A Michau

F Maury

F Schuster

R Boichot

M Pons

E Monsifrot
email: eric.monsifrot@dephis.com

Chromium carbide growth at low temperature by a highly efficient DLI-MOCVD process in effluent recycling mode

Keywords:

MOCVD process Bis(arene)chromium

The effect of direct recycling of effluents on the quality of Cr x C y coatings grown by MOCVD using direct liquid injection (DLI) of bis(ethylbenzene)chromium(0) in toluene was investigated. The results are compared with those obtained using non-recycled solutions of precursor. Both types of coatings exhibit the same features. They are amorphous in the temperature range 673-823 K. They exhibit a dense and glassy-like microstructure and a high hardness (> 23 GPa). Analyses at the nanoscale revealed a nanocomposite microstructure consisting of free-C domains embedded in an amorphous Cr 7 C 3 matrix characterized by strong interfaces and leading to an overall composition slightly higher than Cr 7 C 3 . The stiffness and strength of these interfaces are mainly due to at least two types of chemical bonds between Cr atoms and free-C: (i) Cr intercalation between graphene sheets and (ii) hexahapto η 6 -Cr bonding on the external graphene sheets of the free-C domains. The density of these interactions was found increasing by decreasing the concentration of the injected solution, as this occurred using a recycled solution. As a result, "recycled" coatings exhibit a higher nanohardness (29 GPa) than "new" coatings (23 GPa). This work demonstrates that using bis(arene)M(0) precursors, direct recycling of effluents is an efficient route to improve the conversion yield of DLI-MOCVD process making it cost-effective and competitive to produce protective carbide coatings of transition metals which share the same metal zero chemistry.

Introduction

For a better control of production cost of manufactured objects that comprise CVD coatings, the economic performance of deposition processes is an important need. The increasing use of metalorganic precursors is a way to reduce the cost of large-scale CVD process because this greatly lowers the deposition temperatures leading to substantial energy savings. This is evidenced for instance by the growth of metallic Cr at 673 K by DLI-MOCVD [START_REF] Michau | Evidence for a Cr metastable phase as a tracer in DLI-MOCVD chromium hard coatings usable in high temperature environment[END_REF] in comparison with the industrial chromizing method of pack cementation which operates at about 1273 K.

A way to reduce the cost of CVD products is to repair the coating or to recycle the substrate. Indeed, though the coating and the substrate generally form strong and inseparable pairs there are examples where the substrate can be separated and recycled in CVD process to reduce the production cost. For instance in diamond coated cutting tools the worn coating was removed to apply a new one by the same CVD process [START_REF] Liu | Recycling technique for CVD diamond coated cutting tools[END_REF] and, in the graphene CVD synthesis the Cu substrate used as catalyst was recycled after delamination because it is an expensive substrate [START_REF] Wang | Electrochemical delamination of CVD-grown graphene film: toward the recyclable use of copper catalyst[END_REF].

Another way to improve economic performance of CVD is to implement the recycling of effluents. Recycling in CVD processes is only mentioned in a basic book on the technique although it is important for applications [START_REF] Rees | Introduction[END_REF]. When expensive molecular precursors are used, as for deposition of precious metals, the by-products are collected at the exit of the CVD reactor then leading recyclers and traders develop in parallel complex effluent treatments either to refine and reuse the collected precursor or to transform by-products and reuse pure metal [START_REF]Recycling International, Japanese recycling process for ruthenium precursors[END_REF]. This approach is also applied in high volume CVD production facilities. For instance a hydrogen recycle system was proposed recently for CVD of poly-Si [START_REF] Revankar | CVD-Siemens reactor process hydrogen recycle system[END_REF]; in this case it is the carrier gas which is recycled. Also in the growth of Si for solar cells the exhaust gases (H 2 , HCl, chlorosilanes) were collected, separated and recycled [START_REF]Poly plant project, off-gas recovery & recycling[END_REF]. Generally these strategies reduce the production cost but they did not act directly on the CVD process itself since the precursor is not directly recycled in a loop.

One of the advantages of CVD processes is the deposition of uniform position temperature, high quality of the coatings…. Direct recycling of effluent using metalorganic precursors was not reported because the growth occurs at lower temperature than in hydride and halide chemistry and, in this condition, the quality of the layer strongly depends on the metal source which motivates many studies on molecular precursors [START_REF] Rees | Introduction[END_REF][START_REF] Maury | Selection of metalorganic precursors for MOCVD of metallurgical coatings: application to Cr-based coatings[END_REF][START_REF] Jones | CVD of Compound Semiconductors[END_REF][START_REF] Kodas | The Chemistry of Metal CVD[END_REF]. Furthermore, these compounds generally undergo complex decomposition mechanisms producing many unstable metal-containing by-products. Kinetics plays a major role and the growth occurs far from thermodynamic equilibrium. Examples of complexity of decomposition pathways of Cr precursors are reported in [START_REF] Rees | Introduction[END_REF][START_REF] Kodas | The Chemistry of Metal CVD[END_REF][START_REF] Maury | Evaluation of tetra-alkylchromium precursors for OMCVD: Ifilms grown using Cr[CH 2 C(CH 3 ) 3 ] 4[END_REF]. The bis(arene)M(0) precursors, where M is a transition metal in the oxidation state zero of the columns 5 and 6 are an important family of CVD precursors for low temperature deposition of carbides, nitrides and even metal coatings. This is supported by several works using these precursors on carbides of V [START_REF] Abisset | Low temperature MOCVD of V-C-N coatings using bis(arene) vanadium as precursors[END_REF], Nb [17], Ta [17], Cr [START_REF] Anantha | Chromium deposition from dicumene-chromium to form metal-semiconductor devices[END_REF][START_REF] Maury | Structural characterization of chromium carbide coatings deposited at low temperature by LPCVD process using dicumene chromium[END_REF][START_REF] Schuster | Influence of organochromium precursor chemistry on the microstructure of MOCVD chromium carbide coatings[END_REF][START_REF] Polikarpov | Chromium films obtained by pyrolysis of chromium bisarene complexes in the presence of chlorinated hydrocarbons[END_REF], Mo [START_REF] Whaley | Carbonaceous solid bodies and processes for their manufacture[END_REF] and W [START_REF] Whaley | Carbonaceous solid bodies and processes for their manufacture[END_REF], nitrides of V [START_REF] Abisset | Low temperature MOCVD of V-C-N coatings using bis(arene) vanadium as precursors[END_REF] and Cr [START_REF] Schuster | Characterization of chromium nitride and carbonitride coatings deposited at low temperature by OMCVD[END_REF] and metal V [START_REF] Abisset | Low temperature MOCVD of V-C-N coatings using bis(arene) vanadium as precursors[END_REF] and Cr [START_REF] Maury | Low temperature MOCVD routes to chromium metal thin films using bis(benzene)chromium[END_REF][START_REF] Luzin | Chromium films produced by pyrolysis of its bis-arene complexes in the presence of sulfur-containing additives[END_REF], as well as nanostructured multilayer Cr-based coatings [START_REF] Maury | Multilayer chromium based coatings grown by atmospheric pressure direct liquid injection CVD[END_REF].

Chromium carbides are of great interest as tribological coatings for the protection of steel and metallic alloy components owing to their good resistance to corrosion and wear and their high hardness and melting point. They are used in many fields such as transports (automobile, shipping, aeronautic), mechanical and chemical industries and tools [START_REF] Drozda | Tool and manufacturing engineers handbook[END_REF][START_REF] Bryskin | Innovative processing technology of chromium carbide coating to apprise performance of piston rings[END_REF].

Our greater knowledge of the growth mechanisms of Cr-based coatings [START_REF] Maury | Low temperature MOCVD routes to chromium metal thin films using bis(benzene)chromium[END_REF][START_REF] Vahlas | A thermodynamic approach to the chemical vapor deposition of chromium and of chromium carbides starting from Cr(C 6 H 6 ) 2[END_REF], thermodynamic modeling without [START_REF] Vahlas | A thermodynamic approach to the chemical vapor deposition of chromium and of chromium carbides starting from Cr(C 6 H 6 ) 2[END_REF] and with direct liquid injection (DLI) to feed the reactor [START_REF] Douard | Thermodynamic simulation of Atmospheric DLI-CVD processes for the growth of chromium based hard coatings using bis(benzene) chromium as molecular source[END_REF], the determination of a kinetic model and the simulation of the CVD process [START_REF] Michau | Chromium Carbide Growth by Direct Liquid Injection Chemical Vapor Deposition in Long and Narrow Tubes, Experiments, Modeling and Simulation[END_REF] led us to study the effect of direct recycling of effluents on the quality of chromium carbide (Cr x C y ) coatings grown by DLICVD using bis(ethylbenzene)chromium(0) as representative of this family. The results are compared with those obtained using a non-recycled solution of precursor. Both types of coatings exhibit the same features (composition, structure, hardness), demonstrating that using this specific chemical system, direct recycling of effluents is an efficient route to improve the conversion yield of DLI-MOCVD process making it very competitive to develop industrial applications. The barely significant difference of hardness is commented and selection criteria for molecular precursors are also discussed so that they can be implemented in CVD processes with recycling of effluent.

Experimental

Deposition process

The growth was carried out at low temperature by direct liquid injection of metalorganic precursors in a CVD reactor (namely DLI-MOCVD process). It is a horizontal, hot-wall, Pyrex tubular reactor (300 mm long and 24 mm in internal diameter) with an isothermal zone around 150 mm. Stainless steel (304 L) plates and Si(100) wafers passivated by an amorphous SiN x thin layer acting as a barrier were used as substrates. They were placed on a planar horizontal sample-holder in the isothermal zone. More details are reported elsewhere [START_REF] Michau | Chromium Carbide Growth by Direct Liquid Injection Chemical Vapor Deposition in Long and Narrow Tubes, Experiments, Modeling and Simulation[END_REF]. The total pressure was automatically monitored and kept constant at 6.7 kPa and deposition temperature was set at 723 K.

Commercial bis(ethylbenzene)chromium (BEBC) from Strem (CAS 12212-68-9) was used as chromium precursor. It is in fact a viscous liquid mixture of several bis(arene)chromium compounds with the general formula [(C 2 H 5 ) x C 6 H 6-x ] 2 Cr where x = 0-4 and BEBC is the major constituent. A solution in anhydrous toluene (99.8%) from Sigma-Aldrich (CAS 108-88-3) was prepared under inert atmosphere with a concentration of 3 × 10 -1 mol•L -1 (4 g of BEBC in 50 mL of toluene). This precursor solution was injected in a flash vaporization chamber heated at 473 K using a Kemstream pulsed injector device. A liquid flow rate of 1 mL•min -1 was set by adjusting the injection parameters in the ranges: frequency 1-10 Hz and opening time 0.5-5 ms. Nitrogen was used as carrier gas with a 500 sccm mass flow rate and was heated at approximately 453 K before entering the flash vaporization chamber to prevent condensation.

In this paper, "new" coatings refer to coatings elaborated using a freshly prepared liquid solution of as-received precursor and solvent, while "recycled" coatings concern coatings deposited using directly a recycled liquid solution of precursor, by-products and solvent. The same experimental parameters were used for new and recycled coatings: temperatures, pressure, injection parameters, carrier gas flow rate (the deposition time was about 1 h to inject about 160 mL of solution). The only difference was that the precursor concentration of the recycled solution was significantly lower due to consumption during previous runs. As a result, the growth rate in recycling mode was significantly lower. No attempt was made to change the deposition parameters in order to compare the characteristics of the coatings under identical growth conditions. The main CVD conditions are reported in Table 1.

The recycling mode investigated at this stage was based on an openloop. This means gaseous by-products going out of the CVD reactor were forced to pass through a liquid nitrogen trap. Thus undecomposed molecules of BEBC and solvent were condensed in a tank with the reactions by-products (except light hydrocarbons as C 2 species whose trapping is not effective because of their high volatility). After returning to room temperature, a homogenous "daughter" liquid solution was obtained and stored in a pressurized tank under argon before further use. Several CVD runs were required to recover a sufficient amount of "daughter" solution for a recycled run. For example, we succeeded in obtaining a 1.5 μm thick coating with a recycled solution originating from two "new" deposition experiments which had each produced a 5 μm thick coating.

Each trapped solution could also be analyzed, for instance by UV coatings on 3D components with a high conformal coverage. This is achieved when the process operates in the chemical kinetic regime, i.e. at low pressure and low temperature. However, under these particular conditions the conversion efficiency of reactants is low (typically < 30%). Consequently to develop large-scale CVD processes using expensive reactants recycling of precursor become necessary to achieve a high conversion yield. For instance in the CVD production of boron fibers the selective condensation of unconverted BCl 3 is reused directly in the growth process [START_REF] Rees | Introduction[END_REF]. Also the gas mixture CH 4 /H 2 recycling for diamond growth was reported [START_REF] Lu | Economical deposition of a large area of high quality diamond film by a high power DC arc plasma jet operating in a gas recycling mode[END_REF] and a closed gas recycling CVD process has been proposed for solar grade Si [START_REF] Noda | Closed recycle CVD process for mass production of SOG-Si from MG-Si[END_REF]. Furthermore it was shown that a recycle loop is very useful for the management of the axial coating thickness uniformity of poly-Si in a horizontal low pressure CVD reactor [START_REF] Collingham | Effect of recycling on the axial distribution of coating thickness in a low pressure CVD reactor[END_REF], in agreement with the fact that the regime of the reactor is close to a Continuous Stirred Tank Reactor as previously demonstrated [START_REF] Jensen | Modeling and analysis of low pressure CVD reactors[END_REF]. In these few examples the precursor is a hydride or a halide. Metalorganic precursors become very important CVD sources thanks to the diversity of their molecular structures which allows controlling their chemical, physical and thermal properties. This allows satisfying the stringent requirements for the CVD process, e.g. low de-for new and recycled coatings, typically rms = 18 ± 2 nm. Both types of coatings exhibit a very good conformal coverage on substrates with high surface roughness and on non-planar surfaces (edges, trenches…; not shown here). This is a great advantage of low pressure DLI-MOCVD which combines both a high diffusivity of gaseous reactants and decomposition at low temperature leading to a growth in the reactioncontrolled regime. The only difference at this stage is the growth rate: recycled coatings were deposited with a lower growth rate than new coatings because of the lower precursor concentration of the recycled solution. Of course, this can be adjusted later.

Structure

A typical XRD pattern of a new coating 3.4 μm thick grown at 723 K on Si substrate is presented and compared to a recycled coating in Fig. 2. In both cases, there is no evidence of diffraction peaks of polycrystalline phases. The pattern of the new coating is characteristic of an amorphous material with 4 weak and broad bumps, around 2θ = 13.8°, 28.6°, 42.5°and 79.0°(better seen in the inset zoom). No difference was found for the recycled coating, except that due to its lower thickness, a broad peak originating from the substrate (amorphous SiN x barrier layer) is observed at about 69°and the small bump at 79°is not as well marked.

The last two bumps from the coating at 42.5°and 79.0°corresponds to amorphous chromium carbides such as Cr 7 C 3 (JCPDS 00-036-1482) and Cr 3 C 2 (JCPDS 35-804) which both exhibits their main diffraction peaks in these two angular ranges. The FWHM of the most intense bump at 42.5°gives an average size of coherent domains close to 1 nm using Scherrer's formula, confirming the amorphous character of this carbide phase.

In carbon-containing materials the presence of graphite crystallites spectrophotometry to determine its precursor concentration. Indeed, BEBC exhibits a characteristic absorption band around 315 nm that could be used to measure the concentration according to the Beer-Lambert law (Supplementary material, Fig. S1). Also, as an improvement of the process, a closed-loop recycling system could also be installed and automated (currently in progress).

Coating characterization

The surface morphology and cross sections of coatings were characterized by scanning electron microscopy (SEM; Leo-435VP), and by electron probe micro-analysis (EPMA; Cameca SXFive) for the chemical composition. The crystalline structure was investigated at room temperature and ambient atmosphere by X-ray diffraction (XRD) in 2θ range [8-105°] using a Bruker D8-2 diffractometer equipped with a graphite monochromator (Bragg-Brentano configuration; Cu K α radiation). The microstructure of the coatings was also studied by transmission electron microscopy (TEM; Jeol JEM 2100 equipped with a 200 kV FEG, and a Bruker AXS Quantax EDS analyzer). For TEM observations, the samples were cut and thinned perpendicular to the surface by mechanical polishing, then prepared by a dimpling technique and thinned using a precision ion polishing system (PIPS, Gatan). By this method, the electron transparent area that can be observed is a cross section including the coating and the interface with the substrate.

The chemical environment of each element of the coating was investigated by X-ray photoelectron spectroscopy (XPS; Thermo Scientific K-Alpha) equipped with a monochromatic Al X-ray source and a low energy Ar + gun (1 keV) for surface cleaning and depth profile analysis. Raman spectroscopy (Horiba Jobin Yvon LabRAM HR 800 with a 532 nm laser) was also used to analyze chemical bonding, in particular CeC bonds.

Hardness and Young's modulus were determined by nanoindentation using a Nano Scratch Tester (CSM Instruments). The loading/unloading cycle was respectively from 0 to 30 mN at 60 mN•min -1 , a pause of 30 s, then from 30 to 0 mN at 60 mN•min -1 . With this cycle the indenter penetration depth was lower than 1/10 of the coating thickness. For the thickest coatings, Vickers hardness was also measured using a BUEHLER OmniMet 2100 tester.

Results

General appearance and morphology

New and recycled coatings grown in the conditions reported in Table 1 exhibit the same glassy and dense microstructure, typical of an amorphous material. Interestingly, no grain boundary is observed by SEM, even at higher magnification than in Fig. 1, and it will be confirmed by TEM analysis in the Section 3.3. They have a metallic glossy appearance and a mirror-like surface morphology. Surface roughness of Cr x C y coatings on Si substrates measured by AFM gave similar values is evidenced by the diffraction of the (002) plane expected at 2θ = 26.6°in well crystallized graphite. However, disorder in hexagonal graphite structure (e.g. inside and between basal planes, stacking order between the graphene sheets, slipping out of alignment, folding…) leads to broadening and shifting of this peak. For instance pyrolytic carbon can exhibit a more or less disordered turbostratic or graphitic structure. Consequently the bump at 28.6°is assigned to pyrolytic carbon nanoparticles, namely free-C.

The first bump at 2θ = 13.8°is not related to amorphous chromium carbides or to free-C. At this small angle (large interplanar spacing), this could be a compound with a lamellar structure derived from graphite, such as graphite oxide (GO). Indeed it has been reported that GO (001) plane diffracted from 2θ = 2 to 12°depending on the presence of oxygen-containing groups [START_REF] Blanton | Characterization of X-ray irradiated graphene oxide coatings using X-ray diffraction, X-ray photoelectron spectroscopy, and atomic force microscopy[END_REF]. However, the oxygen content of our coatings is lower than 5 at.% (Table 2), which is too low to support this hypothesis. Therefore the first bump at 13.8°was assigned to another derivative from graphite: the intercalation of Cr atoms between two graphene sheets as in graphite intercalation compound (GIC). Recently there are several reports dealing with the interactions between Cr and different forms of carbon, including graphene, nanotubes and fullerenes. For instance, a structural feature of the functionalization of graphene surface is the grafting of Cr which recreates locally the same type of bonding than in bis(arene)chromium, i.e. with an η 6 -bonding to the aromatic cycles [START_REF] Bui | Graphene-Cr-Graphene intercalation nanostructures: stability and magnetic properties from density functional theory investigations[END_REF][START_REF] Sarkar | Organometallic chemistry of[END_REF]. On the other hand, in sandwich graphene-Cr-graphene nanostructures, the representative distance of an ordered staking is twice the spacing between two consecutive graphene sheets, i.e. 6.556 Å, because Cr cannot be intercalated between two consecutive interlayer spaces as described in [START_REF] Bui | Graphene-Cr-Graphene intercalation nanostructures: stability and magnetic properties from density functional theory investigations[END_REF][START_REF] Sarkar | Organometallic chemistry of[END_REF]. This corresponds to a diffraction angle 2θ = 13.5°considering the (001) plane, which is very close to our 13.8°experimental value (Fig. 2).

Microstructure

From approximately the same magnification as in SEM images (Fig. 1) and to larger values up to high resolution, TEM images showed the same glassy microstructure for Cr x C y coatings, as it can be seen on Fig. 3a. Again no significant difference was found for recycled coatings. A high resolution TEM analysis (Supplementary material, Fig. S2) has revealed a dense and very finely granular structure with homogeneous and monodisperse distribution of contrasted domains. The average size of these domains is of the order of magnitude of 1 nm, in good agreement with the value found by XRD.

The selected area electron diffraction pattern of the micrograph shown in Fig. 3a revealed two diffuse rings (Fig. 3b) as for high resolution TEM analysis in Fig. S2. They are centered on interplanar distances 2.097 Å and 1.223 Å. In accordance with the Bragg relation, they correspond to theoretical XRD diffraction angles at 2θ = 43.1°and 78.1°. Therefore, the inner and outer rings on TEM diffraction patterns correspond to the first and the second Cr x C y bumps on XRD pattern found at 2θ = 43.5°and 79.0°, respectively (Fig. 2). This is supported by the fact that both crystalline Cr 7 C 3 and Cr 3 C 2 phases show their strongest XRD contributions in the 2θ range 39-44°and they also exhibit a second bunch of peaks with a lower intensity around 2θ = 80°. The two bumps on XRD pattern at 2θ = 13.8°and 28.6°assigned to GIC and free-C respectively, were not seen on TEM diffraction pattern likely because they were too weak and diffuse for this technique.

Chemical composition

Atomic composition determined by EPMA of new and recycled coatings is reported in Table 2. No significant difference is observed between both coatings; it is typically Cr 064 C 0.33 O 0.03 . The level of oxygen contamination is slightly higher in recycled coatings but it does not exceed about 5 at.%. This was attributed to the handling of the recycled solution that was stored in a pressurized tank. Although handled under Ar atmosphere this container had to be opened and closed several times in order to recover enough solution after several deposition experiments for a further use in a recycling CVD run. By neglecting traces of oxygen, the total carbon content of these carbide coatings (C:Cr = 0.50 ± 0.02) is intermediate between Cr 7 C 3 (C:Cr = 0.43) and Cr 3 C 2 (C:Cr = 0.67) but it is closer to Cr 7 C 3 . The overall atomic composition is consistent with a nanocomposite structure consisting of an amorphous carbide matrix a-Cr x C y and free-C, as supposed from XRD and TEM results, and the ratio y:x in the matrix is lower than 0.50, i.e. even closer to the Cr 7 C 3 stoichiometry.

XPS analyses did not reveal significant difference between both types of coatings, except a higher contribution of oxygen bonded to Cr for recycled coatings, in agreement with EPMA data. As-deposited samples exhibit Cr 2p 3/2 peaks characteristic of Cr(III) oxide (575.8 eV) and CreOH (576.8 eV), O 1s peaks were assigned to Cr 2 O 3 (530.8 eV) and CeO/OH (532.0 eV), and C 1s core level was characteristic to adventitious carbon with the components CeC/CeH at 284.8 eV and OeC]O at 288.0 eV. A depth profile analysis of C 1s showed that Ar + ion etching of the sample for about a minute at 1 keV removes readily the surface contamination without significant secondary effect of sputtering (Fig. 4a). The C 1s region of as-deposited sample shows the main features of adventitious carbon with the CeC/CeH and OeC]O components (Fig. 4b). After removal of the contamination layer by ion etching for 220 s the C 1s peak reveals two forms of carbon present in the coating: the carbide (282.8 eV) and the free-C (~284 eV) as The coatings being of metallic nature, Raman spectra should not be a priori readily informative. Raman signal originates only from surface oxides and carbon components; no response was expected from a metallic matrix. Fig. 5 compares the Raman spectra of a new and a recycled coating for the 200 to 1800 cm -1 spectral range. Overall, the large width of the bands reveals the absence of long-range order, either because of the amorphous character of the phases or because of defects. At first glance the spectra appear very different but they are both constituted of two zones with different intensities. The bands in the first zone (200-1000 cm -1 ) are essentially due to chromium oxides on the surface of the sample [START_REF] Iliev | Raman spectroscopy of ferromagnetic CrO 2[END_REF][START_REF] Yu | Phase control of chromium oxide in selective microregions by laser annealing[END_REF][START_REF] Barshilia | Structure and optical properties of pulsed sputter deposited Cr x O y /Cr/Cr 2 O 3 solar selective coatings[END_REF] while the bands in the second zone (1000-1800 cm -1 ) are characteristic of carbon in different environments [START_REF] Ferrari | Raman spectroscopy of amorphous, nanostructured, diamond-like carbon, and nanodiamond[END_REF]. In comparison with the new coating (Fig. 5a), the higher intensity of the bands in the first zone for the recycled coating is consistent with a substantially greater oxidation (Fig. 5b), in good agreement with EPMA and XPS analyses. Spectra deconvolution on Fig. 5c and d reveals the D band at 1340 cm -1 and the G band at 1570 cm -1 which are representative of CeC bonds [START_REF] Ferrari | Raman spectroscopy of amorphous, nanostructured, diamond-like carbon, and nanodiamond[END_REF]. G stands for graphite and D for disorder in graphitic structures. The bands at 1225 and 1455 cm -1 were assigned to transpolyacetylene, a strong polymeric chain e(C 2 H 2 ) n e where carbon adopts sp 2 configuration [START_REF] Ferrari | Raman spectroscopy of amorphous, nanostructured, diamond-like carbon, and nanodiamond[END_REF]. Because of overlaps with the bands of trans-polyacetylene, the presence of C sp 3 cannot be ruled out since it is expected at 1180 and 1332 cm -1 in nanocrystalline and cubic diamond respectively, and at 1500 cm -1 in DLC (disordered sp 3 hybridization) [START_REF] Chu | Characterization of amorphous and nanocrystalline carbon films[END_REF].

In carbon materials, correlations were established between the development of the disorder from sp 2 structural model (graphite) to sp 3 (diamond) and the variation of the intensity ratio I(D)/I(G) as well as the FWHM and the position of the G band [START_REF] Ferrari | Raman spectroscopy of amorphous, nanostructured, diamond-like carbon, and nanodiamond[END_REF][START_REF] Chu | Characterization of amorphous and nanocrystalline carbon films[END_REF][START_REF] Cançado | Quantifying defects in graphene via Raman spectroscopy at different excitation energies[END_REF]. Fig. 5 shows that the intensity ratio I(D)/I(G) significantly increases from new coatings (~0.6) to recycled coatings (~1.2), suggesting that the disorder within graphitic nanostructures is higher for the recycled sample. The fact that the relative intensity of the bands at 1220 and 1460 cm -1 assigned to trans-polyacetylene and possibly to C sp 3 does not change from a new coating to a recycled one suggests that the evolution of the disorder cannot be interpreted in terms of C sp 3 proportion. As a result, it is more appropriate to consider interactions between Cr and free-C as structural defects (both in-plane and between graphene sheets) that cause increasing disorder when their number increases.

The average size of graphitic nanoparticles in the basal plane determined from the FWHM of the G band, namely L a [START_REF] Ferrari | Raman spectroscopy of amorphous, nanostructured, diamond-like carbon, and nanodiamond[END_REF], remains constant at around 35 nm for both new and recycled coatings. On the other hand, a disorder measurement L D can be determined from the intensity ratio I(D)/I(G); it represents the average distance between two point defects in graphene planes [START_REF] Cançado | Quantifying defects in graphene via Raman spectroscopy at different excitation energies[END_REF]. As suggested above, Cr grafting on graphene sheet can be considered as a defect which induces disorder. Thus, L D can be considered as the average distance between two grafted Cr and it was found decreasing from 15.5 to 6.2 nm for new and recycled coatings respectively. This means that despite an identical overall atomic composition, recycled coatings exhibit a higher defect density, as interactions at the free-C/a-Cr 7 C 3 interfaces, e.g. both as hexahapto-η 6 -Cr grafting on external graphene sheets and Cr intercalation between graphene layers.

Hardness

For the thickest Cr x C y new coatings (~5 μm), Vickers hardness around 2300 HV was measured which is quite high for CVD chromium carbide coatings. Values in the range 700-1600 HV were previously reported for polycrystalline MOCVD chromium carbides coatings [START_REF] Aleksandrov | Vapor-phase deposition of coatings from bis-arene chromium compounds on aluminum alloys[END_REF][START_REF] Yurshev | Surface hardening of tools by depositing a pyrolytic chromium carbide coating[END_REF] while electrodeposited Cr x C y coatings did not exceed 1300 HV 100 [START_REF] Zeng | Tribological and electrochemical behavior of thick Cr-C alloy coatings electrodeposited in trivalent chromium bath as an alternative to conventional Cr coatings[END_REF][START_REF] Protsenko | Improving hardness and tribological characteristics of nanocrystalline Cr-C films obtained from Cr(III) plating bath using pulsed electrodeposition[END_REF] and electroplating hard Cr is lower than 1200 HV [START_REF] Lausmann | Electrolytically deposited hardchrome[END_REF][START_REF] Liang | Structure characterization and tribological properties of thick chromium coating electrodeposited from a Cr(III) electrolyte[END_REF]. Only PVD processes manage to reach even higher hardness, from 2000 to 3000 HV [START_REF] Aubert | Hard chrome and molybdenum coatings produced by physical vapour deposition[END_REF][START_REF] Cholvy | Characterization and wear resistance of coatings in the Cr-C-N ternary system deposited by physical vapour deposition[END_REF][START_REF] Wang | Synthesis of Cr 3 C 2 coatings for tribological applications[END_REF].

For the thinnest Cr x C y coatings (≤3.5 μm), hardness was determined by nanoindentation on five samples corresponding to three new and two recycled coatings. The values of hardness (H) and Young's modulus (E) reported in Table 3 are While there is no difference on the Young's modulus between new and recycled coatings (285 and 295 GPa, respectively), the nanoindentation hardness of recycled coatings could be considered slightly higher (29 GPa) than that of new coatings (23 GPa), despite the standard deviations. This will be discussed in the next section. The ratio H 3 /E 2 is often referred as a durability criteria; it is proportional to the contact loads needed to induce plasticity and consequently it characterizes the resistance to induce plastic deformation [START_REF] Tsui | Nanoindentation and nanoscratching of hard carbon coatings for magnetic disks[END_REF][START_REF] Musil | Hard and superhard Zr-Ni-N nanocomposite films[END_REF]. Comparison of both types of coatings revealed a better behavior of recycled coatings as a result of their higher hardness, the ratio H 3 /E 2 being increased almost 3-fold.

Discussion

Chromium carbide coatings were successfully deposited using directly recycled solutions in the same DLI-MOCVD conditions as using new bis(ethylbenzene)chromium solutions. The growth rate in effluent recycling mode was found lower (around 0.5-1 μm•h -1 instead of 5-10 μm•h -1 ) essentially because the recycled BEBC solution in toluene was less concentrated due to consumption in previous runs. Chemical and structural characterizations of both types of coatings did not reveal significant differences. The coatings exhibit a smooth surface morphology, a dense and glassy-like microstructure and an amorphous structure (XRD and TEM analyses). The overall atomic composition was found to be Cr 0.64 C 0.33 O 0.03 (Table 2). Interestingly, both coatings exhibit a high conformal coverage on non-planar surfaces at relatively low deposition temperature (723 K).

A high hardness

This section discusses on the one hand the high values of hardness of the coatings whatever the precursor solution injected (new or recycled) and, on the other hand, on the difference of nanohardness between recycled and new coatings, assuming therefore that the difference is significant enough.

Both Vickers hardness (2300 HV) and nanoindentation hardness (23-29 GPa) have revealed high values, at the level of those previously reported for PVD Cr x C y coatings [START_REF] Su | Effect of chromium content on the dry machining performance of magnetron sputtered Cr x C coatings[END_REF][START_REF] Esteve | Cathodic chromium carbide coatings for molding die applications[END_REF][START_REF] Romero | Nanometric chromium nitride/chromium carbide multilayers by R.F. magnetron sputtering[END_REF]. A great advantage of DLI-MOCVD Cr x C y coatings is that they are amorphous, without grain boundaries, while those deposited by other processes are polycrystalline. It is generally reported that crystalline chromium carbides coatings grown by PVD [START_REF] Esteve | Cathodic chromium carbide coatings for molding die applications[END_REF], cathodic arc evaporation [START_REF] Esteve | Cathodic chromium carbide coatings for molding die applications[END_REF] and electrodeposition [START_REF] Zeng | Tribological and electrochemical behavior of thick Cr-C alloy coatings electrodeposited in trivalent chromium bath as an alternative to conventional Cr coatings[END_REF] are harder than amorphous ones. Interestingly the hardness of our amorphous coatings is already at the level of these polycrystalline Cr x C y coatings. Consequently, their amorphous structure cannot explain their high hardness. We are aware that Cr 3 C 2 is the hardest phase of CreC system and therefore its presence, even in the amorphous state, should significantly increase the hardness. However, we have shown that the matrix of our coatings has the stoichiometry a-Cr 7 C 3 . Also it is known that for nanocrystalline Cr x C y coatings, the nanohardness increased by decreasing the average grain size [START_REF] Protsenko | Improving hardness and tribological characteristics of nanocrystalline Cr-C films obtained from Cr(III) plating bath using pulsed electrodeposition[END_REF]. Without evidence for nanocrystalline structure this claim does not hold for our coatings. It was also reported that a high hardness was achieved for high Cr contents [START_REF] Su | Effect of chromium content on the dry machining performance of magnetron sputtered Cr x C coatings[END_REF], or that stoichiometric CreC phases must be privileged, meaning a C excess must be avoided [START_REF] Romero | Nanometric chromium nitride/chromium carbide multilayers by R.F. magnetron sputtering[END_REF]. We will discuss below that in our case a C excess, compared to Cr 7 C 3 stoichiometry, on the contrary plays a key role.

Among the factors that influence the hardness of coatings, residual stresses are probably the most important [START_REF] Nowak | The effect of residual stresses on nanoindentation behavior of thin W-C based coatings[END_REF]. The influence of other factors as coating thickness, growth conditions, micro-and -1.20 and -1.25 GPa were found for the 4.0 and 6.0 μm thick coatings, respectively. Reliable data could not be obtained by this method for recycled coatings because they were too thin. Assuming a rigid substrate, the maximum thermal stress can be calculated according to the equation:

= ′ - ∆ σ E (α α ) T. t f f s
where ∆ T is the variation of temperature and α i the thermal expansion coefficient (α s = 18.3 × 10 -6 K -1 and α f = 10.1 × 10 -6 K -1 ). For ∆ T = 430 K, calculated thermal stresses are -1.32 GPa. These values are generally found for ceramic coatings as TiN on stainless steel substrates [START_REF] Wu | Modified curvature method for residual thermal stress estimation in coatings[END_REF]. These results are confirming the dominant contribution of thermal stresses to residual stresses. The comparison of hardness of new (23 ± 2 GPa) and recycled (29 ± 4 GPa) coatings reveals a small but significant difference, taking into account the standard deviations, which raises the question: why should recycled coatings be harder? This could be discussed in terms of residual stresses but no data are available for recycled coatings. However, the residual stresses are largely dominated by thermal stress which has the same value for both coatings since they were deposited at the same temperature and their thickness was relatively close (3.5 and 1.0 μm, respectively). Consequently this is likely not a major factor to explain the difference of hardness. One of the best ways to comment on this difference in hardness is to focus on the specific nanocomposite structure of these hard coatings since it is known that this influences the hardness.

A specific amorphous nanocomposite structure

The morphology, microstructure and composition of new and recycled coatings are the same. Furthermore, XRD, XPS and Raman analyses gave evidence for an amorphous nanocomposite structure. The only significant differences between both types of coatings were found by Raman spectroscopy (Fig. 5).

Basically, the microstructure is composed of 2 phases with interfaces acting as strong interphases. The dominant phase is an amorphous carbide matrix with the Cr 7 C 3 stoichiometry (namely a-Cr 7 C 3 ). Nanometric free-C domains (L a = 35 nm in-plane correlation length) are embedded in this amorphous carbide matrix. They are related to pyrolytic C, which means they exhibit a disordered graphitic structure (turbostratic stacking) with likely some covalent bonding between graphene sheets via open cycles at the edges (generating C sp 3 sites). Furthermore, some graphene sheets are also connected by trans-polyacetylene chains at the edges of these C domains. The relative amount of free-C does not exceed 20% of the total carbon (XPS and EPMA data). An important finding was to identify signatures by Raman and XRD revealing interactions between Cr and free-C. This particular nanostructure is shown schematically in Fig. 6. Due to their layered structure, the free-C domains exhibit two types of interfaces with the a-Cr 7 C 3 matrix: the one which is parallel to the graphene sheets (parallel interface) and that which is roughly perpendicular to the stacking of graphene planes (perpendicular interface). Cr atoms from the amorphous carbide matrix can be grafted on external graphene sheets as hexahapto η 6 -Cr complexes of graphene [START_REF] Sarkar | Organometallic chemistry of[END_REF]. These specific bonds are very similar to those in the BEBC precursor. They contribute to the strengthening of the parallel interfaces. Also, individual Cr atoms can be intercalated between consecutive graphene sheets as in graphite intercalation compounds as supported by XRD data (Fig. 2) [START_REF] Bui | Graphene-Cr-Graphene intercalation nanostructures: stability and magnetic properties from density functional theory investigations[END_REF][START_REF] Sarkar | Organometallic chemistry of[END_REF]. All these Cr interactions can be considered as point defects in ideal graphene sheets. Despite a graphitic base structure, these interactions and interconnections between free-C and a-Cr 7 C 3 , through C sp 3 , transpolyacetylene and η 6 -Cr bonding rigidify the free-C domains, strengthen the interfaces and consolidate a 3D structural network between the carbide matrix and free-C through strong interphases.

The defect density on the external graphene sheets of free-C nanostructures has been estimated from Raman data as the average distance L D between two point defects in graphene sheet (Fig. 6). Interestingly L D was found to decrease from 15.5 to 6.2 nm for new and recycled coatings, respectively, while the average size of graphene sheet given by in-plane correlation length L a is constant (35 nm). This means the defect density, i.e. the density of interactions between the carbide matrix and free-C is significantly higher for recycled coatings than for the new ones. This trend suggests a correlation with the higher hardness of recycled coatings (29 ± 4 GPa) compared to the new coatings (23 ± 2 GPa). The nanohardness would increase with the density of chemical bonds both within graphene sheets of the free-C nanostructures and between these free-C domains and the amorphous carbide matrix.

Basically the growth mechanism is the same for "new" and "recycled" coatings. A simple chemical mechanism reduced to 4 limiting reactions (1 homogeneous, 3 heterogeneous) was proposed for kinetic nanostructure has been reported but they act both on the stresses and the hardness, and so their effect is difficult to decouple.

For two new coatings 3.5 and 35.0 μm thick the nanohardness and Young modulus were found constant at 23.6 ± 2.0 GPa and approximately 293 GPa, respectively. This means the nanohardness is independent of the thickness for values higher than 3.5 μm. At this stage no data is available for thinner coatings to comment on the possible influence of thicknesses lower than 3.5 μm. It is noteworthy that our experimental value of the Young's modulus is in good agreement with the Cr 7 C 3 theoretical value of 302 GPa [START_REF] Xiao | Mechanical properties and chemical bonding characteristics of Cr 7 C 3 type multicomponent carbides[END_REF].

In coating-substrate systems prepared by CVD, residual stresses (σ r ) originate from the sum of thermal stresses (σ t ) induced by the mismatch of thermal expansion between the coating and the substrate, and intrinsic stresses (σ i ) induced by the growth mechanism. The residual stresses were determined for two "new" coatings (thickness t f = 4.0 and 6.0 μm) deposited on 304 L steel strip 0.5 mm thick (t s ). As the ratio t f /t s (or E f ′ t f /E s ′ t s ) is ≤ 1% [START_REF] Klein | How accurate are Stoney's equation and recent modifications[END_REF], the deformation of the substrate can be negligible. The E i ′ are the biaxial moduli (E i / (1ν i )), t i the thicknesses, ν i the Poisson's ratio and the subscripts s and f respectively denotes the substrate and the film, which leads to E f ′ = 363 GPa and E s ′ = 278 GPa using ν CrC = 0.2 and E CrC = 290 GPa (average data of Cr 7 C 3 and Cr 3 C 2 ). The Stoney's equation is applicable with an error which does not exceed 5%. From the measurement of the change of curvature before and after deposition, compressible residual stresses of reactive, thermally stable in the deposition temperature range, and therefore does not participate in the growth mechanism as found using toluene and cyclohexane [START_REF] Maury | Multilayer chromium based coatings grown by atmospheric pressure direct liquid injection CVD[END_REF][START_REF] Douard | Dépôt de carbures, nitrures et multicouches nanostructurées à base de chrome sous pression atmosphérique par DLI-MOCVD: nouveaux procédés et potentialités de ces revêtements métallurgiques[END_REF].

The solution recovered in the cold trap at the exit of the CVD reactor contains undecomposed BEBC, toluene (solvent) and a mixture of organic by-products originating from the released ligands and the heterogeneous decomposition of a small part of them producing ethylbenzene, diethylbenzene, benzene, ethyltoluene, toluene [START_REF] Travkin | Thermal decomposition of bisarene compounds of chromium[END_REF] as well as lighter and non-aromatic hydrocarbons and hydrogen [START_REF] Maury | Low temperature MOCVD routes to chromium metal thin films using bis(benzene)chromium[END_REF]. The lighter hydrocarbons and hydrogen are not efficiently trapped because of their high volatility. The organic by-products originating from the ligands are of the same family as the solvent. Consequently, the trapped solution that can be directly recycled contains unreacted BEBC and solvents constituted of a mixture of several aromatic hydrocarbons. The major difference with the new solution is that the BEBC concentration in the recycled solution is lower due to its consumption. Finally, direct recycling of all effluents can be implemented using this chemical system in a close-loop to reach a conversion rate of the precursor near 100% (currently in progress).

Conclusions

The impact of the high cost of metalorganic precursors on the economic viability of MOCVD can be overcome by maximizing the conversion yield. It was demonstrated that direct recycling of effluent is possible using appropriate bis(arene)Cr(0) precursors.

Chromium carbide coatings were deposited by DLI-MOCVD using either a new bis(ethylbenzene)chromium solution in toluene or a recycled solution recovered at the exit of the reactor. Chemical and structural characteristics of both types of coatings are very similar. They are amorphous with a composition slightly higher than Cr 7 C 3 . The nanohardness is particularly high with values in the range 23-29 GPa. This high hardness is essentially due to the nanocomposite microstructure, without grain boundary, and strong interphases between free-C domains embedded in an amorphous Cr 7 C 3 matrix. The slightly higher hardness of recycled coatings was assigned to a higher density of chemical bonds both within the C domains (C sp 3 and trans-polyacetylene bridges) and at the interfaces with the amorphous carbide matrix (Cr grafting and intercalation). A gradual filling of prismatic and octahedral C sites of the matrix also likely plays a role in strengthening the interphase.

It is a breakthrough for MOCVD because the process can be extended to metals of columns 5 and 6 for which the same M(0) chemistry can be implemented and the carbides also have many practical applications as protective metallurgical coatings. Recycling in a closed-loop is currently in progress to reach a conversion rate near 100% in a onestep CVD run. modeling and simulation of the process. It is based on site competition reactions [START_REF] Michau | Chromium Carbide Growth by Direct Liquid Injection Chemical Vapor Deposition in Long and Narrow Tubes, Experiments, Modeling and Simulation[END_REF]. The lower concentration of recycled BEBC solutions would induce a lower supersaturation of BEBC near the growing surface. As a result, this would favor a higher mobility of adsorbed chemical species or would influence adsorption competition and finally would facilitate locally the formation of chemical bonds both within the C domains (C sp 3 , trans-polyacetylene bridges) and at the parallel and perpendicular interfaces with the amorphous carbide matrix (Cr grafting and intercalation, respectively). Subsequently the nanostructure of the coating is overall strengthened and its nanohardness is increased.

Another hypothesis about strong interphases instead of sharp and weak interfaces, not supported here by experimental data, is to be aware that in the crystallographic structure of Cr 7 C 3 carbon atoms are in trigonal prisms connected in chains while it was reported that in amorphous Cr 1-x C x for x > 33% carbon progressively filled octahedral interstitial sites as the C content increased, suggesting that C coexisted in both prismatic and octahedral sites [START_REF] Bauer-Grosse | Thermal stability and crystallization studies of amorphous TM-C films[END_REF]. It is reasonable to assume that at the interface a-Cr 7 C 3 /free-C a carbon enrichment of the carbide matrix is possible by the gradual occupation of both prismatic and octahedral sites. For instance in C-rich amorphous Cr x C y grown by PVD, C atoms were located in a mixture of prismatic and octahedral sites with a distribution depending on the total C content [START_REF] Magnuson | Electronic structure and chemical bonding of amorphous chromium carbide thin films[END_REF]. These polyhedral units are characterized by strong covalent Cr 3d-C 2p bonding. Locally, at the a-Cr 7 C 3 /free-C interface, the proportion of C atom filling octahedral sites probably depends on growth conditions. If the growth rate of the a-Cr 7 C 3 matrix is slow enough, for instance because the mole fraction of precursor is low, in a competitive pathway C can diffuse to fill octahedral sites and thus strengthen the interphase.

At this stage it is not reasonable to speculate more on the difference of hardness between "new" and "recycled" coatings because the difference is not so large and it must be confirmed by other experiments. However, it can be retained that both the density of interactions between the carbide matrix and free-C (grafting and intercalation of Cr), supported by experimental data, and the assumed gradual occupation of prismatic and octahedral sites of the carbide by the carbon generate strong interphases which influence the mechanical properties.

Key points making recycling possible: selection of precursor

A barrier in the implementation of MOCVD recycling is that the decomposition of the metalorganic precursor is complex and often produces many by-products which, if recycled, significantly affect the composition and microstructure of the coatings. Consequently a tedious and expensive separation of the by-products is necessary to recover the precursor which has not reacted. The key is therefore to use metalorganic precursors whose decomposition mechanism is very simple, and which do not produce metalorganic by-products that could modify the growth mechanism. This is the case of bis(arene)M(0) compounds where the metal M is in the zero valence state, as in the deposited metal or carbide coatings. This important family of precursors was used for low temperature MOCVD of carbides of V [START_REF] Abisset | Low temperature MOCVD of V-C-N coatings using bis(arene) vanadium as precursors[END_REF], Nb [17], Ta [17], Cr [START_REF] Anantha | Chromium deposition from dicumene-chromium to form metal-semiconductor devices[END_REF][START_REF] Maury | Structural characterization of chromium carbide coatings deposited at low temperature by LPCVD process using dicumene chromium[END_REF][START_REF] Schuster | Influence of organochromium precursor chemistry on the microstructure of MOCVD chromium carbide coatings[END_REF][START_REF] Polikarpov | Chromium films obtained by pyrolysis of chromium bisarene complexes in the presence of chlorinated hydrocarbons[END_REF], Mo [START_REF] Whaley | Carbonaceous solid bodies and processes for their manufacture[END_REF] and W [START_REF] Whaley | Carbonaceous solid bodies and processes for their manufacture[END_REF]. In the deposition process the metal, and in particular Cr, stays in the zero valence state. For instance no hexavalent Cr(VI) compound is formed which entirely satisfies European regulation REACH or related rules. The ligands are stable aromatic molecules; they are readily released by selective bond breaking during the deposition process without undergoing significant pyrolysis [START_REF] Maury | Low temperature MOCVD routes to chromium metal thin films using bis(benzene)chromium[END_REF][START_REF] Travkin | Thermal decomposition of bisarene compounds of chromium[END_REF]. It is then recommended in DLI-MOCVD to use solvent of the same family as the ligands (e.g. toluene for BEBC) to avoid uncontrolled side-reactions.

The main characteristics of the coatings are independent of the nature of the ligands as deduced from the use of Cr(C 6 H 6 ) 2 , Cr (C 6 H 5 i Pr) 2 and Cr(C 6 H 5 Et) 2 [START_REF] Maury | Structural characterization of chromium carbide coatings deposited at low temperature by LPCVD process using dicumene chromium[END_REF][START_REF] Schuster | Influence of organochromium precursor chemistry on the microstructure of MOCVD chromium carbide coatings[END_REF]. As a result, a mixture of different bis(arene)Cr precursors can be used as in [START_REF] Devyatykh | Composition of impurities in bis-ethylbenzene chromium produced according to the Friedel-Crafts method[END_REF][START_REF] Gribov | Super-pure materials from metal-organic compounds[END_REF] and in this work. Also the nature of the solvent is not very important provided that it is non-

Fig. 1 .

 1 Fig. 1. Cross section of Cr x C y coatings grown at 723 K and 6.7 kPa on Si substrates using (a) a new BEBC solution in toluene and (b) a recycled solution. The lower thickness in (b) originates from the lower concentration of the recycled solution.

Fig. 2 .

 2 Fig. 2. Typical XRD pattern of a "new" Cr x C y coating grown by DLI-MOCVD with a new solution of BEBC in toluene (black) compared to that of a "recycled" coating (grey) grown in the same conditions.

Fig. 3 .

 3 Fig. 3. (a) TEM micrograph of a new Cr x C y coating observed in cross section; (b) corresponding selected area electron diffraction showing two diffuse rings of the amorphous carbide phase.

  minority component (Fig.4c). After the surface cleaning by ion etching, the O 1s intensity is significantly decreased and only one component is found at 530.8 eV (CreO). Regarding Cr 2p 3/2 region, the oxygenated components have almost disappeared and the peak is shifted to 574.0 eV as for Cr metal or carbide (CreC bonds). This XPS analysis confirms the presence of free-C and a carbidic form in the coatings as observed by XRD. After in situ surface cleaning the atomic composition of the surface analyzed by XPS is Cr 0.57 C 0.33 O 0.10 that is in good agreement with EPMA data (Table2). From the relative intensity of the two components of C 1s peak of Fig.4cthe proportion of free-C to the total-C is approximately 20%. On the other hand, considering the EPMA composition Cr 0.64 C 0.33 O 0.03 (Table2) as a representative formula and neglecting oxygen content, comparison with the stoichiometric Cr 7 C 3 phase reveals a carbon excess as free-C of 18 at.%. These two values of relative content of free-C determined by XPS and EPMA are in good agreement and confirm the presence of free-C nanostructures identified in XRD. Due to the presence of free-C, it is confirmed that the matrix a-Cr x C y has the composition a-Cr 7 C 3 .

Fig. 4 .

 4 Fig. 4. XPS analysis of a new Cr x C y coating: (a) depth profile of C 1s components, (b) C 1s spectra of as-received sample (0 s ion etching time) and (c) C 1s spectra after 220 s ion etching time.

  the average of at least ten successful indentations per sample. Considering standard deviations of measurements, new and recycled coatings have substantially comparable H and E values. Nanoindentation measurements confirmed Vickers hardness tests. With values in the range 23-29 GPa for coatings on 304 L thicker than 1 μm, both types of coatings exhibit nanoindentation hardness as high as those of Cr x C y coatings deposited by PVD, e.g. 24.2 GPa [50], 22 GPa [51] and 21 GPa [52], as well as MOCVD, 25 GPa [53]. It is noteworthy that our coatings are amorphous, whereas those in the cited references were polycrystalline.

Fig. 5 .

 5 Fig. 5. Raman spectra of Cr x C y coatings grown on a Si substrate with a new BEBC solution (a) (c), and a recycled one (b) (d): spectral range 200-1800 cm -1 (a) (b), and the C region (c) (d). The proposed deconvolution of the C bands is commented in the text.

Fig. 6 .

 6 Fig. 6. Schematic representation of the amorphous and nanocomposite microstructure of Cr x C y coatings deposited by DLI-MOCVD showing the main structural features at the interface between free C nanostructures embedded in an amorphous Cr 7 C 3 matrix (Cr atoms are the red circles). The L a and L D distances shown are discussed in the text. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 1

 1 Experimental DLICVD conditions of new Cr x C y coatings. The growth conditions in recycling mode are the same except that the BEBC solution injected resulting from cryogenic trapping during previous CVD runs had a lower concentration.

	T (K)	P (kPa)	BEBC in toluene (mol/L)	BEBC gas flow rate (sccm)	Toluene gas flow rate (sccm)	N 2 gas flow (sccm)	Frequency (Hz)	Opening time (ms)
	723	6.7	0.3	9	216	500	1-10	0.5-5

Table 2

 2 Atomic composition of coatings grown with a new and recycled solution (EPMA data).

Table 3

 3 Nanoindentation results for new and recycled coatings deposited on 304 L stainless steel substrate.

	Coating	Thickness (μm)	H (GPa)	E (GPa)	H 3 /E 2 (GPa)
	New Cr x C y Recycled Cr x C y	3.5 1.0	23 ± 2 29 ± 4	285 ± 20 295 ± 40	1.4 × 10 -1 3.9 × 10 -1

Acknowledgements

This work was supported by the Centre of Excellence of Multifunctional Architectured Materials "CEMAM" [grant number AN-10-LABX-44-01]. We thank Sofiane Achache and Raphaël Laloo for their help in hardness measurements, Jerome Esvan and Olivier Marsan for their assistance in XPS and Raman spectroscopies.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https:// doi.org/10.1016/j.surfcoat.2017.06.077.


Thomas Chatain

Maurice Comlan

David Delfieu

Loïg Jezequel

Olivier H Roux

Pomsets and Unfolding of Reset Petri Nets

Reset Petri nets are a particular class of Petri nets where transition firings can remove all tokens from a place without checking if this place actually holds tokens or not. In this paper we look at partial order semantics of such nets. In particular, we propose a pomset bisimulation for comparing their concurrent behaviours. Building on this pomset bisimulation we then propose a generalization of the standard finite complete prefixes of unfolding to the class of safe reset Petri nets.

Introduction

Petri nets are a well suited formalism for specifying, modeling, and analyzing systems with conflicts, synchronization and concurrency. Many interesting properties of such systems (reachability, boundedness, liveness, deadlock,. . . ) are decidable for Petri nets. Over time, many extensions of Petri nets have been proposed in order to capture specific, possibly quite complex, behaviors in a more direct manner. These extensions offer more compact representations and/or increase expressive power. One can notice, in particular, a range of extensions adding new kinds of arcs to Petri nets: read arcs and inhibitor arcs [START_REF] Baldan | Contextual Petri nets, asymmetric event structures and processes[END_REF][START_REF] Montanari | Contextual nets[END_REF] (allowing to read variables values without modifying them), and reset arcs [START_REF] Araki | Some decision problems related to the reachability problem for Petri nets[END_REF] (allowing to modify variables values independently of their previous value). Reset arcs increase the expressiveness of Petri nets, but they compromise analysis techniques. For example, boundedness [START_REF] Dufourd | Boundedness of reset P/T nets[END_REF] and reachability [START_REF] Araki | Some decision problems related to the reachability problem for Petri nets[END_REF] are undecidable. For bounded reset Petri nets, more properties are decidable, as full state spaces can be computed.

Full state-space computations (i.e. using state graphs) do not preserve partial order semantics. To face this problem, Petri nets unfolding has been proposed and has gained the interest of researchers in verification [START_REF] Esparza | Unfoldings -A Partial-Order Approach to Model Checking[END_REF], diagnosis [START_REF] Benveniste | Diagnosis of asynchronous discreteevent systems: a net unfolding approach[END_REF], and planning [START_REF] Hickmott | Planning via Petri net unfolding[END_REF]. This technique keeps the intrinsic parallelism and prevents the combinatorial interleaving of independent events. While the unfolding of a Petri net can be infinite, there exist algorithms for constructing finite prefixes of it [START_REF] Esparza | An improvement of McMillan's unfolding algorithm[END_REF][START_REF] Mcmillan | Using unfoldings to avoid the state explosion problem in the verification of asynchronous circuits[END_REF]. Unfolding have the strong interest of preserving more behavioral properties of Petri nets than state graphs. In particular they preserve concurrency and its counterpart: causality. Unfolding techniques have also been developed for extensions of Petri nets, and in particular Petri nets with read arcs [START_REF] Baldan | Efficient unfolding of contextual Petri nets[END_REF].

Our contribution: Reachability analysis is known to be feasible on bounded reset Petri nets, however, as far as we know, no technique for computing finite prefixes of unfolding exists yet, and so, no technique preserving concurrency and causality exists yet. This is the aim of this paper to propose one. For that, we characterise the concurrent behaviour of reset Petri nets by defining a notion of pomset bisimulation. This has been inspired by several works on pomset behaviour of concurrent systems [START_REF] Best | Concurrent bisimulations in Petri nets[END_REF][START_REF] Van Glabbeek | Equivalence notions for concurrent systems and refinement of actions[END_REF][START_REF] Vogler | Bisimulation and action refinement[END_REF]. From this characterization we can then express what should be an unfolding preserving the concurrent behaviour of a reset Petri net. We show that it is not possible to remove reset arcs from safe reset Petri nets while preserving their behaviours with respect to this pomset bisimulation. Then we propose a notion of finite complete prefixes of unfolding of safe reset Petri nets that allows for reachability analysis while preserving pomset behaviour. As a consequence of the two other contributions, these finite complete prefixes do have reset arcs.

This paper is organized as follows: We first give basic definitions and notations for (safe) reset Petri nets. Then, in Section 3, we propose the definition of a pomset bisimulation for reset Petri nets. In Section 4 we show that, in general, there is no Petri net without resets which is pomset bisimilar to a given reset Petri net. Finally, in Section 5 -building on the results of Section 4 -we propose a finite complete prefix construction for reset Petri nets.

Reset Petri nets

Definition 1 (structure). A reset Petri net structure is a tuple (P , T , F , R) where P and T are disjoint sets of places and transitions, F ⊆ (P × T ) ∪ (T × P ) is a set of arcs, and R ⊆ P × T is a set of reset arcs.

An element x ∈ P ∪ T is called a node and has a preset • x = {y ∈ P ∪ T : (y, x) ∈ F } and a postset x • = {y ∈ P ∪ T : (x, y) ∈ F }. If, moreover, x is a transition, it has a set of resets x = {y ∈ P : (y, x) ∈ R}.

For two nodes x, y ∈ P ∪ T , we say that: x is a causal predecessor of y, noted x ≺ y, if there exists a sequence of nodes x 1 . . . x n with n ≥ 2 so that ∀i ∈ [1..n-1], (x i , x i+1 ) ∈ F , x 1 = x, and x n = y. If x ≺ y or y ≺ x we say that x and y are in causal relation. The nodes x and y are in conflict, noted x#y, if there exists two sequences of nodes x 1 . . . x n with n ≥ 2 and ∀i ∈ [1..n -1], (x i , x i+1 ) ∈ F , and y 1 . . . y m with m ≥ 2 and ∀i ∈ [1..m -1], (y i , y i+1 ) ∈ F , so that x 1 = y 1 is a place, x 2 = y 2 , x n = x, and y m = y.

A marking is a set M ⊆ P of places. It enables a transition t ∈ T if ∀p ∈ • t, p ∈ M . In this case, t can be fired from M , leading to the new marking

M = (M \ ( • t ∪ t)) ∪ t • .
The fact that M enables t and that firing t leads to M is denoted by M [t M .

Definition 2 (reset Petri net).

A reset Petri net is a tuple (P , T , F , R, M 0 ) where (P , T , F , R) is a reset Petri net structure and M 0 is a marking called the initial marking. A marking M is said to be reachable in a reset Petri net if there exists a sequence M 1 . . . M n of markings so that: ∀i ∈ [1..n -1], ∃t ∈ T , M i [t M i+1 (each marking enables a transition that leads to the next marking in the sequence), M 1 = M 0 (the sequence starts from the initial marking), and M n = M (the sequence leads to M ). The set of all markings reachable in a reset Petri net N R is denoted by [N R .

A reset Petri net with an empty set of reset arcs is simply called a Petri net.

Definition 3 (underlying Petri net). Given N R = (P , T , F , R, M 0 ) a reset Petri net, we call its underlying Petri net the Petri net N = (P , T , F , ∅, M 0 ).

The above formalism is in fact a simplified version of the general formalism of reset Petri nets: arcs have no multiplicity and markings are sets of places rather than multisets of places. We use it because it suffices for representing safe nets.

Definition 4 (safe reset Petri net). A reset Petri net (P , T , F , R, M 0 ) is said to be safe if for any reachable marking M and any transition

t ∈ T , if M enables t then (t • \ ( • t ∪ t)) ∩ M = ∅.
The reader familiar with Petri nets will notice that our results generalize to larger classes of nets: unbounded reset Petri nets for our pomset bisimulation (Section 3), and bounded reset Petri nets for our prefix construction (Section 5).

In the rest of the paper, unless the converse is specified, we consider reset Petri nets so that the preset of each transition t is non-empty: • t = ∅. Notice that this is not a restriction to our model: one can equip any transition t of a reset Petri net with a place p t so that p t is in the initial marking and • p t = p • t = {t}. One may need to express that two (reset) Petri nets have the same behaviour. This is useful in particular for building minimal (or at least small, that is with few places and transitions) representatives of a net; or for building simple (such as loop-free) representatives of a net. A standard way to do so is to define a bisimulation between (reset) Petri nets, and state that two nets have the same behaviour if they are bisimilar.

The behaviour of a net will be an observation of its transition firing, this observation being defined thanks to a labelling of nets associating to each transition an observable label or the special unobservable label ε.

Definition 5 (labelled reset Petri net). A labelled reset Petri net is a tuple (N R , Σ, λ) so that: N R = (P , T , F , R, M 0 ) is a reset Petri net, Σ is a set of transition labels, and λ : T → Σ ∪ {ε} is a labelling function.

In such a labelled net we extend the labelling function λ to sequences of transitions in the following way: given a sequence t

1 . . . t n (with n ≥ 2) of tran- sitions, λ(t 1 . . . t n ) = λ(t 1 )λ(t 2 . . . t n ) if λ(t 1 ) ∈ Σ and λ(t 1 . . . t n ) = λ(t 2 . . . t n ) else (that is if λ(t 1 ) = ε).
From that, one can define bisimulation as follows.

Definition 6 (bisimulation). Let (N R,1 , Σ 1 , λ 1 ) and (N R,2 , Σ 2 , λ 2 ) be two labelled reset Petri nets with N R,i = (P i , T i , F i , R i , M 0,i ). They are bisimilar if and only if there exists a relation ρ ⊆

[N R,1 × [N R,2 (a bisimulation) so that: 1. (M 0,1 , M 0,2 ) ∈ ρ, 2. if (M 1 , M 2 ) ∈ ρ, then (a) for every transition t ∈ T 1 so that M 1 [t M 1,n there exists a sequence t 1 . . . t n of transitions from T 2 and a sequence M 2,1 . . . M 2,n of markings of N R,2 so that: M 2 [t 1 M 2,1 [t 2 . . . [t n M 2,n , λ 2 (t 1 . . . t n ) = λ 1 (t), and (M 1,n , M 2,n ) ∈ ρ (b) the other way around (for every transition t ∈ T 2 . . . ) p1 t1 p2 p3 t2 p4 N R,1 p1 t1 p2 p3 t2 p4 p5 N R,2

Fig. 2. Two bisimilar nets

This bisimulation however hides an important part of the behaviours of (reset) Petri nets: transition firings may be concurrent when transitions are not in causal relation nor in conflict. For example, consider Figure 2 where N R,1 and N R,2 are bisimilar (we identify transition names and labels). In N R,1 , t 1 and t 2 are not in causal relation while in N R,2 they are in causal relation.

To avoid this loss of information, a standard approach is to define bisimulations based on partially ordered sets of transitions rather than totally ordered sets of transitions (the transition sequences used in the above definition). Such bisimulations are usually called pomset bisimulations.

Pomset bisimulation for reset Petri nets

In this section, we propose a definition of pomset bisimulation for reset Petri nets. It is based on an ad hoc notion of processes (representations of the executions of a Petri net, concurrent counterpart of paths in automata).

Processes of reset Petri nets

We recall a standard notion of processes of Petri nets and show how it can be extended to reset Petri nets. As a first step, we define occurrence nets which are basically Petri nets without loops.

Definition 7 (occurrence net). An occurrence net is a (reset) Petri net (B, E, F O , R O , M O 0 ) so that, ∀b ∈ B, ∀x ∈ B ∪ E: (1) | • b| ≤ 1, (2) x is not in causal relation with itself, (3) x is not in conflict with itself, (4) {y ∈ B ∪E : y ≺ x} is finite, (5) b ∈ M O 0 if and only if • b = ∅.
Places of an occurrence net are usually referred to as conditions and transitions as events. In an occurrence net, if two nodes x, y ∈ B ∪ E are so that x = y, are not in causal relation, and are not in conflict, they are said to be concurrent. Moreover, in occurrence net, the causal relation is a partial order.

There is a price to pay for having reset arcs in occurrence nets. With no reset arcs, checking if a set E of events together form a feasible execution (i.e. checking that the events from E can all be ordered so that they can be fired in this order starting from the initial marking) is linear in the size of the occurrence net (it suffices to check that E is causally closed and conflict free). With reset arcs the same task is NP-complete as stated in the below proposition.

Proposition 1. The problem of deciding if a set E of events of an occurrence net with resets forms a feasible execution is NP-complete.

Proof. (Sketch) Graph 3-coloring reduces to executability of an occurrence net.

The branching processes of a Petri net are then defined as particular occurrence nets linked to the original net by homomorphisms.

Definition 8 (homomorphism of nets). Let N 1 and N 2 be two Petri nets such that

N i = (P i , T i , F i , ∅, M 0,i ). A mapping h : P 1 ∪ T 1 → P 2 ∪ T 2 is an homomorphism of nets from N 1 to N 2 if ∀p 1 ∈ P 1 , ∀p 2 ∈ P 2 , ∀t ∈ T 1 : (1) h(p 1 ) ∈ P 2 , (2) h(t) ∈ T 2 , (3) p 2 ∈ • h(t) ⇔ ∃p 1 ∈ • t, h(p 1 ) = p 2 , (4) p 2 ∈ h(t) • ⇔ ∃p 1 ∈ t • , h(p 1 ) = p 2 , (5) p 2 ∈ M 0,2 ⇔ ∃p 1 ∈ M 0,1 , h(p 1 ) = p 2 . Definition 9 (processes of a Petri net). Let N = (P , T , F , ∅, M 0 ) be a Petri net, O = (B, E, F O , ∅, M O 0
) be an occurrence net, and h be an homomorphism of nets from

O to N . Then (O, h) is a branching process of N if ∀e 1 , e 2 ∈ E, ( • e 1 = • e 2 ∧ h(e 1 ) = h(e 2 )) ⇒ e 1 = e 2 . If, moreover, ∀b ∈ B, |b • | ≤ 1, then (O, h) is a process of N .
Finally, a process of a reset Petri net is obtained by adding reset arcs to a process of the underlying Petri net (leading to what we call below a potential process) and checking that all its events can still be enabled and fired in some order.

Definition 10 (potential processes of a reset Petri net). Let N R = (P , T , F , R, M 0 ) be a reset Petri net and N be its underlying Petri net, let O = (B, E, F O , R O , M O 0 ) be an occurrence net, and h be an homomorphism of

nets from O to N R . Then (O, h) is a potential process of N R if (1) (O , h) is a process of N with O = (B, E, F O , ∅, M O 0 ), (2) ∀b ∈ B, ∀e ∈ E, (b, e) ∈ R O if and only if (h(b), h(e)) ∈ R. Definition 11 (processes of a reset Petri net). Let N R = (P , T , F , R, M 0 ) be a reset Petri net, O = (B, E, F O , R O , M O 0
) be an occurrence net, and h be an

homomorphism of nets from O to N R . Then (O, h) is a process of N R if (1) (O, h) is a potential process of N R , and (2) if E = {e 1 , . . . , e n } then ∃M 1 , . . . , M n ⊆ B so that M O 0 [e k1 M 1 [e k2 .
. . [e kn M n with {k 1 , . . . , k n } = {1, . . . , n}. Notice that processes of reset Petri nets and processes of Petri nets do not exactly have the same properties. In particular, two properties are central in defining pomset bisimulation for Petri nets and do not hold for reset Petri nets.

Property 1. In any process of a Petri net with set of events E, consider any sequence of events e 1 e 2 . . . e n (1) that contains all the events in E and (2) such that ∀i, j ∈ [1..n] if e i ≺ e j then i < j. Necessarily, there exist markings M 1 , . . . , M n so that

M O 0 [e 1 M 1 [e 2 .
. . [e n M n . This property (which, intuitively, expresses that processes are partially ordered paths) is no longer true for reset Petri nets. Consider for example the reset Petri net of Figure 1 (left). Figure 1 (right) is one of its processes (the occurrence net with the homomorphism h below). As not e 2 ≺ e 1 , their should exist markings

M 1 , M 2 so that M 0 [e 1 M 1 [e 2 M 2 . However, M 0 = {c 1 , c 3 } indeed enables e 1 , but the marking M 1 such that M 0 [e 1 M 1 is {c 2 }, which does not enable e 2 .
Property 2. In a process of a Petri net all the sequences of events e 1 e 2 . . . e n verifying (1) and ( 2) of Property 1 lead to the same marking (i.e. M n is always the same), thus uniquely defining a notion of maximal marking of a process. This property defines the marking reached by a process. As a corollary of Property 1 not holding for reset Petri nets, there is no uniquely defined notion of maximal marking in their processes. Back to the example {c 2 } is somehow maximal (no event can be fired from it) as well as {c 2 , c 4 }.

To transpose the spirit of Properties 1 and 2 to processes of reset Petri nets, we define below a notion of maximal markings in such processes. In other words, the maximal markings of a process are all the marking that are reachable in it using all its events. This, in particular, excludes {c 2 } in the above example.

Abstracting processes

We show how processes of labelled reset Petri nets can be abstracted as partially ordered multisets (pomsets) of labels.

Definition 13 (pomset abstraction of processes). Let (N R , Σ, λ) be a labelled reset Petri net and (O, h) be a process of

N R with O = (B, E, F O , R O , M O 0 )
. Define E = {e ∈ E : λ(h(e)) = ε}. Define λ : E → Σ as the function so that ∀e ∈ E , λ (e) = λ(h(e)). Define moreover < ⊆ E × E as the relation so that e 1 < e 2 if and only if e 1 ≺ e 2 (e 1 is a causal predecessor of e 2 in O). Then, (E , < , λ ) is the pomset abstraction of (O, h). This abstraction (E, < , λ ) of a process is called its pomset abstraction because it can be seen as a multiset of labels (several events may have the same associated label by λ ) that are partially ordered by the < relation. In order to compare processes with respect to their pomset abstractions, we also define the following equivalence relation.

Definition 14 (pomset equivalence).

Let (E, < , λ) and (E , < , λ ) be the pomset abstractions of two processes P and P . These processes are pomset equivalent, noted P ≡ P if and only if there exists a bijection f : E → E so that ∀e 1 , e 2 ∈ E: (1) λ(e 1 ) = λ (f (e 1 )), and (2) e 1 < e 2 if and only if f (e 1 ) < f (e 2 ).

Intuitively, two processes are pomset equivalent if their pomset abstractions define the same pomset: same multisets of labels with same partial orderings. Finally, we also need to be able to abstract processes as sequences of labels.

Definition 15 (linear abstraction). Let (N R , Σ, λ) be a labelled reset Petri net, let P = (O, h) be a process of

N R with O = (B, E, F O , R O , M O 0 )
, and let M be a reachable marking in O. Define λ : E → Σ as the function so that ∀e ∈ E, λ (e) = λ(h(e)). The linear abstraction of P with respect to M is the set lin(M , P) so that a sequence of ω is in lin(M , P) if and only if in O there exist markings M 1 , . . . , M n-1 and events e 1 , . . . , e n so that

M O 0 [e 1 M 1 [e 2 . . . M n-1 [e n M and λ (e 1 .
. . e n ) = ω.

Pomset bisimulation

We now define a notion of pomset bisimulation between reset Petri nets, inspired by [START_REF] Best | Concurrent bisimulations in Petri nets[END_REF][START_REF] Van Glabbeek | Equivalence notions for concurrent systems and refinement of actions[END_REF][START_REF] Vogler | Bisimulation and action refinement[END_REF]. Intuitively, two reset Petri nets are pomset bisimilar if there exists a relation between their reachable markings so that the markings that can be reached by pomset equivalent processes from two markings in relation are themselves in relation. This is formalized by the below definition.

Definition 16 (pomset bisimulation for reset nets). Let (N R,1 , Σ 1 , λ 1 ) and (N R,2 , Σ 2 , λ 2 ) be two labelled reset Petri nets with N R,i = (P i , T i , F i , R i , M 0,i ).

They are pomset bisimilar if and only if there exists a relation ρ

⊆ [N R,1 ×[N R,2
(called a pomset bisimulation) so that:

1.

(M 0,1 , M 0,2 ) ∈ ρ, 2. if (M 1 , M 2 ) ∈ ρ, then (
a) for every process P 1 of (P 1 , T 1 , F 1 , R 1 , M 1 ) there exists a process P 2 of (P 2 , T 2 , F 2 , R 2 , M 2 ) so that P 1 ≡ P 2 and

-∀M 1 ∈ M max (P 1 ), ∃M 2 ∈ M max (P 2 ) so that (M 1 , M 2 ) ∈ ρ, -∀M 1 ∈ M max (P 1 ), ∀M 2 ∈ M max (P 2 ), (M 1 , M 2 ) ∈ ρ ⇒ lin(M 1 , P 1 ) = lin(M 2 , P 2
). (b) the other way around (for every process P 2 . . . ) Notice that, in the above definition, taking the processes P 1 and P 2 bisimilar (using the standard bisimulation relation for Petri nets) rather than comparing lin(M 1 , P 1 ) and lin(M 2 , P 2 ) would lead to an equivalent definition.

Remark that pomset bisimulation implies bisimulation, as expressed by the following proposition. The converse is obviously not true. Proposition 2. Let (N R,1 , Σ 1 , λ 1 ) and (N R,2 , Σ 2 , λ 2 ) be two pomset bisimilar labelled reset Petri nets, then (N R,1 , Σ 1 , λ 1 ) and (N R,2 , Σ 2 , λ 2 ) are bisimilar.

Proof. It suffices to notice that Definition 6 can be obtained from Definition 16 by restricting the processes considered, taking only those with exactly one transition whose label is different from ε.

From now on, we consider that (reset) Petri nets are finite, i.e. their sets of places and transitions are finite.

N 0R b1(p0) b2(p2) e1 (t1) e2 (t3) b4(p0) b5(p1) b6(p2) e3 (t2) b3(p3) b7(p3) F0 R Fig. 3. A remarkable pattern N pat
R and its structural transformation N pat str , a labelled reset Petri net N 0 R including the pattern N R, and a finite complete prefix F0 R of N 0 R . Transition labels are given on transitions.

In this section, we prove that it is, in general, not possible to remove reset arcs from safe reset Petri nets while preserving their behaviours with respect to this pomset bisimulation. More precisely, we prove that it is not possible to build a safe labelled Petri net (while this is out of the scope of this paper, the reader familiar with Petri nets may notice that this is the case for bounded labelled Petri net) without reset arcs which is pomset bisimilar to a given safe labelled reset Petri net. For that, we exhibit a particular pattern -Figure 3 (left) -and show that a reset Petri net including this pattern cannot be pomset bisimilar to a Petri net without reset arcs.

As a first intuition of this fact, let us consider the following structural transformation that removes reset arcs from a reset Petri net.

Definition 17 (Structural transformation). Let (N R , Σ, λ) be a labelled reset Petri net such that N R = (P , T , F , R, M 0 ), its structural transformation is the labelled Petri net (N R,str , Σ str , λ str ) where N R,str = (P str , T str , F str , ∅, M 0,str ) so that:

P str = P ∪ P with P = {p : p ∈ P ∧ ∃t ∈ T , (p, t) ∈ R}, T str = T ∪ T with T = {t : t ∈ T ∧ t = ∅}, F str = F ∪ {(p, t) : t ∈ T , (p, t) ∈ F } ∪ {(t, p) : t ∈ T , (t, p) ∈ F } (1)
∪ {(p, t) : p ∈ P , (t, p) ∈ F } ∪ {(t, p) : p ∈ P , (p, t) ∈ F } (2)
∪ {(p, t) ∈ P × T : (t, p) ∈ F } ∪ {(t, p) ∈ T × P : (p, t) ∈ F } (3) ∪ {(p, t), (p, t), (t, p), (t, p) : (p, t) ∈ R}, (4)
M 0,str = M 0 ∪ {p ∈ P : p / ∈ M 0 },
and moreover, Σ str = Σ, ∀t ∈ T, λ str (t) = λ(t), and ∀t ∈ T , λ str (t) = λ(t).

Intuitively, in this transformation, for each reset arc (p, t), a copy p of p and a copy t of t are created. The two places are so that p is marked if and only if p is not marked, the transition t will perform the reset when p is marked and t will perform it when p is not marked (i.e when p is marked). For that, new arcs are added to F so that: t mimics t (1), the link between p and p is enforced (2, 3), and the resets are either performed by t or t depending of the markings of p and p (4). This is examplified in Figure 3 (left and middle left). Lemma 1. A labelled reset Petri net (N R , Σ, λ) and its structural transformation (N R,str , Σ str , λ str ) as defined in Definition 17 are bisimilar.

Proof. (Sketch) The bisimulation relation is

ρ ⊆ [N R,1 × [N R,2 defined by (M, M struct ) ∈ ρ iff ∀p ∈ P, M (p) = M struct (p) and ∀p ∈ P such that p ∈ P , we have M struct (p) + M struct (p) = 1.
For the transformation of Definition 17, a reset Petri net and its transformation are bisimilar but not always pomset bisimilar. This can be remarked on any safe reset Petri net including the pattern N pat R of Figure 3. Indeed, this transformation adds in N pat str a causality relation between the transition labelled by t 1 and each of the two transitions labelled by t 3 . From the initial marking of N pat str , for any process whose pomset abstraction includes both t 1 and t 3 , these two labels are causally ordered. While, from the initial marking of N pat R there is a process which pomset abstraction includes both t 1 and t 3 but does not order them. We now generalize this result.

Let us consider the labelled reset Petri Net N 0 R of Figure 3 (middle right). It uses the pattern N pat R of Figure 3 in which t 1 and t 3 can be fired in different order infinitely often. In this net, the transitions with labels t 1 and t 3 are not in causal relation. Proposition 3. There is no finite safe labelled Petri net (i.e. without reset arc) which is pomset bisimilar to the labelled reset Petri net N 0 R . Proof. We simply remark that any finite safe labelled Petri net with no reset arcs which is bisimilar to N 0 R has a causal relation between two transitions labelled by t 1 and t 3 respectively (Lemma 2). From that, by Proposition 2, we get that any such labelled Petri net N which would be pomset bisimilar to N 0 R would have a process from its initial marking whose pomset abstraction is such that some occurrence of t 1 and some occurrence of t 3 are ordered, while this is never the case in the processes of N 0 R . This prevents N from being pomset bisimilar to N 0 R , and thus leads to a contradiction, proving the proposition. Lemma 2. Any safe labelled Petri net with no reset arcs which is bisimilar (see definition 6) to N 0 R has a causal relation between two transitions labelled by t 1 and t 3 respectively.

Proof. (Sketch) The firing of t 3 prevents the firing of t 2 ; then t 3 and t 2 are in conflict and share an input place which has to be marked again after the firing of t 1 . This place generates a causality between t 1 and t 3 .

In this section, we propose a notion of finite complete prefixes of unfolding of safe reset Petri nets preserving reachability of markings and pomset behaviour. As a consequence of the previous section, these finite complete prefixes do have reset arcs.

The unfolding of a Petri net is a particular branching process (generally infinite) representing all its reachable markings and ways to reach them. It also preserves concurrency.

Definition 18 (Unfolding of a Petri net). The unfolding of a net can be defined as the union of all its branching processes [START_REF] Esparza | Unfoldings -A Partial-Order Approach to Model Checking[END_REF] or equivalently its largest branching process (with respect to inclusion).

In the context of reset Petri nets, no notion of unfolding has been defined yet. Accordingly to our notion of processes for reset Petri nets and because of Proposition 4 below we propose Definition 19. In it and the rest of the paper, nets and labelled nets are identified (each transition is labelled by itself) and labellings of branching processes are induced by homomorphisms (as for pomset abstraction).

Definition 19 (Unfolding of a reset Petri net). Let N R be a safe reset Petri net and N be its underlying Petri net. Let U be the unfolding of N . The unfolding of N R is U R , obtained by adding reset arcs to U according to (2) in Definition 10. Proof. (Sketch) This extends a result of [START_REF] Van Glabbeek | Petri net models for algebraic theories of concurrency[END_REF], stating that two Petri nets having the same unfolding (up to isomorphism) are pomset bisimilar (for a notion of bisimulation coping with our in absence of resets).

Petri nets unfolding is however unpractical for studying Petri nets behaviour as it is generally an infinite object. In practice, finite complete prefixes of it are preferred [START_REF] Mcmillan | Using unfoldings to avoid the state explosion problem in the verification of asynchronous circuits[END_REF][START_REF] Esparza | An improvement of McMillan's unfolding algorithm[END_REF].

Definition 20 (finite complete prefix, reachable marking preservation). A finite complete prefix of the unfolding of a safe Petri net N is a finite branching processe (O, h) of N verifying the following property of reachable marking preservation: a marking M is reachable in N if and only if there exists a reachable marking M in O so that M = {h(b) : b ∈ M }.

In this section, we propose an algorithm for construction of finite complete prefixes for safe reset Petri nets. For that, we assume the existence of a black-box algorithm for building finite complete prefixes of safe Petri nets (without reset arcs). Notice that such algorithms indeed do exist [START_REF] Mcmillan | Using unfoldings to avoid the state explosion problem in the verification of asynchronous circuits[END_REF][START_REF] Esparza | An improvement of McMillan's unfolding algorithm[END_REF].

Because of Proposition 3, we know that such finite prefixes should have reset arcs to preserve pomset behaviour. We first remark that directly adding reset arcs to finite complete prefixes of underlying nets would not work. Proposition 5. Let U be the unfolding of the underlying Petri Net N of a safe reset Petri net N R , let F be one of its finite and complete prefixes. Let F be the object obtained by adding reset arcs to F according to (2) in Definition 10. The reachable marking preservation is in general not verified by F (with respect to N R ).

The proof of this proposition relies on the fact that some reachable markings of N R are not represented in F . This suggests that this prefix is not big enough. We however know an object that contains, for sure, every reachable marking of N R along with a way to reach each of them: its structural transformation N R,str (Definition 17). We thus propose to compute finite prefixes of reset Petri nets from their structural transformations: in the below algorithm, F str is used to determine the deepness of the prefix (i.e. the length of the longest chain of causally ordered transitions).

Algorithm 1 (Finite complete prefix construction for reset Petri nets) Let N R be a safe reset Petri net, (step 1) compute the structural transformation N R,str of N R , (step 2) compute a finite complete prefix F str of N R,str , (step 3) compute a finite prefix F of U (the unfolding of the underlying net N ) that simulates F str (a labelled net N 2 simulates a labelled net N 1 if they verify Definition 6 except for condition 2.b.), (step 4) compute F R by adding reset arcs from N R to F according to (2) in Definition 10. The output of the algorithm is F R .

Applying this algorithm to the net N 0 R of Figure 3 (middle right) -using the algorithm from [START_REF] Esparza | An improvement of McMillan's unfolding algorithm[END_REF] at step 2 -leads to the reset Petri net F 0 R of Figure 3 (right).

Notice that the computation of F str -step 1 and 2 -can be done in exponential time and space with respect to the size of N R . The computation of F from F str (step 3) is linear in the size of F. And, the addition of reset arcs (step 4) is at most quadratic in the size of F.

We conclude this section by showing that Algorithm 1 actually builds finite complete prefixes of reset Petri nets. Proposition 6. The object F R obtained by Algorithm 1 from a safe reset Petri net N R is a finite and complete prefix of the unfolding of N R .

Proof. Notice that if N R is safe, then N R,str is safe as well. Thus F str is finite by definition of finite complete prefixes of Petri nets (without reset arcs). F str is finite and has no node in causal relation with itself (i.e. no cycle), hence any net bisimilar with it is also finite, this is in particular the case of F. Adding reset arcs to a finite object does not break its finiteness, so F R is finite.

Moreover, F str is complete by definition of finite complete prefixes of Petri nets (without reset arcs). As F simulates F str it must also be complete (it can only do more). The reset arcs addition removes semantically to F only the unexpected sequences (i.e. the sequence which are possible in F but not in F str ). Therefore, F R is complete.

Our contribution in this paper is three-fold. First, we proposed a notion of pomset bisimulation for reset Petri nets. This notion is, in particular, inspired from a similar notion that has been defined for Petri nets (without reset arcs) in [START_REF] Best | Concurrent bisimulations in Petri nets[END_REF]. Second, we have shown that it is not possible to remove reset arcs from safe reset Petri nets while preserving their behaviours with respect to this pomset bisimulation. And, third, we proposed a notion of finite complete prefixes of unfolding of safe reset Petri nets that allows for reachability analysis while preserving pomset behaviour. As a consequence of the two other contributions, these finite complete prefixes do have reset arcs.

Figure 1 (

 1 Figure 1 (left) is a graphical representation of a reset Petri net. It has five places (circles) and three transitions (squares). Its set of arcs contains seven elements (arrows) and there is one reset arc (line with a diamond).

Fig. 1 .

 1 Fig. 1. A reset Petri net (left) and one of its processes (right)

Definition 12 (

 12 maximal markings). Let P = (O, h) be a process with set of events E = {e 1 , . . . , e n } and initial marking M O 0 of a reset Petri net. The set M max (P) of maximal markings of P contains exactly the markings M so that ∃M 1 , . . . , M n-1 , verifying M O 0 [e k1 M 1 [e k2 . . . M n-1 [e kn M for some {k 1 , . . . , k n } = {1, . . . , n}.

Proposition 4 .

 4 Any safe (labelled) reset Petri net N R and its unfolding U R are pomset bisimilar.