [START_REF] Browne | The Effect of Passivation on the Activity of Sulfided Mo and Co-Mo Hydrodesulphurization Catalysts[END_REF][START_REF] Geantet | Situ QEXAFS Investigation at Co K-Edge of the Sulfidation of a CoMo/Al2O3 Hydrotreating Catalyst[END_REF] Recently, due to the use of time-resolved quick XAS (QXAS) and chemometry techniques of data analysis (MCR ALS) unraveling between the intermediate species during the catalyst activation become possible. [START_REF] Plais | First in situ temperature quantification of CoMoS species upon gas sulfidation enabled by new insight on cobalt sulfide formation[END_REF][START_REF] Gajdek | Sulfidation of Supported Ni, Mo and NiMo Catalysts Studied by In Situ XAFS[END_REF] While XAS operando was widely applied to follow the genesis of conventional alumina supported (Co)Mo catalysts, it was never used to systematically study the support effect, neither to characterize the ultradispersed sulfide species.

The present study is focused on the properties of ultradispersed Mo sulfide species formed during the sulfidation and HDS reaction on several widely studied supports, including carbons, SiO2, Al2O3 and TiO2. To obtain insights into the nature of the Mo species evolution, we applied operando quick X-ray absorption spectroscopy (QXAS) technique in combination with chemometric data treatment approach.

Experimental

Preparation of the catalysts

All reactants and solvents were high-purity-grade products purchased from Sigma Aldrich.

Four supports were used: Activated Charcoal Norit (AC), graphene flakes (Sigma Aldrich), TiO2 (PC500 anatase, Evonik) and γ-Al2O3 (Prolabo). The silica-incorporated Mo-KIT-6 sample was homemade (vide infra). Different preparation techniques and different Mo precursors were applied to prepare the catalysts, in order to ensure the highest possible dispersion of Mo oxide species. We supposed that if the initial Mo oxide species are ultradispersed, it would maximize the Mo species dispersion during the following sulfidation. On the other hand if nanoparticles of Mo oxide phase were formed in the initial state, then further sulfidation would probably lead to poorly dispersed sulfide phase. For this reason, conventional impregnation was not applied for carbons and silica that contain low amounts of surface functional groups and tend to stabilize poorly dispersed supported oxide species. To avoid agglomeration, the supports with specific surface areas were applied and the loading of Mo precursor was relatively low, corresponding to the calculated value of 1 wt% Mo (see Table S5 for the chemical analysis results).

Mo/AC and Mo/graphene samples were prepared via equilibrium adsorption route applied in our previous work. [START_REF] Ryaboshapka | Ultradispersed (Co)Mo Catalysts with High Hydrodesulfurization Activity[END_REF] A weighted amount (27.0 mg) of (NH4)2MoS4 was dissolved in methanol and added dropwise to a suspension of 1g carbon in 100 mL of methanol, under continuous stirring.

After 4 h of stirring red color of thiomolybdate solution disappeared and adsorption was considered complete. The samples were then filtered and dried under Ar flow at room temperature. Note that even if a thiosalt was used as precursor, this preparation leads to ultradispersed supported oxide species. [START_REF] Ryaboshapka | Ultradispersed (Co)Mo Catalysts with High Hydrodesulfurization Activity[END_REF] Mo/TiO2 and Mo/Al2O3 were prepared via conventional impregnation techniques that already afforded ultradispersed oxidic Mo species in our previous works (but for higher Mo loadings). [START_REF] Len | Ultradispersed Mo/TiO2 Catalysts for CO2 Hydrogenation to Methanol[END_REF] A calculated amount (18.3 mg) of (NH4)6Mo7O24.4H2O (AHM) was dissolved in a volume of distilled water equal to the pore volume of the support. Then, the solution was added dropwise to 1 g of the support powder and mixed vigorously for several minutes. The solids were dried in oven for 3 h at 110 °C. Reference MoS2/Al2O3 with 10 wt% Mo loading was synthesized by impregnation with AHM with addition of citric acid according our previous work. [START_REF] Afanasiev | Calculation of MoS2 Slabs Morphology Descriptors from Transmission Electron Microscopy Data Revisited. Case Study of the Influence of Citric Acid and Treatment Conditions on the Properties of MoS2/Al2O3[END_REF] It is further designated as 10Mo/Al2O3. 10 wt% Mo references on other supports were prepared using impregnation with AHM followed by drying, as for Mo/Al2O3.

Mo-KIT-6 (Mo -incorporated mesoporous silica) was prepared according to the literature procedure, [START_REF] Chen | Study on the Selective Oxidation of Methane over Highly Dispersed Molybdenum-Incorporated KIT-6 Catalysts[END_REF] where predominance of di-and monomolybdate species was evidenced. In brief, 19 g of 2 M HCl was mixed with 2 g Pluronic P-123 (Mw 5800) and stirred until the P123 full dissolution. Then 33.8 mg of AHM in 50 g deionized water was added to the template solution.

After stirring for 2 h, 2.5 g n-butanol was added and kept stirring for 1 h more. Finally, 6.4 g TEOS was added dropwise. After stirring for 24 h, the solution was transferred to a Teflon-lined autoclave and treated for 24 h at 100 °C. After filtration, drying at 100 °C for 6 h and calcination in air at 550 °C for 6 h, the Mo-KIT-6 solid was obtained.

Before HDS reaction the samples were sulfided in a Pyrex reactor under 15%vol. H2S/H2 gas mixture flow (0.6 L.h -1 H2S, 3 L.h -1 H2) at 350 °C for 1 h and heating rate 5 °C.min -1 . When sulfidation was finished, the H2S flow was turned off and the samples were cooled to room temperature in a 1.8 L.h -1 flow of H2. Afterwards the samples were handled and stored under argon.

The designation of the samples is as follows: (i)stands for the initial (oxide) sample; (s) sulfided, sample after sulfidation; (h)sample taken after HDS reaction, e.g.: Mo/Al2O3-i, Mo/Al2O3-s, Mo/Al2O3-h.

Thiophene HDS test

A weighted amount of a freshly sulfided sample was placed in a flow fixed-bed Pyrex microreactor. A 3 L.h -1 H2 flow passed through a saturator containing liquid thiophene at 0 °C.

The reaction was performed at atmospheric pressure at three different temperatures: 320, 330, 340 °C. Steady-state conversions were estimated after at least 16 hours on-stream. The reaction products were analyzed by gas chromatography on an Agilent 7820A device equipped with a flame ionization detector. The HDS rates per unit of catalyst mass were estimated according to the equation (1),

𝑟 = -( 𝐹 𝑡ℎ𝑖𝑜 𝑚 ) ×Ln(1 -𝑋), (1)
where r is the specific HDS rate, Fthio is thiophene molar flow (mol.s -1 ), m is mass of the catalyst (g) and X is thiophene conversion.

Specific activity values per Mo atom were calculated using equation ( 2):

𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑎𝑐𝑡𝑖𝑣𝑖𝑡𝑦 = 𝑟×𝑁 𝐴 𝑁 , (2)
Where r is specific HDS rate per mass unit (mol.g -1 s -1 ), NA is Avogadro number and N is number of Mo atoms per gram of the catalyst.

Characterizations

Mo content in the samples was determined by inductively coupled plasma-optical emission spectroscopy (ICP-OES, Activa instrument from Horiba Jobin Yvon). The carbon supported samples were dissolved in H2SO4 and HNO3 mixture and heated to 300 °C, while for the samples on the oxidic supports the H2SO4 -HNO3 -HF mixture was used.

𝐹(𝑅) = (1-𝑅) 2 2𝑅 ( 3
)
Where R is reflectance.

X-band (9.5 GHz) EPR spectra were recorded at 77 K on a Bruker ESP 300E spectrometer using a standard rectangular (TE102) EPR cavity (Bruker ER4102ST). Microwave power of 1.6 mW and modulation amplitude of 1 G were used. Visualization of the structures initial Mo species and of their transformation during sulfidation was performed using the Avogadro software (ver.

Operando QXAS study

The coordination environment of Mo was studied by means of X-ray absorption spectroscopy (XAS) in the transmission mode at the SOLEIL synchrotron facility, ROCK (Rocking Optics for Chemical Kinetics) beamline. [START_REF] Briois | The New Quick-EXAFS Beamline at SOLEIL[END_REF] A weighted amount of powder of an initial sample was placed into the cavity in the sample holder between two graphite layers. The sample holder was then inserted into the special cell for operando characterizations. More details on the sample holder and the cell might be found in Ref. [START_REF] Fontaine | X-Ray Absorption Spectroscopy and Heterogeneous Catalysis: Performances at the SOLEIL's SAMBA Beamline[END_REF] The spectra were recorded at Mo K edge using a Si(111) channel-cut monochromator aligned at 5.65° and oscillating at 2Hz frequency with an amplitude of ≈ 0.4°. Harmonics were rejected using Pd mirrors with a 2. Preliminary data treatment was performed using the ATHENA software. [START_REF] Ankudinov | Real-Space Multiple-Scattering Calculation and Interpretation of x-Ray-Absorption near-Edge Structure[END_REF] Fitting of the EXAFS spectra was carried out with VIPER program both in k (BFT) and R spaces. [START_REF] Klementev | Extraction of the Fine Structure from X-Ray Absorption Spectra[END_REF] The R-space fitting was performed in the (Im+Module) mode. During the fitting procedure the values of R (distance), CN (coordination number), σ 2 (distance variance) and ΔE0 (energy difference) were varied. The goodness of the fit was determined by the value of R-factor.

To analyze the intermediate species appearing during the sulfidation we applied a set of chemometric data analysis tools. [START_REF] Rochet | Edges Quick-XAS Study of the Sulphidation Properties of Mo/Al2O3 and CoMo/Al2O3 Catalysts[END_REF] All procedures were performed with Matlab R2021a software.

First, we constructed a matrix of normalized XAS spectra recorded during a specific process (sulfidation, reaction…) with a merge over 15 s of the data recorded successively during the heating ramp and over 300 s for the data recorded during isothermal treatment. The normalization of the raw data was performed using the homemade Normal GUI interface available at the beamline. [START_REF] Lesage | High Pressure Cell for Edge Jumping X-Ray Absorption Spectroscopy: Applications to Industrial Liquid Sulfidation of Hydrotreatment Catalysts[END_REF] The normalized data were assembled as a matrix using a homemade XAS-SVD Matlab GUI toolbox which allows for a preliminary PCA (Principal Component Analysis) estimation of the number of components from the scree plot of eigenvalues. Afterwards, MCR-ALS (Multivariate Curve Resolution -Alternating Least Square) was applied on the matrix obtained in XAS-SVD, to extract pure components and their concentrations. [START_REF] Jaumot | MCR-ALS GUI 2.0: New Features and Applications[END_REF] Evolving Factor Analysis (EFA) allowed the initial estimation of concentration profiles. [START_REF] Blanchet | Méthodes de résolution dédiées à l'étude spectroscopique de processus photoinduits[END_REF] After the initial estimation certain constrains were applied in the ALS algorithm, including unimodality, non-negativity and evolution interval from 0 to 100% for concentration profiles and non-negativity for the spectra. Then MCR-ALS performed reconstruction of the initial data matrix using the obtained pure components.

Results and Discussion

Properties of the sulfide catalysts accessed by physical techniques.

Most characterizations were performed on the freshly sulfided samples, unless otherwise stated. Textural characterizations for bare supports and chemical analysis results for sulfided solids are provided in Fig S1 and Table S1 respectively. The XRD patterns of sulfided samples (not shown) contain only the reflections of bare supports. Raman, EPR and XPS spectroscopies were applied on the samples sulfided ex-situ without air admission. Other ex-situ methods (UV-vis DRS, XRD, conventional TEM and STEM-HAADF), that could be implemented only after a short air exposure, are useful to access dispersion of Mo species, but partial oxidation is possible.

Raman spectroscopy

Raman spectra were recorded in an in-situ reactor, under static Ar. The spectra of bulk or nanoscopic MoS2 usually show two intense peaks at 383 cm -1 and 408 cm -1 , corresponding to the E2g and A1g modes. [START_REF] Wieting | Infrared and Raman Studies of Long-Wavelength Optical Phonons in Hexagonal MoS2[END_REF][START_REF] Yaremko | Theoretical and Experimental Study of Phonon Spectra of Bulk and Nano-Sized MoS2 Layer Crystals[END_REF] For Mo/AC-s and Mo/graphene-s, no signal could be detected between 300 and 500 cm -1 even for long acquisition times, but only two intense peaks were spotted at 1321 and 1598 cm -1 , characteristic of defective carbon (Fig. S2a). For Mo/TiO2-s, the region of interest contains intense lines of TiO2 anatase at 396, 516 and 635 cm -1 (Fig. S2b). Poorly resolved broad lines around 354 cm -1 and 409 cm -1 were observed for Mo/Al2O3-s and around 361 cm -1 and 412 cm -1 for Mo-KIT-6-s (Fig. S2c). These features resemble MoS2, but are shifted and strongly broadened, suggesting that Mo species are highly disordered MoS2 clusters. The peaks located around 215 cm -1 for Mo/Al2O3-s and 210 cm -1 for Mo-KIT-6-s are close to the line at 227 cm -1 , attributed to disorder in the MoS2 nanoslabs. [START_REF] Blanco | Resonance Raman Spectroscopy as a Probe of the Crystallite Size of MoS2 Nanoparticles[END_REF] Weak signals at 970 cm -1 and 840 cm -1 Mo-KIT-6s suggest partial oxidation of Mo sulfidic species. [START_REF] Schrader | In Situ Laser Raman Spectroscopy of the Sulfiding of Moγ-Al2O3 Catalysts[END_REF] A weak line at 970 cm -1 observed for pure KIT-6 corresponds to a Si-OH vibration. [START_REF] Chen | Study on the Selective Oxidation of Methane over Highly Dispersed Molybdenum-Incorporated KIT-6 Catalysts[END_REF] For Mo/Al2O3-s a small peak at 920 cm -1 might be ascribed to Mo=O stretching in the supported molybdate. [START_REF] Jeziorowski | Raman and Ultraviolet Spectroscopic Characterization of Molybdena on Alumina Catalysts[END_REF][START_REF] Catita | Application of Magnetic Resonance Imaging and Raman Imaging to Study the Impact of Phosphorus in Impregnation of Hydrotreatment Catalysts[END_REF] Overall, Raman spectra suggest the presence of highly disordered MoS2-like clusters in Mo-KIT-6-s and Mo/Al2O3-s, whereas for Mo/AC-s, and Mo/graphene-s no signals of Mo sulfide species could be spotted. For Mo/TiO2-s, a strong signal from the support would anyways hide the peaks of the sulfide species.

UV-vis DRS

UV-vis DRS spectra were recorded only for the oxides -supported samples, because carbon-supported samples are black. In bulk MoS2, direct electronic transitions between conduction band and valence band result in two absorption peaks around 610 and 670 nm. [START_REF] Rusciano | Single-Cell Photothermal Analysis Induced by MoS2 Nanoparticles by Raman Spectroscopy[END_REF][START_REF] Forsberg | Exfoliated MoS2 in Water without Additives[END_REF] For the MoS2 catalysts supported on alumina, silica and titania, the UV -vis peaks of MoS2 were observed and easily distinguished from those of the corresponding supports even in air. [START_REF] Wang | Direct growth of molybdenum disulfide on arbitrary insulating surfaces by chemical vapor deposition[END_REF][START_REF] Cesano | Model Oxide Supported MoS2 HDS Catalysts: Structure and Surface Properties[END_REF][START_REF] Afanasiev | MoS2 "Inorganic Fullerenes" Combined with TiO2 in Water-Methanol Suspensions: Highly Active Hydrogen Production Photo Catalysts Operating via Transfer of Accumulated Electrons[END_REF][START_REF] Ghayeb Zamharir | Laser-Assisted Tunable Optical Nonlinearity in Liquid-Phase Exfoliated MoS2 Dispersion[END_REF] In our spectra, beside the absorption bands of supports, several broad peaks centered around 420, 470 and 570 nm were observed, but no bands were present above 600 nm (Fig. S3). This difference might be explained by ultradispersion of MoSx species, leading to modification of their electronic properties, or due to partial oxidation of MoS2 nanoclusters in air. [START_REF] Ko | On-Stack Two-Dimensional Conversion of MoS2 into MoO3[END_REF] Therefore our UV-vis DRS results corroborate the absence of MoS2 slabs.

X-ray photoelectron spectroscopy (XPS)

The XPS spectra were obtained for the sulfided samples without air exposure. For all samples single Mo species were detected, with the binding energy (BE) 3d5/2 varying in the interval [START_REF] Schauble | Uncovering the Effects of Metal Contacts on Monolayer MoS2[END_REF] The S 2p spectra are different depending on the nature of the support. For Mo-KIT-6-s, due to low Mo loading, a strong Si 2s signal and the corresponding plasmon hide the S 2p line. On graphene and AC, S 2-and S2 2-species were identified with BE 2p3/2 around 162.3 and 164.1 eV respectively. [START_REF] Qie | Sulfur-Doped Carbon with Enlarged Interlayer Distance as a High-Performance Anode Material for Sodium-Ion Batteries[END_REF] On the surface of Al2O3 and TiO2 oxides, along with S 2-and S2 2-, oxidized sulfur species with BE 2p3/2 at 168.1 and 166.7 eV were also present (SO4 2-, SO3 2-), despite the fact that all samples were prepared for XPS under Ar. A contribution of SOx species on the oxidic supports might be related to sulfur species on the supports surface. As the Mo loading is low, the S signal coming from the sulfur species adsorbed on the supports becomes non-negligible. The XPS Mo/S atomic ratio in the oxide-supported samples is greater than two (Table S2) and therefore a major part of S species is not associated with molybdenum, but bonded with the oxygenated supports moieties, prone to form oxygenated sulfur species.

Transmission electron microscopy (TEM) and scanning transmission electron microscopy (STEM-ADF).

TEM and STEM-ADF were carried out ex situ for the sulfided samples. For the Mo-KIT-6-s, mesoporous ordered structure of the support was revealed at medium magnifications by TEM and STEM (Fig. S4), but at high magnifications the TEM images were featureless (Fig. S4a,b).

For the Mo/Al2O3-s, only acicular γ-Al2O3 particles were spotted with interplanar distances 0.26 nm, corresponding to the (2 2 0) planes (Fig. S5c,d). Small (5-10 nm) oxide nanoparticles were observed for Mo/TiO2-s, showing the (1 0 1) planes of anatase at 0.35 nm spacing (Fig. S5e,f).

Only graphite (002) planes with 0.338 nm spacing were visible on the Mo/AC-s and Mo/graphenes (Fig. S5g, h; Ref. 12). Meanwhile, EDS analysis confirms that Mo is present in all these samples, uniformly spread over the analysis zones (Fig. S6). Therefore, the Mo species are smaller than the size detectable by conventional TEM. Notably, detection of 1-2 nm length MoS2 slabs in the conventional supported catalysts is a routine task, while using the same microscope and the same observation conditions. [START_REF] Chen | Study on the Selective Oxidation of Methane over Highly Dispersed Molybdenum-Incorporated KIT-6 Catalysts[END_REF] In the TEM and STEM images of 10 wt% Mo references, abundant MoS2 slabs were easily detected (Fig. S7).

The STEM-ADF technique allows distinguishing small Mo-containing species from lighter atoms of supports (C, O, Al, Si, Ti), due to a strong Z-contrast. Mostly single atomic species and agglomerates of several atoms were observed in our samples (Fig. 2 and Fig. S8). The dispersion of Mo species seems to be different as a function of support. Thus, for Mo supported on both carbons and TiO2 only small subnanometer clusters were observed (Fig. 2g, h, c, d, Fig. S9, Ref.

[12]), whereas for Mo/Al2O3-s and Mo-KIT-6-s nanometer-sized Mo clusters were encountered (Fig. 2a, b, e, f). Statistical treatment of images suggests that the Mo species on carbons and TiO2 are more dispersed than on alumina and silica (Fig. S9,S10).

In summary, we evidenced that ultradispersed MoSx species are present in the samples under study. However, only operando characterization allows obtaining reliable structural information under controlled conditions. Therefore, we proceeded with operando QXAS study of our systems, as follows.

Operando QXAS investigation

Operando QXAS study aimed to clarify the structure of the ultradispersed species, their genesis during the sulfidation step and their evolution during the HDS reaction. We inquired whether the obtained sulfide species are similar to the conventional sulfide structures or not, and how the nature of the support influences their genesis. Drastic changes of XAS spectra occurred during the sulfidation, but almost no evolution was observed during the following HDS reaction (Fig. S11,12). Therefore, the discussion is focused mostly on the sulfidation step.

The initial and the final states

Fig. 3 and Table 1 summarize the results of XAS study for the initial samples. For all samples, the initial Mo species are monomers or few-atom oxidic clusters. However, the structure of Mo species varies considerably as a function of support. Monomeric molybdate is predominant on alumina, whereas polymolybdates (probably aluminium hexamolybdate) are present in the 10 wt% Mo/Al2O3-i reference, in agreement with the literature. [START_REF] Carrier | Ligand-Promoted Alumina Dissolution in the Preparation of MoOx /γ-Al2O3 Catalysts: Evidence for the Formation and Deposition of an Anderson-Type Alumino Heteropolymolybdate[END_REF] On both carbon supports dimeric "Mo blue" species were formed as described in our previous work. [START_REF] Ryaboshapka | Ultradispersed (Co)Mo Catalysts with High Hydrodesulfurization Activity[END_REF] On the oxide supports, different species are formed despite the same precursor (AHM) and the same Mo loading, in agreement with earlier studies on the support effect. [START_REF] Breysse | Overview of Support Effects in Hydrotreating Catalysts[END_REF][START_REF] Hensen | The Relation between Morphology and Hydrotreating Activity for Supported MoS2 Particles[END_REF][START_REF] Pratt | Morphology and activity of MoS2 on various supports: Genesis of the active phase[END_REF] Alongside monomers, some silicomolybdic (Keggin) anions are probably present in the Mo-KIT-6-i sample, easily formed by the reaction between the dissolved Mo7O24 6-and silica surface, so expected to be produced in the conditions applied for the preparation. [START_REF] Dou | Targeted Synthesis of Silicomolybdic Acid (Keggin Acid) inside Mesoporous Silica Hollow Spheres for Friedel-Crafts Alkylation[END_REF][START_REF] Nikulshina | MoW Synergetic Effect Supported by HAADF for Alumina Based Catalysts Prepared from Mixed SiMonW12-n Heteropolyacids[END_REF] In the Mo/TiO2-i, polymeric species coexist with monomers, despite low Mo loading and high specific surface area of the TiO2 support. A pre-edge is present in the XANES of all initial samples, attesting a non-centrosymmetric coordination environment (Fig. S13). Detailed characterization and discussion of the oxidic Mo species on different supports is beyond the scope of this study. In general, the state of the initial Mo oxidic species is defined by such properties of the support as the value of isoelectric point (IEP), the nature and abundance of surface functional groups and the specific surface area, as demonstrated in many earlier works using vibrational spectroscopies and XAS. [START_REF] Kim | Molecular Structures and Reactivity of Supported Molybdenum Oxide Catalysts[END_REF][START_REF] Hu | Surface Structures of Supported Molybdenum Oxide Catalysts: Characterization by Raman and Mo L3-Edge XANES[END_REF][START_REF] Tougerti | Synergy between XANES Spectroscopy and DFT to Elucidate the Amorphous Structure of Heterogeneous Catalysts: TiO2-Supported Molybdenum Oxide Catalysts[END_REF][START_REF] Hu | Investigation of the Reducibility of Supported Oxomolybdate Species for Mapping of Active Centers of Partial Oxidation Reaction: In Situ Mo K-Edge XAS and DFT Study[END_REF] The final sulfided state was attained in all samples during the temperature increase step and showed no further evolution during the subsequent temperature plateau (1 h at 350 °C in H2S/H2 flow). The spectra acquired in H2S/H2 at 350 °C and the results of fitting are shown in Fig. 4 and Table 2. The coordination number (CN) values in Table 2 attest that in none of the samples Mo atom had a complete MoS2-like shell of six S atoms. Moreover, the spectra of all low-loaded samples demonstrate drastic differences as compared with the 10 wt% Mo/Al2O3-s reference: the Mo-S and Mo-Mo coordination numbers are considerably lesser, whereas the Mo-S and Mo-Mo distances are slightly shortened vs. the reference, which contains highly dispersed, but TEM-observable MoS2 slabs. [START_REF] Afanasiev | Calculation of MoS2 Slabs Morphology Descriptors from Transmission Electron Microscopy Data Revisited. Case Study of the Influence of Citric Acid and Treatment Conditions on the Properties of MoS2/Al2O3[END_REF] This agrees with formation of ultradispersed species: upon an increase of dispersion the coordination numbers should decrease, whereas contraction of chemical bonds in the small clusters is generally expected vs. the corresponding bulk materials. 68 a coordination number, using S0 2 = 1.0; b radial distance; c distance variance (free variable between 0.003 and 0.010); d energy variance (free variable between -10 and 10 eV), R-factorminimized function of the fitting parameters, characterizes the goodness of the fit. XANES spectra for sulfide samples are similar, as XANES of highly dispersed MoS2 and amorphous MoSx have similar shape and are relatively featureless (no pre-edge and no pronounced white line). [START_REF] Afanasiev | Low-Temperature Hydrogen Interaction with Amorphous Molybdenum Sulfides MoSx[END_REF] In summary, analysis of the EXAFS spectra after sulfidation suggests that dispersion of Mo species changes in the sequence: Mo/AC-s > Mo/TiO2-s ≈ Mo/graphene-s > Mo/Al2O3-s ≈ Mo-KIT-6-s >> 10Mo/Al2O3-s. In the Mo/AC-s and Mo/TiO2-s samples the Mo species significantly differ from the MoS2 reference, whereas in Mo/graphene-s, Mo-KIT-6-s and Mo/Al2O3-s they might be considered as MoS2-like few-atom clusters. Remarkably, there is no correlation between the XAS-derived dispersion of Mo species and physical properties of supports, reported as being important for the MoS2 phase dispersion. Indeed, subdivision into "weakly" and "strongly" interacting ones cannot be a guideline. Thus, carbon, considered as a "weakly interacting" support [START_REF] Kouzu | Catalytic Potential of Carbon-Supported NiMo-Sulfide for Ultra-Deep Hydrodesulfurization of Diesel Fuel[END_REF] gives the most dispersed species, whereas "strongly interacting" titania and alumina give respectively the second highest and the least dispersed Mo sulfide species. On the other hand, our results are in a general agreement with the previous experimental works on conventional sulfide catalysts, where the same supports were compared. Thus, titania-supported MoS2 was found to be better dispersed than alumina -supported one. [START_REF] Ramirez | Hydrodesulphurization Activity and Characterization of Sulphided Molybdenum and Cobalt-Molybdenum Catalysts: Comparison of Alumina-, Silica-Alumina-and Titania-Supported Catalysts[END_REF] By the same token, carbonsupported MoS2 was better dispersed and more active than silica-or alumina-supported one. 73 The nuclearity of the initial oxide species was not correlated to that of the final sulfide ones: the most dispersed MoSx was obtained from the dimers on AC and mixed monomersoligomers on TiO2, whereas the monomers prevailing in the initial Mo/Al2O3, were transformed to a lesser dispersed MoSx. Therefore, during sulfidation, migration of Mo atoms and full reorganization their coordination sphere occur.

Sulfidation pathways

Many experimental and theoretical works on the genesis of MoS2 from oxide precursors in the bulk and supported catalysts, converge on the conclusion that sulfidation is a multistep process The intermediates structure should obviously depend on the nature of the supported species and of the support, as well as on the sulfidation conditions. Low loading allows better revealing the support effect. Indeed, all the atomically dispersed Mo species in both the initial and the final states are involved in some interaction with the support.

Multivariate Curve Resolution with Alternating Least Squares (MCR-ALS) algorithm was used to separate large series of acquired spectra (usually 300-500 in our experiments) containing mixtures of initial/final products and reaction intermediates into their individual components and determine their concentrations during the sulfidation process. Assuming that the measured spectrum of the mixture can be represented as a linear combination of the spectra of pure components scaled by their concentration, the algorithm iteratively alternates between the steps of pure spectra resolution and concentration estimation. At the end we treated the spectra of pure components in the same manner as experimental XAS and analyzed the concentrations profiles vs time or temperature, as follows.

Using the MCR-ALS procedure, four components accounting for the structural evolution of the catalysts under sulfidation conditions were identified, as presented in Fig. 5. Notably, this scheme does not require participation of hydrogen. Indeed, sulfidation with pure H2S or H2S/N2 is known to provide supported MoS2 catalysts with similar catalytic properties as sulfidation with H2S/H2. 35 In the following sections we describe the spectra of pure components as obtained by MCR-ALS on different supports and then compare their evolution.

1 st components: The first component corresponds to the initial oxide species (Fig. 6a), already discussed above. It is worth mentioning difficult interpretation of the shell with the Mo-X distance above 2.3 Å, observed for 10%Mo/Al2O3-i and Mo-KIT-6-i (Table 1). Sometimes reported Mo-Al or Mo-Si intermetallic bonds [START_REF] Leliveld | A Mo-K Edge XAFS Study of the Metal Sulfide-Support Interaction in (Co)Mo Supported Alumina and Titania Catalysts[END_REF] were ruled out, because they cannot be formed in the dried or air-calcined samples. The Mo-O-X (X= Ti, Al, Si) bonds in the heteropolymolybdates are much longer, in the range 3.2-3.5 Å (Fig. S14-16). Finally, we attributed this shell to the Mo-O bonds in polymolybdates, involving μ3-oxygen. Such bonds with a length up to 2.3 -2.4 Å exist in both AHM and AlMo6 anions (Fig. S16).

d components:

The second component is an "oxysulfide" (Fig. 6b, Table S3). Its structure is strongly affected by the nature of the support (Fig. 7). For the "strongly interacting" Al2O3 and TiO2, the Mo-O bonds are preserved in the Mo coordination shell, and fewer sulfur ligands are present, probably due to the persistent Mo-O-X species (X = Al, Ti). For the "weakly interacting" carbon supports, the degree of O to S exchange is higher. For Mo/AC, Mo/graphene and Mo-KIT-6, the second components are similar: Mo possesses four sulfur ligands, one residual oxygen and two types of Mo neighbors. One Mo -Mo distance at 3.38 Å is due to the remaining Mo -O -Mo moieties, while another one at 2.80 Å is typical for the Mo-Mo distances in sulfides, such as MoS3.

Remarkably, for the Mo/Al2O3 sample the initial oxide species are monomers, but Mo-S-Mo bonds appear in the component 2. Therefore, the O to S exchange is accompanied by polymerization (Fig. 6a,b). The Mo=O to Mo-S exchange seems to be an easy process. In solutions, monomolybdate MoO4 2-exchanges oxygen with sulfide ions at room temperature, forming oxothiomolybdates. [START_REF] Mcdonald | Syntheses and Characterization of Ammonium and Tetraalkylammonium Thiomolybdates and Thiotungstates[END_REF] Apparently, at the surface of a catalyst, the Mo=S bonds, once formed, tend to oligomerize with reduction from Mo(VI) to Mo(V) and formation of S2 2-species.

Indeed, MoO3/Al2O3 catalysts exposed to H2S at room temperature become dark and show a strong Mo(V) signal in the EPR spectra (Fig. S17). The fourth component is fully sulfided species, as discussed above. Note again a decreased Mo-S CN and shortened Mo-Mo and Mo-S distances for all ultradispersed catalysts as compared with the 10Mo/Al2O3-s reference and bulk MoS2, which agrees with the prevalence of few-atom clusters. Remarkably, the Mo-Mo and Mo-S bonds shortening is the strongest for the TiO2-and AC-supported catalysts, for which the dispersion is the highest (Table 2). As compared with the component 3, in the final state the Mo-S and Mo-Mo CNs are lesser, suggesting that MoS3-like species lose sulfur, but also form smaller structures (Fig. 7, Tables 2 andS3).

Overall, the structure of two identified intermediates is in a general agreement with other experimental QXAFS results. [START_REF] Rochet | Intermediate Species Revealed during Sulfidation of Bimetallic Hydrotreating Catalyst: A Multivariate Analysis of Combined Time-Resolved Spectroscopies[END_REF][START_REF] Nikulshina | Genesis of Active Phase in MoW/Al2O3 Hydrotreating Catalysts Monitored by HAADF and in Situ QEXAFS Combined to MCR-ALS Analysis[END_REF] and DFT calculations [START_REF] Sahu | Genesis of MoS2 from model-Mo-oxide precursors supported on γalumina[END_REF] for alumina-supported catalysts.

However, Mo ultradispersion and variations of the support lead to significant structural differences.

Evolution of components in function of temperature

For all samples, the component 2 ("oxysulfide") appears already at room temperature (Fig. 8). During the settling down of the gas flows and before the beginning of heating ( 5 In conclusion, the nature of the support has a strong influence on the relative kinetics of the sulfidation steps. Oxysulfide intermediate tends to have higher concentrations and/or to survive longer on the oxide supports. On the contrary, the sulfur-rich component 3 is more stable on the weakly interacting carbon supports. A consecutive-parallel transformation occurs for our samples potentially involving all transitions depicted in Fig. 5.

XAS spectra recorded during the HDS reaction

Operando QXAS of atomically dispersed catalysts gives a unique opportunity to probe onstream evolution of the catalytic sites. Indeed, in the nanoparticulate catalysts only a minor part of Mo atoms belongs to the catalytically active edges, so the evolution of their coordination as probed by XAS could be screened by the majority of in-slab Mo atoms. On the contrary, in the atomically dispersed clusters all Mo atoms are exposed to the gaseous reactants and the variations of their coordination are expected to be much higher. Surprisingly, no significant changes of XAS spectra occurred during the HDS reaction and subsequent exposure to pure H2 at 350 °C. The mass-spectra attested high thiophene conversions that rapidly decay during the first several hours of operando HDS tests (Fig. S18,19). Also, a significant decrease of HDS activity occurred during the first hours on-stream in the laboratory HDS tests (Fig. S20), which is a usual settling-down phenomenon. However, our results suggest that this process does not involve any significant reorganization of the Mo coordination. This finding might seem counterintuitive, but could be easily explained on the basis of our previous works. For a qualitative example, in both MoS2 and MoS3 molybdenum has the same Mo-S CN of 6, i.e. MoSx can lose sulfur without a decrease of the coordination number. As shown earlier, during the initial period of HDS reaction MoS2 phase loses considerable amount of hydrogenating SH groups, before reaching a steady state. [START_REF] Afanasiev | The influence of reducing and sulfiding conditions on the properties of unsupported MoS2-based catalysts[END_REF] These SH groups are the only form of adsorbed hydrogen. [START_REF] Afanasiev | On hydrogen adsorption by nanodispersed MoS2-based catalysts[END_REF] When two adjacent Mo-SH groups are condensed (or a Mo-(S)2-Mo moiety is reduced) toward a Mo-S-Mo one, the HDS activity decreases, but the Mo-S and Mo-Mo coordination numbers remain the same. As a result, the loss of extra sulfur, while important for catalysis, is not detected by operando XAS.

Catalytic activity in thiophene HDS.

The HDS activity per Mo atom changes in the sequence: Mo/graphene-s ≈ Mo/AC-s > Mo/TiO2-s > Mo-KIT-6-s > Mo/Al2O3-s (Fig. 9). The Mo/graphene-s and Mo/AC-s samples demonstrated the highest specific activity around 5.5 h -1 (thiophene molecules per Mo atom), while Mo/Al2O3-s was the least active, with the value of 0.94 h -1 . Interestingly, a decrease of catalytic activity from the beginning of the reaction (0.5 -1 h) to the steady state (16 h) was observed: activity was divided by 2.3 and 2.8 for Mo/graphene-s and Mo/AC-s, by 4.7 for Mo-KIT-6-s and by 2.2 for Mo/Al2O3-s, respectively, but only by 1.3 for Mo/TiO2-s. The HDS rates measured during the first two hours of the reaction are shown in Fig. S20. In general, the stronger is the support -active phase interaction, the lesser is the activity evolution. Indeed, the stronger is the interaction with support, the lesser is the amount of (dangling, free form interaction with the support) extra sulfur species. As explained above, the loss of extra sulfur occurs without decrease of Mo-S coordination number and escapes from the observation by operando XAS. The main products for the oxide-supported catalysts are butenes and butane; low amounts of light hydrocarbons (mainly CH4 and C2H6) were also detected (Fig. S21). Mo/TiO2-s possesses high overall activity and the highest hydrogenating ability in agreement with the previous results. [START_REF] Escobar | Highly Active Sulfided CoMo Catalyst on Nano-Structured TiO2[END_REF][START_REF] Signorile | MoS2 supported on P25 titania: A model system for the activation of a HDS catalyst[END_REF] The products distribution for Mo-KIT-6-s and Mo/Al2O3-s is similar, only the amount of light products (mostly, C1-C2) is higher for KIT-6. The product distribution is different for carbonaceous supports: significant amounts of propane being formed instead of the C1-C2

products. Isobutane was also spotted, nevertheless butenes remain the main products. The amount of light products in all cases diminishes with time. Such variations might be related to secondary processes that occur on the acid sites of supports and are beyond the scope of our study.

Discussion

The relationship between the properties of the support, the Mo dispersion and the HDS catalytic activity appears to be not straightforward. There is no single descriptor property (surface area, acidity, number of OH groups...) that would allow predicting better supports. No correlation between the intrinsic activity and the loading per square nm of support (Table S4) could be inferred. The two carbon-supported samples are the most active and show the highest dispersion of active species. In general, carbon is considered as a "weakly interacting" support which favors mobility and aggregation of Mo species leading to high sulfidation degree and formation large slabs of equilibrium triangular and hexagonal shape. [START_REF] Brorson | The Morphology of MoS2, WS2, Co-Mo-S, Ni-Mo-S and Ni-W-S Nanoclusters in Hydrodesulfurization Catalysts Revealed by HAADF-STEM[END_REF][START_REF] Bara | Surface-dependent sulfidation and orientation of MoS2 slabs on alumina-supported model HDS catalysts[END_REF] However, high specific surface area is favorable for high dispersion of Mo species, whereas weak sulfide-support interaction leads to a greater number of active sites than on alumina. [START_REF] Kogan | 35 S Tracer Study of the Effect of Support on the Nature of the Active Sites of Co(Ni)Mo Sulphide Catalysts Supported on Al2O3 and Activated Carbon[END_REF] As with the difference between two carbon supports, activated carbon possesses functional groups [START_REF] Abotsi | A Review of Carbon-Supported Hydrodesulfurization Catalysts[END_REF] that probably decrease the Mo mobility and favor dispersion on Mo/AC-s even more than in case of Mo/graphene-s. Notably, despite the highest dispersion, the HDS activity of Mo/AC-s is not the highest but is slightly lower than for Mo/graphene-s (Fig. 10). It seems possible that single atomic sulfide species present in the Mo/AC-s sample are not active in HDS, but a minimal oligomeric structure (at least a dimer or a trimer) is necessary for the dissociative adsorption of hydrogen and for the thiophene HDS. Unlikely to carbons, ultradispersed Mo species on TiO2 are stabilized by interaction with the oxide surface. Having similar Mo-Mo CN, the Mo species on the strongly interacting TiO2 are intrinsically less active than on carbons, but more still active than on silica and alumina, in agreement with the previous studies. [START_REF] Wang | Quasi-Single-Layer MoS2 on MoS2/TiO2 Nanoparticles for Anthracene Hydrogenation[END_REF] Beside increased dispersion, the electronic interactions seem to play an important role: theoretical calculations suggest that under HDS conditions, TiO2 surfaces stabilize more sulfur-deficient MoSx clusters than alumina surfaces. [START_REF] Arrouvel | A density functional theory comparison of anatase (TiO2)and γ-Al2O3-supported MoS2 catalysts[END_REF] According to the EXAFS results, MoSx clusters possess similar structure on silica and on alumina, with a dispersion lower than on carbons and titania. Overall, the HDS activity follows a general correlation with dispersion (Fig. 10), but specific chemical interactions seem also to play an important role.

The advantage of ultradispersed catalysts over the corresponding 10%wt Mo references in terms of per Mo atom HDS activity appears to depend on the nature of the support. For carbon supports, the ultradispersed catalysts have more than twice higher intrinsic activity than the conventional 10 % wt. Mo systems (Fig. S22). For the oxide-supported systems this ratio is lower, being 1.7 for TiO2 and 1.4 for Al2O3. If we suppose that all Mo atoms in the highly-loaded catalysts are present only in the form of slabs, whereas the TOF of exposed Mo atoms at the edges and in the small clusters is the same, then the expected activity ratio between them should be around 3 (because in the highly dispersed supported MoS2 slabs about 1/3 of Mo atoms are exposed at the edges, [START_REF] Baubet | The Influence of MoS2 Slab 2D Morphology and Edge State on the Properties of Alumina-Supported Molybdenum Sulfide Catalysts[END_REF] whereas we suppose that all Mo atoms are exposed in the small clusters). The observed activity ratios in Fig S22 are lower than 3, which is naturally explained by the presence of significant amounts of few-atom clusters even in the high-loaded catalysts. The difference of the intrinsic activity between the high-loaded and low-loaded samples in function of the support probably depends on the distribution of Mo species between the few-atom clusters and the slabs.

It would be highly instructive to follow such distribution as a function of the support and of the loading, but that remains beyond the frames of the present study.

Conclusions

We prepared ultradispersed MoSx species on different supports, including carbons, silica, titania and alumina and confirmed atomic dispersion by characterizations (TEM, STEM, Raman, UV-vis DRS). Such catalysts show high intrinsic (per Mo atom) HDS activity in the absence of significant amounts of MoS2 slabs. The intrinsic HDS activity of ultradispersed catalysts is always superior to the conventional 10 wt% Mo references. This gain of activity is the highest for the carbon supports and the lowest for alumina. The ratio of intrinsic activity between low-loaded and 10 %wt Mo is probably dependent on the distribution of molybdenum between the small clusters and the slabs, which is in turn determined by the nature of support, namely by the strength of interaction with Mo species. If follows from our results that for any sulfide catalyst, independently on the support and the loading, TEM-invisible subnanomertic clusters potentially provide a significant contribution to the observed HDS activity. The amount and the stability of ultradispersed species would obviously vary in function of the support nature, the Mo loading and the conditions. Previously overlooked, the contribution of ultradispersed species is therefore accountable for a considerable part of the phenomena known as "support effect". Though specific chemical support effects seem to be significant, a general correlation exists between the dispersion of MoSx species and their intrinsic HDS activity.

By means of operando QXAS study we demonstrated that evolution of ultradispersed MoSx species during the sulfidation process occurs via a subsequentparallel pathway including oxysulfide and MoS3-like intermediates. However, the structure of these intermediates and their relative stability is affected by the nature of support. The structure of fully sulfided MoSx species depends on the support. While MoS2 -like clusters are formed on silica and alumina, the MoSx species are significantly different on carbons and titania. As the kinetics of the components transformation is substantially different for different supports, in the further work it seems possible to build reaction kinetic models for these components and to extract energies of transformations, in order to compare them with the ab initio calculations.

  8 mrad grazing incidence compared to the pink and monochromatic beams delivered by the toroidal collimating mirror and the channel-cut monochromator, respectively. A set of 3 ionization chambers filled with 50:50 Ar:N2 mixture was used for measuring the sample transmission and the transmission of a Mo metallic foil used as internal absolute energy calibration. The spectra were recorded during the samples treatment in the following sequence: (1) the initial sample was put under Ar flow; (2) H2 and H2S were introduced into system and the sulfidation/reduction was performed via heating up to 350 °C, heating rate 5 °C.min -1 ; (3) the sample was maintained at 350 °C for 1 h; (4) H2S flow was switched off and thiophene was introduced to perform HDS reaction for 4 h; (5) thiophene was switched off and the sample cooled to room temperature in H2 flow, rate 20 °C.min -1 ; (6) H2 flow was switched off and the system was purged with Ar.

Figure 1 .

 1 Figure 1. XPS spectra for Mo-KIT-6-s (a-b), Mo/Al2O3-s(c-d), Mo/TiO2-s (e-f), Mo/graphene-s (gh).

Figure 2

 2 Figure 2 STEM-ADF images of Mo-KIT-6-s (a,b), Mo/TiO2-s (c,d), Mo/Al2O3-s (e,f), Mo/graphene-s (g,h). Selected Mo clusters and SA are marked with circles.

Figure 3

 3 Figure 3 Illustration of the possible structures for the initial Mo oxide species on (a) carbon; (b) TiO2; (c) Al2O3; (d) 10% Mo/Al2O3-i reference; (e) KIT-6.

Figure 4

 4 Figure 4 Experimental EXAFS spectra recorded in H2S/H2 at 350 °C, compared with 10Mo/Al2O3-s and bulk MoS2 references.

  involving intermediates. As suggested by Weber et al., sulfidation of crystalline MoO3 occurs via an oxysulfide low-temperature intermediate formed via exchange of terminal O 2-ligands of the oxide for S 2-. In the subsequent reactions, bridging S2 2-ligands and Mo 5+ centers are formed. Then, above 200°C, reduction with H2 occurs, leading to oxygen elimination with formation of MoS2.[START_REF] Weber | Basic Reaction Steps in the Sulfidation of Crystalline MoO3 to MoS2, As Studied by X-Ray Photoelectron and Infrared Emission Spectroscopy[END_REF] For supported systems, formation of amorphous sulfur-rich MoS3 -like intermediate was considered beside the oxysulfide one. Rochet et al studied sulfidation of alumina-supported molybdate by QXAS/MCR-ALS and concluded on the subsequent formation of two intermediates.[START_REF] Lesage | High Pressure Cell for Edge Jumping X-Ray Absorption Spectroscopy: Applications to Industrial Liquid Sulfidation of Hydrotreatment Catalysts[END_REF] The first oxysulfide intermediate transforms into the MoS3like second intermediate, that further decomposes to final MoS2 phase.[START_REF] Texier | Activation of Alumina-Supported Hydrotreating Catalysts by Organosulfides or H2S: Effect of the H2S Partial Pressure Used during the Activation Process[END_REF][START_REF] De Jong | Sulfidation Mechanism by Molybdenum Catalysts Supported on Silica/Silicon(100) Model Support Studied by Surface Spectroscopy[END_REF] Recently, Raybaud and coll. studied the sulfo-reduction mechanisms of Mo oxide oligomers supported on γ-alumina by DFT and computed activation free energies of key chemical and diffusion steps. Interfacial O-atoms (Mo-O-Al) appeared as the most reluctant sites to be exchanged with S-atoms. S-and O-removal steps are kinetically too expensive on small-size Mooxysulfide oligomers. Formation of Mo trisulfide oligomers occurs prior to S-removal. An optimal path to MoS2 via Mo trisulfide oligomers was proposed.[START_REF] Sahu | Genesis of MoS2 from model-Mo-oxide precursors supported on γalumina[END_REF]

Figure 5

 5 Figure 5 Scheme of sulfidation steps (as inspired by Ref. [79]) and the corresponding MCR-ALS components (see Fig. 7, 8).

Figure 6 .

 6 Figure 6. Illustration of the possible steps of sulfidation process as exemplified by Mo/Al2O3: (a) initial molybdate monomers (component 1); (b) oxysulfide species with (S2 2-) bridges (component 2); (c) sulfur-rich MoS3-like species (component 3); (d) MoS2 -like structures (component 4). The structures are MM-optimized and given for visual presentation only.

  -10 min) it raises to 20-40 % relative intensity. For Mo/AC, Mo/graphene and Mo/TiO2 samples, the maximum fraction of the component 2 is significantly lower than the one for the component 3, suggesting that the component 3 is formed more rapidly (directly from the initial oxide, or from component 2). On the other hand, for Mo/TiO2 the component 2 remains present even when the component 3 is completely consumed, which means that the component 4 ("MoS2") is formed directly from the component 2 ("oxysulfide"). With this respect, there is a striking difference between "weakly interacting" carbons and "strongly interacting" TiO2, on which oxysulfide persists almost up to the end of the temperature ramp. The MoS2-like component 4 appears already above 100°-150° C for all samples, except carbons, where it appears above 200° C. In the previous work by Rochet et al,[START_REF] Rochet | Edges Quick-XAS Study of the Sulphidation Properties of Mo/Al2O3 and CoMo/Al2O3 Catalysts[END_REF] sulfidation of the 5 wt% Mo/Al2O3 catalyst occurred as almost perfect 1 => 2 => 3 => 4 transformation, in which MoS2 was produced from the component 3 only, above 200 °C. In our case, consecutiveparallel transformations were observed, in which the component 4 appeared when the components 1 and 2 were still important. Only for carbon supports almost pure component 3 is formed at 220 °C and then rapidly transformed into 4 above 300-320° C. It suggests that for the "weakly interacting" carbons relatively free-standing MoS3 is more stable than similar species smeared over an oxide.

Figure 8 .

 8 Figure 8. Evolution of molar fractions of the components vs temperature (a) Mo/TiO2 (b) Mo/Al2O3 (c) Mo-KIT-6 (d) Mo/graphene (e) Mo/AC (f) ref 10Mo/Al2O3.

Figure 9 (

 9 Figure 9 (a) Specific activity (per Mo atom) in HDS of thiophene at 320 °C; (b) Thiophene HDS rates (per g of catalyst) at the beginning of the reaction and in the steady state.

Figure 10

 10 Figure 10 Dependence between the catalytic activity and Mo-Mo coordination number, determined from EXAFS of sulfided samples.

  Typical BE 3d5/2 for Mo(IV) in MoS2 is around 229 -229.2 eV[START_REF] Plais | Catalyseurs d'hydrotraitement Mo/Al2O3 doublement promus preparés à partir d'hétéropolyanions d'Anderson: Suivi de la sulfuration par Quick-EXAFS[END_REF][START_REF] Wang | XPS Studies of MoS2 Formation from Ammonium Tetrathiomolybdate Solutions[END_REF] and here we observe the same energies for Mo/AC-s and Mo/graphene-s. The BE was slightly decreased for Mo/Al2O3-s, Mo-KIT-6-s and Mo/TiO2-s. The lower binding energy values correspond to the MoS2-x sulfur-deficient species.

228.0 -229.3 eV (228.8 eV for Mo-KIT-6-s, 228.6 eV for Mo/Al2O3-s, 229.3 eV for Mo/graphenes, 229.2 for Mo/AC-s, 12 and 228.6 eV for Mo/TiO2-s). A variance in BE exists as a function of support (up to 0.6 eV).

Table 1 EXAFS fitting results for the initial oxide samples

 1

	Scatterer	CN a	R(Å) b	σ 2 (Å 2bc	ΔE0 (eV) d
	atom
			Mo/graphene-i R= 0.09
	O	2.1(2)	1.70(3)	0.004(1)	1(2)
	O	1.6(5)	1.93(2)	0.004(1)	1(2)
	Mo	1.7(2)	3.26(2)	0.006(1)	0(2)
			Mo/AC-i R=0.17
	O	2.2(3)	1.71(2)	0.004(1)	-1(2)
	O	1.5(3)	1.92(2)	0.005(1)	0(3)
	Mo	1.8(2)	3.29(3)	0.005(1)	-1(2)
			Mo/TiO2-i R=0.19
	O	1.7(3)	1.72(3)	0.004(1)	5(3)
	O	1.6(3)	1.95(2)	0.005(1)	1(2)
	Mo	1.0(2)	2.91(2)	0.006(1)	1(2)
	Mo	0.7(2)	3.34(2)	0.007(1)	-1(2)
			Mo/Al2O3-i R= 0.07
	O	3.7(4)	1.75(2)	0.004(1)	0(1)
			Mo-KIT-6-i R= 0.18
	O	1.4(2)	1.69(2)	0.004(1)	0(2)

Table 2 EXAFS fitting results for the sulfided samples (final state at 350 °C)

 2

	Scatterer	CN	R (Å)	σ 2 (Å 2 )	ΔE0 (eV)
	atom
			Mo/graphene-s R= 0.09
	O	0.5(2)	1.94(3)	0.004(1)	3(2)
	S	4.4(5)	2.39(2)	0.007(1)	3(2)
	Mo	0.7(2)	3.09(2)	0.006(1)	-3(2)
			Mo/AC-s R=0.09
	O	0.8(2)	2.00(4)	0.005(1)	-2(2)
	S	3.1(3)	2.38(2)	0.009(1)	4(3)
	Mo	0.2(2)	3.10(3)	0.005(2)	-3(3)
			Mo/TiO2-s R=0.11
	O	0.5(2)	1.94(3)	0.004(1)	-4(3)
	S	4.1(4)	2.36(2)	0.007(1)	-3(2)
	Mo	0.7(2)	2.78(2)	0.007(1)	-1(2)
			Mo/Al2O3-s R= 0.10
	O	0.5(2)	1.95(3)	0.006(1)	-5(3)
	S	4.4(5)	2.39(2)	0.007(1)	2(1)
	Mo	1.0(2)	3.11(2)	0.006(1)	-4(3)
			Mo-KIT-6-s R= 0.09
	O	0.1(2)	1.90(2)	0.004(1)	-2(2)
	S	4.2(5)	2.39(2)	0.007(1)	0(2)
	Mo	1.0(2)	3.11(2)	0.007(1)	2(2)
			10Mo/Al2O3-s reference R=0.10
	O	-	-	-	-
	S	5.2(3)	2.41(2)	0.006(1)	3(2)

Acknowledgements

This work was supported by a public grant overseen by the French National Research Agency (ANR) as a part of the "Investissements d'Avenir" (ref: ANR-10-EQPX-45) provided for the building of the ROCK beamline.

d components:

The EXAFS spectra of the third components are similar for all samples.

There is only a small contribution of oxygen (CN ≤ 0.6) at 1.65 -1.71 Å due to the residual Mo-O bonds. The number of S ligands increases significantly (Table S3), whereas the Mo-O-Mo fragments disappear. The Mo-Mo distance is 2.76 Å, with CN ranging from 0.8 to 1.5. A sulfurrich intermediate with MoS3-like structure is therefore formed on different supports, in agreement with the previous studies. [START_REF] Rochet | Edges Quick-XAS Study of the Sulphidation Properties of Mo/Al2O3 and CoMo/Al2O3 Catalysts[END_REF] 4 th components:


Alexandre Goldsztejn
email: alexandre.goldsztejn@univ-nantes.fr

Fr Éd Éric Goualard
email: frederic.goualard@univ-nantes.fr

Box Consistency through Adaptive Shaving

Keywords: G.1.0 [Numerical Analysis]: General-interval arithmetic Algorithms, Performance Local consistency, constraint programming, Newton method

The canonical algorithm to enforce box consistency over a constraint relies on a dichotomic process to isolate the leftmost and rightmost solutions. We identify some weaknesses of the standard implementations of this approach and review the existing body of work to tackle them; we then present an adaptive shaving process to achieve box consistency by tightening a domain from both bounds inward. Experimental results show a significant improvement over existing approaches in terms of robustness for difficult problems.

INTRODUCTION

Box consistency [START_REF] Benhamou | CLP(Intervals) revisited[END_REF] is a local consistency notion for continuous constraints. Stronger consistencies such as bounds consistency build on box consistency to achieve better domain narrowing, which makes it pivotal in solving difficult constraint systems. Box consistency is usually enforced on unary projections of constraints by a combination of binary search and interval Newton steps to isolate leftmost and rightmost "quasi-zeroes" in the domain of a variable.

The canonical algorithm to enforce box consistency, described in details by Van Hentenryck et al. [START_REF] Van Hentenryck | Solving polynomial systems using a branch and prune approach[END_REF], is quite robust and efficient on many problems. However, as we will see in Section 2, it is an intrinsically optimistic algorithm that always attempts to reduce large subdomains first during the dichotomic process without taking into account information on the successes of its previous attempts. As experimental evidences presented in Section 4 show, this behavior makes it a largely suboptimal algorithm for many problems.

Early on, McAllester et al. [START_REF] Mcallester | Three cuts for accelerated interval propagation[END_REF] proposed a different view of the original algorithm to enforce box consistency in which it is no longer described as a recursive dichotomic process but as some kind of shaving process where a strict enforcement of box consistency is no longer the ultimate goal. In practice, this approach works well, except for those difficult problems for which it pays off to spend more time narrowing domains further in the contracting operator and defer less to the binary search process of the solver.

Following this track, we present in Section 3 an algorithm that enforces box consistency by an adaptive shaving process that takes into account past difficulties encountered in narrowing a domain down. Experimental evidences reported in Section 4 show that our new algorithm has more regular performances than McAllester et al.'s algorithm, even more so for difficult problems, where the later may flounder.

BOX CONSISTENCY

Interval Constraint Programming makes cooperate contracting operators to prune the domains of variables (intervals with floating-point bounds from the set I) with smart propagation algorithms to ensure consistency among all the constraints. Exploration algorithms recursively split domains until a desired precision is achieved.

The amount of pruning obtained from one constraint is controlled by the level of consistency enforced. Box consistency considers interval extensions of the constraints. It is therefore parameterized by the interval extension used. In this paper, we only consider the natural interval extension of constraints, though our approach is not limited in any way to it. The same holds for the kind of constraints considered: while we limit ourselves to equations for the sake of simplicity, box consistency-and the algorithms presented-may be applied to inequations as well.

With these restrictions in mind, the definition of box consistency is as follows (using the notations for interval arithmetic advocated by Kearfott et al. [START_REF] Kearfott | Standardized notation in interval analysis[END_REF], where interval quantities are boldfaced): In order not to lose any potential solution, an algorithm that enforces box consistency of a constraint with respect to a variable xi and a box of domains must return the largest domain I ′ i ⊆ Ii that verifies Eq. ( 1). Given a real function f : R n → R and a box of domains B = I1 × • • • × In ∈ I n , we are hereafter only concerned with interval projections g : I → I of f with respect to B, of the form:

8 < : 0 ∈ f (I1, . . . , Ii-1, [Ii, Ii + ], Ii+1, . . . , In ) and 0 ∈ f (I1, . . . , Ii-1, [Ii -, Ii], Ii+1, . . . , In ), (1)
g(x) = f (I1, . . . , Ii-1, x, Ii+1, . . . , In ), i ∈ {1, . . . n}.
Algorithm bc3revise, presented by Alg. 1, enforces box consistency by searching the leftmost and rightmost canonical domains 1 in the domain I of x for which g evaluates to an interval containing 0. It is parameterized by the two procedures lnar and rnar that perform the search for the left and right bounds respectively.

Algorithm bc3revise first tries to move the right bound of I to the left, and then proceeds to move its left bound to the right (this order could be reversed with no impact on performances, though).

In the original algorithm [START_REF] Benhamou | CLP(Intervals) revisited[END_REF][START_REF] Van Hentenryck | Solving polynomial systems using a branch and prune approach[END_REF], the search within lnar and rnar is performed by a dichotomic search aided with Newton steps to accelerate the process. Algorithm 2 describes the search of a quasi-zero to update the left side of I. The procedure rnar vhmak to update the right bound works along the same lines and is, therefore, omitted.

The Newton procedure in Alg. 2 computes a step of the Interval Newton algorithm, that is, at iteration j + 1:

I (j +1) ← I (j ) ∩ " κ(I (j ) ) - g(κ(I (j ) )) g ′ (I (j ) ) « (2)
with κ(I) being a floating point number in I. The notation Newton (⋆) denotes the computation of Newton steps until reaching a fixpoint; we also use later the notation Newton (k) to denote an unspecified number of Newton steps (determined by the context). The "hull " notation ✷ (S) denotes the smallest interval (in the sense of set inclusion) containing the set S; the notation m(I) denotes the midpoint of the interval I.

1 A non-empty interval [a, b] is canonical if a + b.
5 if I = ∅: 6 return ∅ 7 if 0 ∈ g([I, I + ]): # Left bound is box consistent? 8 return [I, r] 9 else: 10 (I 1 , I 2 ) ← split(I) # I1 ← [I, m(I)], I2 ← [m(I), I] 11 I ← lnar vhmak (g, I 1 ) 12 if I = ∅: 13 return ✷ `lnar vhmak (g, I 2 ) ∪ {r} 14 else: 15 return ✷ `I ∪ {r} énd
Geometrically, an Interval Newton step reduces a domain I for an interval function g by computing the intersection between the x-axis and the area that contains all the lines passing through points in the segment (κ(I), g(κ(I)))-(κ(I), g(κ(I))) with slopes ranging in g ′ (I) (see Fig. 1 for an illustration). By the Mean Value Theorem, this area always encloses the graph of g on I, and thence its intersection with the x-axis contains all the solutions.

The Newton step expression (2) uses an extended version of the division (hereafter noted "⊘") to return a union of two open-ended intervals whenever g ′ (I (j ) ) contains 0. The subtraction and the intersection operators are modified accordingly: The intersection operator is applied to an interval (I (j ) ) and a union of two intervals (result of the subtraction), and returns an interval.

The lnar vhmak and rnar vhmak methods of the original algorithm use the midpoint m(I (j ) ) of I (j ) for κ(I (j ) ). Benhamou et al. [START_REF] Benhamou | CLP(Intervals) revisited[END_REF] considered using the left bound for lnar and the right bound for rnar but found that it could lead to slow convergence phenomena. On the other hand, as noted by McAllester et al. [START_REF] Mcallester | Three cuts for accelerated interval propagation[END_REF], a Newton expansion on a bound always yields some reduction of the domain.

For example, Fig. 2(a) presents a situation where the Newton method cannot tighten a domain when expanding on the center of the domain (green area), while expanding on the left bound allows to tighten the domain on the left significantly (red area). However, Fig. 2(b) shows that this ability to always achieve some reduction may be more a curse than a blessing for problems where the derivative varies widely on the domain considered: no reduction occurs on the original domain [-8, 10], leading to splitting it and investigating separately the left part [-8, 1] (red line below the x-axis); from there, each step of the Newton method with an expansion on the left bound is able to shave the domain ever so slightly, leading to a very slow convergence (the reductions due to Newton steps are shown by the dashed lines below the x-axis). On the other hand, expanding on the center of the domain exhibits a different, more efficient behavior: no reduction occurs on the original domain, as in the previous case; the left half sub-domain [-8, 1] (lowest red line above the x-axis) is investigated; no reduction occurs be- " κ(I), g(κ(I)) " " κ(I), g(κ(I)) " cause the smallest and largest slopes corresponding to the derivative of the function on [-8, 1] are so steep that only a small portion in the middle of [-8, 1] could be removed; since we restrict ourselves to intervals, the whole domain is kept unchanged. The algorithm then splits the domain and considers [-8, -3.5], which can be discarded immediately by a simple evaluation of the function on it; the right part [-3.5, 1] is then considered, and so on. Note that, on this example, the algorithm with an expansion on the center exhibits a quite inefficient behavior of its own in that Newton steps are performed but can never achieve any reduction due to the steepness of the slopes on the [0, 1] subdomain.

y = g ′ (I)(x -κ(I)) + g(κ(I)) y = g ′ (I)(x -κ(I)) + g(κ(I)) y = g ′ (I)(x -κ(I)) + g(κ(I)) y = g ′ (I)(x -κ(I)) + g(κ(I))

BOX CONSISTENCY BY SHAVING

McAllester et al. [START_REF] Mcallester | Three cuts for accelerated interval propagation[END_REF] consider the enforcement of box consistency as a shaving process where the bounds of a domain are moved inward by trying to discard "slices" of varying sizes on the left and right of the domain. The result is an algorithm whose most natural presentation is no longer recursive (cf. Alg. 2). We present in Alg. 3 a generic shaving algorithm for the left bound that uses the interval Newton operator 2 . The slice to be processed is defined by choosing a value p (Line 4). The reduction process is then applied to the slice guess = [I, p].

To shave a domain [a, b] on, say, the left side, we must choose a slice [a, p] ⊆ [a, b] that we will try to discard, and a value for κ([a, p]). The larger [a, p] the better, provided we are able to discard it. On the other hand, a large [a, p] 2 Note, however, that McAllester et al.'s algorithm slightly differs from Alg. 3 in that their purpose is only to move the left and right bounds inward by a "reasonable" amount (in their paper, they consider that reducing a domain by 10% or more is a reasonable goal to achieve for a contracting operator), which means that they may not enforce box consistency. All these situations are presented in Figures 3(a) and 3(b): Given the initial domain [-8, 10], choosing a small slice, such as [-8, -3.5], is often a sure-fire way of achieving some reduction, which may be large compared to the size of the slice, though it is proportionally small compared to the initial domain. Here, we obtain approximately [-3.68, -3.5] for the expansion point on the center (dark red cone in Fig. 3(a)), and [-4, -3.5] for an expansion on the left bound (dark red cone in Fig. 3(b)). For this problem, choosing a small slice leads to a large reduction (relatively to the size of the slice) for both expansion points. For some "well-behaved" functions, however, larger slices have the potential of larger reductions. The same dilemma arises in choosing the expansion point: Comparing Fig. 3(a) and Fig. 3(b), we see that for the slice [-8, -3.5], an expansion point on the center offers a better reduction than on the left bound; The reverse is true for the slice [-8, 0] (light red cones in Fig. 3(a) and Fig. 3(b)), where an expansion on the center leads to no reduction at all because the part to discard is strictly included in the domain considered. On the other hand, the expansion on the left bound always leads to some reduction, as promised above.

To shave the left part of a domain [a, b], the algorithm presented by McAllester et al. [START_REF] Mcallester | Three cuts for accelerated interval propagation[END_REF] first considers the slice formed by the whole domain. If the reduction obtained by a Newton step does not lead to a reduction by 10% or more, it considers the left half [a, (a + b)/2]; again, if the reduction is insufficient, it considers the leftmost quarter of the initial domain, [a, (3a+b)/4]. Lastly, if the reduction is still insufficient, it considers the leftmost eighth, [a, (7a+b)/8]. At that point, it returns whatever domain [l, b] was obtained, even if its left bound is not box consistent (i.e. 0 ∈ g([l, l + ])). For all Newton steps performed, it uses the center of the domain under scrutiny as the expansion point, in a bid to achieve large reductions.

Following McAllester et al.'s work, we have designed an algorithm that uses a different shaving process to address its shortcomings (see discussion below). The corresponding left narrowing procedure lnar sbc3ag is presented in Alg. 4 (The rnar sbc3ag procedure is not shown, since it can easily be obtained from lnar sbc3ag by symmetry). An important difference with McAllester et al.'s algorithm is that ours enforces box consistency every time, while theirs usually stops the narrowing process before that point. As Section 4 will show, spending more time to effectually enforce box consistency is the key to good performances for some difficult problems.

To shave the left part of a domain [a, b], we proceed as follows (see Alg. 4): We consider a slice whose width is γ times the width W of the initial domain, and we apply one Newton step on it; if the reduction obtained is smaller than some threshold σ b , we update γ to β b γ, with β b < 1; if the reduction is greater than some threshold σg, we update γ to with > 1. Otherwise, we keep γ as it is. We then consider the next slice with a size equal to the new γ times W . In short, when we are able to shave large parts, we take larger bids; on the other hand, we decrease the size of the slices considered when things go wrong. This method follows the line of well-known numerical algorithms such as Trust Region Methods.

As shown in Alg. 4, the amount of reduction is computed by comparing the size of the previous domain with the size of the tightened domain where the right bound has been reset to the one of the previous domain (that is, we do not consider the possible displacement of the right bound as a useful reduction since it is not taken advantage of afterwards).

Another difference with McAllester et al.'s algorithm is that we use the left bound as an expansion point when shaving the left side (and the right bound when shaving the right side). There are two reasons for this: Firstly, this is a surefire strategy since it always lead to some reduction, as said previously; Secondly, we have to compute g([I, I + ]) (Lines 3 and 20 in Alg. 4) before each Newton step to check whether the left bound is already box consistent. This value can be reused instead of g([I, I]) in the computation of the Newton step Line 11 at a negligible cost in terms of the width of intervals involved (the correction of the process is obviously preserved since we use a wider interval), while avoiding the cost of one function evaluation. Unreported experimental evidences show that speed-ups up to 30% are attainable that way.

For the problem presented in Fig. 3, the original bc3revise algorithm [START_REF] Van Hentenryck | Solving polynomial systems using a branch and prune approach[END_REF] (that is, the one instantiated with lnar vhmak and rnar vhmak ) obtains the largest box consistent domain (approximately [-0.487, 10]) after 50 Newton steps; McAllester et al.'s algorithm stops with the non box consistent domain [-3.5, 10] after only 6 Newton steps, and our algorithm enforces box consistency after 27 Newton steps.

EXPERIMENTS

To evaluate the impact of our algorithm, we have selected 16 instances of 14 classical test problems. Some are polynomial and others are not. The characteristics of these test problems are summarized in Table 1. All problems are structurally well constrained, with as many equations as variables. Column "Constraints" indicates whether all constraints are polynomial (quadratic, if no polynomial has a degree greater than 2). A problem is labelled "non-polynomial" if at least one constraint contains a trigonometric, hyperbolic or otherwise transcendental operator. Column "Source" gives a (not necessarily primary) reference to the literature where the problem is presented. The size of the problem and the initial domains for the variables are given in Table 2. All experiments were conducted on an Intel Core2 Duo T5600 1.83GHz. The Whetstone test for this machine reports 1111 MIPS with a loop count equal to 100, 000.

All algorithms have been implemented in our own C++ constraint solver, with the not-yet-released version 4.0 of gaol4 as the underlying interval arithmetic library.

Table 2 reports the time spent in seconds to isolate all solutions of the test problems in domains with a width smaller than 10 -8 , starting from the initial domains given in Column "Domains". The number preceded by a dot in Column "Problems" gives the number of equations and variables of the problem. An entry "TO" indicates a time-out (fixed to more than one hour, here). An entry "OV" indicates Except for "bb" and "comb," Alg. sbc3ag b is faster than McAllester et al.'s algorithm, sometimes strikingly so (e.g., on extended-powell). Alg. mavhkm outperforms it on Broyden Banded because this one is an easy problem with only one solution where it pays to be bold in the size of the slices considered for discarding, which is exactly what Alg. mavhkm does. As evidenced by the number of splits on this benchmark (the same for all algorithms), the fact that Alg. mavhkm may defer to the binary search process of the solver while stopping computation before box consistency is reached has no part in the performances of this algorithm here. On the other hand, Alg. mavhkm achieves good performances on problems "com" and "te" while requiring, respectively, 5 times and 2.5 times more splittings than the other methods. This validates the argument by McAllester et al. that spending less time in each contracting operator may pay off.

Nevertheless, some problems (feigenbaum and extendedpowell, among others) show that it sometimes pays off to try hard to tighten the domains at the constraint operator level rather than deferring to the dichotomic process: sbc3ag b is much faster than mavhkm on these benchmarks, and, tellingly, mavhkm fares worse than vhmak as well.

Times-out for Alg. mavhk b correspond to slow convergence phenomena as the one described in Fig. 2

(b).

A comparison of Columns sbc3ag b and sbc3agm, and of Columns mavhkm and mavhk b shows that the choice of point for a Newton expansion (middle point vs. bounds) does not explain alone the good performances of sbc3ag b ; even more, we see that using McAllester's et al. algorithm with an expansion on the bounds leads to an algorithm that may converge very slowly. With the extended-powell problem, we also see that an expansion on the bounds leads to regu-lar performances, provided we either do not try to reach box consistency in order to avoid slow convergence (compare Alg. mavhk b with mavhkm), or we use adaptive guesses (Algs. sbc3ag b and sbc3agm).

CONCLUSION

As Tab. 2 shows, and contrary to the statement by Benhamou et al. [START_REF] Benhamou | CLP(Intervals) revisited[END_REF], using the bounds as expansion points for the Newton method is efficient, provided that the Newton operator is applied on a carefully chosen subpart of the initial domain and not on the whole of it as Benhamou et al. presumably tested.

Algorithm sbc3ag b is significantly faster than McAllester et al.'s algorithm on a subset of the problems only. However, its strength lies elsewhere, viz. it is the most dependable algorithm among the ones tested here to enforce box consistency in that it is always only slightly slower than the best algorithm, while never exhibiting bad performances as McAllester et al.'s algorithm does on difficult problems.

Definition 1 (

 1 Box consistency). The constraint f (x1, . . . , xn) = 0 is box consistent with respect to a variable xi and a box of domains B = I1 × • • • × In if and only if:

Algorithm 2 3 return ∅ 4 I

 234 Computing a box consistent left bound [lnar vhmak ] in: g : I → I; in: I ∈ I # Returns an interval included in I # with the smallest left bound l such that 0 ∈ g([l, l + ]) begin 1 r ← I # Preserving the right bound 2 if 0 / ∈ g(I) : # No solution in I ← Newton (⋆) (g, g ′ , I) # Interval Newton steps

Figure 1 :

 1 Figure 1: Interval Newton step with I = [-1.55, -1.2] and κ(I) = m(I)

y

  Newton linearization at center Newton linearization at left bound Extreme slopes on [-25, 25] (a) Expansion on bound vs. expansion on center

Figure 2 :

 2 Figure 2: Box consistency enforcement. Interval functions have magenta borders and a light blue surface. In Figure 2(b), red domains are obtained by dichotomy; blue domains are discarded by an evaluation of the function; green domains are narrowed by a Newton step.

Figure 3 :Algorithm 3 3 I 4 p ← a in I 5 guess##

 33345 Figure 3: Shaving process for the left side. Initial domain is [-8, 10]; Slices considered are [-8, -3.5] (dark red area) and [-8, 0] (light red area).

Algorithm 4 1 γ ← γ init 2 W: 6 I

 4126 Enforcing box consistency on the left bound with adaptive guesses [lnar sbc3ag ] in: g : I → I; in: I ∈ I # Thresholds σg and σ b , and inflating/reducing factors # βg and β b chosen experimentally. const: γ init = 0.25, σg = 0.25, σ b = 0.75, βg = 1.5, β b = 0.7 begin ← w(I) # W fixed to avoid asymptotic behavior 3 d ← g([I, I + ]) 4 # I is not box consistent on left? 5 while I = ∅ ∧ 0 / ∈ d← [I + , I] # We already know 0 / ∈ [I, I + ] 7 guess ← [I, min(I, I + γW )] Performing one Newton step 11 reduced ← guess∩ " guessd ⊘ g ′ (guess) " 12 if w([reduced, guess]) < σgw(guess): 13 # Good reduction?. . . 14 γ ← γβg # . . . Let us consider larger guesses 15 elif w([reduced, guess]) > σ b w(guess): 16 # Bad reduction?. . . 17 γ ← γβ b # . . . Let us consider smaller guesses 18 # We keep the right bound unchanged: 19 I ← ✷ `reduced ∪ (I \ guess) 20 d ← g([I, I + ]) 21 return I end

  # Returns the largest interval included in I that is # box consistent with respect to g. The algorithm is # parameterized by the "lnar " and "rnar " methods used. = [I k , I k ] are intervals with floating-point bounds, a + (resp., a -) is the smallest floating-point number greater than (resp., the largest floating-point number smaller than) the floating-point number a, and f is the natural interval extension of f .A constraint f (x1, . . . , xn) = 0 is box consistent with respect to B if it is box consistent with respect to x1, . . . , xn and B. A constraint system is box consistent if all constraints are box consistent.

	begin
	1 if 0 / ∈ g(I):
	2	return ∅
	3 return lnar(g, rnar(g, I))
	end
	where I k

Algorithm 1 Computing a box consistent interval [bc3revise(lnar,rnar )] in: g : I → I; in: I ∈ I

Table 1 :

 1 Test problems

	Name	Code Constraints	Source
	Broyden-banded	bb	quad.	[6]
	Broyden tridiagonal	bt	quad.	[5]
	Combustion	comb	poly.	[6]
	DBVF	dbvf	poly.	[5]
	Extended Freudenstein	ef	poly.	COPRIN 3
	Extended Powell	ep	poly	COPRIN 3
	Feigenbaum	fe	quad	[2]
	i4	i4	poly.	[6]
	Mixed Algebraic Trig.	mat	non-poly.	COPRIN 3
	Moré-Cosnard	mc	poly.	[6]
	Trigexp	te	non-poly.	COPRIN 3
	Trigexp 3	te3	non-poly.	COPRIN 3
	Troesch	tro	non-poly.	COPRIN 3
	Yamamura	yam	poly.	COPRIN 3

Table 2 :

 2 Experimental ResultsTimes on an Intel Core2 Duo T5600 1.83GHz (whetstone 100 000: 1111 MIPS). Time-out set to 1 hour.that an overflow (stack or heap) prevented the computation to complete. Column "vhmak" stands for Alg. 1 instantiated with lnar vhmak and rnar vhmak[START_REF] Van Hentenryck | Solving polynomial systems using a branch and prune approach[END_REF]; Column "mavhkm" corresponds to McAllester et al.'s algorithm[START_REF] Mcallester | Three cuts for accelerated interval propagation[END_REF]; Column "mavhk b " corresponds to McAllester et al.'s algorithm where the Newton expansion is done on the left and right bounds instead of the middle; Column "sbc3ag b " corresponds to our adaptive algorithm (that is, bc3revise[lnar sbc3ag ,rnar sbc3ag ]); lastly, Column "sbc3agm" corresponds to our adaptive algorithm where the Newton expansion is done in the middle of the interval considered.

	Times (in seconds)

http://www-sop.inria.fr/coprin/logiciels/ALIAS/ Benches/benches.html

http://sf.net/projects/gaol/