P Castellani

E Quarez

C Nicollet

O Joubert

N Gautier

P Pers

G Taillades

A Le Gal

La Salle

A Le Gal

Cold-Sintering

I. INTRODUCTION

Solid oxides cells have been increasingly investigated in the past decades, owing to their high faradic efficiencies [START_REF] Turner | Sustainable Hydrogen Production[END_REF][START_REF] Hanif | Recent progress of perovskite-based electrolyte materials for solid oxide fuel cells and performance optimizing strategies for energy storage applications[END_REF]. Solid oxide cells are most commonly based on fluorite-type oxide-ion conducting materials such as yttria-stabilized zirconia. To limit ohmic losses in the cells, they typically operate at high temperature (700 -900 °C), which causes accelerated degradation of performance [START_REF] Bi | Steam electrolysis by solid oxide electrolysis cells (SOECs) with protonconducting oxides[END_REF]. More recently, new classes of ceramics were studied for their fast proton transport ability in lower temperature ranges (450 -600 °C) [START_REF] Fan | Nanomaterials and technologies for low temperature solid oxide fuel cells: Recent advances, challenges and opportunities[END_REF][START_REF] Iwahara | Proton conduction in sintered oxides and its application to steam electrolysis for hydrogen production[END_REF]. Since the discovery of protonic conduction in strontium cerates perovskite by Iwahara and coworkers [START_REF] Iwahara | Proton conducting ceramics and their applications[END_REF], many efforts concentrated on more stable perovskite materials such as barium cerate and barium zirconate [START_REF] Ryu | Chemical stability and proton conductivity of doped BaCeO3-BaZrO3 solid solutions[END_REF].

Barium zirconate BaZrO 3 crystallizes in a perovskite structure and is already in use in the industry for its thermal refracting properties, since its melting point is around 2620 °C [START_REF] Wendler | Investigation about the reason of limited grain growth of Y-doped barium zirconate[END_REF].

The material also shows excellent mechanical resistance to yield stress, high chemical and thermal stability and compatibility with a wide range of elements. Doped Barium cerate, particularly with yttrium, Ba(Ce,Y)O 3-δ is also a well-known material that shows high proton conductivity at intermediate temperature [START_REF] Babilo | Enhanced Sintering of Yttrium-Doped Barium Zirconate by Addition of ZnO[END_REF] . However, this material shows poor thermal and chemical stability, especially when exposed to carbon dioxide [START_REF] Kim | Effect of cerium on yttriumdoped barium zirconate with a ZnO sintering aid: Grain and grain boundary protonic conduction[END_REF]. Therefore, solid solutions of the two perovskites were also considered, with the aim to combine the chemical stability of BaZrO 3 with the high proton conductivity of doped BaCeO 3 [START_REF] Gao | Structural and electrochemical properties of yttrium-doped barium zirconate by addition of CuO[END_REF][START_REF] Ricote | Conductivity study of dense BaCexZr(0.9-x)Y0.1O(3-δ) prepared by solid state reactive sintering at 1500 °C[END_REF]. In this study, a composition with high Zr content is considered: BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ (BZCY-721) [START_REF] Nasani | Synthesis and conductivity of Ba(Ce,Zr,Y)O3-δ electrolytes for PCFCs by new nitrate-free combustion method[END_REF].

Therefore, this zirconium-rich composition shows good proton conduction and stability in presence of CO 2 . Its properties make it a suitable material as an electrolyte for Proton Ceramic Fuel Cell or electrolyser (PCFC or PCEC). However, the sinterability remains the major issue of this highly refractory ceramic material. To limit processing costs and thermal stresses during sintering treatments, there is a clear incentive to reduce the temperatures required to densify those ceramics.

In the early 60s, Coble [14] proposed a detailed study explaining the different phenomenon occurring during sintering. He also stated that transparent ceramics (crystalline state of the material) could be obtained by completely removing the porosity, thus reaching the complete densification of the material. For some specific materials such as Al 2 O 3 , it is possible to naturally reach such relative density. However, for more refractory materials, using alternative strategies such as sintering aids is almost mandatory. For example, it is possible to use metal oxides such as ZnO, NiO, FeO, MoO 3 , CuO, etc. as sintering aids to enhance both densification and grain growth [START_REF] Gao | Structural and electrochemical properties of yttrium-doped barium zirconate by addition of CuO[END_REF][START_REF] Kim | Effect of cerium on yttriumdoped barium zirconate with a ZnO sintering aid: Grain and grain boundary protonic conduction[END_REF][START_REF] Hostaša | Sintering aids, their role and behaviour in the production of transparent ceramics[END_REF][START_REF] Li | Stable and easily sintered BaCe0.5Zr0.3Y0.2O3-δ electrolytes using ZnO and Na2CO3 additives for protonic oxide fuel cells[END_REF]. In 2005, ZnO was first used as sintering aid for BaZr 0.8 Y 0.15 O 3-δ , and the sintering temperature was reduced from 1700 °C to 1300 °C. Another study presented by Babar et al. [START_REF] Babar | Sintering behavior of BaCe0.7Zr0.1Y0.2O3-δ electrolyte at 1150 °C with the utilization of CuO and Bi2O3 as sintering aids and its electrical performance[END_REF] showed the possibility to reduce the sintering temperature to 1150 °C using a CuO sintering aid. More recently Y. Li et al. [START_REF] Li | Improvement of sinterability of BaZr0.8Y0.2O3-δ for H2 separation using Li2O/ZnO dual-sintering aid[END_REF] have demonstrated that the potential of the simultaneous use of lithium and zinc oxides is effective to enhance the densification of BaZr 0.8 Y 0.2 O 3-δ (1400 °C for 8 h). Another effective way to reduce the energy consumption during sintering is to use intermediate strategies such as Cold-Sintering Process (CSP). The densification mechanisms of the CSP have been extensively studied by Guo et al. [START_REF] Guo | Cold Sintering: Progress, Challenges, and Future Opportunities[END_REF] for simple oxides. CSP uses a small amount of aqueous solution added to the ceramic powder during pressing. The solvent enables densification via a dissolution-precipitation process, while heating the material at boiling temperatures. Recently, CSP has been also applied in our group to BZCY type oxide with a high concentration of cerium, BaZr 0.1 Ce 0.8 Y 0.1 O 3-δ [START_REF] Thabet | Application of the cold sintering process to the electrolyte material BaCe0.8Zr0.1Y0.1O3-δ[END_REF]. The challenge here is to develop low temperature sintering for the refractory ceramic composition BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ . This study compares the influence of dual sintering aids (lithium and zinc oxides) and of CSP on the densification, the morphology and the conductivity of the BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ electrolyte materials.

II. EXPERIMENTAL

Commercial BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ powder (Cerpotech) was used and characterized for this study. Conventional preparation of the electrolyte membrane consists in pressing the powder at 375 MPa for 10 min followed by a sintering treatment at 1600 °C for 15 h. For CSP protocol, 5 wt. % of water was added to the powder and mixed in a mortar. A simple hotpressing dye is used to enable both heating and pressing [START_REF] Thabet | Application of the cold sintering process to the electrolyte material BaCe0.8Zr0.1Y0.1O3-δ[END_REF]. A 375 MPa pressure is applied to the 10 mm diameter pellet and the temperature is kept at 180 °C during 1 h. The pellet is then dried overnight in an oven at 120 °C. Then, the pellet is further treated at 1600 °C during 15 h. In the protocol using sintering aids, a dual sintering aid containing 20 mol.% of lithium oxide and 80 mol.% of zinc oxide is investigated. The sintering aid powder was prepared by impregnation of a lithium nitrate solution (in ethanol) on a commercial zinc oxide powder (Aldrich). The product was dried at room temperature overnight and calcined at 700 °C for 10 h. Then, 1 wt% of the sintering aid powder is added to the BZCY-721 powder and mixed in a mortar manually. In this study, four densification strategies were used: i) the conventional sintering protocol, ii) the sintering aid protocol and iii) the CSP protocol.

The grain size distribution of the powders was measured by laser-granulometry (Masterize2000) under pressurized air and in an ultrasonic bath of absolute ethanol.

Calibration of the measurement was done using the commercial powder before conducting 15 stable measurements for statistical purposes. The specific surface areas were determined by N 2 adsorption using the Brunauer, Emmett and Teller (BET) method in a Micrometrics 3flex apparatus. X-ray diffraction patterns (XRDP) of the powder and sintered commercial BZCY-721 pellet were recorded at room temperature in air using a Bruker D8 advance diffractometer in Bragg-Brentano geometry. A copper anode source of wavelength λ Cu Kα1 = 1.5406 Å is used, isolated with a germanium monochromator. The data is recorded in the range 10-80° 2θ with a step size of 0.018° and 0.42 s at each step. Rietveld refinements were performed with the JANA2006 software using Pmm and Rc space groups [START_REF] Petříček | Crystallographic Computing System JANA2006: General features[END_REF].

Crystallite size and isotropic microstrain were determined using the fundamental approach as implemented in JANA2006 [START_REF] Cheary | Axial Divergence in a Conventional X-ray Powder Diffractometer. I. Theoretical Foundations[END_REF]. Berar corrections were applied to obtain more realistic standard uncertainties [START_REF] Bérar | s and estimated probable error obtained in Rietveld refinements with local correlations[END_REF]. The lithium doped zinc oxide sintering aid was characterized using X-ray photoelectron spectroscopy (XPS) on Surface Science Instruments using an Al K Alpha source with a spot size of 500µm. 501 energy steps were made using a step size of 0.1 eV. For each measurement ten scans were made for a total acquisition time of 4 min and 10 seconds. The spectra were analysed and fitted using the CasaXPS software. The densification behaviour of the electrolyte material was characterized by dilatometry using a NETZSCH Dil402C dilatometer, from room temperature to 1600 °C at a rate of 3 °C•min -1 . The raw powder was characterized using simultaneous Thermogravimetric Analysis (TGA) and Differential Scanning Calorimetry (DSC) up to 1550 °C in dry air using a NETZSCH JUPITER 4. The morphology of the pellets and powders were characterized using Scanning Electron Microscopy (JEOL 7600 SEM). Energy Dispersive X-ray spectroscopy (EDX) was performed on polished cross sections. Samples were fractured and thermally etched at 1400 °C for 10 h. The ImageJ software was used to determine the volume density of grain boundaries after thermal etching of the cross-section. Scanning transmission electron microscopy (STEM) experiments were carried out using Themis Z G3 Cs-probe corrected microscope (Thermo Fisher Scientific) operated at 300 kV and equipped with a high-angle annular dark field (HAADF) detector, with 30mrad convergence angle. Elemental maps were acquired in STEM mode using a Super-X emission X-ray spectrometer consisting of four windowless silicon-drift detectors (SDDs) providing a large collection solid angle of 0.7 rad.

In order to limit electron beam damages the samples were cooled at 96K.

The relative density of the sample before and after densification is obtained using the ratio between measured and theoretical density. The theoretical density is calculated using the equation 1.

(

) 1
where Z is the coordination number of the cubic lattice, M the molar mass of the material, N a the Avogadro constant and V the theoretical lattice volume of the material. The measured density is determined geometrically by measured the mass and volume of the pellets. The relative density is the ratio between measured density and theoretical density.

For electrochemical measurements, symmetrical cells were prepared [START_REF] Khan | Design of efficient and durable symmetrical protonic ceramic fuel cells at intermediate temperatures via B-site doping of Ni in BaCe0.56Zr0.2Ni0.04Y0.2O3-δ[END_REF]. One ink of Ba 0.5 Sr 0.5 Co 0.8 Fe 0.2 O 3-δ (BSCF) and a second one of a composite BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ / Ba 0.5 Sr 0.5 Co 0.8 Fe 0.2 O 3-δ (50/50 wt%) were prepared. The inks were prepared by mixing the oxide powder and ethyl-cellulose (as binder) in α-terpineol. The inks were further mixed in a three-roll mill for better homogeneity. A multi-layer electrode was deposited by screenprinting. The first layer was the composite material deposited as an interface layer to enhance the mechanical resistance of the electrode and reduce the resistive contribution of the interface to the conductivity. The second layer was made of pure BSCF. After optimization of current collection considering several current collection materials (La 0.6 Sr 0.4 Co 0.2 Fe 0.8 O 3-δ , La 0.5 Sr 0.5 MnO 3 , Silver and Platinum), and Pt was selected. Finally, the three layers were sintered at 1100 °C for 1 h. Impedance diagrams were recorded using the Bio-logic SAS model VSP-300 potentiometer. Measurements were taken every 50 °C while decreasing the temperature from 750 °C to 100 °C. Measurements were repeated in dry and wet (3 %) synthetic air (N 2 and O 2 , Alphagaz TM 2 grade, Air Liquide). Validity of the measurements were controlled using the built-in Kramer-König test from the EC-LAB software (Biologic SAS). Frequency sweeps were performed from 1 MHz to 100 mHz using an amplitude of 100 mV, chosen after optimizing the signal to noise ratio while addressing the linearity requirements. In order to quantify the electrolyte contributions, the impedance diagrams of the symmetrical cells were fitted [START_REF] Bondevik | Disagreements between space charge models and grain boundary impedance data in yttrium-substituted barium zirconate[END_REF] with appropriate equivalent circuits using the ZVIEW software (Scribner Associates).

III. Results and discussion

The electrolyte material conductivity depends on the grain growth and densification i.e. high relative density yields high conductivity. The study compares the impact of several sintering strategies on the morphology and electrochemical properties of the electrolyte material.

III.1 Structural analysis of the electrolyte

BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ crystallizes in a perovskite-type structure. The XRDP recorded at room temperature of the powder as received and of the sintered pellet were refined in the possible space groups Pmm and Rc [START_REF] Mather | Phase Transitions, Chemical Expansion, and Deuteron Sites in the BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ Proton Conductor[END_REF]. The XRDP of the powder as received shows small impurity peaks corresponding to BaCO 3 (Figure 1a) while the XRDP of the sintered pellet shows a pure perovskite phase (Figure 1b). The presence of BaCO 3 suggests that the commercial powder is slightly deficient in barium which would explain the slightly smaller volume than that of the sintered pellet. Indeed, in the latter, barium from BaCO 3 can be dissolved in the perovskite lattice, thus increasing its volume. The results of the Rietveld refinement are reported in Table 1. The refinements in the Pmm and Rc space groups give almost identical results with respect to R-factors, microstructural parameters, and isotropic displacement parameters. However, the Pmm space group gives slightly better R Bragg than in the Rc space group. This result, coupled with the lack of visible superstructure peaks in the patterns, leads to consider refinements in the Pmm space group. Figure 1 shows the final plots of the Rietveld refinements in the Pmm space group. As expected from the temperature increase, the crystallite size of the sintered pellet (103 nm) is larger than that of the commercial powder (36 nm). The line broadening of the diffraction peaks is also affected by microstrain which is due to multiple substitutions in the same 1a site of the cubic structure. The microstrain decreases significantly, from 0.2 to 0.04%, after sintering.

III.2 Specific surface area and grain size distribution of the powder

As powder morphology is critical for describing sintering behavior, the specific surface area and grain size distribution of the commercial electrolyte were studied. Figure 2 shows the particle size distribution measured by laser granulometry. The presence of agglomerates in the powder results in a bimodal distribution. The median repartion d50 of the particles can be separated in two sections at the inflection point. Particles of the first section are measured between 800 nm and 2 µm and represent 28 % of the particle volume density. Particles size of the second section are measured between 5 and 11 µm and represent 25 % of the particle volume density . The particle size measured by granulometry is higher than the cristallite size calculted using Rietveld refinement. This difference can be explained by the presence of agglomerates in the powder during the granulometry measurments [START_REF] Wallis | Structural and electrical properties of BaZr0.7Ce0.2Y0.1O3-δ proton conducting ceramic fabricated by spark plasma sintering[END_REF]. BET measurements shows that the powder specific surface area decreases from 20 to 0.08 m 2 •g -1 after heat treatment at 1600 °C for 15 h. This behavior is consistent with grain coarsening during the densification process. However, the sensitivity of N 2 adsorption might be to low for sintered grains, which limits the precision of the measurement, even with a large amount of powder.

III.3 Characterization of the dual sintering aid

The phase composition of the pure lithium doped (20 wt%) ZnO powder was studied by X-ray diffraction. and interstitial Li atoms (52.9 eV) [START_REF] Shawuti | Complex Impedance Analyses of Li doped ZnO Electrolyte Materials[END_REF] was detected by XPS. Therefore, the XPS results confirm that 66.58 % of lithium substitutes well the Zinc atoms. Binding Energy (eV)

III.4 Conventional sintering, cold sintering and sintering aids processes

The presence of open pores inside the electrolyte layer compromises the gas tightness of the electrolyte, leading to cross diffusion of gases from the fuel and air compartment of the cell. Therefore, it is mandatory to reduce the porosity of the electrolyte by enhancing the densification during sintering. In this section, the morphology of pellets is compared and discussed with respect to the densification protocol. Three cases are considered: pellets prepared with conventional sintering, CSP and sintering aids. To investigate the impact of the different sintering protocols on the grain growth and the grain boundary density, thermal etching [START_REF] Babilo | Enhanced Sintering of Yttrium-Doped Barium Zirconate by Addition of ZnO[END_REF][START_REF] Haile | The role of microstructure and processing on the proton conducting properties of gadolinium-doped barium cerate[END_REF][START_REF] Bach | Effect of thermal treatment conditions on the solid-state synthesis of barium zirconate from barium carbonate and monoclinic zirconia[END_REF] was performed on cross sections of the samples to reveal the grain boundaries. SEM images of the cross sections of samples prepared with the three protocols are shown in Figure 5. Grain size and concentration calculated using ImageJ software are applied to the SEM images (Table 3). For conventional sintering (Figure 5a), the high number of grains and their small size shows a limited grain growth. Figure 5b shows SEM images of sample fabricated using CSP. The cross section shows a grain morphology characterized by the heterogeneity of the grain shape and size. However, the grain number is lower than that of the conventional sintering sample, suggesting that CSP facilitates grain coarsening. In contrast, using sintering aid (Figure 5c) gives the highest relative density value, with low porosity and large grain, suggesting that the sintering aids increases grain coarsening. Then, the crystalline structure of the densified samples is studied by X-ray diffraction (Figure 6). For samples densified conventionally and with a sintering aid, the material structure is single-phase with narrow diffraction peaks. For the CSP strategy, the X-ray diagram shows that each diffraction peak has a shoulder systematically to its right, indicating the presence of two phases: a majority phase corresponding to the main peaks, and a minority phase corresponding to the shoulders, whose composition is close to that of the majority phase and whose lattice parameter is slightly lower than that of the majority phase. The addition of water to the powder could influence the structure via the formation of hydroxides during the hot-pressing stage, which, being complex to decompose, would slow the crystallization of the structure, and prevent the formation of a single-phase material. densification also starts around 1000 °C. However, the disruption occurs at temperatures lower than for the conventional sintering, beginning at 1100 °C, and ending at 1400 °C. This disruption is more pronounced than that of the conventional sintering. Finally, the relative density reaches 96 % at 1600 °C. Here, the sintering mechanism is linked to the reactivity of ZnO-Li with the Barium oxide present at the surface of the grains. The sintering aid reacts to form a mixed Barium-Zinc oxide phase at the grain boundary [START_REF] Babilo | Enhanced Sintering of Yttrium-Doped Barium Zirconate by Addition of ZnO[END_REF]. By tuning the chemistry of the grain surface, the melting temperature of the grain boundary is decreased, owing to the formation of an eutectic (1099 °C) [START_REF] Zhu | High-performance Ba(Zr 0.1 Ce 0.7 Y 0.2 )O 3-δ asymmetrical ceramic membrane with external short circuit for hydrogen separation[END_REF]. Moreover, the addition of lithium oxide decreases the formation temperature of the eutectic [START_REF] Peng | Fabrication of yttrium-doped barium zirconate ceramic by high pressure sintering[END_REF], thus increasing the diffusion kinetics. The cation diffusion in the liquid phase being high as compared to solid state diffusion, sintering aids leads to grain coarsening and growth. After densification, high concentration of ZnO are usually measured inside the grain boundary, whereas low concentrations are measured inside the bulk [START_REF] Babilo | Enhanced Sintering of Yttrium-Doped Barium Zirconate by Addition of ZnO[END_REF]. For the sample including CSP in its fabrication protocol (c), the relative density of the green pellet is higher, around 50 %. Similarly, to the conventional sample, the sintering process of the CSP sample begins around 1000 °C and yields a final relative density of 79 % at 1600 °C. The disruption is still observable at 1450 °C, but at a smaller magnitude. These results confirm that the various sintering strategies have a significant impact on the densification mechanisms. The energy loss linked to the observed disruption could be associated to the decomposition of the Barium carbonate impurity. The reaction produces carbon dioxide, which is released in the furnace atmosphere and barium oxide. Moreover, barium oxide is incorporated inside the structure [START_REF] Miller | Mass-independent fractionation of oxygen isotopes during thermal decomposition of divalent metal carbonates: Crystallographic influence, potential mechanism and cosmochemical significance[END_REF][START_REF] Qiao | Properties of barium zirconate sintered from different barium and zirconium sources[END_REF]. It is also shown that using sintering aids decreases the activation energy of the decomposition reaction and facilitates the reincorporation of barium oxide thus improving the densification.

The decomposition of the barium carbonate and the barium oxide incorporation phenomenon, inside the various electrolyte pellets, is studied in Figure 7b, showing TGA/DSC measurements conducted in similar conditions to dilatometry, coupled to a mass spectrometer. During the measurements the mass spectrometer only identified the presence of molecular weight between 43 and 45u which corresponds to the emission of CO 2 . The mass spectrometer does not detect lithium or barium, which confirm that there is no cation evaporation during sintering. For the conventional sample (red), TGA measurements shows 4 % mass loss below 800 °C, related to evaporation of the residual humidity. DSC shows that the onset of the exothermic phenomenon occurs around 1400 °C, which can be correlated to the disruption observed by dilatometry. The refractive behaviour of this sample implies that the temperature is not high enough to reach the energetic feature observable on the other samples. It also means that the decomposition of the barium carbonate is not complete. The absence of the impurity decomposition could explain the very low relative density of the sample after dilatometry measurements. The CSP sample (yellow) is also marked by a loss of 3 % mass under 1000 °C. The absence of the evaporation step can be linked to the drying step of the CSP pellet after the hot pressing. The DSC show the disruption in between two exothermic steps, the first one around with an onset 1318 °C could indicates the carbonate decomposition followed by diffusion of barium and grain coarsening over 1500 °C. The pellet densified with sintering aids (wine) loses 3 % of mass between room temperature and 400 °C, linked to the evaporation of residual humidity. Between 400 and 800 °C, the pellet loses an additional 2.5 % mass, that could be explained by the decomposition of the lithium carbonate occurring after 690 °C [START_REF] Liu | Preparation of lithium carbonate by thermal decomposition in a rotating packed bed reactor[END_REF]. The remaining 2.5 % mass loss corresponds to the barium carbonate decomposition between 800 and 1300 °C. Moreover, an energetic feature similar to the CSP sample can be observed, where the exothermic response has an onset around 1102 °C follow by a transition marked by an endothermic phenomenon before beginning the densification process. The reaction enthalpy for the decomposition of barium carbonate is measured around 227 kJ.mol -1 , comparable to the 230 kJ.mol -1 found in the literature [START_REF] Arvanitidis | A Study of the Thermal Decomposition of BaC03[END_REF].

Since the carbonates were decomposed, the barium in excess needs to be reincorporated into the structure of the material. As the decomposition of the barium carbonate impacts the mass and thermal response of the electrolyte material, it could interfere with the densification process, thus slowing it down. Figure 7 supports well this hypothesis, mainly for the sintering aided and CSP samples where the sintering process begins only after complete decomposition of the Barium carbonate impurity. This delay of the sintering comes in addition of the natural refractive behaviour of the electrolyte material and is exemplified for the conventional pellet where the measured temperature range is not wide enough to observe the complete phenomenon.

III.5 Bulk and grain boundary conductivity

In order to complete the densification and reach relative densities superior to 90 % the electrolytes were sintered at 1600 °C for 15 h. In dense ceramics, conductivity depends on intrinsic factors such as charge concentration and mobility in the crystal lattice, and on extrinsic factors such as diffusion across grain boundaries [START_REF] Haile | The role of microstructure and processing on the proton conducting properties of gadolinium-doped barium cerate[END_REF][START_REF] Huang | A review of AC impedance modeling and validation in SOFC diagnosis[END_REF]. In proton conducting ceramics, it has been shown that proton transport across grain boundaries can be orders of magnitude slower than that of the bulk, and limit the total conductivity [START_REF] Merkle | Oxides with Mixed Protonic and Electronic Conductivity[END_REF]. Because the various protocols studied here, yields dramatically different microstructures, their conductivity are measured by impedance spectroscopy and compared. (

) 2
where Q CPE is the pseudo capacitance and p is the depression angle parameter. The equivalent capacitance, in F cm -2 , is calculated using the equation ( 3) [START_REF] Bondevik | Disagreements between space charge models and grain boundary impedance data in yttrium-substituted barium zirconate[END_REF][START_REF] Haile | The role of microstructure and processing on the proton conducting properties of gadolinium-doped barium cerate[END_REF][START_REF] Huang | A review of AC impedance modeling and validation in SOFC diagnosis[END_REF].

(

) 3
where C eq is the equivalent capacitance, and R meas the measured resistance of the R-CPE parallel circuit. The Nyquist plot displayed in Figure 8a is composed of three different semicircles, where C 0 = 10 -11 F.cm -2 and C 1 = 10 -9 F.cm -2 correspond to the bulk and grain boundary contribution, respectively. The presence of a C 2 = 10 -7 F.cm -2 shows the evolving electrode contribution and the beginning of the charge transfer phenomenon. As the temperature increases, all contributions are shifted to higher frequencies and it becomes impossible to discriminate the bulk and grain boundary contributions. Therefore, in Figure 8b, between 350 and 100 °C. The bulk conductivity (Figure 9a) is similar for all sintering processes, which confirm that it is primarily intrinsic, i.e. not affected by microstructure. The activation energy for the bulk conductivity is around E A = 0.5 ± 0.2 eV for all processing routes, which is consistent with proton transport in perovskite type compounds [START_REF] Merkle | Oxides with Mixed Protonic and Electronic Conductivity[END_REF]. Figure 9b shows the temperature dependence of the grain boundary conductivity, which is clearly affected by the various sintering strategies. Compared to the conventional sintering route, the

R

III.6 TEM and EDX mapping of the grain boundary

Since the grain boundary conductivity is the limiting factor at low temperature, an indepth analysis of the specific chemical and morphological features of grain boundaries was performed for each sintering protocol.

The Figure 10a shows a TEM image of the grain boundary of the conventional sintering sample. A specific section of this image was analysed by EDX mapping (mapping of the Yttrium element repartition is shown on Figure 10b) in order to determine the elemental distribution of cations in the grain boundary when using the conventional route. Figure 10c shows the atomic distribution of each element as a function of the position in the area. The HAADF curve corresponds to the intensity of the signal measured at a given pixel. From the HAADF curve, it is possible to see that the grain boundary is located between the 11 th and the 20 th nm. The mean thickness value of this grain boundary is 8.3 nm. This measurement shows no variation at the grain boundary, meaning that the element repartition is stable. The stoichiometry calculated from the measured elemental concentrations yields a unit formula of The same analysis is performed on a sample fabricated using the CSP sintering strategy.

Figure 11 shows TEM images taken on FIB-SEM-thinned lamellae of the electrolyte (Figure11a). A closer image (Figure 11b) reveals the EDX mapping of the grain boundary (yttrium element). Figure 11c shows the atomic distribution of the various elements present at the grain boundary plotted as a function of position on the sample. The grain boundaries of the CSP sample are thinner (7.1 nm) than the conventional one (8.3 nm). Figure 11c shows that, the distribution of the elements is also homogeneous. For the CSP sample, the stoichiometry calculated from the measured elemental concentrations yields a unit formula of Ba 0.89 Zr 0.74 Ce 0.22 Y 0.15 O 3-δ , which also corresponds to a Barium deficiency. This observation suggests that the Barium deficiency originates from the initial composition of the commercial powder, rather that of an effect of the sintering strategies. There is no measurable change of cation concentrations close or in the grain boundary. Previous results showed that grain boundary has a dominant influence on the total conductivity. However, conventional and CSP densification protocols do not seem to influence the grain boundary chemistry. Figure 12a shows the atomic matrix of the grain boundary, the change in structure orientation at the intersection between the two crystals can be observed. Figure 12b andc shows the EDX maps for yttrium and zirconium elements. Finally, Figure 12d shows the atomic distribution of the constituting elements present at the grain boundary, plotted as a function of position on the sample. EDX analysis confirms the presence of the doping element Zn (the technique is not sensitive to lithium). The thickness of the grain boundary reaches 2.9 nm, which is much smaller than for the other routes (7.1 and 8.3 nm). Figure 12b andc shows that, while the distribution of barium, cerium and zinc is homogeneous, an excess of yttrium and a slight deficiency of zirconium is observed at the grain boundary, between 2.5 and 5.5 nm. The the assumption that the barium was entirely re-dissolved inside the structure was proposed.

The TEM analysis shows that this assumption might is not true, and that the initial stoichiometry of the commercial powder used might be deficient in barium.

TEM analysis of the different routes shows variations in morphology (different thicknesses) and chemistry at the grain boundary. To discriminate the two effects, the specific grain boundary conductivity is calculated. The specific grain boundary conductivity Figure 13 has been determined based on the brick layer model [START_REF] Haile | The role of microstructure and processing on the proton conducting properties of gadolinium-doped barium cerate[END_REF][START_REF] Peters | Grain-Size Effects in YSZ Thin-Film Electrolytes[END_REF] using equation 4 in the case of C bulk < C gb and where σ gb < σ bulk .

(

) 4
where, g is the grain boundary thickness calculated using TEM images and G the grain thickness given in Table 3. Moreover, σ gb values (displayed in Figure 9b) are obtain by fitting the corresponding Nyquist diagrams and normalized regarding the geometry of the sample.

The values of g and G used to calculate the specific grain boundary conductivity are detailed in Table 4. Figure 13 shows the temperature dependence of the specific grain boundary conductivities of the different samples, which better represents the grain boundary transport by suppressing volumetric effect. Figure 13 shows that the presence of ZnO-Li has a positive impact on the grain boundary as compared to conventional and CSP sintering, increasing by an order of magnitude the specific conductivity of this sample. On the contrary, the CSP route yields a specific grain boundary conductivity closer to conventional sintering, linked to their similar mean grain boundary thicknesses (respectively 7.1 and 8.3 nm). As already observed by Iguchi et al. [START_REF] Iguchi | The relationship between chemical composition distributions and specific grain boundary conductivity in Y-doped BaZrO3 proton conductors[END_REF], cation segregations and depletions can readily occurs at the grain boundaries of zirconate ceramics. These imbalances cause a variation of charged defects concentrations in the grain boundary cores, which in turn results in variation of the space charge barriers that protons need to overcome to cross the grain boundary. Then, the wide range between the specific grain boundary conductivity of the ZnO-Li sample and the other strategies could be explained by the observed segregation of Yttrium and depletion of zirconium. Whereas, the specific conductivity similarity between the conventional and CSP routes can be explained by their closed values of grain boundary thickness. A first hypothesis would be to imagine that the chemistry of the grain boundary has a dominant influence on its conductivity.

III.7 Total conductivity of the electrolytes

Additionally, the total conductivities are compared over a wider range of temperatures in wet and dry atmospheres (Figure 14 a,b andc). Below 600 °C, in wet air, all the samples show higher conductivity, compared to the dehydrated ones (under dry atm), in accordance with the presence of protons in the lattice. At low temperature, thermal energy is not sufficient for the oxide ion to diffuse easily through the grain boundary due to their high ionic radii.

Moreover, the activation energy range from 0.7 to 1 eV of the oxygen-ion diffusion requires higher temperatures (From 600 °C to 1100 °C) to increase the charge mobility in the electrolyte structure [45][START_REF] Wang | Sintering Behavior and Conductivity Study of Yttrium-Doped BaCeO 3 -BaZrO 3 Solid Solutions Using ZnO Additives[END_REF][START_REF] Wallis | The influence of the sintering temperature on BaZr0⋅7Ce0⋅2Y0⋅1O3δ proton conductors prepared by Spark Plasma Sintering[END_REF]. For protonic species, smaller than the oxide ions, the activation energy for diffusion is lower (0.4-0.55 eV), thus increasing the conductivity at lower temperatures. At temperatures above 600 °C, proton concentration decreases as a result of dehydration [START_REF] Quarez | Water incorporation and proton conductivity in titanium substituted barium indate[END_REF], which reduces the protonic conductivity accordingly. Therefore, the conductivity is primarily that of the oxygen-ion vacancies, which is consistent with the observed change of activation from 0.5 eV to 0.7-0.8 eV and an overlap between the conductivities in wet and dry atmospheres. For the sample with sintering aid (Figure 14c), the overlap between protonic and oxygen-ion conduction occurs at a temperature below 600 °C.

This behaviour suggest that the chemistry of the grain boundary has also an influence on the dehydration processes.

III.8 Towards decreasing the sintering temperature

One of the motivations for this study was to better understand the densification behaviour of the BaZr x Ce 0.9-x Y 0.1 O 3-δ electrolyte material in order to reduce its sintering temperature. Figure 15 shows the total conductivity of a sample, fabricated using the CSP protocol, densified at 1500 °C for 15 h, compared to the former CSP sample densified at 1600 °C for 15 h. At 500 °C, it is possible to observe constant conductivity for both samples around 2•10 -3 S•cm -1 . The conductivity remains constant even though the relative density of the sample respectively going from 91 to 88 %. CSP samples have also been fabricated using lower temperature (1400 -1300 -1200 °C), however, at this temperature the densification process is not sufficient and the pellets are too brittle to be measurable.

IV. Conclusion

The densification mechanisms of zirconium-rich BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ electrolytes based on commercial powders using different protocols have been studied. XRD diagrams of the powder material showed the presence of a barium carbonate impurity, which completely decomposes at 1100 °C, leaving a pure crystalline phase of the electrolyte material after a high temperature treatment (1600 °C). From this powder, electrolyte pellets have been synthetized using conventional pressing, CSP, or 1 wt.% of ZnO-Li sintering aid. The three samples were post-sintered at 1600 °C for 15 h. The densification process was characterized by dilatometry. A volume increase, associated to the decomposition of the barium carbonate is observed between 1100 °C and 1450 °C. TGA and DSC results confirm that the disruption is due to the decomposition of the barium carbonate impurity and the dissolution of the barium back in the perovskite lattice. The characterization of the morphology shows that improved densification method such as CSP or the addition of sintering aids have a direct impact on the grain growth, thus giving higher relative density. The conductivity of the samples has been characterized using EIS. At temperatures below 350 °C, the grain boundary is the limiting factor to the conductivity of the electrolyte. Concerning CSP and conventional strategies, the specific grain boundary conductivity showed similar results. However, the ZnO-Li sample showed an improved specific grain boundary conductivity. The TEM analysis conducted shows that, even if it favours a higher conductivity of the material through a better densification (grain boundary thickness equals 2.9 nm), using sintering aids such as ZnO-Li influences the chemistry of the grain boundary. Higher concentrations of yttrium and a zirconium deficiency can be observed at the grain boundary. Moreover, the TEM analysis of the conventional and CSP sample shows a thicker grain boundary (respectively 8.3 and 7.1 nm). However, the latter demonstrates a homogeneous repartition of the elements at the grain boundary. Finally, for each sample, a general barium deficiency is observed.

At temperature between 350 and 600 °C under humid atmosphere, the study showed that the protonic behaviour of the material increases by one order of magnitude the total conductivity. The sintered aided electrolyte (95 % relative density) yields a conductivity of 2•10 -3 S•cm -1 at 450 °C in wet air and 8•10 -4 S•cm -1 in dry air. The specific protonic conductivity is then extracted yielding 1.2•10 -3 S•cm -1 . Finally, at high temperature (750 °C), conductivity is dominated by oxide ion diffusion, and all electrolyte samples yields total conductivities of 2•10 -2 S•cm -1 , regardless of the processing route.

Finally, while using the sintering aid strategy, sintering temperature of the sample has been decreased from 1600 °C during 15 h to 1300 °C during 25 h. While using this new protocol, the sample showed equivalent relative density (93 %) and total conductivity (3•10 -3 S•cm -1 at 500 °C)

V. Acknowledgments

Figure 1 -

 1 Figure 1 -X-ray diffraction patterns, and Rietveld refinements in the Pmm space group of commercial BZCY721 powder (a) as received, and (b) after sintering at 1600 °C for 15 h

Figure 2 -

 2 Figure 2 -Grain size distribution of the Cerpotech commercial powder measured by laser granulometry

Figure 3

 3 shows the XRD patterns of ZnO and Li doped ZnO samples. The positions of the Bragg peaks match well with the reported XRD pattern of the Wurtzite structure of ZnO (JCPDS: 89-1397). No additional phases with zinc and/or lithium compound were detected from the XRD patterns. However, small shifts in the position of the diffraction peaks towards higher angles are observed and indicate a contraction of the unit cell, consistent with Li substitution for Zn in the ZnO lattice. Indeed, Li 1+ has a smaller ionic radius (0.60 Å) in comparison to Zn 2+ (0.74 Å) ion. It was found that the hexagonal lattice parameters (a = 3.24 Å and c = 5.21 Å) are close to those reported in the literature [29].

Figure 3 -

 3 Figure 3 -X-ray diffraction patterns of commercial Zinc oxide powder and the Lithium doped Zinc Oxide powderAdditionally, the oxidation states of lithium and zinc were studied by XPS. The binding energy level of Carbon C 1s (285 eV) was used as a reference energy level and all XPS spectra were shifted according to C 1s energy level. In Figure4ait is possible to see that the carbon peak is influenced by two chemical compositions. The C-C reference at 285 eV and the presence of carbonates (CO 3 ) at 290 eV. Figure4shows the full-scale measurement of the 20 mol% Li doped ZnO sample (figure4a), Zn 2p (figure4b) and Li 1s (figure4c) binding energy were analyzed. The Zn 2p and Li 1s peaks were fitted using the CasaXPS software and all the fitting parameters are referenced in Table2. In the Zn 2p region, the XPS spectra

Figure 4 -

 4 Figure 4 -(a) X-Ray photoelectron spectrometry measurements of Li doped ZnO, (b) Scan for Zn 2p and (c) Li 1s binding energy levels

Figure 5 -

 5 Figure 5 -SEM images of BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ pellets cross sections, densified by (a) conventional sintering, (b) ZnO-Li 1wt% sintering aid (c) CSP at 180 °C before sintering

Figure 6 -

 6 Figure 6 -XRD diagrams of the BZCY-721 samples densified following the three strategies.Because there are two morphology properties affected by the processing protocol, namely densification and grain growth, dilatometry measurements were performed to further characterized the densification mechanisms. Figure7ashows the sintering curves measured by dilatometry. Green pellets for conventional sintering (a), and sintering aid (b) protocols shows similar initial relative densities around 42 %. For the pellet prepared by conventional sintering, densification begins around 1000 °C before reaching a disruption at 1400 °C. This disruption is overcome around 1500 °C, after what the pellet density increases again, to reach a relative density of 66 % at 1600 °C. For the sample fabricated using sintering aid (b),

Figure 7 -

 7 Figure 7 -Temperature dependence (up to 1600 °C) of the relative density and geometrical shrinkage of the of BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ pellets measured by (a) dilatometry based on the different sintering strategies, (a) conventional sintering, (b) ZnO-Li 1 wt% sintering aid and (c) CSP. Evolution of the BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ pellet, (b) characterized using TGA and DSC (up to 1550 °C), fabricated using the different sintering strategies

Figure 8

 8 Figure 8 shows, as an example, the Nyquist diagrams of the Pt//BSCF//BCZY-BSCF//BCZY symmetrical cell, measured at 200 °C and 500 °C in humidified air ( = 0.02 atm). The diagrams show a series of depressed semi-circles that can be fitted by an appropriate equivalent circuit displayed above each of them. The impedance of the CPE element is given by equation (2).

  electrolyte corresponds to bulk and grain boundary contributions. Moreover, C 2 and C 3 , of respective capacitance of 10 -7 and 10 -5 F.cm -2 correspond to the charge transfer and adsorption/dissociation reaction of oxygen and water occurring inside the electrodes. The conductivity of the electrolytes prepared by the different sintering routes was derived from impedance diagrams by determining R 0 and R 1 values, which are not affected by the electrode contribution. After fitting of the impedance data to obtain the resistive contribution of each phenomenon, the conductivities of the samples were calculated by taking into account the geometry of the samples.

Figure 8 -Figure 9

 89 Figure 8 -Nyquist plot of the Pt//BCZY-BSCF//BCZY//BCZY-BSCF//Pt symmetrical cell at 200 °C (a) and 500 °C (b) under humid air and corresponding equivalent circuit. BCZY sample has been sintered at 1600 °C for 15 h Figure 9 shows the temperature dependence on the bulk and grain boundary conductivities of the different sintering processes, on an Arrhenius plot. The grain interior and grain boundary (a,b) have been characterized at low temperatures in wet (3%) synthetic-air

  use of sintering aids seems to improve the grain boundary conductivity, while the CSP route yields lower grain boundary conductivity. The contribution of the grain boundary transport is influenced by both the chemistry of the grain boundary (segregation of charged defects, space BZCY-BSCF//BSCF//Pt at 500°C BZCY-BSCF//BSCF//Pt at 500°C BZCY-BSCF//BSCF//Pt at 500°C ), and by their concentration. Indeed, more grain boundaries will yield higher resistance.

Figure 9 -

 9 Figure 9 -Temperature dependence on the (a) bulk and (b) grain boundary conductivities of the different sintering processes, on an Arrhenius plot

Ba 0 .

 0 85 Zr 0.75 Ce 0.24 Y 0.16 O 3-δ , which corresponds to a Barium deficiency as compared to the nominal formula of BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ .

Figure 10 -

 10 Figure 10 -(a) TEM image of a cross section of BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ electrolyte fabricated using conventional process (b) EDX mapping of the conventional sample grain boundary {Y} and (c) evolution of the atomic distribution as a function of the map position on the sample

Figure 11 -

 11 Figure 11 -(a) TEM image of a cross section of BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ electrolyte fabricated using CSP, (b) EDX mapping of the CSP sample grain boundary {Y} and (c) evolution of the atomic distribution as a function of the map position on the sample Figure 12 shows TEM images of the electrolyte fabricated using ZnO-Li sintering aids.

Figure 12d ,

 12d Figure12d, confirms that the distribution of elements inside the bulk structure (from 0 to 2.5 nm and 5.5 to 8.5 nm), corresponds to a Ba 0.91 Zr 0.73 Ce 0.22 Y 0.13 O 3-δ stoichiometry (the atomic fraction of the elements has been normalized by the presence of zinc) close to the desired one (BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ ). At the grain boundary, from 2.5 to 5.5 nm, the element distribution corresponds to a stoichiometry with yttrium enrichment: BaZr 0.61 Ce 0.19 Y 0.2 O 3-δ .

Figure 12 -

 12 Figure 12 -(a) TEM image of a cross section of BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ electrolyte fabricated using ZnO-Li, (b and c) EDX mapping of the ZnO-Li sample grain boundary {Zr and Y} and (d) evolution of the atomic distribution as a function of the map position on the sample The fine characterization of the grain boundary by TEM and EDX mapping provides information to better explain the discrepancy in grain boundary conductivities revealed by impedance measurements between the different samples. Both Conventional and CSP samples show a similar chemical behaviour at the grain boundary. The elements are well distributed and the stoichiometry of the grain boundary is close to the desired one. In the sample densified with sintering aid, the presence of zirconium deficiencies and high yttrium concentrations influences charge diffusion in this zone. The sintering aids, namely ZnO andLi 2 O, because of their low solubility in the perovskite lattice, will segregate at the grain boundary and promote proton diffusion by reducing the space charge barrier. Their presence could also favours the yttrium and zirconium segregation. Higher concentrations in yttrium indicates higher concentrations of oxygen vacancies, thereby increasing the height of the space charge. Therefore, it is easier for the ionic charges to diffuse inside a grain boundary that is parallel to the charge flow. It should also be noted that no excess zinc is observed in the grain boundary zone, which suggests that the beneficial effect of the sintering aid is correlated with densification mechanisms, and not with the change in composition at the grain boundaries. Finally, for every sintering routes, a constant barium deficiency in the structure is observable. During the decomposition of the barium carbonate impurity studied previously,

Figure 13 -

 13 Figure 13 -Conductivity dependence of the specific grain boundary over the temperature range 100 -350 °C in humid (3 %) atmosphere, of the BZCY-721 electrolytes densified using the different sintering strategies

Figure 14 -

 14 Figure 14 -Temperature dependence on the total conductivities of the different sintering processes; (a) conventional, (b) CSP and (c) ZnO-Li sintering aid on an Arrhenius plot from 750 to 350 °C in dry and wet atmospheres Table 5 compares the values of total conductivities of the different samples. The conventionally sintered electrolyte yields a conductivity of 1.6•10 -3 S•cm -1 at 450 °C in wet

Figure 15 -

 15 Figure 15 -Total conductivities of densified BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ electrolyte using CSP, at decreasing temperatures, 1600 °C for 15 h and 1500 °C for 15 h In view of the promising results obtained with the ZnO-Li aid, several BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ samples were densified with this aid at lower temperatures (1500 -1400 -1300 °C) while maintaining this temperature the longer the lower the temperature (15 -20 -25 h respectively). Figure 16 shows SEM images of cross-sections of the various samples. These images show that the structure is homogeneous and completely dense (around 93 % relative density). It is therefore possible to achieve complete densification of the BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ material using a suitable heat treatment (1300 °C for 25 h) and the sintering aid of ZnO-Li.

Figure 16 -

 16 Figure 16 -SEM images of a cross section of the ZnO-Li sample sintered at (a) 1500 °C for 15 h, (b) 1400 °C for 20 h and (c) 1300 °C for 25 h Figure 17 shows the evolution of the total conductivity of the electrolyte under humid air (3 %) as a function of the densification temperature, compared with that of the initial sample densified at 1600 °C for 15 h. At 500 °C, all samples show a total conductivity of 3•10 -3 S•cm -

Figure 17 -

 17 Figure 17 -Total conductivities of densified BaZr 0.7 Ce 0.2 Y 0.1 O 3-δ electrolyte using the ZnO-Li sintering aid, at decreasing temperatures,1600 °C for 15 h, 1500 °C for 15 h, 1400 °C for 20 h and 1300 °C for 25 h

  This work is part of ARCADE Franco German ANR/BMBF project (ANR-19-ENR-0003) "Advanced and Robust metal supported Cell with proton conducting ceramic for electrolysis Applications in defossilized Energy systems".gion and the European Regional Development Fund of Pays de la Loire and the CIMEN Electron Microscopy Center in Nantes, are greatly acknowledged.

Table 1 .

 1 Structural parameters obtained from Rietveld refinement of the powder and sintered pellet of commercial BZCY721. For refinements, the Ba site occupancy is set to full, the Zr, Ce and Y occupancies are set to 0.7, 0.2 and 0.1, respectively and the O site occupancy is set to obtain the formula BaZr 0.7 Ce 0.2 Y 0.1 O 2.95 . a V c is

	a								b
	 BaCO 3					Observed									Observed Observed
							Calculated									Calculated Calculated
							Difference									Difference Difference
	unit)						Bragg		unit)	unit)						Bragg Bragg
	(arb.								(arb.	(arb.
	Intensity								Intensity	Intensity
	10	20	30	40	50	60	70	80	10	20	30	40	50	60	70	80
										10	20	30	40	50	60	70	80
				2q (°)								2q (°) 2q (°)

b the volume per unit formula. b For the x-coordinate of oxygen in the Rc space group, the value was set to 0.5 due to instability.

Isotropic displacement parameter U iso (Å 2 ), Wyckoff position and x-coordinate for O in the Rc space group

Table 2 .

 2 In the Zn 2p region, the XPS spectra shows two peaks located at 1021.5 eV and 1044.9 eV corresponding to the Zn 2p 3/2 and Zn 2p 1/2 states. The spin-orbit splitting of 24 eV for Zn-2p 3/2 and Zn-2p 1/2 confirms that the Zn atoms are in a completely oxidized state in the sample. Li 1s signal appearing at a binding energy of 55.1 eV also shows two chemical contributions. The first contribution corresponds to Li 2 O (66.58 %) fitted at 54.98 eV. The second contribution corresponds to LiCO 3(33.42 %) and is fitted at 55.88 eV, its presence being supported by the carbonate contribution to the C 1s peak. In this study, no binding energy positions due to metallic lithium cluster (52.3 eV)

Table 2 .

 2 XPS measurement fitting parameters used in Figure4b and 4c

		Li 1s		Zn 2p
		Li 2 O	Li 2 CO 3	1/2	3/2
	Fwhm	1.36703	1.81595	2.60566	2.69236
	Area	876.9	440.2	48116.1	23458.1
	Position	54.98	55.88	1022.28	1045.38
	Line Shape LA(1.3.230) LA(1.3.230) LA(1.3.230) LA(1.3.230)

Table 3 .

 3 Grain density values calculated with ImageJ from SEM cross section images after thermal etching

	Sintering protocol	Conventional	CSP	ZnO-Li
	Grain count (grain)	50	86	26
	Count Area (µm²)	4.52	26.1	409
	Mean grain diameter (µm)	0.35	0.55	3.3
	Grain Density (grain.µm -²)	11	3.3	0.06

As shown in table 3, conventional sintering yields low grain coarsening, exemplified by a high grain density (11 grain•µm -²). The presence of lithium doped zinc oxide increases grain coarsening dramatically, yielding a grain density one order of magnitude lower than that of the conventional sintering (0.2 grain•µm -²). The CSP yields a grain density of 3.3 grain•µm -².

Table 4 -

 4 Grain size calculated from SEM images and grain boundary thickness calculated from TEM images

		Conv	CSP	ZnO-Li
	G (cm)	3.5•10 -5	5.5•10 -5	3.5•10 -4
	g (cm)	8.3•10 -7	7.1•10 -7	2.9•10 -7

Table 5 .

 5 Relative density values calculated from the lattice volume and geometrical parameters of the different samples. Activation energies of the total conductivities under dry and humid (3 %) atmospheres, and total conductivities of the samples at 450 °C and 750 °C.

		Relative	E A wet	E A dry	σ tot 450 °C	σ tot 450 °C	σ tot 750 °C
		density (%)	atm (eV)	atm (eV)	wet (S•cm -1 )	dry (S•cm -	(S•cm -1 )
						1 )
	Conv	90	0.50	0.78	1.6•10 -3	4.7•10 -4	1.8•10 -2
	CSP	91	0.55	0.79	1•10 -3	3.9•10 -4	1.6•10 -2
	ZnO-Li	95	0.48	0.77	2•10 -3	8•10 -4	1.9•10 -2


Chakib Fettal
email: chakib.fettal@etu.u-paris.fr

Lazhar Labiod
email: lazhar.labiod@u-paris.fr

Mohamed Nadif
email: mohamed.nadif@u-paris.fr

Simultaneous Linear Multi-view Attributed Graph Representation Learning and Clustering

Keywords: CCS CONCEPTS, Computing methodologies → Unsupervised learning;, Information systems → Clustering attributed networks, clustering, multi-view

Over the last few years, various multi-view graph clustering methods have shown promising performances. However, we argue that these methods can have limitations. In particular, they are often unnecessarily complex, leading to scalability problems that make them prohibitive for most real-world graph applications. Furthermore, many of them can handle only specific types of multi-view graphs. Another limitation is that the process of learning graph representations is separated from the clustering process, and in some cases these methods do not even learn a graph representation, which severely restricts their flexibility and usefulness. In this paper we propose a simple yet effective linear model that addresses the dual tasks of multi-view attributed graph representation learning and clustering in a unified framework. The model starts by performing a first-order neighborhood smoothing step for the different individual views, then gives each one a weight corresponding to its importance. Finally, an iterative process of simultaneous clustering and representation learning is performed w.r.t. the importance of each view, yielding a consensus embedding and partition of the graph. Our model is generic and can deal with any type of multi-view graph. Finally, we show through extensive experimentation that this simple model consistently achieves competitive performances w.r.t. state-of-the-art multi-view attributed graph clustering models, while at the same time having training times that are shorter, in some cases by orders of magnitude.

INTRODUCTION AND RELATED WORK

Attributed graphs are graphs that contain features in their nodes. Under different approaches, they are used to model a wide variety of structured data [START_REF] Fettal | Subspace Co-clustering with Two-Way Graph Convolution[END_REF][START_REF] Fettal | Scalable Attributed-Graph Subspace Clustering[END_REF][START_REF] Riverain | Semi-supervised Latent Block Model with pairwise constraints[END_REF] with applications in recommender systems [START_REF] Fan | Graph Neural Networks for Social Recommendation[END_REF][START_REF] Salah | Social regularized von Mises-Fisher mixture model for item recommendation[END_REF][START_REF] Ying | Graph convolutional neural networks for web-scale recommender systems[END_REF], computer vision, [START_REF] Qi | 3D Graph Neural Networks for RGBD Semantic Segmentation[END_REF][START_REF] Victor | Few-Shot Learning with Graph Neural Networks[END_REF][START_REF] Xu | Scene Graph Generation by Iterative Message Passing[END_REF], and physical systems [START_REF] Sanchez-Gonzalez | Graph networks as learnable physics engines for inference and control[END_REF]. However, capturing topological (or structural) information at the same time as capturing information about node-level features presents challenges, and in the last few years a number of methods have been proposed for tackling problems such as attributed graph representation learning and attributed graph clustering.

In some real-world applications, data is collected from a variety of different sources, which means that it can be characterized using different sets of information or views. This is the premise of multiview learning [START_REF] Xu | A survey on multi-view learning[END_REF], an area of considerable interest among the data mining and machine learning communities [START_REF] Boutalbi | Implicit consensus clustering from multiple graphs[END_REF]. In the context of attributed graphs, a multi-view attributed graph is simply a set of attributed graphs; each attributed graph counts as a single view. For example, in the case of a recommender system, the relationship between users can be characterized using a two graphs, one representing their "friendship", and one representing their mutual interests; as for the features, one set of features could represent personal information and another set their past transactions.

The task of multi-view attributed graph clustering has lately received a lot of attention. The methods that have been proposed can be separated into two broad approaches. In the first approach, a consensus graph partition is learned directly from the data without explicitly learning an embedding of the graph. Methods adopting this approach include MvAGC [START_REF] Lin | Graph Filter-based Multi-view Attributed Graph Clustering[END_REF], where a graph filter is proposed to perform the graph clustering, and MAGC [START_REF] Lin | Multi-view attributed graph clustering[END_REF], a similar method to MvAGC that uses a graph filter to learn a consensus graph before doing the clustering. The second approach is more flexible; it consists in learning a consensus representation or embedding before applying a simple single-view attributed graph clustering method. For example, DMGI [START_REF] Park | Unsupervised attributed multiplex network embedding[END_REF] is an unsupervised network embedding method for attributed multiplex networks that uses the concept of mutual information, while O2MAC [START_REF] Shaohua Fan | One2multi graph autoencoder for multi-view graph clustering[END_REF] is based on the graph autoencoder [START_REF] Thomas | Variational graph auto-encoders[END_REF], it learns clustering-friendly embeddings through integrating a clustering loss in its objective.

These different methods have their shortcomings. First, the more flexible approaches that learn a consensus representation generally tackle the problems of representation learning and clustering separately, i.e. they learn representations that are not specifically tailored to clustering. Second, they often have unnecessarily complex architectures in comparison to simpler strategies. Finally, some of these methods are not generic, in the sense that they require the multi-view graph to be of a certain type: for example, a multi-view graph with a multiple structures and a single set of features (but not the other way around) like for DMGI, or a graph with exactly two views, etc. As a way of addressing these shortcomings, we propose LMGEC (for Linear Multi-view Graph Embedding and Clustering), a simple yet effective linear model. LMGEC starts by applying a linear graph filter corresponding to a one-hop neighborhood propagation step in each individual view, and then applies a weighting scheme so that views are attended to in order of their perceived importance. This is followed by an iterative process of simultaneous clustering and representation learning, which gives rise to a consensus embedding and partition of the graph. The model is generic in the sense that it can deal with any number of graph structures and/or any number of feature sets. A high-level schematic representation of the model is shown in figure 1. Our contributions may be summarized as follows:

• We introduce a simple yet effective generic linear model for performing multi-view attributed graph representation learning simultaneously with clustering. The model is based on (1) a onehop neighborhood propagation corresponding to a linear graph filter, (2) a view weighting scheme reminiscent of the attention mechanism in neural networks, and (3) a graph clustering and representation learning linear component that addresses both tasks via a unified framework. • We carry out a theoretical study of the linear graph filtering, formulate the problem that we are seeking to solve, and propose an algorithm that we subject to a detailed computational complexity analysis. • We showcase the efficiency and effectiveness of this model against the state of the art through extensive experimentation. We show that our model is both competitive and several magnitudes more efficient than current state-of-the-art multi-view attributed graph clustering. • We release our code for reproducibility 1 .

PRELIMINARIES 2.1 Definitions and Notations

An attributed graph is defined as a quadruple G = (V, E, A, X) where V represents the vertex set, E the edge, X ∈ R 𝑛×𝑑 its node features matrix and A its adjacency matrix of size 𝑛 × 𝑛. A multiview attributed graph is represented as a sequence of attributed graphs

M := G 𝑣 = (V 𝑣 , E 𝑣 , A 𝑣 , X 𝑣 ) 𝑣=𝑉 𝑣=1
. Matrices are denoted by boldface uppercase and vectors by boldface lowercase letters, 1 represents a column vector of ones. I denotes the identity matrix. Where X is a matrix, x 𝑖 is its 𝑖-th row. A matrix referenced as X 𝑣 means that it belongs to the 𝑣-th attributed graph, and its 𝑖-th row is referenced as x 𝑣 𝑖 .

Graph Filters and the Simple Graph Convolutional Network

Let G = (V, E, A, X) be an attributed graph whose symmetrically normalized Laplacian matrix is

L sym = I -D -1 2 AD -1 2
where D is the diagonal matrix of degrees of the graph such that 𝑑 𝑖𝑖 = 𝑗 𝑎 𝑖 𝑗 . A graph signal can be seen as a vector f = [𝑓 (𝑣 1 ), . . . , 𝑓 (𝑣 𝑛 )] such that 𝑓 : V → R is a real-valued function on the vertex set V.

1 https://github.com/chakib401/LMGEC For any graph signal f, we can quantify its smoothness using the Laplacian quadratic form [START_REF] Zhou | A regularization framework for learning from graph data[END_REF]

S(f) = f ⊤ L sym f = 1 2 𝑛 ∑ 𝑖,𝑗 𝑎 𝑖 𝑗 𝑓 𝑖 √ 𝑑 𝑖𝑖 - 𝑓 𝑗 √ 𝑑 𝑗 𝑗 2 . (1)
Now let L sym = UΛU ⊤ be the eigendecomposition of the Laplacian and {u 𝑙 } 𝑛 𝑙=1 and {𝜆 𝑙 } 𝑛 𝑙=1 the sets of eigenvectors and eigenvalues of L sym . Since L sym is symmetric, these eigenvectors form a basis for R 𝑛 , and we can therefore write f = Uc = 𝑛 𝑙=1 𝑐 𝑙 u 𝑙 . This implies that we can write the Laplacian quadratic form as

S(f) = f ⊤ L sym f = c ⊤ U ⊤ UΛU ⊤ (Uc) = c ⊤ Λc = 𝑛 ∑ 𝑙=1 𝑐 2 𝑙 𝜆 𝑙 , (2)
and that diagonal operators applied to the spectrum of the Laplacian modulate the smoothness of the signal; consequently, the eigenvalues can be seen as the frequencies of the signal [START_REF] Defferrard | Convolutional neural networks on graphs with fast localized spectral filtering[END_REF]. Accordingly, if we wish to make a graph signal smoother, we should minimize this measure through removing frequencies that correspond to larger eigenvalues. This is done using a low-pass filter.

To low-pass filter a graph using a polynomial filter whose frequencyresponse function is 𝑔, we use the graph convolution operation which is defined as

f filtered = 𝑔 L sym f = U𝑔(Λ)U ⊤ f (3)
such that 𝑔(Λ) = diag (𝑔(𝜆 1 ), ..., 𝑔(𝜆 𝑛 )). For example, in the case of a graph filter corresponding to a GCN [START_REF] Kipf | Semi-Supervised Classification with Graph Convolutional Networks[END_REF] with 𝑝 layers, or its simplified version [START_REF] Wu | Simplifying graph convolutional networks[END_REF] using a 𝑝-th order feature propagation, its frequency-response function is given as 𝑔(𝜆) = (1-𝜆) 𝑝 , or in matrix form as 𝑔(L sym ) = (I -L sym ) 𝑝 = A 𝑝 , which is a polynomial filter that is low-pass for odd values of 𝑝 and somewhat low-pass for even values of 𝑝 since the filtering function is not strictly decreasing on the interval of definition of the eigenvalues 𝐼 = [0, 2]. Note that the GCN also introduces added self-loops into the adjacency matrix A.

For more details about graph filtering and graph signal processing in general, we refer the reader to [START_REF] Ortega | Graph signal processing: Overview, challenges, and applications[END_REF][START_REF] David I Shuman | The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains[END_REF].

PROPOSED MODEL

Now that we have introduced the necessary background, we can formulate our problem.

First-order Neighborhood Propagation and Linear Graph Filtering

As previously mentioned, the graph neighborhood propagation performed in the GCN acts as a filter on the graph signal and removes high-frequency noise. We argue, however, that these steps of neighborhood propagation as performed in the GCN are unnecessary and even counterproductive because of a risk of over-smoothing, which is when the signal becomes uniform over the different nodes.

To support our argument, we would point to the performance of the linear graph autoencoder [START_REF] Salha | Simple and effective graph autoencoders with one-hop linear models[END_REF], which was competitive w.r.t more complex GCN-based models. In this paper we are seeking to show that a first-order (or one-hop) neighborhood propagation, when applied properly, is also sufficient for the task of simultaneous graph clustering and embedding. Given an attributed graph be the adjacency matrix with 𝛽 added self-loops and D its diagonal matrix of degrees. We define our propagation matrix as

G = (V, E, A, X), let à ← A + 𝛽I (4)
S ← D-1 Ã. ( 5
)
The generalized Laplacian L rw that corresponds to this propagation matrix is a random walk normalized Laplacian with added selfloops. The linear propagation operation we propose is

H ← SX (6)
where H are the new filtered features. The frequency-response function associated with this filter is 𝑔(𝜆) = 1 -𝜆 or, in matrix form, 𝑔(S) = I -L rw , which is clearly a linear function that is decreasing in the interval of definition of the eigenvalues 𝐼 = [0, 2].

Simultaneous Multi-view Attributed Graph Representation Learning and Clustering

Given a multi-view attributed graph represented as a set of attributed graphs M := G 𝑣 } 𝑣=𝑉 𝑣=1 , we define a preliminary version of the problem of simultaneous multi-view graph representation learning and clustering as min

G,F W 1 ,...,W 𝑉 ∑ 𝑣 (∥H 𝑣 -H 𝑣 W 𝑣 W ⊤ 𝑣 ∥ 2 reconstruction term +∥ H 𝑣 W 𝑣 -GF∥ 2 clustering term individual view loss ) multi-view loss 𝑠.𝑡 . ∀𝑣 W 𝑣 W ⊤ 𝑣 = I, G ∈ {0, 1} 𝑛×𝑘 , G1 = 1 (7)
where W 𝑣 ∈ R 𝑑 𝑣 ×𝑓 such that 𝑑 𝑣 is the dimension of the features in view 𝑣, 𝑓 is the dimensionality of the consensus representation we wish to learn and 𝑘 is the number of clusters. The loss corresponding to each view consists of two terms, namely a reconstruction term and a clustering term:

• The reconstruction term can be seen as reconstructing the filtered graph signal of the 𝑣-th view H 𝑣 using a semi-orthogonal matrix W 𝑣 similar to what is done in principal component analysis. We may draw a parallel with autoencoders, where multiplying by W encodes the data and W ⊤ decodes it. • The clustering term is similar to the 𝑘-means objective applied to the embeddings learned from the reconstruction process H 𝑣 W 𝑣 . G is a partition matrix and F is the centroids matrix.

Matrices G and F are the same for each matrix and represent the consensus partition and centroids respectively. There are, however, exactly 𝑉 W 𝑣 matrices, due the fact that the features in the different views do not necessarily have the same dimensionality. Problem [START_REF] Fettal | Scalable Attributed-Graph Subspace Clustering[END_REF] can be rewritten in a way that combines the two terms as follows min

G,F,W 1 ,...,W 𝑉 ∑ 𝑣 ∥H 𝑣 -GFW ⊤ 𝑣 ∥ 2 s.t. ∀𝑣 W 𝑣 W ⊤ 𝑣 = I, G ∈ {0, 1} 𝑛×𝑘 , G1 = 1. (8)
This formulation is more intuitive, since we can see it as trying to minimize the discrepancy between each input vector h 𝑣 𝑖 and its corresponding centroid learned in the latent embedding space after reconstructing g 𝑖 FW ⊤ 𝑣 . A proof of this is available in [START_REF] Yamamoto | A general formulation of cluster analysis with dimension reduction and subspace separation[END_REF].

Paying Attention to the Individual Views

Not all views have the same importance, and for this reason it is not optimal to directly add the losses from each view without applying some kind of importance weighting scheme. To address this issue we introduce a new set of parameters {𝛼 𝑣 } 𝑣=𝑉 𝑣=1 such that 𝑣 𝛼 𝑣 = 1 where 𝛼 𝑣 represents the relative importance of each view 𝑣. With this, we obtain the final formulation of our problem min

G,F,W 1 ,...,W 𝑉 ∑ 𝑣 𝛼 𝑣 ∥H 𝑣 -GFW ⊤ 𝑣 ∥ 2 s.t. ∀𝑣 W 𝑣 W ⊤ 𝑣 = I, G ∈ {0, 1} 𝑛×𝑘 , G1 = 1. (9)
Note, however, that we do not introduce 𝛼 𝑣 as a parameter, since this would cause a solution to pay attention to only a single view, i.e., the view with the smallest individual view loss. With this in mind, we define 𝛼 as the softmax 𝛼 of the negative inertia of each view, and we add a temperature parameter for greater flexibility. The formula for each 𝛼 is then

𝛼 𝑣 ← 𝑒 -𝐼𝑣 𝜏 𝑤=𝑉 𝑤=1 𝑒 -𝐼𝑤 𝜏 (10)
where 𝐼 stands for inertia. For the 𝑣-th view 𝐼 𝑣 is computed as follows: 𝐼 𝑣 = ∥H 𝑣 -G 𝑣 F 𝑣 ∥ such that W 𝑣 is obtained through a truncated singular value decomposition (SVD) on X 𝑣 , and G 𝑣 and F 𝑣 are the results of a 𝑘-means applied on the embeddings of the 𝑣th view X 𝑣 W 𝑣 . When the temperature 𝜏 is sufficiently high, only the best view in terms of inertia is selected, and when it is sufficiently low, all the views have the same weight.

OPTIMIZATION AND COMPLEXITY

Even though solving LMGEC exactly may be NP-hard, a solution can be computed reasonably efficiently via the use of heuristics.

To this end, we propose using a Block Coordinate Descent (BCD) scheme that boils down to iteratively solving sub-problems where we alternately solve for one of W 1 , . . . , W 𝑉 , G, F while keeping the others fixed. All optimizations are described below.

Optimizing for G

When solving for G and fixing the other matrices, we obtain the following problem min

G ∑ 𝑣 𝛼 𝑣 ∥H 𝑣 W 𝑣 -GF ∥ 2 𝑠.𝑡 G ∈ {0, 1} 𝑛×𝑘 , G1 = 1.
(11) This problem is hard to solve, so instead we propose solving the following relaxation obtained using the Cauchy-Schwarz inequality:

min G ∑ 𝑣 𝛼 𝑣 H 𝑣 W 𝑣 -GF 2 𝑠.𝑡 . G ∈ {0, 1} 𝑛×𝑘 , G1 = 1. (12)
In this way we can efficiently minimize the objective of this problem with the assignment step

𝑔 𝑖 𝑗 ← 1 if 𝑗 = arg min 𝑙 ∥ ( 𝑣 𝛼 𝑣 H 𝑣 W 𝑣 ) 𝑖 -f 𝑙 ∥ 2 0 otherwise. ( 13
)

Optimizing for F

When optimizing for F we retrieve the same criterion as for G, the difference lying in the constraints min

F ∑ 𝑣 𝛼 𝑣 H 𝑣 W 𝑣 -GF 2 . ( 14
)
This problem is an instance of an ordinary least-squares problem whose exact solution is easy to find and is given as

F = (G ⊤ G) -1 G ⊤ ∑ 𝑣 𝛼 𝑣 H 𝑣 W 𝑣 . ( 15
)
4.3 Optimizing for W 1 , . . . , W 𝑉 Optimizing for the different W 𝑣=𝑉 𝑣=1 matrices results in 𝑉 sub-problems. For a specific W 𝑣 , the resulting problem is min W 𝑣 ∥H 𝑣 -GFW ⊤ 𝑣 ∥ 2 which can be solved using a singular value decomposition as follows:

W 𝑣 = UV ⊤ s.t. U, Σ, V = SVD(X ⊤ 𝑣 GF) (16
) For more details on the derivation we refer the reader to [START_REF] Fettal | Efficient Graph Convolution for Joint Node Representation Learning and Clustering[END_REF][START_REF] Yamamoto | A general formulation of cluster analysis with dimension reduction and subspace separation[END_REF].

Optimization Algorithm

The main steps are summarized in Algorithm 1. The loss function might not be strictly decreasing due to the use of relaxations for the sub-problems, but it should nevertheless have an overall decreasing trend, as shown in figure 2 where we observe that our algorithm does not need many iterations to converge.

EXPERIMENTATION

In this section we present the experimental setup and results. We start by introducing the datasets and the evaluation metrics used in relation to clustering, the methods used for comparison with LMGEC, and the experimental settings. We then present the results in terms of quality of clustering, followed by analysis where we consider embedding, complexity, sensitivity, and robustness in the face of noisy views.

Datasets and Metrics

In order to demonstrate the generic nature of our model we looked at the three possible types of multi-view attributed graph datasets:

(1) Datasets with Heterogeneous Graph Topology. These datasets have the same set of features X but multiple graph topologies, i.e. multiple adjacency matrices {A 𝑣 } 𝑣=𝑉 𝑣=1 . They include ACM, DBLP and IMDB.

(2) Datasets with Heterogeneous Features. These datasets have the same graph structure A but multiple sets of features {X 𝑣 } 𝑣=𝑉 𝑣=1 . The example we chose was Amazon Photos.

(3) Datasets with Both. These datasets have multiple graph structures {A 𝑣 } 𝑣=𝑉 𝑣=1 as well as multiple sets of features {X 𝑣 } 𝑣=𝑉 𝑣=1 , The only such dataset is Wiki for which we create the additional views from the initial data, we initially have a single topology and features, we then generate a second topology using a nearest neighbor graph based on the cosine distance and a second set of features by using a log-scale of the original ones. The characteristics of these datasets are given in table 1. In quantifying the quality of a clustering we use four metrics: clustering accuracy (CA), clustering F1-score (F1) [START_REF] Van Rijsbergen | Information retrieval[END_REF], normalized mutual information (NMI) [START_REF] Strehl | Cluster ensembles-a knowledge reuse framework for combining multiple partitions[END_REF] and adjusted rand index (ARI) [START_REF] Hubert | Comparing partitions[END_REF].

Baselines

Below we list all the methods evaluated in our proposal.

• LINE [START_REF] Tang | Line: Large-scale information network embedding[END_REF]: A single-view graph embedding method. It is applied on each view and the best results are reported. • GAE [START_REF] Thomas | Variational graph auto-encoders[END_REF]: Another single-view graph embedding method based on the autoencoder. • X-avg: To utilize multiple views of a network we apply the X method to learn node representations on each single view, then average all learned representations. • LINE-avg and GAE-avg: By this we mean that the node representations learned for each view using LINE and GAE are averaged and clustered as such.

• MNE [START_REF] Zhang | Scalable multiplex network embedding[END_REF]: A scalable multi-view network embedding model. Only the graph structure information (adjacency matrix) of each view is input into this model. • PMNE [START_REF] Liu | Principled multilayer network embedding[END_REF]: Encompasses three multi-view graph embedding methods, including network aggregation PMNE (n), results aggregation PMNE (r) and layer co-analysis PMNE (c).

• RMSC [START_REF] Xia | Robust multi-view spectral clustering via low-rank and sparse decomposition[END_REF]: A multi-view spectral clustering method that uses Markov chains and low-rank decomposition. • PwMC and SwMC [START_REF] Nie | Self-weighted Multiview Clustering with Multiple Graphs[END_REF]: PwMC is a parameter-weighted multiview graph clustering method, while SwMC is a self-weighted multi-view graph clustering method. • O2MA and O2MAC [START_REF] Shaohua Fan | One2multi graph autoencoder for multi-view graph clustering[END_REF]: O2MAC is also an autoencoder-based multi-view graph clustering method. O2MA is a simplified version of O2MAC with the clustering loss removed from the objective function.

• DMGI [START_REF] Park | Unsupervised attributed multiplex network embedding[END_REF]: This is an unsupervised embedding method for attributed multiplex network embedding. This approach is only applicable to datasets with one set of features and multiple graphs.

• MvAGC [START_REF] Lin | Graph Filter-based Multi-view Attributed Graph Clustering[END_REF]: Performs graph filtering to do multi-view attributed graph clustering. • MAGC [START_REF] Lin | Multi-view attributed graph clustering[END_REF]: A multi-view graph clustering method that utilizes both node attributes and graphs.

Experimental Settings

We use the clustering results reported in the original papers where possible. When performing our own tests we tried to follow the setups prescribed by the authors of the different models as faithfully as possible. For our model, we set the maximum number of iterations to 30, the tolerance to 0 and 𝑓 = 𝑘 + 1 in all experiments.

We performed experiments with different hyper-parameter values for 𝛽 and 𝜏. We did a tf-idf normalization of the inputs to our model, and we also centered the data after neighborhood propagation. For the values of 𝛽 and 𝜏, we try 𝛽 ∈ {.2, 1, 2} and 𝜏 ∈ {1, 10, 100} and we report the best results. The different experiments were run on the same machine with two Intel(R) Xeon(R) CPU @ 2.20GHz and 13GB RAM. Most of the source codes in the official repositories of the baselines were not optimized for GPU. Note that the results reported for our method are the averages of five runs.

Experimental Results

Below we study in detail the results from our comparisons and highlight the interest of LMGEC.

Clustering Results. Tables 2 and3 show the results of our experiments for the clustering task. Some of the results for ACM, DBLP, IMDB and Amazon Photos are taken from [START_REF] Shaohua Fan | One2multi graph autoencoder for multi-view graph clustering[END_REF][START_REF] Lin | Graph Filter-based Multi-view Attributed Graph Clustering[END_REF][START_REF] Lin | Multi-view attributed graph clustering[END_REF], while results for Wiki are those that we obtained in our experiments. The pattern that emerges is that methods combining multi-view information tend to outperform those using single-view information. Our model LMGEC consistently achieves competitive performances, outperforming other models on ACM, DBLP, IMDB, Amazon Photos and Wiki on most metrics, and being competitive on IMDB, where it has the second best results on two out of four performance metrics. Overall, our model offers the best results in 15 out of 20 cases and has the best or second best results in 18 out of 20 cases, showing that it is competitive despite its simple nature. Note that 3). for datasets with multiple features, in our experiments we used only the baselines that were the best performing (on average). We can see the benefits of our model from tables 2 and 3 and from figure 4, where the performance of LMGEC on individual views is shown against the consensus performance; the consensus performance is consistently better than for the individual views. In some instances LMGEC applied to the individual views even outperforms state-of-the-art models, for example, on ACM, Amazon Photos and Wiki.

Embedding Results. Figure 3 depicts the embeddings produced by LMGEC on the different datasets by projecting them onto a 2dplane using t-SNE. A clustering structure is visible on all datasets apart from IMDB, where the embeddings are not very well separated.

Efficiency Results. We report the training times of our method in table 4 as well as those of the best performing (on average) baselines in our experiments. Our model is consistently much faster than other models, improving on the training time of the second fastest model (MvAGC) by 75%, 89%, 85%, 48% and 86% on ACM, DBLP, IMDB, Wiki and Amazon Photos respectively. Sensitivity Analysis. In our experiments we tried various values for the 𝛽 and 𝜏 hyperparameters. Figure 5 illustrates the performance of LMGEC for different pairs of these parameter values on the different datasets and for the different clustering metrics. We see that the on most datasets the performance remains fairly constant, which is an indication of LMGEC's robustness. The exception is ACM, where LMGEC is sensitive to the temperature parameter 𝜏 because of the presence of uninformative views. We discuss this in greater detail in the following paragraph. As a rule of thumb, we suggest taking 𝜏 = 10 and 𝛽 = 1.

Robustness in the face of noisy views. In real applications noisy data sources are not uncommon and can impair the performance of multi-view models. Since LMGEC takes into account the importance of the different views via a weighting scheme based each view's inertia, it is able to filter out noisy views by increasing the temperature 𝜏 of the softmax function used in computing {𝛼 𝑣 } 𝑣=𝑉 𝑣=1 . Tables 2 and3 and figure 4 report the clustering performance of LMGEC on each individual view for the different datasets, as well as the consensus performance. In the case of DBLP we may consider that the first view is noisy, since LMGEC performs considerably less well on this view than on the second and third views. We remark, however, that the consensus performance is not influenced by the presence of this noisy view, which shows the robustness of LMGEC in the face of noise. The same is true of Wiki as regards the first view, albeit less significantly than for DBLP, since the performance gap is not as flagrant w.r.t the other views.

CONCLUSION

In this paper we proposed a simple linear additive model that addresses the dual tasks of multi-view attributed graph representation learning and multi-view attributed clustering in a unified framework. This model is more generic than most state-of-the-art approaches in the sense that it can deal with any number of graph structures and/or any number of feature sets. Experiments showed that our model is competitive with more complex state-of-the-art models, outperforming these models on most benchmarks in terms of both performance and computation time.

Figure 1 :

 1 Figure 1: Schematic representation of LMGEC.

Figure 2 :

 2 Figure 2: Evolution of the loss value across iterations using BCD for LMGEC.

Figure 3 :

 3 Figure 3: Two-dimensional projections of the LMGEC embeddings using t-SNE colored according to the real class labels.

Figure 4 :

 4 Figure 4: Performance of LMGEC on each individual view vs. its consensus performance when considering all views on ACM, DBLP and IMDB (for the other datasets see table3).

Figure 5 :

 5 Figure 5: Sensitivity analysis of the parameters of LMGEC on the graph topology heterogeneous datasets.

  Algorithm 1: Block Coordinate Descent (BCD) for LMGEC 𝑑×𝑓 1 ∀𝑣 H 𝑣 ← A 𝑣 X 𝑣 ; 2 ∀𝑣 Initialize W 𝑣 through a truncated SVD on H 𝑣 ; 3 ∀𝑣 Compute 𝛼 𝑣 using formula (10); 4 Initialize G and F through a k-means on 𝑣 𝛼 𝑣 H 𝑣 W 𝑣 ; 5 while change in loss > 𝜖 and max_iter not reached do 𝑣 𝛼 𝑣 ∥H 𝑣 -GFW ⊤ For simplicity, we suppose that 𝑓 ∈ 𝑂 (𝑘), that 𝑑 1 , . . . , 𝑑 𝑉 ∈ 𝑂 (𝑑) and that |E 1 |, . . . , |E 𝑉 | ∈ 𝑂 (|E |). We also suppose that 𝑘 << 𝑛, 𝑑, which is almost always the case in real-world attributed graph datasets. Note that in what follows, the multiplication of matrix G with another matrix amounts only to a re-indexing of this other matrix, because of the structure of G. Updating {W 𝑣 } 𝑣=𝑉 𝑣=1 . Like their initialization steps, this step takes roughly 𝑂 (𝑣𝑛𝑑 log(𝑘)). • Updating G. The rule associated with this step takes 𝑂 (𝑛𝑘 2 ) computations. • Updating F. This rule is computed in 𝑂 (𝑣𝑛𝑑𝑘), as a result of computing the embeddings for the different views H 𝑣 W 𝑣 . • Objective Value Calculation. The computation here can be performed in 𝑂 (𝑛𝑑). • Overall Complexity. Altogether, the total computation time for our algorithm is 𝑂 (𝑣 |E |𝑑 + 𝑡𝑣𝑛𝑑𝑘), where 𝑡 is the number of iterations of LMGEC.

	𝑣 ∥;
	10 end
	4.5 Complexity Analysis
	• First-order Neighborhood Propagation. This step consumes
	roughly 𝑂 (𝑣 |E |𝑑) operations.
	• Initializing {W 𝑣 } 𝑣=𝑉 𝑣=1 . This step takes 𝑂 (𝑣𝑛𝑑 log(𝑘)) when us-
	ing a randomized SVD algorithm.
	• Computing {𝛼 𝑣 } 𝑣=𝑉 𝑣=1 . Here it is necessary to compute the inertia
	as well as the sets {G 𝑣 } 𝑣=𝑉 𝑣=1 and {F 𝑣 } 𝑣=𝑉 𝑣=1 for each view, totalling
	around 𝑂 (𝑣𝑛𝑑𝑘) operations.

Input : -Sequence of views {(A 𝑣 , X 𝑣 )} 𝑣=1,...,𝑉 -Number of clusters 𝑘 -Embedding dimension 𝑓 -Temperature 𝜏 -Tolerance 𝜖 -Maximum number of iterations max_iter Output : -Consensus membership indicator G ∈ {0, 1} 𝑛×𝑘 -Consensus embedding centers F ∈ R 𝑘×𝑓 -Consensus embedding matrices W ∈ R 6 ∀𝑣 Update W 𝑣 using formula (16); 7 Update G using formula (13); 8 Update F using formula (15); 9 𝑙𝑜𝑠𝑠 ← • Initializing G and F. Computing the summation 𝑣 𝛼 𝑣 H 𝑣 W 𝑣 and the application of 𝑘-means on it amounts to 𝑂 (𝑛𝑑𝑘) operations, where the number of iterations of k-means is held constant.

•

Table 1 :

 1 Characteristics of the Datasets. For wiki, there are two topologies and two features matrices leading to four possible combinations/views.

	Dataset	Multi-view type #Views #Nodes #Features	#Edges	#Clusters
	ACM [28]	Topology	2	3,025	1,830	29,281 2,210,761	3
						11,113
	DBLP [28]	Topology	3	4,057	334	5,000,495	4
						6,776,335
	IMDB [28]	Topology	2	4,780	1,232	98,010 21,018	3
	Amazon Photos [23]	Features	2	7,487	745 7,487	119,043	8
	Wiki [34]	Both	4	2405 2405	4973 4973	24,357 12,025	17

Table 2 :

 2 Clustering results on ACM, DBLP and IMDB. Best results are highlighted in bold and the second best results in italic.

	Model	CA	F1	ACM NMI	ARI	CA	F1	DBLP NMI	ARI	CA	F1	IMDB NMI	ARI
	LINE-avg 0.6479 0.6594 0.3941 0.3433	0.875	0.866	0.6681 0.7056 0.4719 0.2985 0.0063 -0.009
	GAE	0.8216 0.8225 0.4914 0.5444 0.8859 0.8743 0.6925	0.741	0.4298 0.4062 0.0402 0.0473
	GAE-avg	0.699	0.7025 0.4771 0.4378 0.5558 0.5418 0.3072 0.2577 0.4442 0.4172 0.0413 0.0491
	MNE	0.637	0.6479 0.2999 0.2486		out of memory error		0.3958 0.3316 0.0017 0.0008
	PMNE(n) 0.6936 0.6955 0.4648 0.4302 0.7925 0.7966 0.5914 0.5265 0.4958 0.3906 0.0359 0.0366
	PMNE(r)	0.6492 0.6618 0.4063 0.3453 0.3835 0.3688 0.0872 0.0689 0.4697 0.3183 0.0014 0.0115
	PMNE(c) 0.6998 0.7003 0.4775 0.4431		out of memory error		0.4719 0.3882 0.0285 0.0284
	RMSC	0.6315 0.5746 0.3973 0.3312 0.8994 0.8248 0.7111 0.7647 0.2702 0.3775 0.0054 0.0018
	PwMC	0.4162 0.3783 0.0332 0.0395 0.3253 0.2808	0.019	0.0159 0.2453 0.3164 0.0023 0.0017
	SwMC	0.3831 0.4709 0.0838	0.018	0.6538 0.5602	0.376	0.38	0.2671 0.3714 0.0056 0.0004
	O2MA	0.888	0.8894 0.6515 0.6987	0.904	0.8976 0.7257 0.7705 0.4697 0.4229 0.0524 0.0753
	O2MAC	0.9042 0.9053 0.6923 0.7394 0.9074 0.9013 0.7287	0.778	0.4502 0.4159 0.0421 0.0564
	DMGI	0.8973 0.8985 0.6974 0.7296 0.8722 0.8691 0.6931 0.7034 0.5827 0.4253 0.1317 0.1457
	MvAGC	0.8975 0.8986 0.6735 0.7212 0.9277 0.9225 0.7727 0.8276 0.5633 0.3783 0.0371 0.0940
	MAGC	0.8806 0.8835 0.6180 0.6808 0.9282 0.9237 0.7768 0.8267 0.6125 0.4551 0.1167 0.1806
	LMGEC 0.9302 0.9311 0.7513 0.8031 0.9285 0.9241 0.7739 0.8284 0.5893 0.4267 0.0632 0.1294

Table 3 :

 3 Clustering results on Amazon Photos and Wiki. Additionally, we report the performance of LMGEC on each individual view (for the other datasets see figure4). Note that Amazon Photos has only two views, while Wiki has four.

	Model	CA	Amazon Photos F1 NMI	ARI	CA	F1	Wiki	NMI	ARI
	LMGEC (view 1) 0.6726 0.6451 0.5903 0.4865 0.4757 0.4154 0.4772 0.2944
	LMGEC (view 2) 0.6835 0.6164 0.5971 0.4896 0.5181 0.4463 0.5079 0.3226
	LMGEC (view 3)	-	-	-	-	0.5202 0.4333 0.5383 0.3401
	LMGEC (view 4)	-	-	-	-	0.5264 0.4384 0.5362 0.3455
	MAGC	0.4511 0.3359 0.4297 0.1127 0.4972 0.4084 0.5139 0.2707
	MvAGC	0.6775 0.6397 0.5237 0.3968 0.3297 0.2432 0.3531 0.0864
	LMGEC	0.7117 0.6500 0.6114 0.5123 0.5333 0.4501 0.5408 0.3496

Table 4 :

 4 Training times. Best results are in bold, second best results are in italic. DMGI is only applicable to datasets with one set of features.

	Model	ACM	DBLP IMDB Wiki Amazon Photos
	DMGI	943.19 3117.87 843.96	-	-
	MvAGC	14.48	26.28	32.97	34.73	139.14
	MAGC	139.93 242.99 395.45 150.93	1661.64
	LMGEC 3.49	3.07	4.96	18.06	19.42

ACKNOWLEDGMENTS

This work has been funded by Informatique Caisse des Dépôts et Consignations (ICDC), Association Nationale de la Recherche et de la Technologie (ANRT), and Idex-Spectrans of UP Cité.