Florent Uny

Sofiane Achache

Salim Lamri

Jaafar Ghanbaja

Evelyne Fischer

Michel Pons

Elisabeth Blanquet

Frédéric Schuster

Frédéric Sanchette

Deposition and characterization of (Ti, Al)N coatings deposited by thermal LPCVD in an industrial reactor

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

Introduction

For several decades, increasing productivity in many industrial domains has led to a significant and ever increasing interest for protective and hard coatings. In this context, nitrides and particularly titanium nitrides were widely used, especially for applications such as machining operations, for which chemical and mechanical stability at high temperatures are needed. An enhancement of the thermal stability and mechanical properties can be achieved by adding aluminum as an alloying element in the metastable fcc-structure. The deposition of Ti1-xAlxN coatings near to equilibrium conditions leads to the formation of stable fcc-TiN and hcp-AlN [START_REF] Tomashik | Aluminium-Nitrogen-Titanium[END_REF]. Moreover, the thermodynamic data also shows that fcc-TiN and the metastable fcc-AlN phase are im-miscible over almost the whole range of Al content, and that the de-position of fcc-Ti1-xAlxN solid solution is thermodynamically unexpected [START_REF] Zhou | Phase equilibria, thermodynamics and microstructure simulation of metastable spinodal decomposition in c-Ti1-xAlxN coatings[END_REF][START_REF] Shulumba | Anharmonicity Changes the solid Solubility of a Random Alloy at High Temperatures[END_REF]. It was then demonstrated that the growth of metastable fcc-Ti1-xAlxN coatings with various Al contents mainly results from kinetic factors [START_REF]Transition metal mono nitrides, carbides, oxides[END_REF][START_REF] Anderbouhr | Depot Chimique en Phase Vapeur de Couches Minces de (Ti,Al)N à Composition Variable[END_REF]. These coatings show particular spinodal decomposition, leading to the formation of fcc-TiN and fcc-AlN domains during annealing and then to a hardness enhancement due to the lattice parameter mismatch between these two phases [START_REF] Zhang | Metastable phases and spinodal decomposition in Ti1-xAlxN system studied by ab initio and thermodynamic modeling, a compar-ison with the TiN-Si3N4 system[END_REF][START_REF] Mayrhofer | Self-organized nanostructures in the Ti-Al-N system[END_REF][START_REF] Kyrylov | Correlation between plasma con-ditions and properties of (Ti, Al) N coatings deposited by PECVD[END_REF]. Aluminum addition also allows an increase of the oxidation onset temperature of about 300 °C; from ≈500 °C for TiN coatings to ≈800 °C for Ti1-xAlxN coatings [START_REF] Chen | Thermal stability and oxidation re-sistance of Ti-Al-N coatings[END_REF]. Researches have generally focused on coatings with ever-increasing Al content in the fcc structure. While fcc-Ti1-xAlxN coatings obtained by various Physical Vapor Deposition (PVD) techniques seem to be limited up to an aluminum content x of about 0.67-0.7 [START_REF] Shimizu | Pressure dependence of (Ti, Al)N film growth on inner walls of small holes in high-power impulse magnetron sputtering[END_REF][START_REF] Grossmann | Phase composition and thermal stability of arc evaporated Ti1 -xAlxN[END_REF][START_REF] Mayrhofer | Influence of the Al distribution on the structure, elastic properties, and phase stability of supersaturated Ti[END_REF], the development of Low Pressure Chemical Vapor Deposition (LPCVD) techniques has shown x values up to 0.8-0.9, achieving hardness values higher than 30 GPa and oxidation resistance up to 1000 °C [START_REF] Endler | Novel aluminum-rich Ti1-xAlxN coatings by LPCVD[END_REF]. The growth of nanolamellae microstructures for high aluminum contents, obtained at pressure below 3.5 kPa and high carrier gas flow rates in order to minimize the residence time of the species were thus reported by numerous studies. Keckes et al. first show the deposition of fcc-Ti0.05Al0.95N coatings with spontaneous growth of hcp-AlN/fcc-TiN nanolamellae [START_REF] Keckes | Self-organized periodic soft-hard nanolamellae in polycrystalline TiAlN thin films[END_REF]. The fcc-Al(Ti)N (solid solution of Ti in the fcc-AlN phase) nanolamellae microstructure with a modulation of the Al and Ti contents and showing grains oriented along the [111] direction was later reported [START_REF] Zalesak | Peculiarity of self-assembled cubic nanolamellae in the TiN/AlN system: epitaxial self-stabilization by element deficiency/excess[END_REF]. These coatings show promising mechanical prop-erties and oxidation resistance. More recently, the deposition of (111)-and (100)-oriented coatings with high thermal stability was reported [START_REF] Paseuth | Improvement of mechanical properties and cutting performance of modified MT-TiCxN1-x coating by moderate temperature chemical vapor deposition[END_REF]. These studies mainly concern Al-rich coatings. Relatively poor literature concerns the microstructural changes related to various Al contents in CVD thin films. Ishigaki et al. have thus deposited fcc-Ti1-xAlxN coatings with various Al content at a pressure of 5 kPa without mentioning the carrier gas flow rate (H2) [START_REF] Ishigaki | Influence of the Al content on mechanical properties of CVD aluminum titanium nitride coatings[END_REF]. The proposal of this work is to investigate the microstructural and morphological changes obtained by varying x in Ti1-xAlxN coatings, at 4 kPa and low carrier gas flow rates (H2 flow rate). Indeed, a low H2 flow rate favors the Al content homogeneity in our reactor. The measured Al content in the coatings is then compared with thermodynamic calculations. Mechanical properties and oxidation resistance are also discussed and compared to the state-of-the-art LPCVD Ti1-xAlxN coatings. Ti1-xAlxN coatings have been deposited with various Al contents WC/Co-11%, Al2O3 and M2 substrates (Fig. 1).

Deposition was performed at a pressure of 4 kPa and a temperature of 800 °C. The process is re-producible under these conditions, even if a slight inhomogeneity in Al-content (< 15%) was found between the upper and bottom zones of the reactor. The deposition chamber is about 1250 mm in height and 300 mm in diameter. It consists of graphite plates on which the substrates are positioned. The gas injection is provided by mean of a graphite pipe passing through the trays and distributing gas onto each plate. Coatings were obtained from a mixture of TiCl4, AlCl3, NH3, N2 and H2. Titanium chloride is generated by dosing and vaporizing TiCl4 liquid. AlCl3 results from the in-situ chlorination of aluminum pellets by using an adjusted HCl flow. Temperature and pressure used for chlorination are adjusted to generate almost only AlCl3 and its dimer Al2Cl6. H2 is used as a carrier gas and NH3 is injected directly into the reactor to avoid unwanted reactions. The AlCl3/TiCl4 molar ratio was varied by adjusting the AlCl3 and TiCl4 flows to keep the total molar amount constant. The other gas flows were fixed in order to have a NH3/ (TiCl4 + AlCl3) molar ratio of 0.88 and a NH3/H2 molar ratio of 0.02. These ratios were determined considering a chlorination yield (amount of AlCl3 formed to the inlet HCl amount) of 1. Table 1 shows the main parameters used for Ti1-xAlxN deposition.

Pumping is carried out by a LRVP (Liquid Ring Vacuum Pump). Process by-products are treated in an alkaline solution (Fig. 1). Thermodynamic calculations were carried out using the FactSage software with TiCl4, AlCl3, NH3, H2 and N2 gas as input parameters. The molar amount of reactants used for these calculations was the same that those used for deposition, except for AlCl3, for which the molar amount was fixed at one third of the HCl value, assuming that all the HCl is consumed to form AlCl3 during chlorination.

Coatings morphology was studied using Scanning Electron Microscopy (SEM) analysis in a Hitachi SU-8030 FE-SEM system and the composition was determined by mean of Electron Probe MicroAnalysis (EPMA) analysis, using a Jeol JXA 8530-F. Crystallographic structure was investigated by Grazing Angle X-Ray Diffraction (GAXRD), using a Bruker D8 Advance diffractometer with the Kα1 radiation of Cu (λ = 0.1541 nm). The system is equipped with a 2.0 mm collimator and an Axial Soller slit with an axial divergence of 2.5°. The incidence angle during the GAXRD analysis was determined experimentally to ensure that no signal from the WC/Co substrate was observed. Assuming the density of the coating plays a role in the X-rays absorption, the incidence angle was varied from 3° for the Ti-rich coatings to 1° for the Al-rich coatings. The structure and microstructure of the coatings were also studied by Transmission Electron Microscopy (TEM) and Scanning Transmission Electron Microscopy (STEM) analyses, using a JEM-ARM 200F cold FEG TEM/STEM system. Elemental distribution in the TEM samples was analyzed using Energy Dispersive X-ray spectroscopy (EDX). An almost homogeneous distribution of Nitrogen in the coating is always found and the corresponding cartography is thus not presented. Prior to TEM analysis, the samples were prepared by Focus Ion Beam (FIB), using a MEB-FIB FEI Helios Nanolab 600i. The phases identification of the oxidized samples was also per-formed by GAXRD with an incidence angle of 5°, in order to increase the signal intensity.

Hardness measurements were carried out on the sur-face of M2 pins by mean of a nanoindenter XP, MTS system corporation, using a Berkovitch diamond tip with 300 nm of penetration depth. Prior to the hardness measurements, the surface of the samples was polished with a silicon carbide paper (granulometry P4000). Twenty indents were performed on each sample and the average hardness value was considered. Oxidation resistance was investigated between 500 °C and 1000 °C in ambient air and during 1 h with a heating rate of 15 °C/ min and natural cooling to room temperature. Chemical analyses of our coatings are summarized in Table 2. The Al content, x, in Ti1-xAlxN coatings, is defined as the Al/(Al + Ti) (at.%) ratio. By varying the AlCl3/TiCl4 ratio from 0.52 to 2.8, the Al contents in the coatings are in the range from x = 0.08 to x = 0.86. The coatings thickness is about 5 μm for the coating with the lowest Alcontent (Ti0.92Al0.08N) to about 2.3 μm for the highest Al-containing coating (Ti0.14Al0.86N). The Cl contents (≤1.2 at.%) was found to be higher in coatings having the highest Al content. These results are consistent with those found in literature. This higher Cl content in the coatings (about 1.2 at.%) can result from an incomplete dissociation of AlCl3. As AlCl3 is considered as less reactive with NH3 than TiCl4, an increase of the AlCl3 amount in the gas phase can result in higher Cl contents in the coatings [START_REF] Wagner | Deposition of Ti-Al-N coatings by thermal CVD[END_REF].

2.2.

Morphology, structure, and microstructure SEM analyses are shown in Fig. 2 and give the surface coatings morphology for different aluminum contents. The Ti0.92Al0.08N film (Fig. 2-a) shows a globular morphology while a morphology showing top of cubes is found in the Ti0.35Al0.65N coating (Fig. 2-b). By further increasing the Al content, cube-shaped grains seem to appear in Ti0.16Al0.84N and become predominant for the Ti0.14Al0.86N coating (Fig. 2-c and2-d, respectively). Although the microstructural and morphological evolution is relatively poorly described in literature, the observation of globular grains for the Ti-rich coatings is consistent with the results obtained by Ishigaki et al. [START_REF] Ishigaki | Influence of the Al content on mechanical properties of CVD aluminum titanium nitride coatings[END_REF]. However, they reported the presence of this globular morphology on the full range of Al content. The growth of LPCVD films having tetrahedral and octahedral morphologies has recently been observed by some researchers [START_REF] Paseuth | Improvement of mechanical properties and cutting performance of modified MT-TiCxN1-x coating by moderate temperature chemical vapor deposition[END_REF][START_REF] Todt | Al-rich cubic Al0.8Ti0.2N coating with self-organized nano-lamellar microstructure: thermal and mechanical properties[END_REF]. Their Al-rich Ti0.2Al0.8N and Ti0.18Al0.82N coatings were found to have columnar grains with oriented growth of fcc-Al(Ti)N lamellae in the [111] and [100] directions, respectively. However, there was no mention of the presence of cube-shaped grains as those found in Fig. 2-c and-d. The reason of the growth of these cubes is not clear. The number of these cubes seems to increase in the coating containing the highest amount of aluminum. Further investigations are needed in order to establish hypotheses regarding the growth of these cubes. Fig. 3-A-a shows the STEM -High Angle Annular Dark Field (STEM-HAADF) image corresponding to the cross-section of the Ti0.92Al0.08N coating. The EDX analyses performed on this Ti0.92Al0.08N coating are presented in Fig. 3-A-b and-A-c and show the distribution of aluminum and titanium respectively. The dark domains observed between the globular grains (Fig. 3-A-a) show the presence of cubes, that look like those observed in the high Al containing Ti0.14Al0.86N coating (see Fig. 2-d). These dark domains contain a higher Al content than in the globular grains (Fig. 3-A-b and3-A-c). These observations are consistent with the morphology, showing that the cube-shaped grains are observed in the coatings having a high Al content (Fig. 2-c and2-d).

It is also clearly shown that the coating is composed of a stack of globular grains constituted of bright and dark lamellae (Fig. 3-A-a), which correspond to a modulation of the composition inside these globular grains (Fig. 2-c andd). The dark and bright lamellae observed on the Bright Field (BF) STEM image (Fig. 3-A-b and3-A-c) are a modulation of Ti-rich (97 at.%) and Ti-depleted (90 at.%) lamellae. The XRD analysis performed on the Ti0.92Al0.08N coating (Fig. 4-a) shows mainly the {111}, {200} and {220} peaks of the fcc-Ti(Al)N phase which is assumed to be the solid solution of Al in fcc-TiN. However, the peaks corresponding to this phase appear exactly at the position of the pure fcc-TiN peaks (PDF file 00-038-1420). The Al content in these grains is probably too low to create a significant shift of the fcc-Ti(Al)N peaks towards the high angles on our GAXRD experiments. This fcc-Ti(Al)N phase can then correspond to the globular grains shown in Fig. 3-A-a. A small peak is also detected at 38.3° and can be attributed to the fcc-Al(Ti)N {111} planes. Only a very weak reflection of the {1010} planes of the hcp-AlN is observed at a 2θ angle of 33.2°. The BF TEM micrograph of this Ti0.92Al0.08N coating is shown in Fig. 5-A-a. The associated Selected Area Electron Diffraction (SAED) pattern seems to be representative of the globular grains in the coating. It only shows the presence of the fcc-Ti(Al)N phase while fcc-Al(Ti)N as well as hcp-AlN are not observed by this TEM analysis. Thus, this coating seems to be mainly constituted of the fcc-Ti(Al)N phase, even if some traces of the hcp-AlN and fcc-Al(Ti)N phases are detected by GAXRD. These phases should correspond to the dark domains observed in Fig. 3-A-a. Fig. 5-A Several researchers reported the lamellae microstructure with modulation of the composition of aluminum and titanium but it was observed for coatings with higher Al contents [START_REF] Keckes | Self-organized periodic soft-hard nanolamellae in polycrystalline TiAlN thin films[END_REF][START_REF] Ishigaki | Influence of the Al content on mechanical properties of CVD aluminum titanium nitride coatings[END_REF][START_REF] Todt | Al-rich cubic Al0.8Ti0.2N coating with self-organized nano-lamellar microstructure: thermal and mechanical properties[END_REF]. The present results seem to mean that this particular lamellae microstructure is not dependent on the Al content. A fcc-Al(Ti)N nanolamellae micro-structure with a modulation of the Al and Ti contents was found for coatings with an overall Al content between x = 0.62 to x = 0.82 [START_REF] Ishigaki | Influence of the Al content on mechanical properties of CVD aluminum titanium nitride coatings[END_REF][START_REF] Todt | Al-rich cubic Al0.8Ti0.2N coating with self-organized nano-lamellar microstructure: thermal and mechanical properties[END_REF], while an hcp-AlN/fcc-TiN modulation was found for coatings with an Al content of x = 0.95 [START_REF] Keckes | Self-organized periodic soft-hard nanolamellae in polycrystalline TiAlN thin films[END_REF]. We can expect that the observation of this lamellae microstructure in our coating having an Al content of only x = 0.08 is due to the deposition conditions used here, which are closer to the equilibrium conditions than those found in literature. The relatively high pressure and the low H2 flow rate, thus leading to a higher residence time of the species, allow a better mobility of the atoms at the surface and then promote the demixing of the solid solution associated with the growth of a modulated microstructure. In summary, the dark domains should be hcp-AlN or fcc-Al(Ti)N and the globular grains are fcc-Ti(Al)N with a modulation of composition. Fig. 3-B-a shows the STEM HAADF image of the Ti0.35Al0.65N coating cross section. The EDX cartographies corresponding to aluminum and titanium are shown in Fig. 3-B-b and3-B-c. Bright grains and grains constituted of bright and dark lamellae are observed (Fig. 3-B-a). The bright grains are almost exclusively com-posed of titanium (and nitrogen) (Fig. 3-B-c). The XRD pattern of the Ti0.35Al0.65N coating (Fig. 4-a) shows that the peaks corresponding to the fcc-Ti(Al)N phase with a very low Al content are still predominant. Thus, it is assumed that the structure of these grains is fcc-Ti(Al)N with a very low Al content. Fig. 3-B-b and -B-c also show that the bright and dark lamellae are respectively aluminum-rich and titanium-rich. The composition of the bright lamellae are 23 at.% Al and 30 at.% Ti, while that of the dark lamellae are 49 at.% Al and 5 at.% Ti. XRD analysis shows, in addition to the fcc-Ti(Al)N peaks, a strong reflection that can correspond to the {1011} peak of the hcp-AlN phase or to the {111} peak of the fcc-Al(Ti) N. The shift of this peak, compared to the position of the pure fcc-TiN peak, corresponds to an Al content x of about 0.67. This value is consistent with the overall Al content found in this coating. So, this peak is probably associated with the fcc-Al(Ti)N phase. Moreover, a reflection that can be attributed to the fcc-Al(Ti)N phase is also observed at 44°. The BF TEM image and the associated diffraction pattern obtained on a grain constituted of bright and dark lamellae (Fig. 5-B) highlights only a fcc structure. These observations suggest the presence of a coherent fcc-Al(Ti)N lattice constituted of lamellae with a modulation of the Al and Ti contents. Thus, the following assumptions can be made. (i) The XRD peak that can be indexed as the {1011} plane of hcp-AlN is actually the {111} peak of the fcc-Al(Ti)N phase, forming a nanostructured lattice with compositional modulation of aluminum and titanium. (ii) It is also possible that hcp-AlN domains, which are evidenced by a peak at 33.2° in the GAXRD pattern (Fig. 4), are distributed elsewhere in the coating. Unfortunately, these zones were not detected by these TEM analyses. In summary, Ti0.35Al0.65N contains bright grains (Fig. 3-B), which are fcc-Ti(Al)N and nanolamellae, which are fcc-Al(Ti)N with a mod-ulation of Al and Ti contents. Some traces of hcp-AlN are detected by XRD. Fig. 3-C-a shows the BF STEM image of the Ti0.14Al0.86N coating cross section. The coating seems to be constituted of a stack of particles composed of bright and dark domains. Regarding the square shape of the particle in the upper right frame on Fig. 3-C-a, it can be assumed that these particles can correspond to the cube-shaped grains observed on the surface morphology analyses (Fig. 2-d).

This Ti0.14Al0.86N coating shows a strong compositional modulation of aluminum and titanium (Fig. 3 shows the HRTEM image of a particle that can correspond to a cube-shaped grain (Fig. 2-d) at the surface of the Ti0.14Al0.86N coating (upper left frame). The FFT pattern associated with this HRTEM image reveals the presence of the {111} and {002} planes of a fcc structure, with a [110] zone axis. Interplanar distances of ≈0.23 nm and ≈0.20 nm were found for the {111} and {200} planes, respec-tively. This particle can be attributed to the fcc-Al(Ti)N metastable phase with a low Ti content.

It could be possible that the higher amount of cube-shaped grains in the Ti0.14Al0.86N compared to the Ti0.16Al0.84N (Fig. 2-c and-d) is re-lated to the presence of the fcc-Al(Ti)N highlighted by XRD as well as by HRTEM analyses in the highest Al containing coating (x = 0.86).

Coatings deposited by Atmospheric Pressure CVD (APCVD) showing a mixture of fcc-TiN, fcc-AlN and hcp-AlN were also found by Wagner et al. with an Al content of x = 0.40 and x = 0.45 [START_REF] Wagner | Deposition of Ti-Al-N coatings by thermal CVD[END_REF]. At higher Al contents, a mixture of the stable phases fcc-TiN and hcp-AlN was found. This result is consistent with those found in our work, except that the hcp-AlN phase seems to be significantly present above an Al content of x = 0.65 in our coatings. These observations clearly confirm that high pressure and/or low carrier gas flow rates promote the growth of the stable phases.

The growth of the lamellae microstructure is assumed to be associated to spinodal decomposition or as the result of kinetic growth, where it should be preferable for the system to stabilize the fcc structure by adjusting the composition of the lamellae [START_REF] Zalesak | Peculiarity of self-assembled cubic nanolamellae in the TiN/AlN system: epitaxial self-stabilization by element deficiency/excess[END_REF][START_REF] Paseuth | Improvement of mechanical properties and cutting performance of modified MT-TiCxN1-x coating by moderate temperature chemical vapor deposition[END_REF]. However, these hypotheses are not sufficient to explain the micromodulation forming the Al depleted domains ("bright" domains) and Al-rich domains ("dark" domains) in the Ti0.14Al0.86N coating. It could be due to local competition, which would occur between the consumption of TiCl4 and AlCl3 near the substrate surface. If, for example, the reaction of TiCl4 with NH3 is favored compared to that of AlCl3, this can lead to a local fluctuation of the AlCl3/TiCl4 ratio near the surface and finally leads to the formation of domains showing a modulation of the Al and Ti contents. This hypothesis can also explain the growth of Al-rich domains in the Ti0.92Al0.08N coating and the growth of the fcc-Ti(Al)N grains ("bright" grains) in the Ti0.35Al0.65N coating. In summary, the Ti0.14Al0.86N coating contains bright and dark do-mains, which are a micromodulation of the Al and Ti contents. This coating is mainly composed of the hcp-AlN phase with some evidences of the fcc-Ti(Al)N and fcc-Al(Ti)N phases. The distribution of these phases is difficult to clarify. Fig. 6 gives a diagram summarizing the structures observed in this work.

Thermodynamic calculations

Thermodynamic calculations were performed using the FactSage software to find the stable phases at given parameters (temperature, pressure, molar amount of each reactant in the gas mixture) and to estimate the Al content in the coatings. The phase stability was calcu-lated by minimizing the Gibb's free energy. Calculations were carried out with the same molar quantity as those used during experimental deposition and at 800 °C and 4 kPa. Fig. 7 shows the evolution of the calculated Al content in the coating (x in Ti1-xAlxN) as a function of the AlCl3/(AlCl3 + TiCl4) ratio. The Ti-Al-N system being immiscible in almost the entire aluminum content range, the growth of the metastable fcc-Ti1-xAlxN is considered as the result of a kinetic issue more than of a thermodynamic metastability [START_REF] Anderbouhr | Depot Chimique en Phase Vapeur de Couches Minces de (Ti,Al)N à Composition Variable[END_REF]. It is thus assumed that only the stable fcc-TiN and hcp-AlN phases will be formed experimentally. Data for the different phases is based on the work of Chen and Sundman [START_REF] Chen | Thermodynamic assessment of the Ti-Al-N system[END_REF], defining the thermodynamic parameters of the stable phases in the Ti-Al-N system. The calculated Al contents (solid black curve of Fig. 7, Table 2) are compared to the experimental values (dashed orange curve of Fig. 7). The experimental as well as calculated Al contents are found to have the same evolution. The solid blue line (Fig. 7) is the calculated theoretical Al content in the coating in the case of all AlCl3 being used to form AlN.

As the AlCl3/(AlCl3 + TiCl4) ratio increases, the calculated Al con-tent in the coating progressively gets closer to the theoretical maximum value that can be obtained. Except for the lowest Al content, it is clear that the experimental values are above the other values (Fig. 7).

Yield calculations for the formation of hcp-AlN (not presented here) were carried out in order to explain the calculated curve evolution. Hcp-AlN yield is defined as the ratio of the AlCl3 inlet to the amount of hcp-AlN formed. Results show that, at an AlCl3/(AlCl3 + TiCl4) flow ratio of 0.34, only 26% of the injected AlCl3 in the reactor reacts to form AlN, while at a ratio of 0.74, about 81% of the AlCl3 is used to form hcp-AlN. These yield calculations thus explain that calculated Al contents progressively approach the theoretical value given by the blue line.

The experimental Al contents are higher than the calculated values for the coatings with an AlCl3/(AlCl3 + TiCl4) ratio above 0.46. As we use a relatively low H2 flow rate and considering that the reactivity of TiCl4 with NH3 is higher than that of AlCl3 (the mixing enthalpy at 800 °C for the formation of fcc-TiN from NH3 and TiCl4 is about -100 KJ/mol while that of hcp-AlN from AlCl3 and NH3 is about -48 KJ/mol), fcc-TiN may already be formed in the injection pipe. Then, a lower TiCl4 amount is available for deposition, leading to higher Al contents in the coatings. This assumption is supported by the observation, after the deposition process, of a yellow layer on the inner wall of the injection pipe.

Concerning the lowest Al-containing coatings (with the corresponding ratio: AlCl3/(AlCl3 + TiCl4) of 0.34), it is assumed that, even if a small amount of fcc-TiN is already deposited in the injection gas pipe, the gas mixture in the deposition zone will still be mainly com-posed of TiCl4. So, the amount of AlCl3 may be insufficient to react strongly with NH3. A low Al content is then expected in the coating. This can explain that the experimental value is almost the same as the calculated one (for low Al contents).

For the Al-rich coating (AlCl3/(AlCl3 + TiCl4) of 0.74), the experi-mental value is closer to the calculated curves. It can be considered in this case that the amount of TiCl4 in the gas mixture is not sufficient to deposit on the wall of the injection pipe.

Hardness

Fig. 8 shows the evolution of the coatings hardness as a function of Al content. Hardness value is 26.6 GPa for the Ti0.92Al0.08N coating and a weak hardness increase is observed for x = 0.65. Above this value, a hardness drop is observed, with a value of 20.0 GPa for Ti0.16Al0.84N. Finally, hardness seems to increase slightly for Ti0.14Al0.86N. According to the low Al content in the Ti0.92Al0.08N coating, its hardness (26.6 GPa) seems to be quite high compared to the typical values reported for fcc-TiN coatings (about 20 GPa) [START_REF] Wagner | Deposition of Ti-Al-N coatings by thermal CVD[END_REF][START_REF] Quinto | High temperature microhardness of hard coatings produced by physical and chemical vapor deposition[END_REF]. This relatively high value can be attributed to the nanolamellae micro-structure. The coherent lattice along lamellae associated with a slight modulation of composition can generate stresses at the interfaces be-tween these lamellae and then reduce dislocations motion, leading to higher hardness [START_REF] Sanchette | Nanostructured hard coatings deposited by cathodic arc deposition: from concepts to applications[END_REF]. An increase of the Al content in the fcc-Ti1-xAlxN phase generally leads to an increase in hardness. Here, the slight rise in hardness between the Ti0.92Al0.08N and the Ti0.35Al0.65N coatings could be related to the higher amount of Al in the Ti0.35Al0.65N coating. In-deed, no fcc-Ti(Al)N peak shift is detected for Ti0.92Al0.08N, whereas a shift is observed for Ti0.35Al0.65N. However, this apparent increase in hardness could also be due to the high deviation of the hardness values. The hardness drop observed by further increasing the Al content in the coatings is related to the predominant formation of hcp-AlN, as shown by the XRD analyses. The increase of hardness observed for the Ti0.14Al0.86N coating could be related to the presence of fcc-Al(Ti)N domains in this coating which is not observed in Ti0.16Al0.84N. The fcc-Al(Ti)N phase is notably reported to have a high hardness and can thus be responsible for this increase in hardness. However, this higher hardness for the Ti0.14Al0.86N coating could also be due to the high deviation of the hardness values.

The hardness measurements of this work are compared with literature data. Ishigaki et al. have recently deposited Ti1-xAlxN coatings with various Al contents [START_REF] Ishigaki | Influence of the Al content on mechanical properties of CVD aluminum titanium nitride coatings[END_REF]. The same hardness evolution (black dotted curve) was observed for their coatings. Nevertheless, higher hardness values were obtained due to the presence of the fcc-structure in a wider Al content range as well as to the formation of the nanolamellae microstructure up to an Al content of x = 0.82 [START_REF] Ishigaki | Influence of the Al content on mechanical properties of CVD aluminum titanium nitride coatings[END_REF]. In a general way, the nanolamellae microstructures are associated with a significantly higher hardness than that of our coatings, with values of about 35 GPa [START_REF] Todt | Al-rich cubic Al0.8Ti0.2N coating with self-organized nano-lamellar microstructure: thermal and mechanical properties[END_REF][START_REF] Paseuth | Deposition and analysis of Al-rich c-Al x Ti 1-x N coating with preferred orientation[END_REF]. In our case, the presence of fcc-Ti(Al)N phase with a very low Al content (bright grains) in the Ti0.35Al0.65N can ex-plain the lowest hardness. Most of the other hardness values found in literature are higher than those found in our work. A value of 35 GPa was found in a Ti0.18Al0.82N coating showing the presence of a fcc-Al (Ti)N nanolamellae microstructure with modulated Al and Ti contents (grey triangle mark) [START_REF] Todt | Al-rich cubic Al0.8Ti0.2N coating with self-organized nano-lamellar microstructure: thermal and mechanical properties[END_REF]. Values between 28 GPa and 32 GPa were reported in single phase Ti1-xAlxN coatings (empty blue triangle marks) [START_REF] Endler | Novel aluminum-rich Ti1-xAlxN coatings by LPCVD[END_REF], and a hardness of about 27 GPa was found in Ti0.05Al0.95N coatings showing the presence of fcc-TiN/hcp-AlN nanolamellar grains (yellow square mark) [START_REF] Keckes | Self-organized periodic soft-hard nanolamellae in polycrystalline TiAlN thin films[END_REF]. Among the results found in literature, only coatings deposited at atmospheric pressure (blue empty square mark) were found to have lower hardness, confirming the strong influence of deposition pressure on the properties of the Ti1-xAlxN coatings [START_REF] Wagner | Deposition of Ti-Al-N coatings by thermal CVD[END_REF].

Oxidation resistance

Oxidation resistance was investigated between 500 °C and 1000 °C. Fig. 9 shows an overall view of coated Al2O3 samples after 1 h of oxidation at 700 °C, 850 °C and 950 °C. At a temperature of 700 °C, no sample shows significant oxidation. At 850 °C, the coating with the lowest Al content (Ti0.92Al0.08N) is completely oxidized and delamination is observed. The coatings with higher Al contents do not show strong oxidation, with only a thin white layer that seems to appear on the Ti0.35Al0.65N. After 1 h of oxidation at 950 °C, the coatings with an Al content up to x = 0.65 are completely oxidized whereas the two films with higher Al content demonstrate a higher resistance against oxidation. These observations are confirmed by the XRD patterns (Fig. 10) after 1 h of oxidation at 850 °C. The XRD pattern of Ti0.92Al0.08N shows intense peaks corresponding to the iron oxide Fe2O3. The XRD pattern also reveals that the oxidized layer is mainly rutile titanium oxide (r-TiO2). Weak reflections corresponding to Al2O3 are also detected.

Additional GAXRD experiments (not presented here) were performed on this coating revealing that the first signs of r-TiO2 occurs from 500 °C. Complete oxidation of the coating was observed after oxidation at 800 °C. Ti0.35Al0.65N shows better oxidation resistance, as the fcc-Ti(Al) N peaks at 36.6° and 42.6° are still present. Anyway, Al2O3 and r-TiO2 reflections remain the predominant phases. After 1 h at 800 °C, coatings with the highest Al contents (Ti0.16Al0.84N and Ti0.14Al0.86N) mainly show the hcp-AlN phase with some Al2O3 reflections. GAXRD analyses performed between 750 °C and 950 °C (not presented here) show that oxidation seems to start from 800 °C for these coatings. After oxidation at 950 °C, the presence of the hcp-AlN phase is still detected and the coatings are not completely oxidized. Thus, these high Al content coatings show a higher oxidation resistance than coatings with a low Al content, in accordance with the results found in literature [START_REF] Chen | Thermal stability and oxidation re-sistance of Ti-Al-N coatings[END_REF]. The better oxidation resistance of Al-rich Ti1-xAlxN coatings is generally associated with the formation of Al2O3 as a dense and diffusion barrier layer that prevents the coating from further oxidation. Vaz et al. conducted a complete study of the oxidation of Ti1-xAlxN coatings with various Al contents, from x = 0.38 to x = 0.81 [START_REF] Vaz | Thermal oxidation of TiAlN coatings in air[END_REF]. They found, by mean of Rutherford Backscattering Spectrocopy (RBS) ana-lysis, that coatings with low Al contents were mainly composed of r-TiO2 and Al2O3 oxides after oxidation at temperatures above 750 °C. These results are consistent with our observations of Ti0.92Al0.08N. As our coating is mainly composed of titanium nitride, we obviously found almost only r-TiO2 in the Ti0.92Al0.08N after 1 h of oxidation at 850 °C. The presence of this phase from 500 °C and the complete oxidation of the coating after annealing at 800 °C are consistent with the results obtained for fcc-TiN coatings [START_REF] Chen | Oxidation behavior of titanium nitride films[END_REF]. The low Al content in our Ti0.92Al0.08N seems to be insufficient to significantly increase the oxidation resistance compared to fcc-TiN. Rutile-TiO2 is known as a non-protective and porous oxide. The growth of this oxide can cause the appearance of cracks, due to the difference in molar volume between r-TiO2 and fcc-TiN [START_REF] Chen | Thermal stability and oxidation re-sistance of Ti-Al-N coatings[END_REF][START_REF] Vaz | Thermal oxidation of TiAlN coatings in air[END_REF]. These cracks act as preferential pathways for oxygen diffusion and then lead to a rapid oxidation of the coating. The presence of such mixed oxides as Al2TiO5, found in our Ti0.92Al0.08N film, has also been reported by Chen et al. after oxidation of Ti1-xAlxN coatings at 1450 °C [START_REF] Chen | Thermal stability and oxidation re-sistance of Ti-Al-N coatings[END_REF]. RBS analysis led by Hugon et al. on Ti1-xAlxN coatings up to x = 0.4 also confirms that mixed aluminum-titanium oxide can be formed during Rapid Thermal Annealing in oxygen at 7509°C [START_REF] Hugon | Narrow resonance profiling study of the oxidation of reactively sputtered TiAlN thin films[END_REF]. The growth of Al2O3 layer can prevent oxygen diffusion, and thus explains that coating material diffraction peaks are still pre-sent for x = 0.65 to x = 0.86 after annealing at 850 °C during 1 h. The diffusion barrier character of the Al2O3 layer is notably demonstrated by the presence of the fcc-TiN peaks in the Ti0.35Al0.65N coating, while coating with higher titanium content is completely oxidized under the same conditions. Concerning the chemical composition, it was found that the evolution of Al content of the films is consistent with thermodynamic calculations, although the experimental Al contents are higher than the calculated values. This difference is attributed to the fcc-TiN deposition prior the deposition zone, leading to a higher AlCl3/TiCl4 ratio near the substrates and consequently to coatings with a higher Al-content.

A change in morphology was observed by increasing the Al content, from globular grains to cube-shaped grains. A modulation of Al and Ti contents is found in all coatings. In the Ti0.92Al0.08N and Ti0.35Al0.65N, this modulation occurs mainly by forming fcc-nanolamellar grains. The formation of these lamellae even in the coating having the lowest Al content is attributed to the deposition conditions (relatively high working pressure, low carrier gas flow), which are closer to the thermodynamic equilibrium than the deposition conditions described in literature. The coatings with the highest Al contents seem to be mainly composed of the hcp-AlN phase and show a micromodulation of Al and Ti contents, forming Al-rich and Al-depleted domains. The formation of these domains can result from a competitive consumption of AlCl3 and TiCl4 near the substrate surface, leading to a local variation of the AlCl3/TiCl4 ratio.

An increase of the Al content in the coatings allow the formation of a dense and protective Al2O3 layer, leading to higher oxidation resistance up to 950 °C in the films having the highest Al contents. Higher hardness (up to 27 GPa in the coating with an Al content x = 0.65) is obtained for coatings showing the nanolamellar grains. This relatively low value compared to those found in literature is associated to the growth of fcc-Ti(Al)N grains with a very low Al content in the Ti0.35Al0.65N coating. A drop in hardness (down to 20 GPa) is observed for the coatings with the highest Al content, for which the hcp-AlN is predominant. The hardness values are lower than those measured in literature (up to ≈35 GPa). These coatings are generally elaborated at lower pressure and/or higher carrier gas flow than our coatings, al-lowing the formation of grains only composed of fcc-Al(Ti)N nanolamellae with modulated Al and Ti contents.

Fig. 1 .

 1 Diagram of the reactor layout

  b shows the HRTEM image of the Ti0.92Al0.08N coating and highlights the crystallized microstructure. The associated Fast Fourier Transformation (FFT) image confirms the presence of the fcc structure and shows the {111} and {200} planes with a [110] zone axis. The interplanar distances are evaluated to be ≈0.24 nm and ≈0.21 nm for the {111} and {200} planes respectively, confirming the presence of fcc-Ti (Al)N with a very low Al content.

Fig. 2 .

 2 Fig. 2. Surface morphology of coatings deposited at 800 °C and 4 kPa; a) Ti0.92Al0.08N; b) Ti0.35Al0.65N; c) Ti0.16Al0.84N; d) Ti0.14Al0.86N.

  Fig.3-C-ashows the BF STEM image of the Ti0.14Al0.86N coating cross section. The coating seems to be constituted of a stack of particles composed of bright and dark domains. Regarding the square shape of the particle in the upper right frame on Fig.3-C-a, it can be assumed that these particles can correspond to the cube-shaped grains observed on the surface morphology analyses (Fig.2-d).This Ti0.14Al0.86N coating shows a strong compositional modulation of aluminum and titanium (Fig.3-C-b). EDX analyses actually revealed the presence of Al-rich and Al-depleted zones, with an Al content of about 85 at.% in the bright domains while a Al content of about 95 at.% was found in the dark zones. This observation is consistent with the high Al overall content in the Ti0.14Al0.86N. The XRD patterns obtained for the Ti0.16Al0.84N and Ti0.14Al0.86N films (Fig.4-a) show broad and poorly defined peaks, corresponding mainly to the presence of the hcp-AlN structure with some traces of the fcc-Al(Ti)N and fcc-Ti(Al)N phases. Evidence of the {200} peaks corresponding to both fcc-Ti(Al)N and fcc-Al(Ti)N phases in the Ti0.14Al0.86N coating is shown in Fig. 4-b. The shift of the fcc-Ti(Al)N and fcc-Al(Ti)N peaks, compared to the fcc-TiN and fcc-AlN peaks positions (PDF files 00-038-1420 for fcc-TiN and 00-046-1200 for fcc-AlN), can correspond to an Al content x of about 0.08 in the fcc-Ti(Al)N phase and 0.85 in the fcc-Al(Ti)N phase. This last value is consistent with the Al content in the Ti0.14Al0.86N coating. Fig. 5-Ca shows the TEM image and the associated diffraction pattern obtained on the Ti0.14Al0.86N coating. The diffraction pattern shows the presence of the {1010}, {0002}, {1011}, {1012} and {1120} planes of the hcp-AlN phase and the {200} planes of the fcc-Ti(Al)N phase. These observations confirm the XRD results, showing the pre-dominance of the hcp-AlN phase in the Ti0.14Al0.86N coating. Fig. 5-C-bshows the HRTEM image of a particle that can correspond to a cube-shaped grain (Fig.2-d) at the surface of the Ti0.14Al0.86N coating (upper left frame). The FFT pattern associated with this HRTEM image reveals the presence of the {111} and {002} planes of a fcc structure, with a [110] zone axis. Interplanar distances of ≈0.23 nm and ≈0.20 nm were found for the {111} and {200} planes, respec-tively. This particle can be attributed to the fcc-Al(Ti)N metastable phase with a low Ti content.It could be possible that the higher amount of cube-shaped grains in the Ti0.14Al0.86N compared to the Ti0.16Al0.84N (Fig.2-c and -d) is re-lated to the presence of the fcc-Al(Ti)N highlighted by XRD as well as by HRTEM analyses in the highest Al containing coating (x = 0.86).

Fig. 3 .

 3 Fig. 3. A-a) BF STEM cross section of the Ti0.92Al0.08N coating; A-b) EDX cartography of Al; A-c) EDX cartography of Ti; B-a) BF STEM cross-section of the Ti0.35Al0.65N coating; B-b) EDX cartography of Al; B-c) EDX cartography of Ti; C-a) BF STEM cross-section of the Ti0.14Al0.86N coating, the particle in the upper right frame is one of the cube-shaped grains; C-b) EDX cartography of Al (red) and Ti (green). Coatings were protected by a Platinium

Fig. 5 .

 5 Fig. 5. A-a) BF TEM image of the Ti0.92Al0.08N coating and the corresponding diffraction pattern; A-b) HRTEM image of the Ti0.92Al0.08N coating associated to its FFT pattern; B-a) BF TEM image of the Ti0.35Al0.65N coating and the associated diffraction pattern; C-a) BF TEM image of the Ti0.14Al0.86N and the corresponding diffraction pattern; C-b) HRTEM image corresponding to a cube-shaped grain at the surface of the Ti0.14Al0.86N coating. The upper right frame is the FFT pattern associated to the HRTEM image.

Fig. 6 .Fig. 7 .Fig. 8 .

 678 Fig. 6. Diagram of the microstructures obtained in our coatings as a function of the Al content. The Ti0.92Al0.08N coating is mainly constituted of fcc-Ti(Al)N nanolamellae with a modulation of the Ti content (globular grains). Some traces of the hcp-AlN and fcc-Al(Ti)N phase are also present (dark domains). The Ti0.35Al0.65N coating is constituted of fcc-Ti (Al)N grains (bright grains) and fcc-Al(Ti)N nanolamellae with a modulation of the Al content (bright lamellae and dark lamellae). Some traces of hcp-AlN are also detected in the coating. The Ti0.14Al0.86N coating is constituted of Al-rich and Al-depleted domains. This coating is mainly constituted of the hcp-AlN phase with some evidences of the fcc-Ti(Al)N and fcc-Al(Ti) N phases. The distribution of theses phases cannot be determined.

Fig. 9 .

 9 Fig. 9. Optical images of the Ti1-xAlxN coatings deposited on Al2O3 substrates after oxidation during 1 h in air between 700 °C and 950 °C.

Fig. 10 .

 10 Fig. 10. XRD patterns obtained on Ti1-xAlxN coatings from x = 0.08 to x = 0.86, deposited on M2 tool steel, after oxidation during 1 h at 850 °C.Conclusion

Acknowledgement

The authors would like to acknowledge the CEA Saclay, Direction du Programme Matériaux Avancés and the Groupement d'Intérêt Public Haute-Marne (GIP 52) for their financial support.


Yannick Le Cacheux
email: yannick.lecacheux@cea.fr

Hervé Le Borgne
email: herve.le-borgne@cea.fr

Michel Crucianu
email: michel.crucianu@cnam.fr

Using Sentences as Semantic Representations in Large Scale Zero-Shot Learning

Zero-shot learning aims to recognize instances of unseen classes, for which no visual instance is available during training, by learning multimodal relations between samples from seen classes and corresponding class semantic representations. These class representations usually consist of either attributes, which do not scale well to large datasets, or word embeddings, which lead to poorer performance. A good trade-off could be to employ short sentences in natural language as class descriptions. We explore different solutions to use such short descriptions in a ZSL setting and show that while simple methods cannot achieve very good results with sentences alone, a combination of usual word embeddings and sentences can significantly outperform current state-of-the-art 3 .

Introduction and Related Work

Zero-shot learning (ZSL) is useful when no visual samples are available for certain classes, provided we have semantic information for these classes [START_REF] Lampert | Attribute-based classification for zeroshot visual object categorization[END_REF]. It is then possible to train a model to learn the relations between the visual and semantic features using seen classes, for which both modalities are available; these relations can later be employed to classify instances of unseen classes, for which no visual sample is available during training, based on their semantic class prototypes [START_REF] Norouzi | Zero-shot learning by convex combination of semantic embeddings[END_REF][START_REF] Frome | Devise: A deep visual-semantic embedding model[END_REF][START_REF] Romera-Paredes | An embarrassingly simple approach to zero-shot learning[END_REF][START_REF] Cacheux | From classical to generalized zeroshot learning: A simple adaptation process[END_REF][START_REF] Cacheux | Modeling inter and intra-class relations in the triplet loss for zero-shot learning[END_REF]. The semantic information can consist of vectors of attributes, e.g. binary codes for "has fur", "has stripes", etc. if classes are animal species. In a large-scale setting, it can be impractical to devise and provide attributes for hundreds or even thousands of classes. Word embeddings are then typically used to represent classes. However, a large performance gap still exists between these two types of class representations [START_REF] Cacheux | Webly supervised embeddings for large-scale zero-shot learning[END_REF].

An ideal solution could be to use short natural sentences to describe each class, as this is less time-consuming than providing comprehensive attributes and can be more visually informative than word embeddings derived from generic text corpora. The use of sentences as class descriptions in ZSL is not well studied. Some works already employed sentences for ZSL, but not in a convenient setting. For instance, Akata et al. [START_REF] Akata | Evaluation of output embeddings for fine-grained image classification[END_REF] rely on 10 sentences per image instead of 1 per class. Ba et al. [START_REF] Lei Ba | Predicting deep zero-shot convolutional neural networks using textual descriptions[END_REF] and Elhoseiny et al. [START_REF] Elhoseiny | Link the head to the" beak": Zero shot learning from noisy text description at part precision[END_REF] use tf-idf from large text collections such as Wikipedia articles and different neural architectures to obtain semantic representations. Zhu et al. [START_REF] Zhu | A generative adversarial approach for zero-shot learning from noisy texts[END_REF] generate visual features from noisy text description (again from Wikipedia articles) using Generative Adversarial Network. These works as well as others such as [START_REF] Elhoseiny | Write a classifier: Zero-shot learning using purely textual descriptions[END_REF][START_REF] Qiao | Less is more: zero-shot learning from online textual documents with noise suppression[END_REF] tend to focus on finegrained recognition and explicitly or implicitly assume that classes are similar (consisting for example of birds species) and that class descriptions will contain specific information (for example regarding the beak, the wings, the plumage...). On the other hand, we focus on generic object recognition and make close to no assumption regarding the nature of these descriptions. The closest work to ours is probably Hascoet et al. [START_REF] Hascoet | Semantic embeddings of generic objects for zero-shot learning[END_REF], in which different methods to obtain prototypes from WordNet definitions are evaluated, but reported performance is significantly below that of usual word embeddings.

In this article, we explore several ideas to leverage sentences to build semantic embeddings for ZSL and show that this can significantly outperform previous best reported performance. These proposals are easy to implement and computationally light. We provide the code4 as well as the corresponding embeddings which can be used out-of-the-box as better quality semantic representations.

Proposed Method

To deal with a zero-shot learning (ZSL) task, one considers a set C s of seen classes used during training and a set C u of unseen classes that are available for the test only, with C s ∩ C u = ∅. Each class is also associated to a semantic class prototype s c ∈ R K that characterizes it. Let us consider a training set {(x i , y i ), i = 1 . . . N } with labels y i ∈ C s and visual features x i ∈ R D . The goal of the ZSL task is to learn a compatibility function f : R D × R K → R assigning a similarity score to a visual sample x and a class prototype s. It is usually a parametrized function that is learned by minimizing a regularized loss L:

1 N N i=1 |Cs| c=1 L(f (x i , s c ), y i ) + λΩ[f ] ( 1
)
where Ω is a regularization term weighted by λ. Once f is learned, the testing phase consists in determining the label ŷ ∈ C u corresponding to a visual sample

x such that ŷ = arg max c∈Cu f (x, s c ).
A simple linear model is presented in Section 2.1. Our contribution presented in Section 2.2 consists in leveraging sentences to build semantic prototypes s c 2.1 Linear ZSL model.

We adopt a linear ridge regression model, in which semantic prototypes are projected into the visual features space so as to minimize a regularized least square loss. This choice is motivated by the good and consistent results of this model in [START_REF] Cacheux | Webly supervised embeddings for large-scale zero-shot learning[END_REF], as well as its low computational cost and ease of reproducibility. The choice of the visual space as the projection space is based on [START_REF] Shigeto | Ridge regression, hubness, and zero-shot learning[END_REF], which argues that this choice enables to mitigate the hubness problem [START_REF] Radovanović | Hubs in space: Popular nearest neighbors in high-dimensional data[END_REF]. Results with other models from [START_REF] Hascoet | On zero-shot recognition of generic objects[END_REF] are provided in Section 3.3.

The visual features {x 1 , . . . , x N } are extracted with a pre-trained network, so that x n ∈ R D , D the dimension of the visual space. We thus have a corresponding training matrix X ∈ R N×D . The semantic representations {s 1 , . . . , s C } of the C seen (training) classes lie in R K . We write T = (t 1 , . . . , t N ) ∈ R N ×K the semantic representations associated with each training sample x n with label y n so that t n = s yn .

The resulting loss for the ridge regression with parameters

Θ ∈ R K×D is 1 N X -TΘ 2 F + λ Θ 2 F ( 2
)
where λ a hyperparameter weighing the 2 regularization penalty Θ 2 F , and • F is the Frobenius norm. This minimization problem has a closed-form solution:

Θ = (T T + λN I K ) -1 T X (3)
During the prediction phase, given a testing sample x and the semantic representations {s 1 , . . . , s C } of the C unseen classes, s c ∈ R K , we predict label ŷ corresponding to the class whose prototype's projection is closest to x:

ŷ = argmin c x -Θ s c 2 (4)

Sentence-Based Embeddings

A simple baseline consists in averaging the embeddings of the words in the definition, as is usually done for class names consisting of several words: if a sentence s describing a class has N words with respective embeddings {w 1 , . . . , w N }, then the corresponding semantic representation is s = 1 N N n=1 w n . We call this baseline the Def average approach. However, as illustrated in Fig. 2, not all words are equally important in a short sentence description. We therefore explore the use of attention mechanisms: the sentence embedding is a weighted average of the embeddings of its words, so that more important words contribute more to the result. We consider two ways to achieve this.

Visualness Based Approach

The first, called the Def visualness approach, aims to estimate how "visual" a word is. For a given word w n , we thus collect the 100 most relevant images from Flickr using the website's search ranking. We obtain representations {r i 1 , . . . , r i M } with 2048 dimensions (r m ∈ R 2048 ) using a pre-trained ResNet (see Section 3.1). We then measure the average distance of vectors r i m to the mean vector ri = 1 M M m=1 r i m to obtain the additive inverse of the visualness v i :

v i = - 1 M M m=1 r i m -ri 2 (5)
We hypothesize that words with high visual content have features close to each other, so that the quantity v i from Equation 5 can be used as a qualitative measure of the intuitive concept of how "visual" a given word is. Examples of words with high and low visualness shown in Section 3.4 tend to confirm that this hypothesis is reasonable.

Figure 1 shows the distribution of the inverse of the visualness for the 4059 words from class names and WordNet definitions of classes from [START_REF] Hascoet | On zero-shot recognition of generic objects[END_REF]. Equation 5is then used to obtain weights for each word based on its visualness in order to use a weighted average of the word embeddings from a definition as class prototype. As the initial scale of the average distances / negative visualnesses is arbitrary, a temperature τ is introduced in the softmax to weigh each word, so that the resulting sentence embedding is

s = N n=1 exp(v n /τ ) N k=1 exp(v k /τ ) w n (6)
The value τ is cross-validated on the validation set -see Section 3.1. In practice, this often leads to selecting τ = 5. Learned Attention Based Approach A second approach called Def attention aims to learn to predict the visualness v i of word w i from its embedding w i ∈ R K such that v i = θ w i , where θ are learned parameters. Equation 5 can then be used to create a prototype from a class' definition. As the v i are directly learned, it is no longer necessary to account for their initial scale. We can thus discard the temperature in the softmax by setting τ = 1 in Equation 5. Different ways could be considered to learn θ. A straightforward approach could consist in randomly initializing θ along with Θ from Equation 2, computing visualnesses v i = θ w i , computing class prototypes s c using the v i and Equation 5, computing T and finally computing the loss from Equation 2; we could then use back-propagation and gradient descent to update θ and Θ until convergence.

However, we instead take advantage of the closed-form solution from Equation 3 and proceed as follows: we randomly initialize θ and compute class prototypes s c and T using Equation 5 as previously. We then directly estimate Θ using the closed-form solution from Equation 3, and use this value to compute the loss in Equation 2. We then back-propagate the gradient and perform gradient descent on θ only, the value of Θ being estimated with Equation 3 at each iteration. We repeat this process for 50 "epochs". An illustration of the resulting weights from both approaches is shown in Fig. 2.

The resulting sentence embeddings can then be compared to the standard class prototypes obtained by embedding the class name (Classname approach). Since recent results show that hierarchical and graph relations between classes contain valuable information [START_REF] Hascoet | On zero-shot recognition of generic objects[END_REF], in the Classname+Parent approach we combine the Classname prototype with the prototype of its parent class.

Finally, we experiment with different combinations of the base approaches:

-Classname+Def average , -Classname+Def visualness -Classname+ Def visualness +Parent.

All these combinations simply consist in finding a value µ ∈ [0, 1] such that the combined prototype is s = µs 1 + (1 -µ)s 2 , given s 1 and s 2 the prototypes of two approaches.

3 Experiments When several prototypes s 1 and s 2 are combined with s = (1 -µ)s 1 + µs 2 , µ ∈ [0, 1], for instance in the Classname+Def average or Classname+Def visualness approaches, µ is also considered to be a hyperparameter and its value is selected by cross-validation along with the other hyperparameters. In these cases, it leads to surprising results with a frequently selected value of µ = 0.7, meaning the combined prototype is 70% from the definition prototype and 30% from the classname prototype, even though the definition alone performs far worse than the classname alone. Nonetheless, this still led to consistent results.

On the other hand, when combining a prototype with the prototype of its parent class, cross-validation tended to yield very inconsistent and unstable values. When combining a prototype with its parent prototype, we therefore fixed µ = 0.25 (meaning 75% base class and 25% parent class) for all methods and all word embeddings. Note that in the Classname+Def visualness and Class-name+ Def visualness +Parent approach, the prototype from the parent class is also built with Classname+Def visualness , and the value of µ is the same as the one selected for the child class.

Pre-trained word embeddings. We used widely adopted pre-trained embeddings available on the Internet for Word2vec [START_REF] Mikolov | Distributed representations of words and phrases and their compositionality[END_REF], FastText [START_REF] Bojanowski | Enriching word vectors with subword information[END_REF] and Glove [START_REF] Pennington | Glove: Global vectors for word representation[END_REF]. We used a Word2vec embedding model pre-trained on Wikipedia as we found it gave better results than other version, such as a Word2vec embedding model trained on Google News. For the same reason, we used FastText and Glove models pre-trained on Common Crawl. We used a 300-dimension version for all three. The pre-trained embedding models were downloaded from the following links:

-Word2vec: https://wikipedia2vec.github.io/wikipedia2vec/pretrained/ (English version trained on Wikipedia). -Fasttext: https://fasttext.cc/docs/en/english-vectors.html (version trained on Common Crawl with 600B tokens, no subword information). -Glove: https://nlp.stanford.edu/projects/glove/ (version trained on Common Crawl with 840B tokens).

For Elmo [START_REF] Peters | Deep contextualized word representations[END_REF], we similarly adopted a pretrained version obtained from https://allennlp.org/elmo. We used the original version with 93.6 million parameters, pre-trained on the 1 Billion Word Benchmark. Elmo embeddings have dimension 3 × 1024 (a 1024-dimensional embedding from each of the three layers). For the sake of simplicity we combine the three layers using the same weight of 0.33 to obtain a single 1024-dimensional representation for each word, as weights fine-tuned for our specific task gave similar results.

Visual and semantic features. Visual features are obtained from the last pooling layer of a ResNet [START_REF] He | Deep residual learning for image recognition[END_REF] model pre-trained on ImageNet [START_REF] Deng | Imagenet: A largescale hierarchical image database[END_REF]. They therefore have 2048 dimensions. We use the ResNet-101 implementation of PyTorch [START_REF] Paszke | Pytorch: An imperative style, highperformance deep learning library[END_REF]. Visual features are normalized to have unit 2-norm.

Classname ZSL embeddings are obtained from synsets as follows: a synset in WordNet [START_REF] Miller | Wordnet: a lexical database for english[END_REF] can consist of several lemmas, each lemma consisting of several words. The prototype of a lemma is the mean of the embeddings of its words when such embeddings exist in the pre-trained model. The prototype of the synset and thus of the class is the mean of the prototypes of its lemmas. In the approximately 6 (depending on the pre-trained embedding) cases where no word from any lemma had an embedding in the pre-trained model, we used the embedding from the word "thing" as class prototype. All semantic prototypes are 2-normalized, for the Classname approach as well as all the others. Evaluation protocol We measure ZSL accuracy on the large-scale ImageNet dataset, with synsets as classnames and corresponding WordNet definitions as sentences. We also use the WordNet hierarchy to determine parent classes. We employ the experimental protocol of [START_REF] Hascoet | On zero-shot recognition of generic objects[END_REF], with the same ResNet features and, especially, the same train/test splits since [START_REF] Hascoet | On zero-shot recognition of generic objects[END_REF] evidenced significant structural bias with previous popular splits. Glove embeddings are employed in [START_REF] Hascoet | On zero-shot recognition of generic objects[END_REF]; we also compare with Word2vec and FastText [START_REF] Almeida | Word embeddings: A survey[END_REF], as well as Elmo contextual embeddings using pre-trained embeddings available on the Internet (details are provided in the supplementary material). The ZSL model is a ridge regression from the semantic to the visual space as it gives good, consistent and easily reproducible results (see [START_REF] Cacheux | Webly supervised embeddings for large-scale zero-shot learning[END_REF]).

Results with Linear Model

Results are shown in Table 1. The baseline Def average approach performs poorly compared to the usual Classname approach. Attention provides a slight improvement (both Def visualness and Def attention give comparable results), but still does not match Classname, even though attention weights seem reasonable (some examples are shown in Fig. 2). The use of parent information improves results compared with Classname alone, which is consistent with [START_REF] Hascoet | On zero-shot recognition of generic objects[END_REF] where the best methods make use of hierarchical relations between classes. The combination of Classname and Def approaches brings significantly better scores than either separately. Surprisingly, while any Def alone has lower performance than Classname alone, the best trade-off obtained by cross-validation is 70% definition and 30% classname in every case, meaning that the definition has a much stronger presence than the class name in the resulting embedding.

The word embedding method has an impact on performance; FastText consistently ranks above the other methods. Elmo has surprisingly low performance including with fine-tuned attention, even though one could expect attention to be more effective here as Elmo considers the role of the word in the sentence.

Results with Other Models

We report results with the best performing prototypes for a few other ZSL models in addition to the linear model already described. We specifically report results for models used in [START_REF] Hascoet | On zero-shot recognition of generic objects[END_REF] (except for graph-convolutional models which use a different setting). For ConSE [START_REF] Norouzi | Zero-shot learning by convex combination of semantic embeddings[END_REF], DeViSE [START_REF] Frome | Devise: A deep visual-semantic embedding model[END_REF] and ESZSL [START_REF] Romera-Paredes | An embarrassingly simple approach to zero-shot learning[END_REF], we use the implementation from [START_REF] Hascoet | On zero-shot recognition of generic objects[END_REF]. We report results averaged over 5 runs with different random initializations of parameters for ConSE and DeViSE. Since the other models use closed-form solutions, it is not necessary to report results averaged over several runs as there is no variance in the results. We also provide results with another simple linear model, consisting in a ridge regression from the visual to the semantic space. This model is called Linear V → S , as opposed to Linear S → V which is the ridge regression model from the semantic to the visual space described in Section 2.1.

We report results with the best class prototypes as determined in Table 1 of the main paper, i.e. the ones obtained with the Classname+Def visualness +Parent approach with the FastText embeddings. These prototypes will be made available to the community along with the others. Top-1, top-5 and top-10 accuracies for the models mentioned above with these prototypes are provided in Table 2.

Illustration of visualness

We provide a few examples of the words with the highest and lowest visualness in Figure 3, as well as the corresponding inverse visualness (the mean distance to the mean feature representations for images associated with this word) and the corresponding top image result with no copyright restriction from Flickr.

Conclusion.

To scale zero-shot learning to very large datasets it is important to solve the problem of providing class prototypes for many classes. The use of class name embeddings scales better than the provision manually-defined attributes but the resulting performance is nowhere near. We suggest that low-cost textual content, consisting in one sentence per class, can bring substantial performance improvements when combined with class name embeddings, when they are processed with the proposed approaches. The improved class prototypes for ImageNet are available at https://github.com/yannick-lc/zsl-sentences.

Beyond the case of zero-shot learning, other tasks are known to suffer from an incomplete semantic information, such as image retrieval, classification [START_REF] Znaidia | Multimodal feature generation framework for semantic image classification[END_REF] or annotation [START_REF] Znaidia | Tag completion based on belief theory and neighbor voting[END_REF]. The work we presented in this article can thus benefit to these tasks, in particular when one addresses large-scale datasets. Predicting automatically the important words into sentences with regard to their visualness leads to semantic representation that can be mixed to visual representation to build intrinsic multimedia representations [START_REF] Znaidia | Bag-ofmultimedia-words for image classification[END_REF] that can be used to address crossmodal retrieval [START_REF] Tran | Aggregating image and text quantized correlated components[END_REF] and classification [START_REF] Tran | Cross-modal classification by completing unimodal representations[END_REF].

Fig. 1 .

 1 Fig. 1. Inverse of the visualness (low values correspond to high visualness) for the 4059 words from class names and WordNet definitions.

3. 1

 1 Settings Selecting hyperparameters. Hyperparameter values such as λ in Equation 2 or τ in Equation 5 are selected by cross-validation on seen classes, keeping 200 of the 1000 seen training classes as unseen validation classes. The model is then retrained on all 1000 training classes with the selected values.

Fig. 2 .

 2 Fig. 2. Illustration of attention scores on some test classes. Left: weights from Def visualness after softmax (the temperature is τ = 5 so differences are less pronounced than initially). Right: weights learned with Defattention, with FastText embeddings.

Fig. 3 .

 3 Fig. 3. Top: words with highest visualness. Bottom: words with lowest visualness. The visualness of a word is the inverse of the mean distance (shown in parenthesis) to the mean representation of visual features from the top 100 corresponding images from Flickr, see Section 3.4. Top 1 image with no copyright restriction is displayed.

Table 1 .

 1 Comparison of approaches on ImageNet with a linear model. Best previous reported result is 14.1 in[START_REF] Hascoet | On zero-shot recognition of generic objects[END_REF]; the result marked with * corresponds to the same setting (use of Classname with Glove embeddings).

		Word2vec FastText Glove Elmo
	Classname	12.4	14.8 14.5* 10.9
	Classname+Parent	13.4	15.9	15.4 11.4
	Defaverage	9.7	10.6	10.0 8.7
	Def visualness	10.5	10.9	10.5 9.5
	Defattention	10.5	11.0	10.2 9.5
	Classname+Defaverage	14.6	17.2	16.9 12.2
	Classname+Def visualness	14.8	17.3	16.8 12.1
	Classname+Def visualness +Parent	15.4	17.8 17.3 12.5

Table 2

 2

	.	Top-k	ZSL	accuracy	for	different	models,	using	the
	Classname+Def visualness +Parent prototypes constructed from FastText embeddings.
					top-1	top-5	top-10
			LinearS → V 17.8	43.6	56.7
			LinearV → S	9.1	26.2	36.7
			ESZSL		16.3	40.6	52.4
			ConSE		12.7	31.8	42.4
			DeViSE		14.0	38.3	52.1

https://github.com/yannick-lc/zsl-sentences


F Hernandez

How to model the temporal constraints ?

A set covering formulation for VRPTW Ω a set of feasible trips, xed in time θ k indicates the number of times where trip r k is selected for covering, c k cost of trip r k a ik = 1 if the customer i is visited by r k , 0 else minimize

r k ∈Ω c k θ k r k ∈Ω a ik θ k ≥ 1 (i ∈ V \ {0}) θ k ∈ IN (r k ∈ Ω)
c k θ k r k ∈Ω a ik θ k ≥ 1 (i ∈ V \ {0}) θ k ∈ IN (r k ∈ Ω) r k ∈Ω b tk θ k ≤ U (∀δ t )
one time interval δ t by instant ⇒ b tk are binary combinatorial explosion of constraint number related to temporal precision

r k b (t+2)k =1 temps δ 1 δ 2 δ 3 δ 4 δ 5 δ 6 … δ 0
one time interval δ t by instant ⇒ b tk are binary combinatorial explosion of constraint number related to temporal precision time interval length = minimal duration of possible trips

∆ t+1 ∆ t+3 ∆ t ∆ t+2 r k b (t+2)k =1
∆ t+1 ∆ t+3 ∆ t ∆ t+2 r k b (t+2)k =1 b tk = 0 b (t+3)k =0.2 b (t+1)k =0.8
temps Ω a set of feasible trips, xed in time θ k indicates the number of times where trip r k is selected for covering, c k cost of trip r k a ik = 1 if the customer i is visited by r k , 0 else b tk Let L 2 and L 3 in the same group (their represent the same partial path) reduced cost formula : c r Let L 2 and L 3 in the same group (their represent the same partial path) reduced cost formula : c r

∈ [0, 1] indicates if the trip r k includes the instant ∆ t minimize r k ∈Ω c k θ k r k ∈Ω a ik θ k ≥ 1 (i ∈ V \ {0}) θ k ∈ IN (r k ∈ Ω) r k ∈Ω b tk θ k ≤ U (∀∆ t ) F.
k + t=h k t>h k µ t ≤ c r k c r 1 = 3; c r 3 = 5; c r 2 = c r 3 + 3.5 = 8.5 comparaisons : 5 1 time ∆ 4 ∆ 1 ∆ 3 ∆ 2 3 3 3 3 3 4 3 4 L 1 L 2 L 3 Σμ t = 3.5 F.
k + t=h k t>h k µ t ≤ c r k c r 1 = 3; c r 3 = 5; c r 2 = c r 3 + 3.5 = 8.5 comparaisons : c r 1 + 4 < c r 2 5 1 time ∆ 4 ∆ 1 ∆ 3 ∆ 2 3 3 3 3 3 4 3 4 L 1 L 2 L 3 Σμ t = 4
F. Hernandez Branch and Price August 2012 20 / 33

Let L 2 and L 3 in the same group (their represent the same partial path) reduced cost formula : c r

k + t=h k t>h k µ t ≤ c r k c r 1 = 3; c r 3 = 5; c r 2 = c r 3 + 3.5 = 8.5 comparaisons : 3 + 4 < 8.5 5 1 time ∆ 4 ∆ 1 ∆ 3 ∆ 2 3 3 3 3 3 4 3 4 L 1 L 2 L 3 Σμ t = 4
F. Hernandez Branch and Price August 2012 20 / 33

Let L 2 and L 3 in the same group (their represent the same partial path) reduced cost formula : c r

k + t=h k t>h k µ t ≤ c r k c r 1 = 3; c r 3 = 5; c r 2 = c r 3 + 3.5 = 8.5 comparaisons : 5 1 time ∆ 4 ∆ 1 ∆ 3 ∆ 2 3 3 3 3 3 4 3 4 L 1 L 2 L 3 Σμ t = 4
F. Hernandez Branch and Price August 2012 20 / 33

Let L 2 and L 3 in the same group (their represent the same partial path) reduced cost formula : c r

k + t=h k t>h k µ t ≤ c r k c r 1 = 3; c r 3 = 5; c r 2 = c r 3 + 3.5 = 8.5 comparaisons : c r 1 + 4.5 > c r 3 5 1 time ∆ 4 ∆ 1 ∆ 3 ∆ 2 3 3 3 3 3 4 3 4 L 1 L 2 L 3 Σμ t = 4.5 1 time ∆ 4 ∆ 1 ∆ 3 ∆ 2 3 3 3 3 3 4 3 4 L 1 L 2 L 3 Σμ t = 4.5
Select relative to the possible extensions it must accept all extensions accepted by at least one label of its group all labels of the group have to take into account ⇒ retardation

1 1 2 3 time
Select relative to the possible extensions it must accept all extensions accepted by at least one label of its group all labels of the group have to take into account ⇒ retardation

L p = (c r p , h p , q p , d p , rd p , V 1 p , • • • , V n p )
, where: c r p is the reduced cost of p h p is the starting time of the service of j q p is the carried quantity d p is the arrival time to the depot rd p is the possible retardation of the arrival time of the depot V i p =1 if the customer i is unreachable by p, 0 else Initialisation One label is created at the end of the planning time horizon and one label at the beginning of each time interval with a not null dual value Recall Branch and bound Branch and Price Simplexe Column generation

Branching on arcs select an arc (i,j) with a fractional ow

x ij = 1 ⇒ θ k = 0 if trip k visits customer i or j without using arc (i, j)

x ij = 0 ⇒ θ k = 0 if trip k uses arc (i, j)

Recall Branch and bound Branch and Price Simplexe Column generation

Branching on arcs select an arc (i,j) with a fractional ow

x ij = 1 ⇒ θ k = 0 if trip k visits customer i or j without using arc (i, j)

x ij = 0 ⇒ θ k = 0 if

Ω

  a set of feasible trips, xed in time θ k indicates the number of times where trip r k is selected for covering, c k cost of trip r k a ik = 1 if the customer i is visited by r k , 0 else b tk ∈ {0, 1} indicates if the trip r k includes the instant δ t minimize r k ∈Ω

temps

  one time interval δ t by instant ⇒ b tk are binary combinatorial explosion of constraint number related to temporal precision time interval length = minimal duration of possible trips ⇒ b tk ∈ [0, 1] become the fraction of time interval ∆ t occupied by trip r k

  from j to 0 is represented by the label:

  -i∈V \{0} a ik λ i + ∆t b tk µ t

	Time interval length = minimal duration of possible trips relaxation of the problem Inspiration Simplexe algorithm Only basic varaibles are interesting How does a nonbasic variable becomes a basic variable ? Problem decomposition Restricted master problem : Set covering problem where the integrity constraints are relaxed, reduced to a subset of variables Ω w Subproblem : Find a negative reduced cost variable ⇒ Elementary shortest path problem with resource constraints (ESPPRC) Problem decomposition Restricted master problem : Set covering problem where the integrity constraints are relaxed, reduced to a subset of variables Ω w Subproblem : Find a negative reduced cost variable ⇒ Elementary shortest path problem with resource constraints (ESPPRC) Reduced cost c r k = c k µ t dual value associated to time interval ∆ t labels = L num = (T 1 num , • • • , T n num ) each node has a label list during label extension create a new label and insert it in correponding node label list set resource consumption check that the resource constraints are meet stop when no more label can be extended labels = L num = (T 1 num , • • • , T n num ) each node has a label list during label extension create a new label and insert it in correponding node label list set resource consumption check that the resource constraints are meet stop when no more label can be extended Problem too many generated labels labels = L num = (T 1 num , • • • , T n Objective 1 Objective 1 Objective 1 Objective 1 Label group denition Label group denition num ) take into account the loading times take into account the loading times take into account the loading times take into account the loading times A group of labels is a set of labels that represent the same partial path and A group of labels is a set of labels that represent the same partial path and each node has a label list during label extension The loading times are at the depot before departure The loading times are at the depot before departure The loading times are at the depot before departure whose arrival-to-destination dates belong to the same time interval whose arrival-to-destination dates belong to the same time interval The loading times are at the depot before departure create a new label and insert it in correponding node label list add a customer to a trip ⇒ delay time of service for the previous add a customer to a trip ⇒ delay time of service for the previous add a customer to a trip ⇒ delay time of service for the previous add a customer to a trip ⇒ delay time of service for the previous How to select a representative label How to select a representative label set resource consumption customers customers customers customers two main rules two main rules check that the resource constraints are meet ⇒ extend the label in backward move ⇒ extend the label in backward move ⇒ extend the label in backward move ⇒ extend the label in backward move it can be dominated by an other label if and only if all labels of its it can be dominated by an other label if and only if all labels of its stop when no more label can be extended Problem Objective 2 generate a trip with minimal reduced cost Objective 2 generate a trip with minimal reduced cost Objective 2 generate a trip with minimal reduced cost group are dominated by this label group are dominated by this label Objective 2 it must accept all extensions accepted by at least one label of its group it must accept all extensions accepted by at least one label of its group generate a trip with minimal reduced cost too many generated labels Solution apply a dominance relation after each extension on corresponding label list avoid the time interval associated with a dual value µ t not null take into account all possible departure time avoid the time interval associated with a dual value µ t not null take into account all possible departure time too many labels In our previous studies avoid the time interval associated with a dual value µ t not null take into account all possible departure time too many labels ⇒ group labels that represent the same structure and select a representative To compare the time dependent reduced cost of two dierent labels L k and L k during the dominance relation process we use this formula c r k + t=h k t>h k µ t ≤ c r k
	Hernandez F. Hernandez F. Hernandez	Branch and Price Branch and Price Branch and Price	August 2012 August 2012 August 2012	13 / 33 14 / 33

⇒ if the solution found is not feasible then the problem have to be solved with one time interval δ t by instant Ω is a set of feasible trips, xed in time too many variables Time interval length = minimal duration of possible trips relaxation of the problem ⇒ if the solution found is not feasible then the problem have to be solved with one time interval δ t by instant Ω is a set of feasible trips, xed in time too many variables ⇒ adressed by column generation In minimisation case : only if its reduced cost is negative Method The column generation consists to solve iteratively two problems. the restricted master problem = problem restricted to a sub set of variables (basic and nonbasic) the pricing problem = Find new nonbasic variables that can become basic Stop when the pricing problem don't nd new nonbasic variables that can improve the solution.

λ i dual value associated to customer i

  trip k uses arc (i, j) ⇒ θ k = 0 if trip k visits customer i or j without using arc (i, j) x ij = 0 ⇒ θ k = 0 if trip k uses arc (i, j)

	Implementation Results Results Conclusion
	Recall
	Problem having all arcs with an integer ow (x ij ∈ IN) does not imply that the solution is integer (θ k ∈ {0, 1}) Branch and bound Branch and Price Simplexe Column generation Branching on arcs select an arc (i,j) with a fractional ow Instance of VRPTW customers service time demande time window arcs (i, j) travel time cost Our branching problem structures duration time window arcs (i, j) cost of the structure j Instance of VRPTW customers service time demande time window arcs (i, j) travel time cost Our branching problem structures duration 0 time window arcs (i, j) 0 cost of the structure j Hardware and software Language: C++ Solver: GLPK (open source) Computer: Intel Core2Duo E7300 2.66GHz, 3Gb RAM Benchmarks based on Solomon's instances: R2, RC2, C2 25 customers: 2 vehicles allowed Instances with 50 customers Conclusion Instances with 25 customers Instances % GAP Time (sec) Min Max Avg Min Max Avg C 0.55 4.85 2.27 12 371 170 R 0.44 4.70 2.41 22 3769 1006 RC 3.52 8.98 5.41 9 20038 12537 Master problem: time constraint between trips Instances % GAP Time (sec) C -time aggregation -R201-50 1.58 all solution found with time aggregation are feasible that imply a good 237 R202-50 2.42 relaxation of time constraint 78880 R205-50 2.39 24062 RC201-50 2.24 662 RC202-50 2.40 99346 Sub-problem: time dependent reduced cost appropriate dynamic programming ⇒ representative labels Perspectives x ij = 1 Problem having all arcs with an integer ow (x ij ∈ IN) does not imply that the solution is integer (θ k ∈ {0, 1}) 25/27 instances closed Problem with a great temporal dependence 50 customers: 4 vehicles allowed computation time limited to 30h per instance great variation of computation times and GAPs mainly solved with a structural branching scheme Only 5/27 instances closed great variation of computation times ⇒ improve the branching scheme with temporal branching politic
	⇒ call a repair strategy
	F. Hernandez F. Hernandez F. Hernandez F. Hernandez F. Hernandez	Branch and Price Results Results Results Conclusion	August 2012 August 2012 August 2012 August 2012 August 2012	25 / 33 29 / 33 30 / 33 31 / 33 32 / 33

?

Thank you for your attention orent.hernandez@cirrelt.ca

August 2012

Ω is a set of feasible trips, xed in time too many variables

Select relative to the reduced cost Let L 2 and L 3 in the same group (their represent the same partial path) reduced cost formula : c r

Dominance relation

Label L 1 dominates label L 2 if and only if: q 1 ≤ q 2 (carried quantity) All customers unreachable by L 1 are not by L 2 too h 2 +rd 2 ≤h 1 (starting time of the service) c r

Restricted master problem

Case when the ow matrix is integer and some variables fractional The ow matrix is such that every customer has an unique successor The ow matrix represents a set of structures In the actual fractional solution some structures are represented by several trips with dierent time positions

We consider the following issue: Is it possible to assign a single time position to every structure? Equivalent to determining the existence of an integer solution supported by the integer ow matrix


Ian Aupiais

Pierre Hemme

Marc Allen

Alexis Scida

Xiao Lu

Christian Ricolleau

Yann Gallais

Alain Sacuto

Stanislaus Wong

Rogério De Sousa

Maximilien Cazayous

Impact of the surface phase transition on magnon and phonon excitations in BiFeO 3 nanoparticles

We have performed Raman scattering measurements on BiFeO 3 nanoparticles and studied both magnetic and lattice modes. We reveal strong anomalies between 140 K and 200 K in the frequency of magnon and E(LO 1 ), E(TO 1 ) and A 1 (LO 1 ) phonon modes. These anomalies are related to a surface expansion and are enhanced for nanoparticle sizes approaching the spin cycloidal length. These observations point out the strong interplay between the surface, the lattice, and the magnetism for sizes of BiFeO 3 nanoparticles close to the cycloid periodicity.

Bismuth ferrite (BiFeO 3 ) represents a prototypical multiferroic with outstanding properties. This multifunctional material has been associated with a number of diverse technological applications such as photovoltaics 1,2 , electrocatalysis 3 , nanoelectronics 4,5 , energy harvesting 6 , and even spintronics 7,8 . Also, it has been demonstrated that controlling the dimensionality of this material allows us to modify its properties. For example, controlling the size of BiFeO 3 nanoparticles permits a significant tuning of the magnetic, electrical, and optical properties 9,10 . Moreover, it has been proposed that the confinement in BiFeO 3 nanoparticles should induce a size-dependent bistability of magnetization and polarization useful for memory bits 11 .

In bulk phase, BiFeO 3 is ferroelectric below the Curie temperature T C ∼ 1100 K with a large spontaneous electrical polarization of ∼ 100 µC•cm -2 (see Fig. 1(a)) 12,13 . The electrons present at the Bi sites are responsible for the cationic displacements at the origin of the ferroelectricity 14,15 . In addition, an antiferromagnetic order appears below the Néel temperature, T N ∼ 640 K. This magnetic order corresponds to a G-type antiferromagnetic behavior, modulated by a long-wavelength cycloid of 62 nm 16 . This modulation is attributed to a homogeneous Dzialoshinskii-Moriya interaction activated by the ferroelectric displacement of the Fe sites [17][18][19] . The cycloid is also slighty tilted out of its propagation plane and characterised by a local ferromagnetic moment cancelled out over a period 20 . This additional modulation is due to an inhomogeneous Dzialoshinskii-Moriya interaction activated by the rotation of oxygen octahedra 18,19 . Reducing the size of ferroelectric nanoparticles induces a) corresponding author : maximilien.cazayous@u-paris.fr structural distortions which favor the high-symmetry paraelectric phase. In the case of BiFeO 3 , it has been determined that the paraelectric phase appears below 9-10 nm 21,22 . For antiferromagnetic nanoparticles, the uncompensated antiferromagnetic sublattice at the surface leads to a ferromagnetic moment which increases as the size is reduced 23,24 . Unexpectedly, F. Huang et al. have observed an anomalous behavior of the magnetization for a size close to the cycloid periodicity 25 . In addition, a surface expansion takes place between 140 K and 200 K within the topmost 10 nm of the bulk 26 .

The role of the surface expansion on the dynamical properties of BiFeO 3 nanoparticles, for which the surface/volume ratio increases as the size is reduced, remains to be studied. In this work, we have tracked the effect of the surface expansion by the corresponding size reduction on phonons and magnons. We observe the disappearance of the fingerprint of the cycloidal spin excitations between the sizes of 83 nm and 61 nm. At two different temperatures associated with surface expansion, anomalies in the frequency of magnetic and lattice excitations are observed. Such anomalies are enhanced for a nanoparticle size close to the cycloid periodicity.

We performed Raman spectroscopy measurements on BiFeO 3 nanoparticles. Five sizes have been synthesized by the sol-gel method which has been adapted and optimized from the protocol reported in Ref. 19 : 250 nm ± 100 nm, 158 nm ± 17 nm, 83 nm ± 20 nm, 61 nm ± 9 nm, and 31 nm ± 7 nm. with a thickness of 3 nm have been prepared under very low pressure for Raman measurements. Raman spectra were recorded in a backscattering geometry with a triple spectrometer Jobin Yvon T64000 using the 647.1 nm excitation line from a Ar + -Kr + mixed gas laser. The penetration depth of the laser light is very large because of the low extinction coefficient k = 1.2 10 -4 of BFO 27 . Thus, BFO is nearly transparent for the 647.1 nm light. The excited volume is given by the focus (∼70 µm). The high rejection rate of the spectrometer allows us to detect excitations as low as 10 cm -1 . Temperature is controlled with an open cycle He cryostat. Since nanoparticles are randomly oriented, there are no Raman selection rules. We observe all the phonon modes that characterize the rhombohedral phase (R3c) for each nanoparticle size.

Figure 2(a) shows the low-energy part of the Raman spectra for the five sizes of nanoparticles. For 250 nm, three peaks are observed. In the bulk phase, the magnetic cycloid manifests itself by two series of narrow peaks in the Raman spectra. They are the fingerprint of the magnon modes 28 . They correspond to the spin excitation oscillations in and out of the cycloidal plane measured at zero wave vector. The peaks observed in Fig. 2 nanoparticles) and are associated with the cycloidal spin states. For 158 nm and 83 nm, we distinctly only observe two peaks. No peak is detected for sizes of 61 nm and 31 nm. The optical gap of BiFeO 3 shifts with the nanoparticle size which could explain the disappearance of the peaks 9 . However, the signature of the cycloid has not been detected using any other laser wavelengths. In Fig. 2(b), we have reported the frequency of the magnon at 17 cm -1 as a function of temperature. For 250 nm, the frequency has a similar dependence as a magnetic excitation in the bulk phase 28 . For 158 nm, the frequency presents a jump around 140 K and a plateau around 200 K. A slightly more pronounced trend is measured for a size of 83 nm. In the same range of temperature, a surface phase transition with an associated sharp change in lattice parameter at the surface has been observed 26 . This "skin" effect is estimated to be within the topmost 10nm of BFO 26 . For a nanoparticle size of 80 nm, this effect represents 33% of the volume of the nanoparticle. The observed anomalies in the magnon frequencies of nanoparticles can be attributed to the surface expansion. The effect of this expansion on the spin excitations is enhanced around 83 nm. Hence, our results show that the surface has a significant impact on the magnetism for a size approaching the cycloid periodicity of 62 nm. It is known that BiFeO 3 shows a strong magneto-striction and spin-phonon coupling 17,18,29,30 . Both phenomena might be at the origin of the observed anomalies. This raises the question of the impact of the surface expansion in nanoparticles on the lattice excitations when the size approaches the cycloid periodicity.

Figure 3(a) shows the Raman spectra of the E(TO 1 ) and E(LO 1 ) phonons at 75 cm -1 and 81 cm -1 respectively for the five sizes. The temperature dependence of the E(TO 1 ) phonon is responsible for the variation of the dielectric constant and is a potential candidate to be the soft mode driving the ferroelectric transition 31,32 . It has been observed in infrared spectroscopy that, compared to that of the E(LO 1 ) phonon, the intensity of the E(TO 1 ) phonon decreases for sizes smaller than 83 nm 22 . This behavior has been ascribed to the depolarization of the nanoparticles. This size-dependent intensity can be expressed as a function of the inverse nanoparticle diameter. The critical size below which the nanoparticles become paraelectric can be determined using this relation and the relative intensity of the E(TO 1 ) and E(LO 1 ) 22 . We find a critical size of 12 nm in agreement with previous measurements around 10 nm 21 . The frequencies of the two phonons are reported in Fig. 4. To have access to the frequency of the E(TO 1 ) mode, the E(LO 1 ) is subtracted by a Fano shape. Notice that the E(TO 1 ) mode is too low in intensity to be studied as a function of the temperature for the sizes below 83 nm. In Fig. 4(a) and (c), the temperature dependence of the frequencies for the 250 nm is similar to the one observed in the bulk. For 158 nm, both mode frequencies present a jump at around 140 K followed by a plateau at around 220K (Fig. 4(b) and(d)). This behavior is even more pronounced for 83 nm. At 61 nm, the presence of the jump and the plateau are less obvious in the frequencies of E(LO 1 ). They disappear for a size of 31 nm for which the temperature dependence of the frequency is close to the one at 250 nm. The surface phase transition no longer plays a role for a size of 31 nm.

Figure 3(b) shows the Raman spectra of the A 1 (LO 1 ) and A 1 (LO 2 ) phonons at 148 cm -1 and 177 cm -1 respectively for the five sizes. The temperature dependencies of their frequency are plotted in Fig. 5. The frequencies of the A 1 (LO 2 ) mode present the same size dependent anomalies (jump and plateau) as the ones measured for the E(LO 1 ) mode. Notice that the other phonons, such as the A 1 (LO 2 ) mode, exhibit a classical behavior for each size corresponding to the thermal expansion of the lattice. These measurements show that there is an unusual trend associated with particular sizes. First of all, we note the disappearance of the fingerprint of the cycloidal spin excitations between the sizes of 83 nm and 61 nm. Moreover, the frequency of the magnetic excitations and the frequency of the E(LO 1 ), E(TO 1 ) and A 1 (LO 1 ) phonons present identical anomalies at the same temper-atures related to the surface expansion. In addition, the anomalies are enhanced for 83 nm and disappeared for 31 nm. We remark that the only comparable physical scale to these peculiar sizes is the periodicity of the cycloid (62 nm). The first question that arises is the critical size of the nanoparticles below which the spiral of spins is either strongly modified or suppressed. From an experimental point of view, previous Mössbauer spectroscopy measurements have detected the cycloid state for a size of 54 nm 33 . The modification of the cycloid can come from the uncompensated spins at the surface 23 and the magnetic surface anisotropies. However magnetic surface anisotropies can dominate magnetic properties only in very small nanoparticles 11 and the cycloid should adapt itself. The disappearance of the Raman signature of spin excitations below 83 nm in this work pleads in favor of the suppression of cycloid between 83 and 61 nm. This possibility cannot be totally ruled out. One for the vanishing of the Raman peaks for 61 and 31 nm might be inhomogeneous broadening. Previous theoretiwork from Ref. 11 shows that the cycloid wavevector highly sensitive to size fluctuations for these small sizes. Inhomogeneous broadening is expected to be large, and it would greatly reduce the sensitivity to Raman. From this work, inhomogeneous broadening of magnon peaks might prevent the detection of Raman peaks associated to the cycloid. Other techniques such as Mössbauer and neutron scattering must be used to answer this question definitively.

Our measurements show the impact of the surface expansion on magnetic and lattice modes. The link between both types of excitations is explained by their coupling through magnetic interactions 17,18 and magnetostriction 29,30 . Notice that the surface expansion is only related to lattice properties. Indeed, it should have affected the 31 nm more than all the others sizes because the surface-to-volume ratio is the largest for the 31 nm particle. It has been shown that the ferromagnetic moment and the magneto-electric coupling are enhanced for a size of BiFeO 3 nanoparticles approaching the periodicity of the cycloid. Around this size, the rotation angle of the FeO 6 octahedra (Fig. 1(a)) increases due to lattice strain. We recall that this rotation angle determines the local ferromagnetic moment through Dzialoshinskii-Moriya interaction. Since this local ferromagnetic moment is no longer cancelled out by the cycloid at the surface, it leads to an anomalous increase of the magnetization at the size of 61 nm 25 . The behavior of the frequency of phonons and magnons could be a signature of the interplay between this additional ferromagnetic moment and the surface.

Such a coupling between surface, lattice and magnetic properties in nanoparticles makes BiFeO 3 a promising compound for applications wherein the functionalization of surfaces can improve performance, for example. It would then be possible to envisage a multiferroic nanoparticle with a very strong magneto-electric effect where the electric and magnetic properties, not only of the nanoparticle itself, but also of any material deposited onto its surface could be simultaneously tuned.

In conclusion, we have investigated both the magnetic and structural modes of BiFeO 3 as a function of particle size. We have found compelling evidence for strong frequency anomalies of phonons and magnons at a size of 83 nm associated with an expansion of the surface. This work reveals a cross-correlation between the lattice, the magnetism, and the surfaces in BiFeO 3 nanoparticles at a size close to the cycloid periodicity.

Work on this class of materials in SSW's laboratory is supported by the U.S. National Science Foundation under Grant No. CHE-1807640.

Figure 1 (

 1 b) shows a transmission electron microscopy (TEM) image of nanoparticles with a mean size of 83 nm. The nanoparticles are regularly shaped and approximately spherical.

Figure 1 (

 1 c) presents a magnification of the region defined in Fig. 1(b). The Fourier transform shows the presence of sharp diffraction spots which are indicative of the formation of single crystalline BiFeO 3 . Platelets of nanoparticles

FIG. 1 .

 1 FIG. 1. (a) Schematic of the crystal structure of BFO and the ferroelectric polarization (arrow) (b) TEM image of nanoparticles of about 83 nm. (c) Magnification of the marked region in (a) with the associated Fourier transform.

  (a) correspond to the superposition of the narrow peaks of the bulk (no Raman selection rules in randomly oriented

  FIG. 2. (a) Magnon part of the Raman spectra for the five sizes of nanoparticles. Peaks have been fitted by Lorentzian functions (dashed peaks) (b) Temperature dependence of the magnon frequency with the higher intensity normalised by the value at 10 K for a nanoparticle sizes of 250 nm, 158 nm and 83 nm. Dashed line corresponds to the behavior in the bulk. Vertical dashed lines represent the range where the surface expansion is expected.

FIG. 3 .

 3 FIG.3. Raman spectra at 10 K for the five sizes of nanoparticles of the (a) E(TO1) and E(LO1) phonons and the (b) A1(LO1) and A1(LO2) phonons.

FIG. 4 .

 4 FIG. 4. Temperature dependence of the frequency of the E(LO1) phonon mode (a) for a size of 250 nm and 31 nm and (b) for a size of 158 nm, 83 nm and 61 nm. Temperature dependence of the frequency of the E(TO1) phonon mode (c) for a size of 250 nm and (d) for a size of 158 nm and 83 nm. Dashed lines correspond to the behavior in the bulk. Vertical dashed lines represent the range where the surface expansion is expected. The error bars are visible for some points because they exceed the size of the symbols.

FIG. 5 .

 5 FIG. 5. Temperature dependence of the frequency of the A1(LO1) and A1(LO2) phonons mode (a) for a size of 250 nm and 31 nm and (b) for a size of 158 nm, 83 nm and 61 nm. Dashed lines correspond to the behavior in the bulk. Vertical dashed lines represent the range where the surface expansion is expected.