Harsh A Chaliyawala

Stephane Bastide
email: stephane.bastide@cnrs.fr

Diane Muller-Bouvet

Christophe Pichon

Kadiatou Bah

Anffane Djoumoi

Frédéric Marty

Tarik Bourouina

Encarnacion Torralba
email: encarnacion.torralba-penalver@cnrs.fr

One-step synthesis of Ag-Cu bimetallic nanoparticles on p-Si photoelectrodes for solar-driven CO 2 reduction

Keywords: Metal assisted chemical etching (MACE), bimetallic nanoparticles, photoelectroreduction, ratiometric power saved, synergetic effect

In the present work, we report the one-step synthesis of AgxCu100-x bimetallic nanoparticles (NPs) on p-type silicon (p-Si) supports and their application as photocathodes for the photoelectrochemical CO2 reduction reaction (PEC-CO2RR). Based on the metalassisted chemical etching (MACE) of Si in HF medium, the proposed method allows the formation of NPs homogeneously distributed on the Si surface, with an excellent control of their bimetallic composition, surface coverage and morphology by adjusting the experimental conditions (deposition time, precursor concentrations). Microstructural studies reveal a phaseseparated polycrystalline structure for the bimetallic with a high dispersion of Ag and Cu nanocrystallites, remarkable for Ag/Cu nanostructures synthesized by soft chemistry.

The analysis of the PEC-CO2RR shows a better performance of Ag50Cu50/p-Si than Ag/p-Si and Cu/p-Si, with an onset potential shift of 840 mV and a ratiometric power saved of ~3% compared to an analogous metal electrode in CO2 saturated 0.5 M NaHCO3.

Photoelectrolysis tests under 1 sun illumination coupled with gas chromatography demonstrate the capacity of the bimetallic photocathode to guide the selectivity with the applied potential. Thus, CO2 conversion to CO and CH4 are achieved on Ag50Cu50/p-Si, with a maximum Faradaic efficiency (FE) of 26 % and 18.2% at -0.72 and -0.87 VRHE, respectively. These values are consistent with recent research on Ag-Cu systems on Si for PEC-CO2RR but

Introduction

The electrochemical reduction of carbon dioxide (CO2) into various value-added chemicals and fuels is receiving increasing attention in research and technology. It promises to simultaneous reduce atmospheric CO2 concentrations and store renewable energy, all by converting CO2 into useful low carbon products [START_REF] Durst | Electrochemical CO2 Reduction -A Critical View on Fundamentals, Materials and Applications[END_REF] . Among the various existing technologies, the photoelectrochemical CO2 reduction reaction (PEC-CO2RR) using semiconductors (SCs) is particularly attractive because it utilizes solar radiation, water and CO2, that react at the semiconductor/electrolyte interface, somehow mimicking natural photosynthesis [START_REF] He | Recent Advances in Solar-Driven Carbon Dioxide Conversion: Expectations versus Reality[END_REF][START_REF] Kan | System Engineering Enhances Photoelectrochemical CO2Reduction[END_REF] . The SC acts as a light absorber, for photocarriers generation and collection. Thus, a great deal of effort has been devoted to study the CO2 reduction using various SCs, such as doped gallium nitride (p-GaN), cadmium telluride (p-CdTe), indium telluride (p-InP), silicon carbide (p-SiC) and silicon (p-Si), among others [START_REF] Kumaravel | Photoelectrochemical Conversion of Carbon Dioxide (CO2) into Fuels and Value-Added Products[END_REF][START_REF] Kim | Photoconversion of Carbon Dioxide into Fuels Using Semiconductors[END_REF] . P-type Si is considered a very good photocathode material due to its optimal energy band gap (~ 1.12 eV) in the visible and near-IR region, its high carrier mobility (on the order of ~10 3 cm 2 V -1 s -1 at 300 K), its large abundance and the maturity of its processing technologies [START_REF] Yalamanchili | Integration of Electrocatalysts with Silicon Microcone Arrays for Minimization of Optical and Overpotential Losses during Sunlight-Driven Hydrogen Evolution[END_REF][START_REF] Kempler | Si Microwire-Array Photocathodes Decorated with Cu Allow CO2 Reduction with Minimal Parasitic Absorption of Sunlight[END_REF] . P-Si photoelectrodes has been decorated with sparsely distributed metal nanoparticles (e.g., Cu, Ag or Au) as catalysts for PEC-CO2RR. The nanoparticles (NPs) allow the collection of photogenerated charge carriers without preventing light from reaching the semiconductor. Thus, significant photocurrent densities at low overpotentials can be obtained for PEC-CO2RR due to the catalytic effect of the nanoparticulate metal surface [START_REF] Kempler | Si Microwire-Array Photocathodes Decorated with Cu Allow CO2 Reduction with Minimal Parasitic Absorption of Sunlight[END_REF][START_REF] Hinogami | An Approach to Ideal Semiconductor Electrodes for Efficient Photoelectrochemical Reduction of Carbon Dioxide by Modification with Small Metal Particles[END_REF][START_REF] Roh | Photoelectrochemical CO2 Reduction toward Multicarbon Products with Silicon Nanowire Photocathodes Interfaced with Copper Nanoparticles[END_REF][START_REF] Hinogami | Modification of Semiconductor Surface with Ultrafine Metal Particles for Efficient Photoelectrochemical Reduction of Carbon Dioxide[END_REF] .

The catalyst can be preferentially selected from a well-established group of materials leading to identified products 11 : (i) Au, Pd, Ag, and Zn from which carbon monoxide (CO) as the main product; (ii) Pd, Sn, In, and Bi, which produce formate (HCOOH); and (iii) Cu, which can produce hydrocarbons.

Although Cu is known to be the only metal capable of providing hydrocarbons and alcohols in measurable amounts, its selectivity is not optimal. It requires a large overpotential, and it is ultimately poisoned by CO and byproducts [START_REF] Wang | Insights into the Development of Cu-Based Photocathodes for Carbon Dioxide (CO2) Conversion[END_REF][START_REF] Lee | Defining a Materials Database for the Design of Copper Binary Alloy Catalysts for Electrochemical CO2 Conversion[END_REF] . The addition of a second metal to Cu is one of the most promising approaches to improve its selectivity and reactivity [START_REF] Lee | Defining a Materials Database for the Design of Copper Binary Alloy Catalysts for Electrochemical CO2 Conversion[END_REF][START_REF] Zhu | Bimetallic Electrocatalysts for CO2 Reduction[END_REF] . Among Cu-based bimetallics, several reports have shown that Ag can significantly activate Cu towards the formation of multi-carbon products [START_REF] Choi | Intimate Atomic Cu-Ag Interfaces for High CO2RR Selectivity towards CH4 at Low over Potential[END_REF][START_REF] Huang | Structural Sensitivities in Bimetallic Catalysts for Electrochemical CO2 Reduction Revealed by Ag-Cu Nanodimers[END_REF][START_REF] Lee | Importance of Ag-Cu Biphasic Boundaries for Selective Electrochemical Reduction of CO2 to Ethanol[END_REF] . The faradaic efficiency (FE) and selectivity can be tuned as a function of the AgCu composition and crystalline structure. For example, AgCu nanocrystals with Ag and Cu segregated domains show a 3.4-fold increase in C2H4 production at -1.1 V vs. RHE (VRHE) with respect to pure Cu [START_REF] Huang | Structural Sensitivities in Bimetallic Catalysts for Electrochemical CO2 Reduction Revealed by Ag-Cu Nanodimers[END_REF] , or AgCu NWs can reach 72 % FE for CH4 [START_REF] Choi | Intimate Atomic Cu-Ag Interfaces for High CO2RR Selectivity towards CH4 at Low over Potential[END_REF] .

Despite their interest, very few works have exploited bimetallic nanoelectrocatalysts on p-Si supports for PEC-CO2RR. In 2016, Kong et al. presented a photocathode based on p-Si Nanowires (SiNWs) decorated with Au3Cu by drop casting. A CO2-to-CO FE of 80% was demonstrated at very low overpotentials (-0.2 VRHE) with a photocurrent density of jph of -5 mA/cm 2 18 . In 2020, SiNWs decorated with Ni-In were investigated by Ma et al. [START_REF] Ma | Nickel and Indium Core-Shell Co-Catalysts Loaded Silicon Nanowire Arrays for Efficient Photoelectrocatalytic Reduction of CO2 to Formate[END_REF] . A 97% of FE towards HCOOH with a jph of -7 mA/cm 2 was obtained. With respect to AgCu, in 2019 Gurudayal et al. demonstrated its efficiency integrated at the dark side of p + nn + texturized Si supports, with CO2 conversion to a mixture of C2 and C3 products with 60% FE, at -0.4 VRHE and 10 mA/cm 2 . However, this required a high loading of dendritic Cu modified with Ag NPs, which was incompatible with direct illumination of the Si/Metal side [START_REF] Gurudayal; Beeman | Si Photocathode with Ag-Supported Dendritic Cu Catalyst for CO2 Reduction[END_REF] . Recently, Cu-Ag thin film catalysts (100 nm in Cu and 3 nm in Ag) were deposited on the illuminated side of p-Si photocathodes [START_REF] Dong | Grain Boundary Engineering of Cu-Ag Thin-Film Catalysts for Selective (Photo)Electrochemical CO2 Reduction to CO and CH4[END_REF] . The deposition involved several synthesis and deposition steps including electron beam evaporation, photolithography and lift-off. The photocathodes thus fabricated converted CO2-to-CO and CO2-to-CH4 with a FE of 26.9% and 18.2% at -1.4 and -1.3 VRHE, respectively. The application of a SiO2 passivation layer further increased the FE at the expense of further increasing the fabrication cost. To the best of our knowledge, the PEC-CO2RR on p-Si supports decorated with well-dispersed AgxCu100-x bimetallic NPs has never been investigated.

The present work reports on the development of a one-step synthesis method of AgxCu100-x bimetallic NPs on p-type c-Si by Metal Assisted Chemical Etching (MACE) and their application as photocathodes for PEC-CO2RR. MACE is a well-established method for the nano-and micro-structuring of Si in hydrofluoric acid (HF) by electroless deposition of single NPs (Ag, Au, Cu, etc.) [START_REF] Yae | Nucleation Behavior in Electroless Displacement Deposition of Metals on Silicon from Hydrofluoric Acid Solutions[END_REF][START_REF] Torralba | Tunable Surface Structuration of Silicon by Metal Assisted Chemical Etching with Pt Nanoparticles under Electrochemical Bias[END_REF] . It does not require any masking or lithography step, it is low cost, simple and easy scalable. Moreover, it has not yet been exploited for bimetallics.

The proposed method allows the direct synthesis of AgxCu100-x on p-Si in the whole range of atomic compositions. Morphological and compositional analysis performed by SEM-EDX, structure and a high degree of polycrystallinity, with Ag and Cu homogeneously mixed at the atomic scale. Charge transfer from Cu to Ag in the bimetallic NPs is demonstrated by XPS.

The analysis of the PEC-CO2RR in the as-synthesized materials is performed with Ag50Cu50/p-Si, Ag/p-Si, Cu/p-Si and analogous metallic electrodes. A superiority of Ag50Cu50/p-Si and Ag/p-Si vs. Cu/p-Si is observed under solar illumination, in terms of onset potential, photocurrent density and power saved. Photoelectrolysis coupled to gas chromatography demonstrate the ability of the bimetal to direct selectivity towards high energy density products. A maximum FE for CO and CH4 of 26 % and 17.8% at -0.75 and -0.86 VRHE, respectively, is obtained. These FEs are very similar to those observed with Cu-Ag thin-film catalysts on p-Si [START_REF] Dong | Grain Boundary Engineering of Cu-Ag Thin-Film Catalysts for Selective (Photo)Electrochemical CO2 Reduction to CO and CH4[END_REF] , but at a significantly lower overpotential, which add value to the proposed synthesis method. Finally, the effect of long-term photoelectrolysis on the structural stability of the bimetallic is shown. We believe that these findings can contribute to the development of efficient and affordable photoelectrocatalytic materials for solar-assisted CO2 conversion.

Experimental section

Materials and Reagents

P-type boron doped (100) c-Si polished wafers (ρ = 1-3 ohm cm, thickness ∼350±25 μm), purchased from Siegert Wafer were used as substrates for metallization. 30 wt% H2O2, 96-97 wt% H2SO4, 40 wt% HF, 65 wt% HNO3 and absolute EtOH (VWR Chemicals) were used as received for Si cleaning. Ag (NO3) (99.8%) and Cu(SO4)⸱5H2O (98.5 wt% Cu, Prolabo) were used as a metal salt precursors for the metallization. Dilution and cleaning were performed with 18.2 Mohm cm ultra-pure water (Millipore).

Photocathode preparation

Si pretreatment: Si wafers were diced into 2x2 cm 2 pieces and cleaned for 20 min in Piranha solution (3:1 v/v H2SO4/H2O2) to remove organic residues. After extensive rinsing with ultrapure water, the samples were immersed in 1 mol L -1 HF for 5 min to remove surface oxides and obtain hydrogen passivation. Polished Ag and Cu plates were used as dark electrodes (section 3.3). Before each electrochemical measurement they were polished with alumina powder (1 and 0.3 µm) and washed thoroughly with ultrapure water. The Cu plate was additionally immersed in HCl/water (50:50 v/v) for oxide removal.

Metal deposition: Metallization of silicon was performed by electroless deposition in HF media (MACE). The pretreated silicon sample was immersed in a HF solution containing a mixture of the salt precursors (AgNO3 and CuSO4) at selected concentrations (see Table 1).

The backside of the silicon sample was masked with an adhesive tape to prevent metal deposition on both sides. After MACE, the sample was thoroughly rinsed with water and the adhesive tape was removed. Pure Cu and Ag NPs were synthesized in a similar manner.

Different deposition times (from 1 min to 15 min) and HF concentrations (from 0.1 mol L -1 to 2.0 M L -1 ) were tested to investigate the characteristics of the bimetallic AgxCu100-x NPs obtained by MACE (see Section 3.1 in Results and Discussion). For the samples used as photocathodes, a HF concentration of 100 mmol L -1 and a deposition time of 1 min were chosen. A back contact was formed on the backside of the Si sample by scratching and rubbing In-Ga alloy.

Physical Characterizations

The morphology and elemental composition (bulk properties) were analyzed by Field Emission Scanning Electron Microscopy (FESEM) coupled with Energy Dispersive X-ray Analysis (SEM-EDX) using a Zeiss Merlin FEG microscope equipped with AZtec EDS Advanced system (HKL Nordlys Nano, Oxford Instruments) and an X-Max SDD detector.

Average NP size and surface coverage were determined using ImageJ software. Structural characterization was performed by X-Ray Diffraction (XRD) using a D8 Advanced Bruker diffractometer (Cu Kα radiation). EVA software was used for XRD data processing. Chemical states and surface elemental composition were analyzed by K-alpha X-ray Photoelectron Spectroscopy (XPS) equipped with an ion beam etching from Thermo Fisher Scientific.

Microstructural observations were performed by Transmission Electron Microscopy (TEM)

with a 200 kV FEG TEM (FEI Tecnai F20 equipped with a Gatan CCD camera Orius 1000, point-to-point resolution 0.24 nm), and the chemical analysis is carried out on the same microscope with an EDX windowless detector (60 mm 2 ). For TEM observations, NPs were collected on Ni mesh grids by pouring a few drops of ethanol solutions in which the Si/AgxCu100-x samples had been previously sonicated.

Electrochemical measurements

A homemade O-ring type three-electrode photoelectrochemical cell (volume of 54 mL) equipped with a quartz window was used for the electrochemical measurements (see Fig. S1).

It was coupled to a Metrohm Autolab PGSTAT 30 potentiostat equipped with NOVA software. The working electrode was the metallized p-Si, the counter electrode was a Pt wire, and the reference electrode was Ag/AgCl/3 mol L -1 KCl (0.209 V vs. NHE). All potentials were converted to the RHE reference scale unless otherwise noted (E vs. RHE = E vs.

Ag/AgCl + 0.209 + 0.059 * pH). The electrocatalytic reduction of CO2 was performed in 0.5 mol L -1 NaHCO3 at room temperature (pH = 8.4). Prior to the experiment, 36 mL of electrolyte were added into the cell and purged with Ar for 10 min (pH = 8). The electrolyte was then bubbled with CO2 (Alphagaz N45) for 20 min to ensure CO2 saturation (pH ~7.4,

[CO2] = 33 mmol L -1 ). The CO2 atmosphere was maintained throughout the experiment. The illuminated silicon surface area was 1.76 cm 2 (O-ring diameter = 15 mm). Cyclic voltammetry was used as the electrochemical characterization technique. The illumination source was a Newport LSH7320 solar simulator providing AM1.5G light with a power of 1000 Wm -2 .

Product analysis

The gas products were analyzed by Gas Chromatography (GC). A Shimadzu gas chromatograph (GC 2010 Pro) equipped with a detector and a Micropacked ShinCarbon GC column was used to analyze the gas composition inside the electrochemical cell during the bulk electrolysis. The gas was sampled from the electrochemical cell and injected into the GC at different electrolysis times. An etalon gas mixture purchased from Air Liquide was used to calculate the concentration of the gases produced. He was used as carrier gas. More details on gas injection and product quantification can be found in the SI.

Results and Discussion

Principle of the bimetallic deposition by MACE

AgxCu100-x bimetallic NPs with different compositions were synthesized directly on polished Si substrates by MACE through immersion in a HF solution containing the two ionic metal precursors (Ag + and Cu 2+ , see Experimental section) at different concentrations (Table 1). Several deposition times and HF concentrations were tested to observe their influence on the bimetallic properties. The results are described in the following section.

The principle of the deposition is shown in Fig. 1. As in typical MACE for monometallic deposition, when Ag + and/or Cu 2+ ions get in contact with the Si surface, they are spontaneously reduced by extracting electrons from the Si valence band and deposited on the Si surface as Ag 0 and Cu 0 metal nuclei (Eq. 1). In parallel, Si atoms in the immediate vicinity of the metal ions are oxidized due to hole injection. In HF media, this Si dissolution occurs in the porous silicon formation regime (Eq. 2). The nuclei thus formed become catalytic sites for the subsequent ion reduction and NP growth, with Si atoms close to the NPs providing all the necessary electrons. The whole process (Eq. 3) is spontaneous, electroless, and provides an optimal metal/p-Si contact without any oxides [START_REF] Chartier | Metal-Assisted Chemical Etching of Silicon in HF-H2O2[END_REF] . One Cu or two Ag atoms can be deposited for each dissolved Si atom (Eq. 4). If Ag + and Cu 2+ are at the same concentration and deposited simultaneously, the overall reaction becomes:

Si(s) + ( 1 2 Cu 2+ + Ag + )(aq) + 4HF 2 - (aq) → ( 1 2 Cu +Ag)(s)+ SiF 6 2- (aq) + 2HF(aq) + H2(g) Eq. 4
Eq. 4 indicates that there should be 2 times more Cu 2+ than Ag + in the precursor solution (i.e., 2(100-x) moles of Cu 2+ and x moles of Ag + ) to obtain a given composition AgxCu100-x.

Its accuracy will be examined below.

Physicochemical characterization

Surface morphology and atomic composition

The surface morphology of the synthesized Ag, Cu and AgxCu100-x NPs have been examined by FESEM as shown in Fig. 2. Table 1 shows the molar concentration of the metal precursor solutions and the corresponding atomic composition obtained by EDX together with a plot of the atomic percentage of Cu determined by EDX vs. the molar fraction of Cu 2+ in the precursor solutions (red dots). A solid line indicates the calculated values assuming that, to obtain a given composition AgxCu100-x, there should 2 times more of Cu 2+ than of Ag + in the precursor solution (Eq. 4). The HF concentration for these experiments was 1 mmol L -1 .

The EDX data presented in Table 1 reveal that AgxCu100-x NPs can be synthesized by our method in the whole range of atomic compositions by simply varying the concentration of metal precursors in the deposition solution. Furthermore, the experimental values for the at. Cu% determined by EDX vs. the molar ratio of the metal precursors used overlap very well with the calculated values (solid line), demonstrating the validity of Eq. 4. A clear example is observed for Ag49Cu51, where the precursors are in concentrations 1 and 2 mM for Ag + and Cu 2+ , respectively (Table 1). Therefore, the electroless bimetallic deposition of AgxCu100-x under these conditions (1 min deposition, 1 mmol L -1 HF) on Si supports by MACE is not affected by differences in diffusion coefficients or in reduction kinetics of Ag + and Cu 2+ and obtaining a given composition is straightforward.

Table 1: Molar concentration of the metal precursors used (Ag + and Cu 2+ ) and atomic composition of the synthesized AgxCu100-x NPs determined by EDX. HF concentration: 1 mmol L -1 . Inset: Plot of at% of Cu vs. the Cu molar concentration ratio in the precursor solution. (Dots: experimental values. Line: predicted).

For the sake of simplicity, hereafter we will refer to the averaged atomic composition instead of the exact one if not otherwise stated (e.g. Ag49Cu51 will be referred as Ag50Cu50 or Ag32Cu68 will be referred as Ag30Cu70).

The morphology of the deposits is shown in the FESEM images of Fig. Fig. S2 shows SEM images comparing Ag50Cu50 and Ag20Cu80 obtained with a lower concentration of metal ions in the precursor solution with those of Fig. 2. The main difference when we lower the precursor concentration is a notable decrease in the NP size and a modification of the NP morphology from worm-like shaped to quasi-spherical. These experiments show that the more pronounced worm-like morphology of the NPs in Fig. 2 when increasing Cu content is basically due to the higher initial Cu 2+ concentration (see Table 1), which results in a greater amount of metal deposited. The effect of deposition time and HF concentration on the growth of bimetallic NPs, analyzed by using the Ag50Cu50 composition as reference, is shown in the SI (Fig. S3 andS4). The increase of the deposition time highly increases the surface coverage, until a nanoparticulate film is obtained at 5 min deposition (Fig. S3). On the other hand, an elongation of the NP ligaments together with a decrease in surface coverage is observed when increasing the HF concentration (Fig. S4). These 220) and (311) planes [START_REF] Lin | Effects of Accumulated Energy on Nanoparticle Formation in Pulsed-Laser Dewetting of AgCu Thin Films[END_REF][START_REF] Dutta | Activation of Bimetallic AgCu Foam Electrocatalysts for Ethanol Formation from CO2 by Selective Cu Oxidation/Reduction[END_REF] .

The two powders exhibit face-centered cubic (FCC) structures with a polycrystalline nature (broad XRD peaks, see Fig. 3b). The pattern of the synthesized Ag50Cu50/p-Si powder contains the peaks of both pure metals (together with those of the support Si), showing a typical phase-separated crystalline structure [START_REF] Lin | Effects of Accumulated Energy on Nanoparticle Formation in Pulsed-Laser Dewetting of AgCu Thin Films[END_REF][START_REF] Kushchev | Structure and Mechanical Properties of Ag-Cu Films Prepared by Vacuum Codeposition of Au and Cu[END_REF] . These can be the pure Ag and Cu phases or some α (Ag-rich) and β (Cu-rich) phases (fcc α and β solid solutions can reach at high temperature a maximum solubility of 14.1 at% Cu and 4.9 at% Ag, respectively [START_REF] Subramanian | The Ag-Cu (Silver-Copper) System[END_REF] ). A closer look at the region of the most intense peaks is shown in Fig. 3 crystallites, respectively). Thus, co-depositing Ag and Cu reduces the crystallite size to some extent [START_REF] Kushchev | Structure and Mechanical Properties of Ag-Cu Films Prepared by Vacuum Codeposition of Au and Cu[END_REF][START_REF] Loza | Structure, Properties, and Applications of Bimetallic Nanoparticles of Noble Metals[END_REF] .

Further analysis of the structure and bulk composition of the bimetallic NPs was performed by transmission electron microscopy (TEM). S1 andS2). This confirms the presence of both Ag and Cu face-centered crystallites in the NPs, i.e., their phase-separated nature.

To investigate the atomic distribution in the AgCu NPs, high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) analysis coupled with EDX mapping was performed on the Ag50Cu50 sample (Fig. 5). HAADF-STEM, which is very sensitive to atomic numbers, confirmed the presence of the two different elements in the same NP. The elemental composition of Ag46Cu54 is consistent with that obtained from the SEM-EDX data (Ag49Cu51, see Table 1). Along with some Cu-rich spots (Fig. 5(c)), the EDX maps of Ag and Cu atoms show a homogeneous distribution of the two elements in the bimetallic NPs (Fig. 5(b-c)), with no apparent distinct phases (Fig. 5(d)). The reason for this is probably the superposition of several Ag and Cu nanocrystallites in the NP volume through which the electron beam passes. Nevertheless, Ag/Cu nanostructures synthesized by soft chemistry routes such as co-reduction or electrodeposition are most often phase segregated (core-shell, crescent, etc. [START_REF] Huang | Structural Sensitivities in Bimetallic Catalysts for Electrochemical CO2 Reduction Revealed by Ag-Cu Nanodimers[END_REF][START_REF] Freire | Natural Arrangement of AgCu Bimetallic Nanostructures through Oleylamine Reduction[END_REF] ). Specific techniques have been developed to synthesize homogeneously dispersed Ag-Cu bimetallic NPs similar to those obtained in this work [START_REF] Choi | Intimate Atomic Cu-Ag Interfaces for High CO2RR Selectivity towards CH4 at Low over Potential[END_REF][START_REF] Yang | Overcoming Immiscibility toward Bimetallic Catalyst Library[END_REF] .

Chemical state and surface composition

The chemical state and surface composition of the AgxCu100-x NPs at different selected compositions were investigated by X-ray photoelectron spectroscopy (XPS). Table 2 summarizes the values measured by XPS and SEM-EDX. The XPS compositional analysis indicates an Ag-rich surface, this characteristic being more pronounced for a bulk Ag content below 50 at%. This difference between bulk and surface compositions can be related to a partial displacement of Cu atoms by Ag + ions in solution (given the more noble character of Ag vs. Cu), according to:

This would not affect the bulk composition of the NPs since diffusion of Ag + and Cu 2+ cations is the limiting process during deposition, i.e. there are always enough electrons (from Si oxidation) to reduce them. However, when the deposition is ended, the displacement could affect the surface composition since the Cu atoms eventually displaced can no longer be reduced. Therefore, if Eqs. 1 -4 correctly describe the essential mechanism of bimetallic deposition by MACE, it must be taken into account that Eq.5 may occur at the NP surface. Fig. 6 shows the core spectra of the Ag 3d and Cu 2p emissions for the different samples synthesized (conditions in Table 1). Two well-defined peaks can be observed in the core spectra of Ag 3d and Cu 2p for all the compositions (spin doublets 3d5/2 -3d3/2 and 2p3/2 The Ag 3d peaks for pure Ag NPs (Ag100 in the figure) are centered at 367.9 and 373.9 eV.

-2p1/2,
They shift to higher binding energies when the Cu content is increased (e.g., maximum shift of -0. This opposite shift of the binding energies indicates a charge transfer from Cu to Ag upon formation of the bimetallic NPs, i.e., the electron density of Ag atoms is modified by the surrounding Cu atoms and vice-versa in the formed immiscible alloy. Similar observations have already been reported for AgPd alloys [START_REF] Sengar | Size and Alloying Induced Shift in Core and Valence Bands of Pd-Ag and Pd-Cu Nanoparticles[END_REF] or AgCu bimetallic nanodimers with phaseseparated structures with improved performance for CO2RR compared to separated metals [START_REF] Huang | Structural Sensitivities in Bimetallic Catalysts for Electrochemical CO2 Reduction Revealed by Ag-Cu Nanodimers[END_REF] .

Photoelectrochemical CO2RR

Photoelectrochemical j/E responses

In order to evaluate interest of the synthesized AgxCu100-x/p-Si as photocathode materials, we first performed the analysis of their j/E response. A simplified schematic representation of the photoelectrochemical cell illustrating the principle of the PEC-CO2RR measurement is shown in Fig. 7 (a). Briefly, when the AgxCu100-x/p-Si photocathode in equilibrium with the electrolyte solution is illuminated with photons of energy equal to or greater than its band gap, h ≥ Eg (= 1.12 eV), electrons from the valence band are excited into the conduction band, leaving positive holes behind. Due to the semiconductor band structure, under the application of an external potential, the generated electron/hole pairs are separated and collected; the photo-excited electrons at the photocathode/electrolyte junction to perform the electrochemical reactions (CO2RR or its competitor hydrogen evolution reaction, HER), and the holes at the back contact of the Si, and from there to the anode to perform the oxygen evolution reaction (OER). More information about PEC and its principles can be found in the reported literature [START_REF] He | Recent Advances in Solar-Driven Carbon Dioxide Conversion: Expectations versus Reality[END_REF][START_REF] Bard | Encyclopedia of Electrochemistry: Inorganic Electrochemistry[END_REF][START_REF] Tan | Principles and Applications of Semiconductor Photoelectrochemistry[END_REF] . The polarization curve of the dark electrodes also shows the typical exponential increase without mass transport limitations. In all cases, the signals appear at significantly more negative potentials than those of the illuminated photocathodes, clearly demonstrating the energy-saving interest of the latter. Ag50Cu50/p-Si shows a remarkable Vonset of 840 mV, closely followed by Ag/p-Si, with Vonset of 727 mV, and Cu/p-Si with 249 mV (vs Ag Cu metals, respectively). The Vonset reported here for PEC-CO2RR with Ag/p-Si and Cu/p-Si photocathodes are in good agreement with those observed in pioneer studies [START_REF] Hinogami | Modification of Semiconductor Surface with Ultrafine Metal Particles for Efficient Photoelectrochemical Reduction of Carbon Dioxide[END_REF] . Vonset for a given photocathode is the difference between the onset potential of this photocathode and that of its homologous dark electrode, e.g. Vonset p-Si/AgCu = Eonset p-Si/AgCu -Eonset n ++ Si/AgCu, where Eonset is taken for a current density of -0.1 mA cm -2 (to avoid the contribution of native metal oxide reduction observed at low cathodic potential). Vonset is interesting for power saving metrics, but should not be confused with the photovoltage of a given photocathode, i.e. EOCP, light -EOCP, dark, which was close to 40 mV in all cases.

As for the jph values, the jph, plateau is directly related to the efficiency of the PEC system (catalyst / photoabsorber / electrolyte) to generate and collect photocarriers. We expect jph, plateau to decrease slightly when metal NPs are deposited on the Si surface due to optical losses, since a small fraction of the incident light is reflected by the metal NPs. In our case, jph is -2.4 mA/cm 2 for bare Si and -2.1 and -2.2 mA/cm 2 for Ag50Cu50/p-Si and Ag/p-Si mA/cm 2 , respectively, in agreement with the above reasoning. For Cu/p-Si, jph is slightly higher (-2.6 mA cm -2 ), which may be due to the low surface coverage of Cu NPs compared to Ag and Ag50Cu50.

The jph values at moderate potentials (-0.5 VRHE) are useful to analyze the catalytic activity. For example, at -0.5 VRHE, a significant current density is obtained for the photocathodes. The highest value is obtained for the Ag50Cu50/p-Si (-1.1 mA/cm 2 ), very close to Ag/p-Si (-0.9 mA/cm 2 ) and Cu/p-Si (-0.5 mA/cm 2 ), which is still five times higher than bare Si. These jph values were recorded at 0.2 sun to avoid distortion of the signal morphology with gas evolution under 1 sun at potentials close to the jph plateau. Polarization curves at 1 sun are shown in Fig. S6 of the SI. They can reach values as high as 11-12 mA/cm 2 , which compares very well with the values reported in the state-of-the-art literature [START_REF] Dong | Grain Boundary Engineering of Cu-Ag Thin-Film Catalysts for Selective (Photo)Electrochemical CO2 Reduction to CO and CH4[END_REF] .

In order to quantify the energy conversion properties of our materials, we have calculated the ratiometric power saved for the three electrode system (ϕsaved, Fig. 7(c)), as described by Coridan et al. [START_REF] Coridan | Methods for Comparing the Performance of Energy-Conversion Systems for Use in Solar Fuels and Solar Electricity Generation[END_REF] . This figure of merit is defined as the ratio between the power saved at a given current, Psaved (I), by using an illuminated photocathode versus its equivalent dark electrode, and the input solar power, Pinput: Elight is the potential required to drive the reaction at a selected current value I for the illuminated photocathode and Edark is the corresponding value for the dark electrode. This metric is designed to quantify the performance of individual photoelectrodes and is independent of the potential drops associated with electrochemical processes at the counterelectrode. We refer the reader to Coridan et al. [START_REF] Coridan | Methods for Comparing the Performance of Energy-Conversion Systems for Use in Solar Fuels and Solar Electricity Generation[END_REF] more details on metrics for energy conversion comparisons in solar fuels and solar electricity generation. ϕsaved is calculated at the maximum saved power, Pmax,saved, as shown in Fig. 7 (c).

𝜙
The as-calculated 𝜙 saved for Ag50Cu50/p-Si is ~3 %. These values are promising since our photocathodes consist of simple p-Si substrates with a p-Si/electrolyte junction. Future research will be oriented to design Si photocathodes with a buried p-n junction to achieve higher performance (p-n junction Si solar cells typically exhibit a photovoltage of ~700 mV and photovoltaic conversion efficiencies of ~20%). The value obtained with the Ag/p-Si photocathode is also not negligible, ~2.1% (calculation shown in the SI, Fig. S7). For Cu/p-Si, 𝜙 saved ~ 0.4 %. This does not mean that Cu/p-Si is incapable of converting light into electricity, but its energy conversion performance compared to a Cu metal electrode under the same conditions is very poor. Theoretically, Ag/Si and Cu/Si junctions should have similar rectifying properties (Cu and Ag work functions are 4.64 and 4.70 eV, respectively). The corresponding photocathodes exhibit similar photovoltages (40 mV), indicating that the nature of the Si/metal NPs junction does not significantly affect the electronic properties of the Si(metal NPs)/electrolyte junction. We believe that the poor performance of Cu/Si vs.

Ag/Si is more related to the reshaping and loss of Cu during the cathodic scan, resulting in a reduced number of catalytic sites (see also section 3.4.) as compared to Ag which is more stable. Nevertheless, Cu is indispensable for the conversion of CO2 into carbon products in significant amounts [START_REF] Lee | Defining a Materials Database for the Design of Copper Binary Alloy Catalysts for Electrochemical CO2 Conversion[END_REF] . In the case of AgCu NPs, Ag stabilizes the Cu grains in the bimetallic and prevents these degradation processes for some time.

These results demonstrate the interest of Ag50Cu50/p-Si and Ag/p-Si for PEC conversion in terms of onset potentials, jph and power saved metrics. This superior performance is related to the nanostructuring of the catalyst, which increases the active surface area and multiplies the number of catalytic sites without compromising the photo-absorber properties, as well as to the intrinsic catalytic properties of the metals. Ag50Cu50/p-Si gives slightly higher values than pure Ag, which is undoubtedly related to the electronic and geometric bimetallic effects, as shown in the next section.

Gas products analysis

Electrolysis coupled to gas chromatography (GC) was performed at various applied potentials ranging from -0.37 to -1.07 VRHE for 1h in the CO2-purged 0.5 M NaHCO3 electrolyte under 1 sun illumination. Fig. 8 shows the current densities and FEs obtained in these measurements. Details of the FE calculations are given in the SI. and moderate potentials where relatively little charge crosses the interface (i.e., -0.72 VRHE, 9.5 C passed). From -0.87 VRHE a significant increase in current density is observed and the j/t signal begins to increase with time. This behavior is associated with an increase in HER (more details below). The product distribution of the bimetallic photocathode is highly dependent on the applied potential. At low overpotentials, the main CO2RR product is CO

(2 e -product), with a FE that increases from 14.5% at -0.37 VRHE to a maximum of 26 % at -0.72 VRHE, and then decreases (FE = 4 % at -1.07 VRHE). In addition, CH4 (8 e -product) starts to be detected at higher overpotentials (i.e. at -0.57 VRHE) with a FE of 1.1%, reaches a maximum FE of 17.8% at -0.87 VRHE and decreases up to 6% FE at -1.07 VRHE. Thus, the potential range at which CO and CH4 are produced is well defined and a peak of selectivity for each product is obtained at different potential values. The higher activation energy required for PEC-CO2RR to CH4 production vs. CO is demonstrated.

As for HER, it is dominant at all the applied potentials except -0.72 VRHE (FECO+CH4 = 41.5 % vs. FEH2 = 35%). At -1.07 VRHE, CH4 and CO production are strongly suppressed, and HER is enhanced, which coincides with the noticeable increase in current density observed in the j/t curve (Fig. 8a). A decrease in the catalyst coverage during electrolysis, implying more naked Si susceptible to HER, could be at the origin of this phenomena (see next section). The potential -0.87 VRHE, corresponding to the maximum FECH4 (Fig. 8b), was selected for the following experiments.

Fig. 8 (c)-(f) show the j/t and FE calculations at -0.87 VRHE for Ag/p-Si, Cu/p-Si and Ag50Cu50/p-Si at different electrolysis times. The gas products were analyzed for all the samples every 15 min intervals for 1 h. As can be seen, the Ag/p-Si photocathode produces CO as the main CO2RR product (with a maximum FE 10%), and a very small amount of CH4 (< 1%, Fig. 8(d)) [START_REF] Kuhl | Electrocatalytic Conversion of Carbon Dioxide to Methane and Methanol on Transition Metal Surfaces[END_REF] . Cu/p-Si produces CH4, with FE ~3% (Fig. 8(e)). In turn, at the selected potential, Ag50Cu50/p-Si is able to convert CO2 into a mixture of CO and CH4 with a ratio CH4:CO ~3.75 during the whole electrolysis time, and a maximum of CO2 conversion at 45 min operation (18.2 % FE for CH4 and 5.2% for CO, Fig. 8(f)). HER is more important for the photocathodes decorated by monometal than for those with bimetallics, in agreement with their lower CO2RR performance. In general, the gas product distribution is stable with time, with CO, CH4 and H2 being the main products. Equivalent plots for bare Si, Cu and Ag metals are given in the SI (Fig. S8) where H2 was the only product for bare Si; and Ag and Cu plates produced H2 and very little CO (FE< 1 % for Ag and FE< 0.5 % for Cu).

Similar results in terms of FEs were recently obtained by Dong. et al. [START_REF] Dong | Grain Boundary Engineering of Cu-Ag Thin-Film Catalysts for Selective (Photo)Electrochemical CO2 Reduction to CO and CH4[END_REF] The results shown in Fig. 8 clearly demonstrate the interest of the bimetallic photocathode to direct the selectivity of the PEC-CO2RR towards the high energy density product CH4, and to produce it at moderate overpotentials compared to mono-metal decorated photocathodes and unsupported metals. The enhanced selectivity of the bimetallic towards CH4 production results from the electronic and geometrical effects generated by the nanoscale coupling of the two metals. This is consistent with the charge transfer from Cu to Ag as evidenced by XPS (Fig. 6), confirming a modification of the electronic structure of the bimetallic with an optimized CO* binding energy with respect to the separated metals [START_REF] Huang | Structural Sensitivities in Bimetallic Catalysts for Electrochemical CO2 Reduction Revealed by Ag-Cu Nanodimers[END_REF][START_REF] Dong | Grain Boundary Engineering of Cu-Ag Thin-Film Catalysts for Selective (Photo)Electrochemical CO2 Reduction to CO and CH4[END_REF] , and the high degree of dispersion of Ag and Cu atoms in the bimetallic, as evidenced by XRD and HRTEM (see Figs. 3 and4), providing an optimum geometry for CO2-to-CH4 conversion [START_REF] Choi | Intimate Atomic Cu-Ag Interfaces for High CO2RR Selectivity towards CH4 at Low over Potential[END_REF][START_REF] Dong | Grain Boundary Engineering of Cu-Ag Thin-Film Catalysts for Selective (Photo)Electrochemical CO2 Reduction to CO and CH4[END_REF] .

There seems to be a consensus on the role of Ag and Cu sites in CO2RR with bimetallic AgCu nanoelectrocatalysts. Ag sites are responsible for the first step (CO2-to-CO), creating a CO-concentrated environment that Cu sites use to continue the reduction towards C1 and C2

products. Once CO is generated, the selectivity between C1 or C2+ products (e.g. CH4 vs.

C2H4) depends on the bimetallic crystal structure, with AgCu NPs exhibiting segregated domains producing predominantly C2H4 (C-C coupling favored on concomitant Cu sites), whereas catalysts with intermixed Ag and Cu domains preferentially produce CH4. DFT calculations suggest a common pathway for both products with the *COH intermediate, which

deviates to CH4 and C2H4 depending on the availability of the catalyst to adsorb H (Had) or CO (COad) and the length scale at which the metal mix favors C-C coupling [START_REF] Choi | Intimate Atomic Cu-Ag Interfaces for High CO2RR Selectivity towards CH4 at Low over Potential[END_REF][START_REF] Huang | Structural Sensitivities in Bimetallic Catalysts for Electrochemical CO2 Reduction Revealed by Ag-Cu Nanodimers[END_REF][START_REF] Chen | Chemical and Structural Evolution of AgCu Catalysts in Electrochemical CO 2 Reduction[END_REF][START_REF] Gao | Selective C-C Coupling in Carbon Dioxide Electroreduction via Efficient Spillover of Intermediates As Supported by Operando Raman Spectroscopy[END_REF] .

Structural evolution of the catalyst

The chemical state of the catalyst after photoelectrolysis was evaluated by SEM and XPS. SEM-EDX analysis revealed a decrease in NP coverage and a loss of Cu after photoelectrolysis. Specifically, the surface composition changed from Ag60Cu40 to Ag83Cu17, as determined by XPS. A significant amount of oxidized Cu was observed on the surface after photoelectrolysis (probably due to a longer air exposure after this experiment). Fig. S9 in the SI shows the XPS analysis of an Ag50Cu50/p-Si photocathode after 3 h of photoelectrolysis under the same conditions. An atomic surface composition of Ag92Cu8 was measured by XPS, confirming a further loss of Cu during prolonged electrolysis. A detachment of Ag nanograins could not be observed after 1 h, but has been reported for long electrolysis times (24 h) [START_REF] Yu | Silver Nanoparticles Embedded within a Silicon Photocathode for Photoelectrochemical Fixation of CO2 to Synthesize Intermediates of Profen Drugs[END_REF] .

The instability of Cu catalysts during CO2RR is well known. Nanoscale degradation mechanisms in Cu surfaces during CO2RR include detachment, dissolution, reshaping, fragmentation and agglomeration [START_REF] Popović | Stability and Degradation Mechanisms of Copper-Based Catalysts for Electrochemical CO2 Reduction[END_REF][START_REF] Wilde | Is Cu Instability during the CO2reduction Reaction Governed by the Applied Potential or the Local CO Concentration?[END_REF] . For example, in recent works on PEC-CO2RR with Cu NPs/p-Si photocathodes, the poor adhesion of Cu to Si has been pointed out as a possible reason for catalysts degradation [START_REF] Roh | Photoelectrochemical CO2 Reduction toward Multicarbon Products with Silicon Nanowire Photocathodes Interfaced with Copper Nanoparticles[END_REF] . Two main factors are thought to be responsible: (i) structural reorganization of the catalysts under the application of a potential and (ii) CO adsorption on Cu -Cu Cu -O Cu sites leading to Cu coordination and detachment [START_REF] Wilde | Is Cu Instability during the CO2reduction Reaction Governed by the Applied Potential or the Local CO Concentration?[END_REF][START_REF] Yang | Operando Studies Reveal Active Cu Nanograins for CO2 Electroreduction[END_REF] . In the specific case of CO2RR on Ag-Cu core-sell NPs, the latter has been identified as the main cause of instability [START_REF] Wilde | Is Cu Instability during the CO2reduction Reaction Governed by the Applied Potential or the Local CO Concentration?[END_REF] . It is possible that a similar phenomenon occurs in our bimetallic photocathode, although the crystalline structure is not identical. Recently, P. Yang et al. have reported a study of the evolution of dimeric and intermixed AgCu on carbon supports during CO2RR [START_REF] Chen | Chemical and Structural Evolution of AgCu Catalysts in Electrochemical CO 2 Reduction[END_REF] . They showed that, regardless of the initial catalyst structure, Cu can move and leach out of the catalysts to be detached/re-deposited elsewhere, which is consistent with our observations [START_REF] Chen | Chemical and Structural Evolution of AgCu Catalysts in Electrochemical CO 2 Reduction[END_REF][START_REF] Yang | Operando Studies Reveal Active Cu Nanograins for CO2 Electroreduction[END_REF] . Systematic studies of catalyst degradation as a function of time and composition are underway in our laboratory. The addition of atomic-scale protective layers (TiO2, SiO2, Al2O3) or the anchoring of NPs to the photoelectrode [START_REF] Yae | New Surface-Activation-Process for Electroless Deposition of Adhesive Metal (Ni, Cu) Films on Si Substrates[END_REF] are being implemented to increase the stability of these systems.

Conclusion

We have developed an innovative method for the deposition of AgxCu100 The performance of the as-synthesized AgxCu100-x/p-Si materials as photocathodes for PEC-CO2RR has been evaluated by analyzing their polarization curves and of product distribution. A better performance of Ag50Cu50/p-Si and Ag/p-Si vs. Cu/p-Si under solar illumination is observed compared to the corresponding dark electrodes (Ag50Cu50/n ++ -Si and Ag metal, respectively) in terms of onset potentials, photocurrent densities and power saving metrics. In particular, a remarkable positive potential shift of ~ 840 mV vs. Ag50Cu50/n ++ Si and a ratiometric power saved of ~ 3% have been obtained for Ag50Cu50/p-Si. Finally, photoelectrolysis coupled to gas chromatography has been performed at different potentials and electrolysis times. Maximum FEs for CO and CH4 of 26 % and 17.8% at -0.75 and -0.86 VRHE, respectively, have been obtained. These FEs compare favorably with those reported in the recent literature. The study demonstrates the ability of the bimetallic photocathode to guide the selectivity of PEC-CO2RR towards high energy density products (i.e., CH4) with respect to the monometal-decorated homologue. The specific electronic and geometric properties, as evidenced by XPS and HRTEM, account for this bimetallic effect. Our study demonstrates the interest of bimetallic deposition by MACE and its potential impact on the rapid and cost-effective preparation of Cu-based bimetallic Si photocathodes for efficient PEC-CO2RR.

Figure 1 :

 1 Figure 1: Schematic representation of the electroless synthesis of AgxCu100-x bimetallic NPs on Si supports by MACE developed in this work.

2 .

 2 Ag NPs are homogeneously distributed on the surface, exhibiting a quasi-spherical shape with an average Feret diameter of 25 nm (Fig.2(a)) and a surface coverage of ~47%. Pure Cu NPs (Fig.2(b)) are also quasi-spherical, with an average Feret diameter of 30 nm (similar to Ag NPs) and a surface coverage (~26%). On the other hand, the bimetallic AgxCu100-x NPs under these deposition conditions exhibit in general a worm-like elongated shape with several interconnected ligaments in the same NP (clearly observed for a Ag content varying between 20-50 at% (Fig2(e-h)), with a surface coverage of around 70% and a ligament size of 80 -90 nm. Interestingly, while the change in morphology from quasi-spherical to worm-like is gradual with decreasing Ag content, it is immediate in the opposite direction (i.e., Ag20Cu80NPs are worm-like shaped while Cu NPs are quasi-spherical).

Figure 2 :

 2 Figure 2: (a-h) Representative FESEM images of AgxCu100-x NPs deposited on Si supports by MACE. Averaged atomic composition given on the figures. Deposition time: 1 min at RT, [HF] = 1 mmol L -1 .

  experiments exemplify the possibilities of tunning the morphology and the surface coverage of the bimetallic NPs by varying key experimental deposition parameters (precursor concentrations and deposition time).

3 . 2 . 2 .

 322 Fig.3(a) shows the X-ray diffraction patterns of Ag, Cu and Ag50Cu50 NPs synthesized by electroless deposition on Si. In order to obtain an acceptable signal-to-noise ratio for XRD analysis, the metallization was performed on Si powder instead of flat Si samples. SEM images of the AgxCu100-x/p-Si powders are given in the SI (Fig.S5), together with the synthesis details and all data extracted from XRD.

Figure 3 :

 3 Figure 3: (a) X-ray diffraction patterns of Ag, Cu and Ag50Cu50/p-Si powders synthesized by MACE. Squares, circles and stars mark the XRD peaks for Cu, Ag and Si, respectively. (b) Closer view of the XRD patterns focusing on the (111) crystallographic plane; and (c) estimated crystallite sizes of the (111) crystallographic plane for Ag, Cu and Ag50Cu50. The XRD peaks of Ag for Ag/p-Si appear at 2θ values of 39º, 44.5º, 65º,75.8º, 82º and 96.7º and belong to the (111), (200), (220), (311), (222) and (400) crystal planes 25 . The diffractogram of Cu in Cu/p-Si shows a similar pattern, with peaks appearing at 2θ values of 43.5º, 50.7º, 74.14º and 90.2º, corresponding to the (111), (200), (220) and (311) planes26,27 .

  (b). Compared to the (111) peak of pure Ag, that of Ag in Ag50Cu50 is not shifted to lower 2θ values, indicating a pure Ag phase in these bimetallics. This is consistent with the fact that Ag and Cu are known to be immiscible in almost all ratios at RT.

Figure 4 :

 4 Figure 4: Bright-Field TEM images of the AgxCu100-x NPs at low (a)-(c) and high (g)-(i) magnifications. (d)-(f): SAED patterns for the high magnification images and indexed (hkl) planes.The XRD peaks for the bimetallic are broader than those of pure Ag and Cu, indicating a smaller crystallite size for the bimetallic[START_REF] Torralba | Electrochemical Study of Carbon Dioxide Reduction at Copper-Palladium Nanoparticles: Influence of the Bimetallic Composition in the CO Poisoning Tolerance[END_REF] . Fig.3(c)shows the crystallite size obtained by applying the Debye-Scherrer formula to the peaks associated to (111) planes (see TableS1in the Supplementary Information for more details and the values obtained for other crystallographic orientations). The estimated values for Ag and Cu in Ag50Cu50 are 15 nm and 19 nm respectively, lower than those of pure metals (c.a. 20 and 30 nm for Ag and Cu

Fig. 4 shows

 4 Fig.4(g-i)) is attributed to moiré patterns, i.e., overlapping of Ag and Cu crystal planes with similar spacing and orientation, as described by Freire et al.[START_REF] Freire | Natural Arrangement of AgCu Bimetallic Nanostructures through Oleylamine Reduction[END_REF] . The SAED patterns of Figs.4(d-f)) show a concentric ring-like structure indicating the polycrystalline nature of the bimetallic NPs. The evaluation of the d-spacing values was performed for the most visible rings with the three compositions (results are given in TableS2of the Supporting Information). The values determined are in very good agreement with those obtained by XRD (TablesS1 and S2). This confirms the presence of both Ag and Cu face-centered crystallites

Figure 5 :

 5 Figure 5: (a) HAADF-STEM image of Ag50Cu50 NPs; (b)-(d) EDX elemental mapping in Ag50Cu50, Ag (green), Cu (pink).

Figure 6 :

 6 Figure 6: (a) Ag 3d and (b) Cu 2p XPS spectra of AgxCu100-x NPs deposited on Si by MACE. The peaks in the core spectra of Cu 2p have been fitted with Cu-Cu and Cu-O contributions (orange and blue lines, respectively). The experimental data recorded are given by red dotted lines for Ag 3d and black dotted lines for Cu 2p.

  3 eV from Ag100 to Ag54Cu46, Fig.6 (a)). The values for Ag-Ag are in good agreement with the bulk values, with little or no oxidized Ag-O observed[START_REF] Sengar | Size and Alloying Induced Shift in Core and Valence Bands of Pd-Ag and Pd-Cu Nanoparticles[END_REF] . The peaks at the Cu 2p core level are centered at 932.2 eV and 952.1 eV for Cu 0 in pure CuNPs (Cu100 in the fig.6 (b)) A contribution of CuO is also observed at 934.05 eV and 953.9 eV (see Fig.6 (b)), probably due to native oxide. With increasing Cu content, a small shift of the emission peaks to lower binding energies is observed (e.g., maximum shift of +0.1 eV from Ag89Cu11 to pure Cu).

Fig. 7 (

 7 Fig. 7 (b) shows the polarization curves (j/E responses) of the different photocathodes under 0.2 sun illumination in CO2 atmosphere, together with the response of their homologous dark electrodes (i.e., metals used as reference). The j/E response of Ag and Cu was recorded by using Ag and Cu metal plates. Ag50Cu50/n ++ -Si obtained by MACE was used as a dark electrode for Ag50Cu50/p-Si (n ++ -Si, ρ = 3 -4 mohm cm). The j/E curve of bare p-Si is also shown for reference. Fig.7(c) shows the power saving analysis for the Ag50Cu50/p-Si versus its analogous dark electrode. The table given in Fig.7(d) presents relevant values to quantify the catalytic activity of the photocathodes extracted from the j/E polarization curves (see figure legend for more details).All the photocathodes show the typical j/E response of a PEC process limited by the photon flux and not by the electrolyte mass transfer (Fig.7(b)), i.e., the photocurrent, jph, increases with the applied potential until a jph plateau is reached when the maximum

Figure 7 :

 7 Figure 7: (a) Scheme of the PEC-CO2RR using AgxCu100-x as photocathode and Pt as anode. (b) Polarization curves of p-Si, Ag/p-Si, Cu/p-Si and Ag50Cu50/p-Si under 0.2 sun illumination in CO2 saturated 0.5 mol L -1 NaHCO3, and the corresponding dark electrodes. (c) Blue and black curves: polarization curves of Ag50Cu50/p-Si and Ag50Cu50/n ++ -Si (same conditions than in Fig 7b). Inset: power saved (calculated from Eqs. 5) vs. applied potential E for Ag50Cu50/p-Si (maximum at ). Orange rectangle: actual power saved at j/E corresponding to Pmax,saved . At this point, Vmax saved = -0.69 V and j = -1.67 mA/cm 2 ; (d) Vonset (= Eonset, photocathode -Eonset, dark electrode), jph at the plateau and at 0.5 VRHE (extracted from Fig. 7(b)). Eonset is chosen at j = -0.1 mA/cm 2 .

  I) = I * (E dark (I)-E light (I)) = I * ΔV saved (I) } Eqs. 5

Figs 8 (

 8 Figs 8 (a) and (b) show the chronoamperograms (j/t) and the FE values obtained for a Ag50Cu50/p-Si photocathode at different potentials. The j/t curves are stable with time for low

Figure 8 :

 8 Figure 8: (a) j/t responses at different potentials (given on the figure) and (b) FE calculations of Ag50Cu50/p-Si in CO2-purged 0.5 M NaHCO3 electrolyte at 1 Sun illumination; (c) j/t

Fig. 9

 9 Fig.9depicts the Cu 2p core-level emission spectra for the Ag50Cu50/p-Si photocathode before and after 1 h of photoelectrolysis (bottom and top rows, respectively), together with representative SEM images.

Figure 9 :

 9 Figure 9: Cu 2p XPS core-level spectra for (a) Ag50Cu50/p-Si before and (b) after 1 h photoelectrolysis at -0.87 VRHE; experimental spectra (black circles) and peak analysis (blue, orange, and red lines). Atomic compositions given on the Figures are the surface compositions measured by XPS. Insets: representative SEM images. The scale bar is 100 nm. Other conditions as in Fig. 6.

  -x bimetallic NPs on Si by MACE. It allows to control the bimetallic composition and to tune the NP morphology (from quasi-spherical to elongated worm-like structures) and the surface coverage using key experimental parameters. XRD, SAED and HAADF-STEM patterns show a phase-separated polycrystalline structure with highly dispersed Ag and Cu nanocrystallites (~15 nm), which is quite unusual for soft-chemistry synthesized AgCu NPs. XPS compositional analysis shows Ag-rich surfaces. Opposite shifts in the binding energies of the Cu 2p and Ag 3d core signals with atomic composition indicate a charge transfer from Cu to Ag in the bimetallics.

Table 2 :

 2 Atomic compositions of synthesized AgxCu100-x NPs determined by XPS and SEM-EDX.

	Cu 0	(s) + 2Ag +	(aq) → Cu 2+	(aq) + 2Ag(s)	Eq. 5
	XPS composition (at%)		EDX composition (at%)

  This shows the interest of bimetallic NP deposition and gives value to this low-cost and straightforward bimetallic deposition method.Interestingly, Dong et al. greatly improved their FE results by depositing an SiO2 layer as a mask on bare p-Si, which strongly suppresses HER. This opens up further scope for improving the photocathode efficiencies presented here[START_REF] Dong | Grain Boundary Engineering of Cu-Ag Thin-Film Catalysts for Selective (Photo)Electrochemical CO2 Reduction to CO and CH4[END_REF] . It is worth noting that although the present work focuses on the analysis of the gas products (the main products expected according to the literature[START_REF] Dong | Grain Boundary Engineering of Cu-Ag Thin-Film Catalysts for Selective (Photo)Electrochemical CO2 Reduction to CO and CH4[END_REF] ), preliminary analysis of liquid products by H-NMR have shown up to 5% FE for HCOOH + EtOH for Ag50Cu50/p-Si. TableS3in the supporting information compares our results with FEs reported by other authors.

	, by using a
	patterned Ag/Cu thin film in a p-Si photocathode fabricated by a multi-step elaboration
	process (physical deposition, photolithography and lift-off). Under similar experimental
	conditions, the photocathode converted CO2 to CO and CO2 to CH4 with a maximum FE of

26.9% and 18.2 % at -1.4 and -1.3 VRHE, respectively. Our results are in good agreement with these numbers, with almost identical FEs values but at lower overpotentials (-0.72 and -0.87 VRHE for CO and CH4, respectively).

Acknowledgment

This work was supported by the French National Research Agency (ANR JCJC SIROCCO 2022-2024) and the I-Site Project Future (Exploratory project EVOLVE 2021-2022, Gustave Eiffel University). H. Chaliyawala acknowledges Paris-East Creteil University (UPEC) for a postdoctoral fellowship (2020-2021). The authors also acknowledge Dr. S. Le Gall from the Electrical Engineering Group of Paris (GeePs), Paris-Saclay University, for fruitful discussions on bimetallic electronic properties, J. Bourgon for HRTEM, SAED and HAADF-STEM studies, and R. Pires Brazuna for technical assistance with the SEM/EDX instrument (Microscopy facilities of the East-Paris Institute of Chemistry and Materials).

Finally, the authors acknowledge the support of the Centre National de la Recherche Scientifique (CNRS) for providing the central facilities to carry out this work.

Funding Sources

French National Research Agency (ANR JCJC SIROCCO 2022-2024), I-Site Project Future (Exploratory project EVOLVE 2021-2022, Gustave Eiffel University).

Conflict of interest

There is no conflict of interest between the authors.

Associated content


Pierre Rublon

Bertrand Huneau

Erwan Verron

Nicolas Saintier

Daniel Berghezan

Effect of crystallization on fatigue crack growth resistance of natural rubber

published or not. The documents may come

Effect of strain-induced crystallization on fatigue crack

growth resistance of natural rubber

INTRODUCTION

The impressive resistance of natural rubber (NR) to fatigue crack growth as compared to others synthetic rubbers, has been reported by many authors in the literature [START_REF] Lake | Mechanical fatigue limit for rubber[END_REF][START_REF] Lake | Fatigue and fracture of elastomers[END_REF], Mars and Fatemi 2004, among others). The srain-induced crystallization (SIC) phenomenon, that takes place in particular in the neighbourhood of a crack tip (Trabelsi et al. 2002, Lee and[START_REF] Lee | Microstructural changes in the crack tip region of carbon-black-filled natural rubber[END_REF] because of large strains in this region, is believed to play a major role and would slow down the crack advance during a fatigue test [START_REF] Mars | A phenomenological model for the effet of R ratio on fatigue of strain crystallizing rubbers[END_REF][START_REF] Mars | Factors that affect the fatigue life of rubber: a literature survey[END_REF][START_REF] Saintier | Cyclic loadings and crystallization of natural rubber: An explanation of fatigue crack propagation reinforcement under a positive loading ratio[END_REF]. However, the influence of the crystallized zone at the crack tip on the crack propagation in fatigue remains unproven today.

In this context, the present paper aims to relate the measurements of fatigue crack propagation rates in NR samples with the characteristics of the crystallized zone around the crack tip in order to explain this good macroscopic behaviour. To do this, two experimental methods and their results are compared:

• On the one hand, classical fatigue crack propaga-tion tests are performed on different pre-cracked "Pure Shear" samples of NR filled with various amounts of carbon black, to evaluate the crack propation rate.

• On the other hand, in-situ fatigue tests are conducted with almost the same loading conditions at the French national synchrotron facility SOLEIL, to measure in real time the crystallinity in the vicinity of the crack tip by mapping this region with Wide-Angle X-ray Diffraction (WAXD) measurements. This original experimental method, described in details in [START_REF] Rublon | In situ synchrotron wide-angle X-ray diffraction investigation of fatigue cracks in natural rubber[END_REF], permits to determine the SIC distribution around the crack during uninterrupted fatigue tests.

After the description of the experimental set-up and methods, the end of the paper discusses the different results, in order to solve the question of the influence of SIC on the fatigue crack growth behaviour of natural rubber.

EXPERIMENTAL METHOD

Materials and samples

Four elastomers with the same gum of NR but with various amounts of N347 carbon black (CB N347) are used (0, 30, 50 and 70 phr) in order to evaluate the influence of fillers. Vulcanization is carried out with 1.6 phr of sulphur, where CBS acts as an accelerator. Each blend also contains ZnO and stearic axid. 6PPD is used as an antioxydant. As a comparison, a synthetic isoprene rubber (IR) filled with 50 phr of CB N347 and with exactly the same amounts of additives NR compounds, is also used. In IR, the SIC is considerably reduced compared to NR, and the results obtained on this material are very helpful to discuss the influence of this phenomenon. The respective formulation of those five materials are given in Table 1. The geometry of the samples is a classical "Pure Shear" geometry (PS), commonly employed for fatigue crack growth tests in the literature [START_REF] Young | Dynamic property and fatigue crack propagation research on tire sidewall and model compounds[END_REF][START_REF] Andreini | Crack growth behavior of styrene-butadiene rubber, natural rubber, and polybutadiene rubber compounds: comparison of pure-shear versus strip tensile test[END_REF], whose dimensions are 157 mm long, 13 mm high and 2 mm thick.

Fatigue crack growth rate tests

Fatigue crack propagation measurements are based on the energy balance approach of Rivlin and Thomas [START_REF] Rivlin | Rupture of rubber[END_REF], who have extended the initial theory of energy release rate formulated by Griffith in his seminal paper [START_REF] Griffith | The phenomena of rupture and flow in solids[END_REF]. Considering thin samples of uniform thickness h 0 and denoting c the crack length, they defined the "tearing energy" T by:

T = - 1 h 0 ∂w ∂c l (1)
where w is the strain energy and the suffix . l denotes differentiation with constant displacement of the boundaries over which forces are applied. In the case of PS samples, they demonstrated the following relation between T , W 0 the strain energy density and l 0 the height of the undeformed sample:

T = W 0 l 0 (2)
All the tests were performed at Michelin (Clermont-Ferrand, France) on a MTS servo-hydrolic fatigue machine. After clamping, the sample is first cycled during 300 cycles at a global stretch ratio (λ= l/l 0 ) of 1.92 at 2Hz, to supress the Mullins effect and lower the residual stretch due to an inherent viscous behaviour. Cyclic tests with uncracked samples are then conducted at different stretch ratios from 1.08 to 1.92 in order to determine the strain energy density W 0 as a function of the global stretch ratio, by integration of the area under the stress-strain curve. After the insertion of three notches (which entails 4

synchrotron WAXD experiments

The in-situ WAXD measurements are conducted with a homemade fatigue machine using two electrical actuators which supply a cyclic sinusoidal displacement to the rubber sample. This machine is fixed on a mobile diffractometer on the DiffAbs beamline at the French national synchrotron facility SOLEIL. The wavelength of the beam is 0.1305 nm and the beamsize is approximately 300 µm x 200 µm (full width at half maximum height). The two-dimensional WAXD patterns are recorded by a MAR CCD X-ray detector (SX-165) with an exposure time of 1s. The transmitted beam is blocked by a Pb beamstop on which a Si PIN photodiode is placed to measure the transmitted intensity through the rubber and thus, estimate the thickness of the sample. After almost the same sample preparation as the one described in the previous section 2.2, except the fact that only one notch is introduced at the edge of the sample, the procedure of WAXD acquisition starts. It consists in a mapping of the neigbourhood of the crack tip with 90 WAXD measurements without stopping cyclic loading, by acquiring diffraction data at the top of each cycle and moving the sample with respect to the beam between two cycles in order to perform the next WAXD acquisition on an other point of the map. Figure 1 summarizes this procedure, which is fully described in [START_REF] Rublon | In situ synchrotron wide-angle X-ray diffraction investigation of fatigue cracks in natural rubber[END_REF]. However, the determination of the index of crystallization is here quite different from the procedure described in [START_REF] Rublon | In situ synchrotron wide-angle X-ray diffraction investigation of fatigue cracks in natural rubber[END_REF], in order to take into account this time, the oriented amorphous phase and the influence of carbon black fillers. It consists in ex- peaks. It is to note that, in order to consider the orientation of the amorphous phase during elongation of the sample, the amorphous intensity is averaged in the polar and equatorial directions.

Those measurements lead finally to the plot of isoχ curves around the identified crack tip, as depicted in figure 3. Moreover, taking into account the value of the transmitted intensity as a measurement of the thickness, it is also possible to evaluate the volume around the crack tip in which the SIC phenomenon occurs.

RESULTS AND DISCUSSION

Fatigue crack growth tests

NR50 vs. IR50

The fatigue crack growth behaviour of a NR filled with 50 phr of CB (NR50) is first compared to the behaviour of an IR filled with the same amount of fillers (IR50). The results are presented in Figure 4. First of all, we notice that for a given tearing energy T , the crack growth rate is lower in the case of NR50, and this is especially true for high values of T . This is consistent with the different results of the literature, in particular those of [START_REF] Lake | Fatigue and fracture of elastomers[END_REF], who found a similar behaviour between a filled NR and a filled SBR.

Influence of carbon black fillers

The influence of CB N347 fillers on fatigue crack growth behaviour is reported in Figure 5. It appears that the resistance to crack advance is improved with the addition of fillers, if we compare the unfilled natural rubber (NR0) with the filled ones. Moreover, the slope of the curve seems to be lower in the case of filled NR. Those results tend to confirm the results already obtained by [START_REF] Papadopoulos | Rate Transitions in the Fatigue Crack Growth of Elastomers[END_REF], who Figure 4: Fatigue crack growth rate behaviour of a NR50 sample (unfilled symbols) and a IR50 sample (filled symbols). With the doubly logarithmic plot, the black straight lines represent simple power law relationship (slopes of the curves: 1.70 for NR50, 2.73 for IR50) Figure 5: Fatigue crack growth rate measured on NR0, NR30, NR50 and NR70 samples. With the doubly logarithmic plot, the black straight lines represent simple power law relationship (slopes of the curves: 3.72 for NR0, 1.73 for NR30, 1.85 for NR50 and 1.34 for NR70) compared the fatigue crack growth resistance of a 23 phr CB filled NR with an unfilled one. However, if one looks only at the filled rubbers, the CB content does not seem to impact significantly the crack growth behaviour and just a small decrease of dc/dn is observed when increasing the amount of fillers.

WAXD experiments results

NR50 vs. IR50

Figure 6 presents the iso-crystallinity curves obtained after the WAXD measurements performed around a fatigue crack tip of NR50 for four differents values of tearing energy T . It emphasizes the noteworthy influence of this parameter on the size of the crystallized zone. The values of the transmitted intensity at each point of the map permit to quantify this result with the evaluation of the volume where crystallites are detected in the neighbourhood of the crack tip, as reported in figure 7 for the four previous values of T . As SIC phenomenon is almost null for IR50 sample, the values of the crysallized volume are put equal to zero in this graph.

Influence of carbon black fillers

The influence of CB content on the size of the crystallized zone at crack tip was also investigated. Figure 8 provides the crystallized volume obtained on four different NR filled with various amounts of fillers, as a function of tearing energy T . One notes that the expected effect of fillers is quite hard to notice and on the contrary, it appears that, whatever the CB content, a given tearing energy T leads to a same value of crystallized volume at fatigue crack tip, revealing a master curve in Figure 8.

Discussion

• First, the results of crack growth measurements shown in Figure 4 and the evolution of the crystallized volume at crack tip with tearing energy plotted in Figure 7 can be related. On the one hand, the resistance to fatigue crack growth in NR is greater than the one in IR, and even more for high values of T . On the other hand, the volume of the crystallized zone at crack tip increases with T . Recalling that SIC is considerably lower in IR, it proves that crystallization at crack tip slows down the crack advance. To sum up this idea: the higher the tearing energy, the larger the crystallized zone at crack tip and the lower the crack growth rate.

• Second, one cannot propose the same kind of conclusion by comparing NR samples with different amounts of CB. Indeed, the differences observed on the crack growth behaviour of the materials in Figure 5 are not explained by the crystallized volume measured at crack tip on the same materials, taking into account the master curve appearing in Figure 8. It is consistent with the fact that the improvement of the resistance to fatigue crack growth with CB fillers is also reported on rubbers which do not crystallize at all [START_REF] Papadopoulos | Rate Transitions in the Fatigue Crack Growth of Elastomers[END_REF]).

• Finally, the tearing energy appears to be a parameter of most importance in the phenomenon of SIC at a fatigue crack tip in NR, and may govern it whatever the carbon black content.

CONCLUSION

An original experimental set-up consisting in the association of a homemade fatigue machine with synchrotron radiation, allows the measurement of the crystallized zone at crack tip of NR during fatigue tests. The comparison of those results with fatigue crack growth rates measurements with almost the same conditions of cyclic stretching permits to understand the effect of SIC on fatigue crack propagation of NR and gives a first explanation of the well-known resistance to fatigue crack growth of NR. Furthermore, the comparison of different samples of NR with various amounts of CB fillers underlines the importance of the tearing energy parameter on the size of the crystallized zone in the crack tip neigbourhood.

Figure 1 :Figure 2 :

 12 Figure 1: WAXD measurements during fatigue tests. Top: each diffraction pattern is recorded at the maximum displacement of each cycle. Bottom: between two measurements, the sample moves with respect to the beam to reach the next point on the prescribed map.

Figure 3 :

 3 Figure 3: Raw data (left) and post-processed data (right) of the index of crystallinity χ at a fatigue crack tip in a NR filled with 50 phr of CB N347, at a global stretch ratio of λ = 1.92 and a corresponding tearing energy of T = 14140J/m 2 . The hatched area represents the crack and dotted line show the limits of the measurement map.

Figure 6 :

 6 Figure 6: Distribution of the index of crystallinity around a fatigue crack tip in NR50 for 4 different values of tearing energy: a) T = 4620J/m 2 b) T = 7290J/m 2 c) T = 10390J/m 2 d) T = 14140J/m 2 .The lowest value of the index, corresponding to the limit of detected crystallisation, is 0.002.

Figure 7 :

 7 Figure 7: Volume where crystallinity is detected as a function of tearing energy for NR50 and IR50. The line represents a power law relationship fitted on the values relative to the NR50 sample.

Figure 8 :

 8 Figure 8: Volume where crystallinity is detected as a function of tearing energy for four NR compounds filled with various amounts of CB. The dotted line just gives the tendency of the data.

Table 1 :

 1

	Materials formulation NR0 NR30 NR50 NR70 IR50
	Ingredients		Content (phr)
	NR	100	100	100	100	-
	IR	-	-	-	-	100
	Carbon black N347	0	30	50	70	50
	Stearic acid	2	2	2	2	2
	Zinc oxyde	2.5	2.5	2.5	2.5	2.5
	CBS accelerator	1.6	1.6	1.6	1.6	1.6
	6PPD	1.9	1.9	1.9	1.9	1.9
	Sulphur	1.6	1.6	1.6	1.6	1.6
	crack tips) in the sample at the edges (2cm-long) and
	in the middle of the test piece (3cm-long), a short
	preliminary cyclic test is performed to blunt the crack
	tips and to transform cutter incisions into fatigue
	cracks. After all, the procedure of fatigue crack
	growth experiments is applied, by cycling the cracked
	sample at a frequency of 2Hz and for different global
	stretch ratios (corresponding to different values of
	T ). The cyclic tests are paused several times to
	measure the mean value of the advance of the 4
	cracks with a binocular microscope, from which is
	deduced the crack growth rate (dc/dn in nm/cycle). A
	conventional way to represent the results is a log-log
	(dc/dn) vs. T plot (Lake 1995).

AKNOWLEDGMENT

The authors thank Dr. Dominique Thiaudiere and Dr. Cristian Mocuta for their great help on this project. All the reported diffraction measurements were performed on DiffAbs beamline at synchrotron SOLEIL, Gif-sur-Yvette, France (proposal 20110204).


S Leroy
email: fsleroy@laas.fr

J P Laumond

T Sim

Multiple path coordination for mobile robots: a geometric algorithm

This paper presents a geometric based approach for multiple mobile robot motion coordination. All the robot paths being computed independently, we address the problem of coordinating the motion of the robots along their own path in such a way they do not collide each other. The proposed algorithm is based on a bounding box representation of the obstacles in the so-called coordination diagram. The algorithm is resolution-complete. Its e ciency is illustrated by examples involving more than 100 robots.

Introduction: Path coordination

This paper addresses the following problem: consider n mobile robots sharing the same workspace and planning their paths independently; n such paths being given we want to devise an algorithm deciding whether coordinated motions exist for the mobile robots along their own paths, so that each robot can reach its own goal without colliding the other ones. The problem is known as the multiple robot path coordination problem Latombe, 1991b].

Path coordination versus Path planning Multiple robot path coordination and path planning are two related issues in robot motion planning. In multiple robot path planning the robot paths are not a priori computed. A solution to the multiple robot path planning problem is a collision-free path in the cartesian product of the con guration spaces of all the robots. A solution to the problem exists i the start and goal con gurations belong to a same connected component of the global collision-free con guration space. Searching such a space is a highly combinatorial problem [START_REF] Hopcroft | [END_REF].

To face this complexity several authors have investigated decoupled schemes1 . The decoupled approach has been introduced in Kant and Zucker, 1986]: the method rst plans the paths of the robots independently and then computes the velocity pro les so that the robots do not collide. The approach has been further revisited in Erdmann and Lozano-P erez, 1986;Buckley, 1989;Warren, 1990;Alami et al., 1995].

The path coordination problem as such has been addressed in O 'Donnell and Lozano-P erez, 1989] where the notion of coordination diagram has been rst introduced. It dealt with two robots, a case which has been also addressed in [START_REF] Bien | [END_REF]Chang et al., 1994]. A strategy based on dynamic programming was proposed more recently in La Valle and [START_REF] Valle | Optimal motion planning for multiple robots having independants goals[END_REF] to address problems involving more than two robots.

Objective, approach and contribution We want to solve problems involving more than 100 robots in realistic situations. The algorithm consists in searching a n-dimensional coordination diagram. The main contribution is to propose a bounding box representation of the diagram obstacles. With respect to the previous works above we do not use any regular grid representation. The algorithm is resolution complete and it is complete for a large class of inputs. Its e ciency inherits from the e ciency of simple geometric operations giving rise to a collision-checker dedicated to mobile robot coordination and summarized in Section 2. After having introduced a cell decomposition of the coordination diagram for the case of two robots (Section 3), we extend the algorithm to the general case (Section 4).

2 Paths SA and geometric tools Paths SA The geometric tools we use are based on the following assumption: the robot paths are sequences of straight line segments (S) and arcs of a circle (A). Such sequences are denoted by SA . This assumption is supported by both theoretical and practical considerations. gorithms (see [START_REF] Svestka | [END_REF] and references therein). From another point of view, cooperation-oriented approaches are based on local informations (potential methods): see for instance [START_REF] Reif | [END_REF][START_REF] Reif | [END_REF][START_REF] Cao | [END_REF] for a recent overview. Techniques for path coordination are out of the scope of all these methods.

First of all, it has been proved that a collision-free admissible path exists i there exists a collision-free admissible path of type SA Laumond, 1986]. Moreover, most of the existing complete motion planners for mobile robots provide solution paths of the type SA (e.g., [START_REF] Laumond | [END_REF]Latombe, 1991a;[START_REF] Svestka | [END_REF][START_REF] Mirtich | [END_REF]). Finally geometric algorithms like boolean operations or swept volume computations are simple and computationally e cient when dealing with arcs of circle and straight line segments. (a 2 ; b 2 ). The computation of the boxes is then done by computing the coordination con gurations (see above).

Path decomposition Let us now consider two SA paths 1 and 2 . Instead of applying the bounding box representation directly in the coordination diagram of 1 and 2 , we rst apply a path decomposition. Each path is decomposed into its elementary pieces consisting of either straight line segments, or arcs of a circle. Let ( 1;i ) and ( 2;j ) the pieces sequences of 1 and 2 respectively. The coordination diagram for 1 and 2 then appear as the union of the coordination diagrams of the various pairs ( 1;i ; 2;j ). For instance, the two paths in Figure 2 Completeness The algorithm is complete i it is complete when applied to the elementary diagrams corresponding respectively to three cases: SjjS, SjjA, AjjA.

For the rst two cases the algorithm is complete. The only way for the bounding box approach to loose a solution is that there exist two vertical and horizontal lines intersecting two obstacles (Figure 3). This is however not possible since at least one robot moves along a straight line segment: indeed, the robot moving along the straight line cannot intersect twice the other (convex) robot remaining at a xed position. Then the bounding box approximation does not a ect the completeness of the algorithm for these rst two cases. Completeness is not necessarily guaranteed in the third case AjjA: we may nd counterexamples where the bounding box approximation of the obstacles may split the free space into two connected components. Figure 4 shows an example where the bounding box transforms the full space into an obstacle. However such cases can be solved by the following resolution complete procedure: both arcs of a circle are recursively split into smaller arcs and each pair of the new elementary pieces is processed with the bounding box approach. Moreover such cases are easily identi ed in the path decomposition step above. This means that, according to the inputs, the algorithm may or not activate the recursive subdivision. A solution to the coordination problem is a collisionfree path between (0; : : :0) to (1; : : :1). Let us consider three paths 1 , 2 and 3 . The cell decomposition of ( 1 ; 2 ) coordination diagram induces a partition of the axis s 2 . Then the cell decomposition of the ( 2 ; 3 ) diagram is re ned according to this partition. More generally, the cell decomposition of a ( i ; j ) diagram induces a re nement of the cell decompositions of the 2(n 1) diagrams ( i ; k ) and ( k ; j ) (see Figure 6). We denote by (i; j)-cell a cell of the ( i ; j ) diagram after re nement. The 2-dimensional (i; j)-cells of all the ( i ; j ) diagrams induce a cell decomposition of the ncube. The cells of the n-cube are denoted by n-cells. The main advantage of the following search is that it does not require an explicit representation of the n-cube.

Let us consider a (collision-free) n-cell reached at a current step of the search. The strategy consists in moving only one robot at once at each step. To do that the algorithm generates the 2n cells adjacent to the ncell through a (n 1)-dimensional hyper-plane. Let us consider a n-cell cell, adjacent to the current collisionfree n-cell and corresponding to an elementary motion of robot i. Due to the cylindrical shape of the obstacles, testing if cell is collision-free is easily performed: each of the (n 1) projections of cell onto the elementary ( i ; : ) diagrams should be a collision-free (i; :)-cell.

The search is performed by an A algorithm whose heuristic function is the shortest Euclidean path to the goal point (1; : : :1) of the n-cube. Our algorithm com-putes coordination paths which are Manhattan paths: only one robot moves at once. If needed, we may overcome this fact by \smoothing" the computed path with the help of optimization techniques as in [START_REF] Svestka | [END_REF]. Completeness Due to the cylindrical shape of the obstacles in the generalized coordination diagram, the algorithm above inherits from the completeness property of the coordination procedure for two robots presented in Section 3.

Interaction graph

The nal extension we propose is supported by a practical assumption. When a high num-ber of robots plan their paths independently the path coordination problems are in general localized in di erent domains of the environment and only concern robot subsets. To reduce the combinatorial complexity of the global problem in practice we rst identify which robot traces intersect another trace. We then build an interaction graph whose nodes are the robots; two robot-nodes are adjacent i both corresponding traces intersect. A simple decomposition of the graph into connected components identi es automatically the various subgroups of robots requiring motion coordination. Then the algorithm above is applied to each subgroup.

Results Figure 7 All the steps of the algorithm have been implemented in C++ and run on Sparc Ultra-1. The following table presents the computation times of each step of the algorithm for the examples in the gure 7 and the gure 8 that involves 150 robots5 . A more complete analysis appears in Leroy, 1998]

Conclusion

The proposed approach permits to solve problems for more than 100 robots in a reasonable time. The key points of the method are the e ciency of computation of the coordination con gurations and the bounding box representation of the obstacles in the elementary coordination diagrams.

Nevertheless we should notice that the performance depends on the decomposition of the interaction graph into connected components. The worst case appears when the interaction graph has only one component (e.g., when the trace of some robot intersects all the other traces). In fact, the complexity of the approach is dominated by the highest dimension of the considered ncubes. In practice the algorithm may explore e ciently n-cubes of dimension up to ten (i.e., involving 10 robots).

Figure 1 :Figure 2 :

 12 Figure 1: Two intersecting robot traces

  (a) both consist of 4 arcs of a circle. Therefore the coordination diagram appears as the union of 16 elementary coordination diagrams (Figure 2(c)). Then, for each elementary coordination diagram, we compute a bounding box representation of the obstacles. Figure 2(d) shows the bounding box representation of the diagram in Figure 2(b).Search Such a representation induces a cell decomposition of the coordination diagram into rectangles. Any classical search algorithm may be used to compute a collision-free path from the origin (0,0) to the goal (1,1). Figure2(d)shows a solution path. For this example, note that the widthest robot R 2 (corresponding to the vertical coordinate in the diagram) should necessarily move forward, backward and then forward along the rst two pieces of its path.

Figure 3 :

 3 Figure 3: This case cannot appear when at least one robot moves along a straight line segment.

Figure 4 :

 4 Figure 4: AjjA special case: bounding boxes would ll the space.

  The activation condition is a function dedicated to the case AjjA and checking the existence of a collision-free vertical or horizontal line in the diagram. The activation cases are seldom seen. For instance they do not appear in the examples displayed in Figures 5, 7 and 8. 4 Coordination for n robots Generalized coordination diagram Let us now consider n robot paths i . The cartesian product of all the n(n 1) 2 elementary ( i ; j ) coordination diagrams is a n-dimensional cube called generalized coordination diagram. A point in the n-cube belongs to an obstacle i at least two robots collide. Therefore, the obstacles in the generalized coordination diagram have a cylindrical shape 4 . As a consequence the topology of the generalized coordination diagram is fully characterized by the topology of the elementary 2-dimensional diagrams. Figure 5(b) shows the 10 elementary diagrams for the path coordination problem of Figure 5(a).

Figure 5 :

 5 Figure 5: The 10 elementary diagrams (b) of the generalized coordination diagram of 5 paths (a).

Figure 6 :

 6 Figure 6: The cell decomposition of a diagram re nes the cell decomposition of other diagrams. Generalized coordination diagram modeling and searching We have seen that the bounding box representation of the coordination diagram for two robots induces a decomposition of the diagram into rectangles.Let us consider three paths 1 , 2 and 3 . The cell decomposition of ( 1 ; 2 ) coordination diagram induces a partition of the axis s 2 . Then the cell decomposition of the ( 2 ; 3 ) diagram is re ned according to this partition. More generally, the cell decomposition of a ( i ; j ) diagram induces a re nement of the cell decompositions of the 2(n 1) diagrams ( i ; k ) and ( k ; j ) (see Figure6). We denote by (i; j)-cell a cell of the ( i ; j ) diagram after re nement. The 2-dimensional (i; j)-cells of all the ( i ; j ) diagrams induce a cell decomposition of the ncube. The cells of the n-cube are denoted by n-cells. The main advantage of the following search is that it does not require an explicit representation of the n-cube.Let us consider a (collision-free) n-cell reached at a current step of the search. The strategy consists in moving only one robot at once at each step. To do that the algorithm generates the 2n cells adjacent to the ncell through a (n 1)-dimensional hyper-plane. Let us consider a n-cell cell, adjacent to the current collisionfree n-cell and corresponding to an elementary motion of robot i. Due to the cylindrical shape of the obstacles, testing if cell is collision-free is easily performed: each of the (n 1) projections of cell onto the elementary ( i ; : ) diagrams should be a collision-free (i; :)-cell.The search is performed by an A algorithm whose heuristic function is the shortest Euclidean path to the goal point (1; : : :1) of the n-cube. Our algorithm com-

  Figure 7: A case with 32 robots: the robots traces (a) and the 496 elementary diagrams. The partition into the 8 robot subgroups is illustrated by the 8 bold triangles.

  (a) shows an example of 32 mobile robots paths (including the traces). The 8 connected components of the interaction graph have been computed automatically. The global coordination diagram appears in Figure 7(b) showing clearly the structure induced by the 8 connected components. A detailed view of the coordination diagram involving a subgroup of 5 robots appears; it includes a display of the computed solution path for this group.

Other schemes for multiple robot path planning have been proposed. For instance some centralized approaches aim at facing the problem complexity with probabilistic al-

The obstacles in Figure2(b) have been computed with a brute force discretization approach used only for display purpose.

In our context the coordinate of a con guration on a path is its curvilinear abscissa s on .

This property has been already noticed in La Valle and[START_REF] Valle | Optimal motion planning for multiple robots having independants goals[END_REF]

The motion planner computing an admissible collisionfree path for each robot is based on the algorithm presented in[START_REF] Laumond | [END_REF]. It is not possible to display the \e ective" motions on pictures; animations related to this work may be seen at http://www.laas.fr/ sleroy.

We just argue that this limitation is not critical in practice. Moreover we do not know any alternative approach allowing to solve the case of Figure 8.