Fabrice Piazza
email: fpiazza75@gmail.com

Kathleen Gough

Marc Monthioux

Pascal Puech

Iann C Gerber

Richard Wiens

Germercy Paredes

Cristhofer Ozoria

Low temperature, pressureless sp 2 to sp 3 transformation of ultrathin, crystalline carbon films

teaching and research institutions in France or abroad, or from public or private research centers.

Low temperature, pressureless sp2 to sp3 transformation of ultrathin, crystalline carbon films

Introduction

Beyond graphene, a new nanocrystalline carbon material consisting of few atomic layers (FL) of diamond or lonsdaleite, also called diamane by Chernozatonskii et al. who first predicted it [START_REF] Chernozatonskii | Diamond-Like C2H Nanolayer, Diamane: Simulation of the Structure and Properties[END_REF], holds great technological promise and is attracting increasing interest [START_REF] Chernozatonskii | Diamond-Like C2H Nanolayer, Diamane: Simulation of the Structure and Properties[END_REF][START_REF] Leenaerts | Hydrogenation of bilayer graphene and the formation of bilayer graphane from first principles[END_REF][START_REF] Zhu | Formation and electronic properties of hydrogenated few layer graphene[END_REF][START_REF] Chernozatonskii | Influence of Size Effect on the Electronic and Elastic Properties of Diamond Films with Nanometer Thickness[END_REF][START_REF] Barboza | Room temperature compression-induced diamondization of few-layer graphene[END_REF][START_REF] Li | Transformation between different hybridized bonding structures in twodimensional diamond based materials[END_REF][START_REF] Chernozatonskii | Determination of ultrathin diamond films by Raman spectroscopy[END_REF][START_REF] Muniz | Opening and tuning of band gap by the formation of diamond superlattices in twisted bilayer graphene[END_REF][START_REF] Li | Layer number-dependant structural evolution of two-dimensional diamond films[END_REF][START_REF] Ruoff | Personal perspectives on graphene: New graphene-related materials on the horizon[END_REF][START_REF] Odkhuu | Conversion of multilayer graphene into continuous ultrathin sp 3 -bonded carbon films on metal surfaces[END_REF][START_REF] Rajasekaran | Interlayer carbon bond formation induced by hydrogen adsorption in few-layer supported graphene[END_REF][START_REF] Machado | Tunable mechanical properties of diamond superlattices generated by interlayer bonding in twisted bilayer graphene[END_REF][START_REF] Kvashnin | Phase diagram of quasi-two dimensional carbon, from graphene to diamond[END_REF][START_REF] Martins | Raman evidence for pressureinduced formation of diamondene[END_REF][START_REF] Gao | Ultrahard carbon film from epitaxial two-layer graphene[END_REF][START_REF] Bianco | A carbon science perspective in 2018: Current achievements and future challenges[END_REF][START_REF] Ruoff | A perspective on objectives for carbon science[END_REF]. In such a material, sp 3 -bonded carbon atoms are arranged in a hexagonal lattice. According to calculations, such sheets would be produced by the chemisorption of hydrogen atoms on the "top" and "bottom" surfaces of two-or three-layer graphene and the subsequent interlayer sp 3 -C bond formation [START_REF] Chernozatonskii | Diamond-Like C2H Nanolayer, Diamane: Simulation of the Structure and Properties[END_REF][START_REF] Leenaerts | Hydrogenation of bilayer graphene and the formation of bilayer graphane from first principles[END_REF][START_REF] Zhu | Formation and electronic properties of hydrogenated few layer graphene[END_REF]. Another possibility would be the conversion of few-layer graphene (FLG) on a metal surface into a sp 3 -carbon film through hydrogenation of the top layer. In this case, strong hybridization between the sp 3 dangling bond orbitals and the metallic surface orbitals would stabilize the sp 3 -bonded carbon layers [START_REF] Odkhuu | Conversion of multilayer graphene into continuous ultrathin sp 3 -bonded carbon films on metal surfaces[END_REF][START_REF] Rajasekaran | Interlayer carbon bond formation induced by hydrogen adsorption in few-layer supported graphene[END_REF]. A last approach, which is not explored in the present work, would be the use of high pressure to induce the sp 2 -to-sp 3 transformation [START_REF] Barboza | Room temperature compression-induced diamondization of few-layer graphene[END_REF][START_REF] Martins | Raman evidence for pressureinduced formation of diamondene[END_REF][START_REF] Gao | Ultrahard carbon film from epitaxial two-layer graphene[END_REF].

A single layer of diamane refers to the full hydrogenation of a single graphene, a material known as graphane, which was tentatively prepared in Ref. [START_REF] Elias | Control of graphene's properties by reversible hydrogenation: evidence for graphane[END_REF]. The stability of graphane was first predicted by Soto et al [START_REF] Sofo | Graphane: A two-dimensional hydrocarbon[END_REF]. Graphane consists of a single layer of hexagonal network of sp 3 -bonded carbon in which every carbon is bonded to hydrogen, alternately above and below the layer. The synthesis of graphane remains elusive. In ref. [START_REF] Elias | Control of graphene's properties by reversible hydrogenation: evidence for graphane[END_REF] full hydrogenation was not achieved as evidenced in the published Raman spectra which contain sp 2 -C features. The authors in Ref [START_REF] Elias | Control of graphene's properties by reversible hydrogenation: evidence for graphane[END_REF] claim that a small residual sp 2 phase dominates their spectra because of the much larger Raman diffusion crosssection for sp 2 -carbon than for sp 3 -carbon. However, in fully fluorinated graphene, a material known as fluorographene, sp 2 -C features are not observed in visible Raman spectra [START_REF] Nair | Fluorographene: A Two-Dimensional Counterpart of Teflon[END_REF]. Graphane could be considered as the first member in the series of diamanes of increasing thickness, as a new class of carbon nanoform which nomenclature is not consensual yet. The chemically-induced phase conversion of FLG into a C(sp 3 ) structure may have been previously obtained from the chemisorption of halogen atoms such as fluorine [START_REF] Touhara | On the structure of Graphite Fluoride[END_REF]. However, to the best of authors' knowledge, diamane structure has not been evidenced.

According to computational studies, diamanes display the electronic structure of semiconductors, with a direct band-gap depending on the film thickness, which make them very attractive for nanoelectronics, band-gap engineering and active laser medium in nanooptics [START_REF] Chernozatonskii | Influence of Size Effect on the Electronic and Elastic Properties of Diamond Films with Nanometer Thickness[END_REF][START_REF] Chernozatonskii | Determination of ultrathin diamond films by Raman spectroscopy[END_REF].

Because of their expected high thermal conductivity, they may also be used in thermal management devices [START_REF] Ruoff | Personal perspectives on graphene: New graphene-related materials on the horizon[END_REF]. Furthermore, the incorporation of nitrogen-vacancy (N-V) centers in diamanes through substitution by nitrogen atoms in the C lattice could be used to configure qbits for use in quantum computing [START_REF] Ruoff | Personal perspectives on graphene: New graphene-related materials on the horizon[END_REF]. Diamanes are also expected to be very strong, which would make them very attractive for ultrathin protective coatings, ultrahigh-strength components in composite materials for aerospace applications for instance, and nano-electromechanical systems [START_REF] Ruoff | Personal perspectives on graphene: New graphene-related materials on the horizon[END_REF]. Because of the expected low friction coefficient of the hydrogenated surface, diamanes may also be used to improve the wear resistance of coated mechanical parts. In addition, diamanes are expected to be biocompatible, so that they may become very important component in bio-devices and bio-sensors.

Heterostructures of graphene and diamane would be attractive for tunnel devices, optical linear waveguides, high efficiency optoelectronic sensors, lithium batteries, and supercapacitors [START_REF] Chernozatonskii | Influence of Size Effect on the Electronic and Elastic Properties of Diamond Films with Nanometer Thickness[END_REF][START_REF] Ruoff | Personal perspectives on graphene: New graphene-related materials on the horizon[END_REF].

Finally, mass production of few layer lonsdaleite, in particular from a low pressure method, would be very desirable for protective thin film and composite materials in mechanical applications, as it is predicted that lonsdaleite is 58 % harder than diamond [START_REF] Pan | Harder than Diamond : Superior Indentation Strength of Wurzite BN and Lonsdaleite[END_REF]. It was claimed once that "lonsdaleite is faulted and twinned cubic diamond and does not exist as a discrete material" [START_REF] Németh | Lonsdaleite is faulted and twinned cubic diamond and does not exist as a discret material[END_REF]. However, this statement was based on the analysis by transmission electron microscopy, at a rather high energy of 200 keV (hence likely to modify the material), of only two non-pure samples, a natural one and a synthetic one. It was also based on the observation that no example of successful synthesis of lonsdaleite was reported in the literature. Therefore, the demonstration is weak. In the absence of any reason why lonsdaleite could not form as an isolated phase, the possibility that suitable conditions could be found one day cannot be excluded. A very similar case can be found with rhombohedral graphite, usually found as stacking faults in hexagonal graphite but which existence as an isolated phase was claimed impossible for decades, until the contrary demonstration was made [START_REF] Lin | High-resolution TEM observations of isolated rhombohedral crystallites in graphite blocks[END_REF].

Even if various studies based on calculations hypothesized that diamane can form when nonsupported 2LG and 3LG are placed in hydrogen cold plasma, to the best of authors' knowledge, there is no report of any experimental evidence of such scenario. Only few experimental studies on the hydrogenation of FLG have been published [START_REF] Elias | Control of graphene's properties by reversible hydrogenation: evidence for graphane[END_REF][START_REF] Luo | Thickness-dependent reversible hydrogenation of graphene layers[END_REF][START_REF] Zheng | Catalytic hydrogenation of graphene films[END_REF][START_REF] Diankov | Extreme Monolayer-Selectivity of Hydrogen-Plasma reactions with Graphene[END_REF][START_REF] Felten | Controlled modification of monoand bilayer graphene in O2, H2 and CF4 plasmas[END_REF]. In particular, no features from diamane were reported in Raman spectra. Rajasekaran et al. reported on carbon-carbon interlayer bond formation in FLG up to 4 layers through deuterium adsorption on FLG surface which resulted in a thin film of diamond-like layer on a Pt(111) substrate as evidenced by synchrotron spectroscopic techniques [START_REF] Rajasekaran | Interlayer carbon bond formation induced by hydrogen adsorption in few-layer supported graphene[END_REF]. The surface adsorption of H induced a hybridization change of carbon from the sp 2 to the sp 3 bond symmetry, which propagated through the graphene layers, resulting in interlayer carbon bond formation. According to the authors of Ref. [START_REF] Rajasekaran | Interlayer carbon bond formation induced by hydrogen adsorption in few-layer supported graphene[END_REF], the structure was stabilized through the termination of interfacial sp 3 -C by the substrate. The structural transformation was believed to occur as a consequence of high adsorption energy. This would be in good agreement with the calculations of ref. [START_REF] Odkhuu | Conversion of multilayer graphene into continuous ultrathin sp 3 -bonded carbon films on metal surfaces[END_REF]. However, crystal structure was not evidenced in ref. [START_REF] Rajasekaran | Interlayer carbon bond formation induced by hydrogen adsorption in few-layer supported graphene[END_REF]. No Raman spectroscopy nor electron diffraction results were reported. The material obtained in ref. [START_REF] Rajasekaran | Interlayer carbon bond formation induced by hydrogen adsorption in few-layer supported graphene[END_REF] could be sp 3 -C-rich amorphous carbon, that is to say tetrahedral amorphous carbon (ta-C) [START_REF] Robertson | Diamond-like amorphous carbon[END_REF][START_REF] Piazza | Hard-hydrogenated tetrahedral amorphous carbon films by distributed electron cyclotron resonance plasma[END_REF]. Further work is necessary to evidence, understand and control diamane synthesis. In particular, it is not clear whether a metallic substrate is necessary or not to stabilize sp 3 bonding in hexagonal lattice.

To achieve the above-presented route to convert FLG into diamane, it is necessary to develop an efficient method to hydrogenate FLG. Yet, since the first experimental description of hydrogenated 1LG and 2LG [START_REF] Ryu | Reversible basal plane hydrogenation of graphene[END_REF], only few experimental studies and methods were reported on this topic. Those methods have been mainly focusing on two approaches: gas phase [START_REF] Elias | Control of graphene's properties by reversible hydrogenation: evidence for graphane[END_REF][START_REF] Luo | Thickness-dependent reversible hydrogenation of graphene layers[END_REF][START_REF] Zheng | Catalytic hydrogenation of graphene films[END_REF][START_REF] Felten | Controlled modification of monoand bilayer graphene in O2, H2 and CF4 plasmas[END_REF][START_REF] Balog | Atomic Hydrogen Adsorbate Structures on Graphene[END_REF][START_REF] Guisinger | Exposure of Epitaxial Graphene on SiC(0001) to Atomic Hydrogen[END_REF][START_REF] Wang | Toward High Throughput Interconvertible Graphane-to-Graphene Growth and Patterning[END_REF][START_REF] Jones | On the mechanism for plasma hydrogenation of graphene[END_REF][START_REF] Burgess | Tuning the electronic properties of graphene by hydrogenation in a plasma enhanced chemical vapour deposition reactor[END_REF][START_REF] Wojtaszek | A road to hydrogenating graphene by a reactive ion etching plasma[END_REF][START_REF] Poh | High-pressure hydrogenation of graphene:towards graphane[END_REF][START_REF] Kataria | Raman imaging on high quality graphene grown by hot-filament chemical vapor deposition[END_REF] or liquid phase hydrogenation [START_REF] Subrahmanyam | Chemical storage of hydrogen in few-layer graphene[END_REF][START_REF] Yang | Birch Reduction of Graphite. Edge and Interior Functionalization by Hydrogen[END_REF][START_REF] Schäfer | On the Way to Graphane-Pronounced Fluoresence of Polyhydrogenated Graphene[END_REF][START_REF] Krishna | Facile synthesis of hydrogenated reduced Graphene oxide via hydrogen spillover mechanism[END_REF][START_REF] Sofer | Highly hydrogenated graphene via active hydrogen reduction of graphene oxide in the aqueous phase at room temperature[END_REF][START_REF] Whitener | Chemical hydrogenation of single-layer graphene enables completely reversible removal of electrical conductivity[END_REF]. A review on the subject is available in Ref. [START_REF] Pumera | Graphane and hydrogenated graphene[END_REF]. In any case, partially hydrogenated graphene has been prepared; never graphane. The liquid phase hydrogenation methods can generate a maximum of ~76 at.% of hydrogenated carbon atoms [START_REF] Subrahmanyam | Chemical storage of hydrogen in few-layer graphene[END_REF] against ~10 at.% in the gas phase methods [START_REF] Burgess | Tuning the electronic properties of graphene by hydrogenation in a plasma enhanced chemical vapour deposition reactor[END_REF]. Even though they have so far not lead to the highest hydrogenation rate, gas phase hydrogenation methods, especially at low temperature and low pressure, remain very attractive on a technology standpoint, in particular in applications where the use of wet-chemistry methods is limited, such as in nanoelectronics, where hydrogenation must be precisely and spatially controlled in a tunable manner. The hot-filament process may constitute a very competitive method for the efficient hydrogenation of FLG and the potential formation of diamane. It has been widely and preferentially used at industrial scale for the elaboration of microcrystalline and nanocrystalline diamond [START_REF] Prelas | Handbook of Industrial Diamonds and Diamond Films[END_REF][START_REF] Butler | The CVD of Nanodiamond Materials[END_REF][START_REF] Zimmer | Aspects of scaling CVD diamond reactors[END_REF]. It has been developed to efficiently produce atomic hydrogen (H), which has been shown to play a critical role for the conventional synthesis of metastable diamond at low pressure from a dilute mixture of a hydrocarbon in H2 [START_REF] Prelas | Handbook of Industrial Diamonds and Diamond Films[END_REF][START_REF] Frenklach | Detailed surface and gas-phase chemical kinetics of diamond deposition[END_REF][START_REF] Stiegler | Activation energy for diamond growth from the carbon-hydrogen gas system at low temperatures[END_REF][START_REF] Redman | H atom production on a hot filament chemical vapour deposition reactor[END_REF]. H is produced heterogeneously by the thermal decomposition of H2 on the hot filament surface, and then rapidly diffuses into the bulk gas. H recombination reactions are sufficiently slow at the typical process pressures (below 100 Torr) that most of the H diffuse to the reactor walls. H is present at super-equilibrium concentration throughout most of the reactor [START_REF] Redman | H atom production on a hot filament chemical vapour deposition reactor[END_REF]. The effects of H2 pressure and flow rate, filament temperature and radial distance from the filament on the relative H concentrations, and the gas temperature profiles were investigated in details, in particular for the case of pure H2 [START_REF] Redman | H atom production on a hot filament chemical vapour deposition reactor[END_REF][START_REF] Redman | Resonance enhanced multiphoton ionisation probing of H atoms in a hot filament chemical vapour deposition reactor[END_REF][START_REF] Schäfer | Atomic hydrogen concentration profiles at filaments used for chemical vapor deposition of diamond[END_REF][START_REF] Chen | Temperature and concentration distribution of H2 and H atoms in hot-filament chemical-vapor deposition of diamond[END_REF][START_REF] Connell | Spatially resolved atomic hydrogen concentrations and molecular hydrogen temperature profiles in the chemicalvapor deposition of diamond[END_REF][START_REF] Zumbach | Experimental investigation and computational modeling of hot filament diamond chemical vapor deposition[END_REF][START_REF] Chenevier | Measurement of atomic hydrogen in a hot filament reactor by two-photon laser-induced fluorescence[END_REF]]. In the hot filament process, as compared to low pressure plasma techniques, the presence of ions accelerated toward the substrate and which can damage the graphene is avoided. Indeed, high kinetic energy ions in the plasma tend to etch the graphene film instead of participating in the hydrogenation process [START_REF] Burgess | Tuning the electronic properties of graphene by hydrogenation in a plasma enhanced chemical vapour deposition reactor[END_REF]. Besides, the hot filament process was used to grow crystalline nanodiamonds at low substrate temperature, below 300 C [START_REF] Piazza | Diamond film synthesis at low temperature[END_REF][START_REF] Piazza | Synthesis of diamond at sub 300 C substrate temperature[END_REF][START_REF] Piazza | Synthesis of diamond nanocrystals on polyimide film[END_REF]. It was used to coat carbon nanotube bundles with diamond and SiC nanocrystals from solid carbon and silicon sources exposed to H at around 190 C [START_REF] Piazza | Carbon nanotubes coated with diamond nanocrystals and silicon carbide by hot-filament chemical vapor deposition below 200 C substrate temperature[END_REF][START_REF] Piazza | Carbon nanotubes conformally coated with diamond nanocrystals or silicon carbide, methods of making the same and methods of their use[END_REF]. The synthesis of hydrogenated graphene by the hot filament process onto a 1000 C-heated copper substrate and from a gas mixture of CH4:H2 was previously reported [START_REF] Kataria | Raman imaging on high quality graphene grown by hot-filament chemical vapor deposition[END_REF]. However, this was evidenced by Raman spectroscopy analysis, which is ambiguous as detailed below.

One important aspect to achieve an efficient hydrogenation of FLG and subsequent conversion into crystalline sp 3 -bonded C sheets is the choice of characterization techniques to unambiguously evidence C-H and sp 3 -C-sp 3 -C bonding. As pointed out in Ref. [START_REF] Pumera | Graphane and hydrogenated graphene[END_REF], only few studies [START_REF] Poh | High-pressure hydrogenation of graphene:towards graphane[END_REF][START_REF] Subrahmanyam | Chemical storage of hydrogen in few-layer graphene[END_REF][START_REF] Yang | Birch Reduction of Graphite. Edge and Interior Functionalization by Hydrogen[END_REF][START_REF] Schäfer | On the Way to Graphane-Pronounced Fluoresence of Polyhydrogenated Graphene[END_REF][START_REF] Sofer | Highly hydrogenated graphene via active hydrogen reduction of graphene oxide in the aqueous phase at room temperature[END_REF] reported direct quantitative data on hydrogenation rate from reliable techniques such as combustion elemental analysis (CHNO) or nuclear magnetic resonance. This may be due to the small amount of material processed in most cases. Furthermore, only a few studies [START_REF] Zheng | Catalytic hydrogenation of graphene films[END_REF][START_REF] Poh | High-pressure hydrogenation of graphene:towards graphane[END_REF][START_REF] Subrahmanyam | Chemical storage of hydrogen in few-layer graphene[END_REF][START_REF] Yang | Birch Reduction of Graphite. Edge and Interior Functionalization by Hydrogen[END_REF][START_REF] Schäfer | On the Way to Graphane-Pronounced Fluoresence of Polyhydrogenated Graphene[END_REF][START_REF] Krishna | Facile synthesis of hydrogenated reduced Graphene oxide via hydrogen spillover mechanism[END_REF][START_REF] Sofer | Highly hydrogenated graphene via active hydrogen reduction of graphene oxide in the aqueous phase at room temperature[END_REF] reported on the direct detection of C-H bonding from Fourier transform infrared spectroscopy (FTIR), which is a highly sensitive and widely accessible technique. The lack of quantitative data and FTIR analysis characterizes most of the reports on gas phase hydrogenation methods. Those studies only rely on Raman spectroscopy analysis. However, Raman spectroscopy is an indirect method to detect C-H bonding since it probes indiscernibly any kind of defects; vacancies, edges and C-H bonding [START_REF] Elias | Control of graphene's properties by reversible hydrogenation: evidence for graphane[END_REF]. In some of those studies, the fact that all of the defectrelated peaks in Raman spectra are suppressed or highly attenuated upon annealing or laser irradiation of hydrogenated graphene materials is used to show that defects generated during the hydrogenation treatment are C-H bonding [START_REF] Elias | Control of graphene's properties by reversible hydrogenation: evidence for graphane[END_REF]. However, Raman spectroscopy using single-point measurement cannot determine whether H is bonded to the surface of the layers or only to the edges.

To the authors' knowledge, this last paramount point has not been addressed in the literature.

Likewise, there is no Raman mapping analysis published for hydrogenated graphene while such analysis has been widely used for graphene research. Empirical analysis of Raman D and G peak intensity ratio from unknown number of single-point measurements has been used to estimate hydrogenation rates [START_REF] Burgess | Tuning the electronic properties of graphene by hydrogenation in a plasma enhanced chemical vapour deposition reactor[END_REF]. As far as we can tell, there is no FTIR mapping data published either. These are the reasons why it has been decided here to use Raman and FTIR mapping. Furthermore, as for visible energies, the Raman scattering of sp 2 -C is a resonant process between 50 and 230 times more effective than that of sp 3 -C [START_REF] Wagner | Resonance effects in Raman scattering from polycrystalline diamond films[END_REF], Raman spectroscopy at 244 nm was employed to obtain a more evenly weighted probe of sp 3 -C and sp 2 -C sites. A review of the literature shows that there is a lack of understanding of the hydrogenation mechanisms of 1LG and FLG. There are contradictory results and interpretation concerning differences in reactivity for 1LG and FLG [START_REF] Elias | Control of graphene's properties by reversible hydrogenation: evidence for graphane[END_REF][START_REF] Luo | Thickness-dependent reversible hydrogenation of graphene layers[END_REF][START_REF] Ryu | Reversible basal plane hydrogenation of graphene[END_REF][START_REF] Jones | On the mechanism for plasma hydrogenation of graphene[END_REF][START_REF] Wojtaszek | A road to hydrogenating graphene by a reactive ion etching plasma[END_REF][START_REF] Diankov | Extreme Monolayer-Selectivity of Hydrogen-Plasma reactions with Graphene[END_REF][START_REF] Felten | Controlled modification of monoand bilayer graphene in O2, H2 and CF4 plasmas[END_REF]. Hence, it is impossible to compare the results published so far. This is due to differences regarding the graphene materials, substrates, method and conditions used for the hydrogenation. Also, various studies lack of direct detection of C-H bonding and all fail to show that hydrogen is effectively bonded to basal plane.

In this work, we studied the hydrogenation of FLG by the chemisorption of H generated from the dissociation of H2 in a hot filament reactor at low temperature and low pressure, and the subsequent FLG structure conversion into a crystalline sp 3 material. FTIR microscopy was used to track the formation of C-H bonds, and visible and UV Raman spectroscopy and mapping were used to track the structure conversion and the extension of the converted domains. We provide, for the first time, evidences of the crystalline structure of the sp 3 -carbon material obtained from the hydrogenation of FLG at low temperature and at low pressure.

Experimental

Graphene pristine materials

As-received CVD FLG films deposited on 3 mm copper transmission electron microscopy (TEM) grids (2000 Mesh) from Graphene Supermarket (SKU # SKU-TEM-CU-2000-025) were used as graphene materials. The FLG films had been grown by CVD from CH4 at 1000C on Ni substrate and transferred onto a commercial TEM grid using a polymer-free transfer method to minimize contamination as described in Ref. [START_REF] Regan | A direct transfer of layer-area graphene[END_REF]. FLG thickness is typically between 0.3-2 nm (1-6 monolayers), however thicker film, up to 20 layers, were observed in TEM. Typical FLG TEM grid coverage is between 60 and 90 %. For some control experiments, as-received bilayer graphene (2LG) deposited on 2.5 m holey silicon nitride TEM grids from Ted Pella (PELCO® 2 Layer Graphene Prod No. 21722-10) was used as graphene material.

Hydrogenation process

A commercial hot filament reactor from Blue Wave Semiconductors (BWI 1000 model) previously described in details [START_REF] Piazza | Carbon nanotubes coated with diamond nanocrystals and silicon carbide by hot-filament chemical vapor deposition below 200 C substrate temperature[END_REF], was used for hydrogenation and subsequent structure conversion of FLG into ultrathin and crystalline sp 3 -bonded carbon sheets. Briefly the reactor is a six-way cross stainless steel vacuum chamber. It is fluid-cooled (15% water, 75% glycol at 18 C) with brazed copper tubing covered with an aluminum foil. The reactor is equipped with a molybdenum filament cartridge that accommodates between one and three 0.5 mm diameter straight metallic wires. In this study, two 5.7 cm length tungsten wires, 1 cm apart, were used. Prior mounting, the filaments were cleaned with acetone to avoid any carbon contamination. Substrates were placed on the movable and fluid-cooled 5 cm diameter stainless steel sample stage. FLG grids were placed vertically on the substrate holder and were pinched by two (100) silicon pieces (500-550 m thickness, 14 mm x 14 mm) to keep their position fixed during operation. The silicon pieces were previously cleaned in an ultrasound acetone bath for 5 minutes, rinsed in deionized water, and dried in nitrogen at room temperature. The substrate holder and FLG temperatures were estimated by a thermocouple tip (K-type; sheathed wires) located on top of the surface of the substrate holder located in front of the filaments. Before synthesis, the reactor was evacuated down to 10 -1 Torr (0.133 mbar) using a hydrocarbon oil pump (DUO 20M from Pfeiffer Vacuum). For hydrogenation, ultra-high purity H2 gas (about 99.999 % pure) was introduced into the chamber via a stainless steel tube located on top of the chamber. Gas flow was regulated by an Omega mass flow controller. The pressure was regulated via an automatic valve located below the substrate holder. Once H2 was introduced into the chamber and steady-state conditions were reached, the current for both filaments was raised and maintained at 55 A, which resulted in a filament temperature of 2330 °C, as monitored with a two-color pyrometer (M90R2 model from Mikron Infrared, Inc.). The pressure and flow were 50 Torr and 1 sccm, respectively. Before hydrogenation process, the tungsten filaments were not exposed to hydrocarbon gas for carburization or conditioning as it is necessary for the conventional growth of diamond by hot filament CVD. This was to avoid carbon contamination from a carburized filament. The distance between the substrate holder and the filaments was 22.8 mm. The resulting maximum substrate holder temperature was ~300C. Taking into account substrate holder temperature variation with the distance between the substrate and the filaments, the maximum FLG temperature was ~325 C, due to the closer proximity of the TEM grid substrate to the filaments. The duration of the hydrogenation process was 6 minutes and 20 seconds.

Material characterization

Multi-wavelength Raman spectroscopy was employed to examine the material structure before and after hydrogenation. Raman spectra were recorded with two Renishaw InVia Reflex Spectrometer Systems equipped with a stigmatic single pass spectrograph including holographic gratings of 2400 grooves.mm -1 (for visible analysis) and 3600 grooves.mm -1 (for UV analysis) and using the 244, 488 and 514.5 nm lines of Ar ion lasers. For the UV analysis, a 85 Lexel Second Harmonic Generation laser was employed. The scattered light was collected in the 180 backscattering geometry. For visible analysis, a ×50 objective was used while for UV analysis, a ×40 objective was employed. For UV Raman spectroscopy, the detection system was a Peltiercooled UV coated Deep Depletion CCD array detector and the entrance slit was set to 50 m. The resolution was of 1 cm -1 . The laser power on the sample and acquisition time were adjusted to obtain optimum signal without any sample modification. Typically, laser power was in the range of 1 mW, and exposure time to the laser was in the range of 1 s. No visible damage and no change of the spectral shape during measurements have been observed unless exposure time was intentionally raised to induce some structure modification in specific experiments as detailed below. Silicon and highly-oriented pyrolytic graphite (HOPG) were used for peak position calibration: silicon for spectra excited with visible radiation, and HOPG for spectra excited with deep UV radiation. Raman mapping was employed using high-speed encoded mapping stage and a 1" CCD to generate high definition 2D chemical images over thousands of square microns. Probed surface area was typically of ~140100 m 2 . Peaks were fitted using Lorentzian function using Wire 4.4 software from Renishaw.

C-H bonding was directly detected using Fourier Transform Infrared (FTIR) spectroscopy.

FTIR images and spectra were recorded with an Agilent Technologies Cary 670 FTIR spectrometer coupled to a Cary 620 FTIR microscope equipped with a 64×64 focal plane array mercury cadmium telluride detector. Attenuated Total Reflection (ATR) mode, with a germanium crystal as internal reflection element, was employed.

It is worth noting that the converted material, as opposed to the behavior of graphene or diamond, appeared to be electron sensitive, possibly because of the high hydrogenation degree of the surface, even for electron energy as low as 80 keV. Therefore, attempts of TEM-related investigation (including electron diffraction and EELS) could not be conclusive and will not be shown.

Computational Details

The atomic structures, the quasi-particle band structures and optical spectra were obtained from Density Functional Theory calculations using the VASP package [START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF][START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF] and the planeaugmented wave scheme [START_REF] Blöchl | Projector augmented-wave method[END_REF][START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF] to treat core electrons. Perdew-Burke-Ernzerhof (PBE) functional [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] was used as an approximation of the exchange-correlation electronic term for all the geometry optimization steps as well as for phonons and frequencies calculations. The cut-off energy was set to 400 eV, with a Gaussian smearing with a width of 0.05 eV for partial occupancies. To build the wave-function, which served as starting point for further G0W0 calculations [START_REF] Hedin | New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem[END_REF][START_REF] Shishkin | Implementation and performance of the frequency-dependent GW method within the PAW framework[END_REF], the Heyd-Scuseria-Ernzerhof (HSE) exchange-correlation functional [START_REF] Heyd | Assessment and validation of a screened Coulomb hybrid density functional[END_REF][START_REF] Paier | Screened hybrid density functionals applied to solids[END_REF] was used. During geometry optimization step all the atoms were allowed to relax with a force convergence criterion below 0.005 eV/Å. The optimized lattice parameter of graphane, bilayer-diamane and three-layer diamane were 2.53, 2.52 and 2.52 Å, respectively. A vacuum height of 20 Å was used to avoid spurious interaction between periodic images of the different slabs. A 21x21x1 grid was set for G0W0 calculations, in conjunction with a total number of bands of 800, and an energy cutoff of 100 eV for the response function, after a careful check of the direct band-gap convergence, to be smaller than 0.1 eV in function of k-points sampling. To estimate phonon dispersion (7x7) supercells were used with a 3x3x1 grid for k-points sampling, in the Density Functional Perturbation Theory (DFPT) framework, using the Phonopy code [START_REF] Togo | First principles phonon calculations in materials science[END_REF]. C-H stretching mode frequency calculations were performed from a finite difference approach and using a displacement of 0.001 Å.

Nomenclature

Although diamane has been described as a full series of materials structurally derived from hydrogenated bilayer graphene (2LG), the first structure that the hydrogenation of a graphene could form is actually graphane, which consists of a single layer of a hexagonal network of sp 3 -bonded carbon atoms in which each carbon is bonded to one hydrogen atom, alternately above and below the layer. Then comes genuine diamane, which should be limited to the two-layer structure strictly speaking, i.e., where half of the carbons from the top layer are hydrogenated while the other half is bonded to carbon atoms from the bottom layer, in which, consequently, only half of the carbon atoms are hydrogenated as well. Two structural configurations are possible, depending on whether the stacking sequence in the starting 2LG is AB or AA. The former will result in the diamond structurebased diamane (called diamane I in [START_REF] Chernozatonskii | Diamond-Like C2H Nanolayer, Diamane: Simulation of the Structure and Properties[END_REF]), while the latter will result in the lonsdaleite structure-based diamane (called diamane II in [START_REF] Chernozatonskii | Diamond-Like C2H Nanolayer, Diamane: Simulation of the Structure and Properties[END_REF]). As the number of layers increases, because the inner layers cannot be hydrogenated, diamane-related materials should be more accurately described as surfacehydrogenated diamond or lonsdaleite, or more generally diamanoïds, in order to include any mixture of diamond polytypes. However, whereas an AB 2LG can generate a diamond structure, an ABA 3LG cannot. But the diamond structure can develop from ABCAetc. (i.e., the stacking sequence of rhombohedral graphite). On the other hand, lonsdaleite structure can develop from AAA… stacking.

As soon as a diamanoïd is built from more than 2 graphenes, its structure can also result from a mixture of diamond and lonsdaleite. Because of this, beside graphane and diamane (I and II), it is more relevant to designate the subsequent hydrogenated multilayer diamanoïd structures with the number of layers equal to 3, 4 and so on up to "few", by the stacking sequence of the starting graphenes, as suggested in [START_REF] Chernozatonskii | Influence of Size Effect on the Electronic and Elastic Properties of Diamond Films with Nanometer Thickness[END_REF], for instance (ABB)D, (ABBA)D, (ABBC)D, (AABBCC)D, and so on (where "D" stands for "Diamanoïd"). Alternatively, as for graphene, mention could be made of the number of stacked layers without precising the stacking sequence, i.e., 3LD, 4LD, 5LD and so on up to FLD or MLD (where F/MLD stands for "few/multi-layer diamanoïd"). What the limit of "few" is cannot be given so far, because calculations for showing above which number of layers diamanoïd properties no longer discriminate from regular diamond or lonsdaleite are yet to be performed. It is worth noting that this nomenclature does not discriminate whether one surface or both (i.e., top and bottom layers) are hydrogenated. In case it is necessary, 1H-FLD and 2H-FLD could be used to designate one-surface-and both-surface-hydrogenated few-layer diamanoïd, respectively. Likewise, in some conditions, hydrogenation may affect one surface only of the twolayer C(sp 3 ) structure (AB or AA), hence it is no longer genuine diamane. In such a case, this material may be described as "surface hydrogenated 2-layer diamond or lonsdaleite", which may be abbreviated as 1H-2D or 1H-2L, respectively. If fluorination is concerned instead of hydrogenation, "H" can then be substituted by "F" in the proposed nomenclature. Likewise for any other atom or functional group the surface carbon atoms could be bonded to instead of H.

Results and discussion

Visible Raman spectroscopy

Different types of spectra were obtained for a given sample. It is supposed that this is due to the variation in the number of layers, their respective orientation and the subsequent variation in hydrogenation rate [START_REF] Elias | Control of graphene's properties by reversible hydrogenation: evidence for graphane[END_REF][START_REF] Luo | Thickness-dependent reversible hydrogenation of graphene layers[END_REF][START_REF] Ryu | Reversible basal plane hydrogenation of graphene[END_REF]. FIG. 1B shows a typical Raman spectrum obtained after H exposure taken at 488 nm. FIG. 1A shows the spectrum before hydrogenation for comparison, which is typical from pristine FLG [START_REF] Casiraghi | Spectroscopy of graphene[END_REF]. It shows the G and 2D peaks [START_REF] Casiraghi | Spectroscopy of graphene[END_REF].

FIG. 1. Typical Raman spectrum (at 488 nm) of FLG; (A) before and (B) after exposure to the hotfilament-promoted hydrogenation process.

For FLG, in comparison with monolayer graphene (1LG), the 2D to G peak relative intensity ratio is lower (below 1) and the full width at half maximum (FWHM) of the 2D peak is higher [START_REF] Casiraghi | Spectroscopy of graphene[END_REF].

Following the hydrogenation process, significant changes in the spectra can be observed (FIG. 1B).

A narrow D peak of high relative intensity, higher than the G peak, dominates the spectra. The D' and D+D' peaks of lower relative intensity are also observed. Such spectra are obtained on material supported by the copper grid or on the material suspended on top of the grid holes. Those changes in Raman spectra have been previously reported for partially hydrogenated graphene [START_REF] Elias | Control of graphene's properties by reversible hydrogenation: evidence for graphane[END_REF]. The D, D' and D+D' peaks are activated by defects, which could be vacancies, edge defects, or C-H binding [START_REF] Elias | Control of graphene's properties by reversible hydrogenation: evidence for graphane[END_REF]. In the case of 1LG, it was shown that the relative intensity of D and D', or the ratio of their height, can be used to determine the type of defects [START_REF] Eckman | Probing the Nature of Defects in Graphene by Raman Spectroscopy[END_REF]. The ratio of the height of the D and D' peaks, hD/hD', is found to be lower than 2.7. In the case of the spectrum shown in FIG. 1B, this ratio is of 2.63. This ratio cannot be used here to determine the type of defects as FLG are concerned instead of 1LG. However the ratio of the height of the D and G peaks, hD/hG, is 2.13, which is comparable to data previously published for hydrogenated 1LG and 2LG [START_REF] Elias | Control of graphene's properties by reversible hydrogenation: evidence for graphane[END_REF]. Assuming that the layers below the surface layers, which contribute to the G peak intensity, do not contribute to the D peak because defects are created at the surface in the absence of ions in the process, this hD/hG indicates a high content of defects on surfaces, which could be C-H bonding.

A region characterized by the spectrum shown on FIG. 1B was purposely exposed to longer laser irradiation time in order to induce a possible change in the structure such as hydrogen exodiffusion in case defects would be C-H bonding (laser power on the sample below 1 mW; exposure time of 60 s). FIG. 2 shows the spectrum, taken at 488 nm, obtained after such treatment.

FIG. 2. Typical Raman spectrum (at 488 nm) of FLG exposed to the hot-filament-promoted hydrogenation process and then to long time laser irradiation during Raman spectroscopy analysis (see text).

There is a drastic change in the spectrum as compared to the spectrum obtained after a shorter laser exposure (FIG. 1B). The spectrum is composed of a wide band between 1100 and 1750 cm -1 , which includes the D and G bands. The 2D and D+D' bands are no longer observed as in FIG. 1B. This wide band between 1100 and 1750 cm -1 is typical of hydrogenated amorphous carbon (a-C:H) [START_REF] Robertson | Diamond-like amorphous carbon[END_REF][START_REF] Piazza | Transpolyacetylene chains in hydrogenated amorphous carbon films free of nanocrystalline diamond[END_REF][START_REF] Piazza | Transpolyacetylene chains in DECR plasma deposited aC:H films[END_REF][START_REF] Piazza | Incorporation of hydrogen and oxygen into (t)a-C:H thin films deposited using DECR plasma[END_REF] (hydrogen content typically between 20 and 37 at.% [START_REF] Piazza | Incorporation of hydrogen and oxygen into (t)a-C:H thin films deposited using DECR plasma[END_REF]). This result strongly indicates that defects generated in FLG upon H exposure are C-H bonding. Hence it is supposed that hydrogenated FLG are converted into a-C:H under longer laser exposure. This result is consistent with the baseline observed in FIGs 1B and 2 which is likely to be related to the well-known luminescence effect due to CH groups [START_REF] Robertson | Diamond-like amorphous carbon[END_REF][START_REF] Casiraghi | Bonding in hydrogenated diamond-like carbon by Raman spectroscopy[END_REF]. Intensity (a. u.)

Raman shift (cm -1 )

13 FIG. 3 displays other typical spectra, taken at 514.5 nm, of the same sample. There is no G peak nor 2D peak from FLG. However, in the 1000-1800 cm -1 Raman shift region (FIG. 3A-C), the spectrum shows two peaks of low relative intensity on a strong luminescence background: a wide peak centered at ~1360 cm -1 (FIG. 3C) which is interpreted as the D peak from disordered sp 2 -C arrangements [START_REF] Tuinstra | Raman spectrum of graphite[END_REF], and a sharp peak centered at ~1060 cm -1 (FWHM of 8 cm -1 ) (FIG. 3B) which is interpreted as the T peak from sp 3 -C-C, previously reported as a wide and low relative intensity band in UV Raman spectra of ta-C [START_REF] Gilkes | Direct quantitative detection of the sp 3 bonding in diamond-like carbon films using ultraviolet and visible Raman spectroscopy[END_REF][START_REF] Ferrari | Resonant Raman spectroscopy of disordered, amorphous, and diamond-like carbon[END_REF]. A high intensity and narrow peak in the visible Raman spectra of transpolyacetylene has been reported at the same position but it is always associated to another band at 1450 cm -1 [START_REF] Schügerl | Optical modes of trans-polyacetylene[END_REF]. Likewise, a high intensity and narrow peak at 1090 cm -1 was detected in visible Raman spectra of diamond clusters of about 5 nm in diameter, which have been shown to be free of graphite-like sp 2 bonded structures [START_REF] Prawer | The Raman spectrum of nanocrystalline diamond[END_REF]. Since electron energy loss spectroscopy studies showed strong surface plasmon, it was hypothesized that this mode was due to surface plasmon.

However this assumption was not confirmed by observing whether the peak moves to lower frequencies as the dielectric constant of the surrounding medium increases, and that it grows in intensity as the crystalline size decreases. Taking into account that, for visible energies, the Raman scattering of sp 2 -carbon is a resonant process about 50 to 230 times more effective than that of sp 3carbon [START_REF] Wagner | Resonance effects in Raman scattering from polycrystalline diamond films[END_REF], both the absence of high relative intensity sp 2 -C related features in visible Raman spectra and the presence of a sharp T peak indicate the conversion of sp 2 -carbon into sp 3 -carbon.

This will be further discussed below. In the 2600-3200 cm -1 Raman shift region, the spectrum displays two bands with low relative intensities, centered at ~2853 and ~2940 cm -1 respectively. The first band was predicted for diamane [START_REF] Chernozatonskii | Determination of ultrathin diamond films by Raman spectroscopy[END_REF] while the second one is interpreted as the C-H band [START_REF] Ferrari | Resonant Raman spectroscopy of disordered, amorphous, and diamond-like carbon[END_REF][START_REF] Casiraghi | Raman spectroscopy of hydrogenated amorphous carbons[END_REF],

which is consistent with the strong luminescence background (FIG. 3).

FIG. 3. Typical visible Raman spectra of FLG exposed to the hot-filament-promoted hydrogenation process (at 514.5 nm); (A) full spectrum between 1000 and 3250 cm -1 ; (B) zoom of the T peak spectral region; (C)

zoom of the 1300-1700 cm -1 spectral region; (D) zoom of the 2600-3200 cm -1 spectral region.

In some region of the same sample, additionally to the T and G peaks, two others peaks centered at ~1290 and ~1317 cm -1 are observed (FIG. 4). Those peaks have been previously ascribed to lonsdaleite in both experimental [START_REF] Smith | UV and VIS Raman spectra of natural lonsdaleites: Towards a recognized standard[END_REF][START_REF] Gogotsi | Pressure-induced phase transformations in diamond[END_REF][START_REF] Spear | Diamond polytypes and their vibrational spectra[END_REF] and calculation works [START_REF] Chernozatonskii | Determination of ultrathin diamond films by Raman spectroscopy[END_REF][START_REF] Knight | Characterization of diamond films by Raman spectroscopy[END_REF].

FIG. 4. Typical visible Raman spectra of FLG exposed to the hot-filament-promoted hydrogenation process (at 514.5 nm).

The features shown in visible Raman spectra shown in FIGs. 3 and 4 are further evidences of the formation of C-H bonds and other structural transformation of FLG as further discussed below.

FTIR microscopy

To directly detect C-H bonding, FTIR mapping was carried out. FIG. 5A shows a FTIR-ATR image processed on the integrated intensity of C-H stretching band. The image was obtained on the plain part of the TEM grid which circles the grid mesh, and which was exposed to H radicals. Prior treatment, this region of the grid contains FLG as evidenced by Raman spectroscopy analysis (FIG.

FIG. 5. (A) Typical FTIR-ATR microscopy image processed on the integrated intensity of C-H stretching band of FLG exposed to the hot-filament-promoted hydrogenation process; (B) Typical absorbance FTIR spectrum (cumulated over 5 spectra) taken in the region arrowed in (A).

FIG. 5B shows that the narrow C-H stretching band, which is centered at around 2846 cm -1 , is composed of only one among the nine possible vibration mode components. It is proposed that this mode corresponds to the C(sp 3 )-H stretching mode. This assignment is based on the following arguments:

1. Taking into account vibration modes in free molecules [START_REF] Fox | Investigations of infra-red spectra. Determination of C-H frequencies (~3000 cm-1) in paraffins and olefins, with some observations on "polythenes[END_REF], within the wavenumber range of 2800 to 2900 cm -1 , it is expected to detect the following vibration modes: C(sp 3 )-H centered at ~2900 cm -1 , C(sp 3 )-H2 symmetrical centered at ~2875 cm -1 and C(sp 3 )-H3 symmetrical centered at ~2850 cm -1 . However, although the infrared (IR) absorption cross-section of the C-H stretching vibration modes is unknown in hydrogenated FLG and in ultrathin and crystalline sp 3 -bonded carbon sheets, it is expected that the C(sp 3 )-H2 symmetrical and C(sp 3 )-H3 symmetrical modes would be simultaneously detected with their antisymmetrical counterpart, which is not the case since the observed C-H stretching band is composed of only one vibration mode component. Therefore, the C(sp 3 )-H2 symmetrical and C(sp 3 )-H3 symmetrical modes are excluded.

2. The C(sp 2 )-H olefinic and C(sp 2 )-H aromatic modes are not considered as they are typically detected at much higher wavenumber, above 2975 cm -1 in the case of free molecules [START_REF] Fox | Investigations of infra-red spectra. Determination of C-H frequencies (~3000 cm-1) in paraffins and olefins, with some observations on "polythenes[END_REF].

3. The C2H2 olefinic symmetrical and antisymmetrical modes are also typically detected at significantly higher wavenumber, above 2950 cm -1 , in the case of free molecules [START_REF] Fox | Investigations of infra-red spectra. Determination of C-H frequencies (~3000 cm-1) in paraffins and olefins, with some observations on "polythenes[END_REF]. Furthermore, they would be accompanied by their anti/symmetrical counterpart, which is not the case here, since the observed C-H stretching band is composed of only one vibration mode component. The detection of the C(sp 3 )-H stretching mode is a remarkable result since the IR absorption crosssection for this mode is expected to be significantly lower than the values for the other modes, as in free molecules [START_REF] Fox | Investigations of infra-red spectra. Determination of C-H frequencies (~3000 cm-1) in paraffins and olefins, with some observations on "polythenes[END_REF]. In free molecules, the IR absorption cross-section for the C(sp 3 )-H stretching mode is between 18.6 and 31.6 times lower than the corresponding value for C(sp 3 )-H3 and C(sp 3 )-H2 stretching modes [START_REF] Fox | Investigations of infra-red spectra. Determination of C-H frequencies (~3000 cm-1) in paraffins and olefins, with some observations on "polythenes[END_REF]. The shift, by around 54 cm -1 , of the position of the C(sp 3 )-H mode peak, as compared to the corresponding position in free molecules [START_REF] Fox | Investigations of infra-red spectra. Determination of C-H frequencies (~3000 cm-1) in paraffins and olefins, with some observations on "polythenes[END_REF], is theorized to result from the structure change generated by the hydrogenation of FLG, the subsequent conversion of sp 2 hybridization into sp 3 hybridization, and the interlayer sp 3 -C bond formation as further discussed below.

The narrow, one component C-H stretching band that carbon atoms are bonded to one hydrogen atom. This single component narrow C-H stretching band has never been reported before in the case of hydrogenated graphene. Known disclosures on FTIR spectroscopy analysis of hydrogenated graphene report on a wide multi-component C-H stretching band including C(sp 3 )-H3 and C(sp 3 )-H2 modes instead of C(sp 3 )-H mode [START_REF] Zheng | Catalytic hydrogenation of graphene films[END_REF][START_REF] Poh | High-pressure hydrogenation of graphene:towards graphane[END_REF][START_REF] Subrahmanyam | Chemical storage of hydrogen in few-layer graphene[END_REF][START_REF] Yang | Birch Reduction of Graphite. Edge and Interior Functionalization by Hydrogen[END_REF][START_REF] Schäfer | On the Way to Graphane-Pronounced Fluoresence of Polyhydrogenated Graphene[END_REF][START_REF] Krishna | Facile synthesis of hydrogenated reduced Graphene oxide via hydrogen spillover mechanism[END_REF][START_REF] Sofer | Highly hydrogenated graphene via active hydrogen reduction of graphene oxide in the aqueous phase at room temperature[END_REF], consistent with graphene domains of reduced size which are hydrogenated only on their edges. Comparatively, the relatively intense one component narrow C-H stretching band shown here is representative of large-size (~150 µm 2 ) graphene hydrogenated on faces so that the related FTIR signal prevails over that generated by the hydrogenated graphene edges.

UV Raman spectroscopy

As pointed out in the Introduction section, in the visible, Raman scattering is much more effective in probing sp 2 -carbon than sp 3 -carbon. UV excitation is necessary to obtain a more evenly weighted probe of both types of hybridized-carbon and to avoid strong luminescence background. broad band [START_REF] Gilkes | Direct quantitative detection of the sp 3 bonding in diamond-like carbon films using ultraviolet and visible Raman spectroscopy[END_REF][START_REF] Ferrari | Resonant Raman spectroscopy of disordered, amorphous, and diamond-like carbon[END_REF] or as a high-intensity and narrow peak in diamond nanoclusters [START_REF] Prawer | The Raman spectrum of nanocrystalline diamond[END_REF] and which was detected under visible excitation as shown above (FIGs. 3A and 3B). The peak centered at ~1323.1 cm -1 could be interpreted as corresponding to two vibrational modes of lonsdaleite [START_REF] Chernozatonskii | Determination of ultrathin diamond films by Raman spectroscopy[END_REF][START_REF] Smith | UV and VIS Raman spectra of natural lonsdaleites: Towards a recognized standard[END_REF][START_REF] Gogotsi | Pressure-induced phase transformations in diamond[END_REF][START_REF] Knight | Characterization of diamond films by Raman spectroscopy[END_REF] which were detected under visible excitation as shown above (FIGs. 4). Those modes are the A1g and E2g modes, corresponding to the triply-degenerate stretching mode of cubic diamond, which are, in lonsdaleite, split into a component vibrating in the plane of the layers and a component vibrating perpendicular to the layers [START_REF] Knight | Characterization of diamond films by Raman spectroscopy[END_REF]. Unlike graphite, the bonding strength of lonsdaleite is comparable whether it is parallel or perpendicular to the layers and the wavenumber separation between the A1g and E2g modes is expected to be small [START_REF] Knight | Characterization of diamond films by Raman spectroscopy[END_REF]. Alternatively, the peak could be interpreted as corresponding to the diamond peak [START_REF] Wagner | Resonance effects in Raman scattering from polycrystalline diamond films[END_REF] which would be shifted because of stress [START_REF] Prelas | Handbook of Industrial Diamonds and Diamond Films[END_REF] or because of confinement effects in few atomic layers or in sub-10 nm nanocrystals [START_REF] Ager | Spatially resolved Raman studies of diamaond films grown by chemical vapor deposition[END_REF][START_REF] Osswald | Phonon confinement effects in the Raman spectrum of nanodiamond[END_REF]. Here, the T peak is narrow and of high relative intensity, indicating that the structure of the C(sp 3 ) in the material is crystalline, as opposed to the FWHM of the T peak in ta-C which is wide because of the amorphous structure. The C(sp 3 ) features in Raman spectra are consistent with the detection of only one component in the FTIR C-H stretching band as shown above (FIG. 5). Another example is provided as FIG. 7C, with the T peak at 1061.6 cm -1 (FWHM of 17.3 cm -1 ) and the diamond/lonsdaleite peak at 1323.5 cm -1 (FWHM of 30.9 cm -1 ). Hence, the Raman spectroscopy results show that it is possible to form crystalline sp 3 -carbon ultrathin sheets from the hydrogenation of FLG at low temperature and low pressure and the subsequent formation of interlayer C(sp 3 ) bonds. The relatively large FWHM value of the diamond/lonsdaleite peak may reveal confinement effects in few atomic layers or in sub-10 nm nanocrystals [START_REF] Ager | Spatially resolved Raman studies of diamaond films grown by chemical vapor deposition[END_REF][START_REF] Osswald | Phonon confinement effects in the Raman spectrum of nanodiamond[END_REF] or diamond/lonsdaleite hybrid structure. The laminar character of diamond/lonsdaleite will be further discussed below.

In most of the regions, the T and diamond/lonsdaleite peaks are simultaneously detected as shown in FIGs.7B to 7E and in FIG. 8. From the analysis of 329 spectra taken from four maps and three samples, variations in peak position are observed. The position of the T and diamond/lonsdaleite peaks is found to vary within a range of 16.4 cm -1 and 17.5 cm -1 , respectively and, more interestingly, both peak positions vary in parallel so that there is a clear correlation between them (FIGs. 8 and9). In addition, the diamond/lonsdaleite peak is found to vary from 1319.5 to 1337 cm -1 , therefore including 1332 cm -1 which is the value for diamond [START_REF] Wagner | Resonance effects in Raman scattering from polycrystalline diamond films[END_REF]. However, those two results do not allow clearly identifying the actual structure, as they are consistent with the occurrence of both lonsdaleite and diamond as well as a variable combination of both. The C(sp 3 ) symmetry obtained upon hydrogenation and subsequent C(sp 2 ) to C(sp 3 ) transformation could indeed depend primarily on the relative orientation of the superimposed graphenes in the primary In some regions of the grid, the T peak and the lonsdaleite or diamond peak are simultaneously detected along with the graphene G peak at ~1582 cm -1 (FIGs. 7C and 7E, respectively). The relative intensity of both the lonsdaleite or diamond peak and that of the G peak varies depending on the region probed by the laser. In some cases, the intensity of the G peak is several time higher than that of the T and diamond/lonsdaleite peaks. Those observations along with FIG. 9 suggest that it is possible to prepare graphene-lonsdaleite/diamond hybrids from the partial conversion of FLG into lonsdaleite, or diamond, or both. It is hypothesized that the variation of the relative intensity of the peaks related to C(sp 2 ) and C(sp 3 ) may be indicative of the sp 2 /sp 3 ratio in the hybrid material. Also, the UV spectra (FIGs. 7B-E) show a peak at 1620 cm -1 , which was previously attributed to a particularly stable defect in diamond consisting of an isolated sp 2 bonded pair [START_REF] Saada | Transformation of diamond (sp 3 ) to graphite (sp 2 ) bonds by ionimpact[END_REF].

In order to verify the physical origin of the T peak, a region of 2LG exposed to the hotfilament promoted hydrogenation process where this peak was the only one detected was overexposed to the UV laser in order to induce a possible structure transformation. In fact the measurement was repeated on the same spot under the same conditions (6 mW power at the entrance of the microscope; duration of 1 s). FIGs. 10A and 10B show the spectrum of that region after the first and repeated measurements, respectively. After the second measurement, the G peak appears in the spectrum (FIG. 10B). This experiment confirms that the T peak is from C(sp 3 )-C(sp 3 ). 11A shows that diamond/lonsdaleite stretching mode Raman peak can be detected in multiples regions in a surface area of ~220x140 m 2 . FIG. 11B and FIG. 11C show some examples of extended regions of ~33x51 m 2 (FIG. 11B), ~21x33 m 2 (FIG. 11C) and ~15x15 m 2 (FIG. 11C), where lonsdaleite or diamane or diamond are continuously detected. These dimensions are consistent with that of the hydrogenated region revealed by FTIR microscopy (FIG. 5).

Although electron microscopy analyses would help confirming this hypothesis, the fact that C(sp 3 )-H3 and C(sp 3 )-H2 vibration modes that should be detected in hydrogenated crystalline sp 3 clusters, were not detected in FTIR spectra although their IR absorption cross-section are expected to be significantly higher than the values for the C(sp 3 )-H mode, as in free molecules [START_REF] Fox | Investigations of infra-red spectra. Determination of C-H frequencies (~3000 cm-1) in paraffins and olefins, with some observations on "polythenes[END_REF], strongly suggest that part of the laminar character of the material has been conserved upon sp 2 to sp 3 conversion. The possibility of producing few-layer crystalline sp 3 -bonded carbon at low temperature and at low pressure, by the exposure of FLG to the hot-filament-promoted hydrogenation process is of great technological interest for a wide range of applications (see section 1) [100]. Our results have also wider scientific implications. For instance, they may help at clarifying the origin and formation mechanism(s) of meteoritic and interstellar diamond dust grains, including those supposed to be presolar [START_REF] Daulton | Extraterrestrial Nanodiamonds in the Cosmos[END_REF]. As reviewed in Ref because of the high reactivity of C24 [START_REF] Da | The formation of fullerenes: clues from new C60, C70, and (possible) planar C24 detections in magellanic could planetary nebulae[END_REF]; also, recently, it has been speculated from laboratory experiments that FLG grains could be formed in space from pyrene [START_REF] Zhao | Pyrene synthesis in circumstellar envelopes and its role in the formation of 2D nanostructures[END_REF].

Conclusion

Thanks to combined Raman spectroscopy and FTIR microscopy investigations, the possibility of obtaining various crystalline C(sp 3 ) structures (diamond and/or lonsdaleite and/or diamane and/or hybrids of those) from a C(sp 2 ) structure (few-layer graphene film) at low temperature and low pressure was demonstrated, by means of a hydrogenation process. This statement is reasonably ascertained from mostly two unprecedented observations: (i) The combined occurrence of both a sharp so-called T peak (at ~1055-1071 cm -1 , assigned to C(sp 3 )-C(sp 3 ) bonds) and a diamond/lonsdaleite stretching mode Raman peak (at 1319-1337 cm -1 , usually assigned to lonsdaleite or diamond) as evidenced by Raman over sample surfaces as large as 1,683 µm 2 ; (ii) the occurrence of a single vibration mode in the C-H stretching band, hence assigned to the prevalent presence of C(sp 3 )-H groups over any other group involving more than one H, over sample surfaces as large as 150 µm 2 .

More in the details, the work also brought a couple of other new experimental observations and findings: (i) a 54 cm -1 shift of the C(sp 3 )-H mode as compared to the case of free molecules is observed, as a consequence of the sp 2 -to-sp 3 conversion upon hydrogenation and subsequent difference in resulting atomic environment with respect to small, isolated molecules; (ii) in addition to the formation of isolated crystalline and ultrathin sp 3 -carbon sheets over surfaces as large as 33x51 µm 2 , the combined signals of graphene and crystalline and ultrathin sp 3 -carbon sheets are also observed, suggesting the formation of crystalline and ultrathin sp 3 -carbon sheets /graphene hybrid materials. Further studies are however needed to ascertain in which extent the film morphology was maintained; (iii) the sp 2 -to-sp 3 transformation and resulting structure (diamond, or lonsdaleite, or hybrid) is assumed to depend on the number of graphenes in the former FLG and the relative orientation of the stacked layers.

Finally, the work presented here makes a significant technological advance by introducing the hot filament process for the efficient production of atomic hydrogen from gaseous H2, applied to the efficient hydrogenation of FLG and the subsequent formation of crystalline and ultrathin sp 3carbon sheets as a new route for the C(sp 2 )-to-C(sp 3 ) allotropic transformation. It also makes a significant scientific advance by providing a new clue for understanding the origin of sp 3 carbon structures in specific (e.g., extraterrestrial) environments.

  1A). FIG. 5A reveals a large-sized circular area of ~150 m 2 containing C-H bonding, which first confirms that defects generated during the hydrogenation process are indeed C-H bonding and, second, indicates that hydrogenation took place in the basal plane of graphene, not only at the edges of graphene domains. FIG. 5B displays one absorbance spectra taken in the region identified by a white arrow in FIG. 5A.

4 .TAB. 1 .

 41 Our experimental value is consistent with calculated C-H stretching mode frequency for graphane, 2H-(AB)D, (AB)D, and 2H-(ABC)D (TAB. 1). The C-H stretching mode frequencies of isolated Hon graphene and on single vacancy in graphene are found to be much higher giving a further evidence of high hydrogenation degree. Only the C-H stretching mode frequency of AB stacking in 2H-(AB)D and ABC stacking in 2H-(ABC)D were calculated as we found that AB stacking is more stable than AA stacking, and that ABC stacking is more stable than AAA and ABB stacking. Calculated C-H stretching mode frequency in various structures.

TAB. 2 3 TAB. 2 .FIG. 6 .

 2326 FIG. 6. Phonon dispersion curve of 2H-(ABC)D.

FIG. 7 .

 7 FIG. 7. Typical UV Raman spectra of FLG (A) before and (B-E) after the exposure to the hot-filamentpromoted hydrogenation process (at 244 nm).

FIG. 8 .

 8 FIG. 8. Comparison of UV Raman maps of (A) the intensity of the T peak with (B) the intensity of thediamond/lonsdaleite peak in the same region (at 244 nm).

FIG. 9 .

 9 FIG. 9. Diamond/lonsdaleite stretching mode Raman peak position as a function of the T peak position inFLG exposed to the hot-filament-promoted hydrogenation process (spectra taken at 244 nm).

FIG. 10 .

 10 FIG. 10. Typical Raman spectra (at 244 nm) of a 2LG film exposed to the hot-filament-promoted hydrogenation process after, (A) a first measurement and (B) a repeated measurement.

m 2 CFIG. 11 .

 211 FIG. 11. Typical UV Raman maps of the intensity of the diamond/lonsdaleite stretching mode Raman peak of FLG exposed to the hot-filament-promoted hydrogenation process (at 244 nm).

  [102], "Opinions regarding the formation mechanism(s) of meteoritic and interstellar diamond grains have changed over the years as new methods of making synthetic diamond were developed…". Our results could reveal the existence of an additional formation route of diamond grains wherein FLG grains are converted into lonsdaleite, diamond or other polytypes by exposure to H flux without requiring high pressure. Those conditions can actually exist in space, for instance in the circumstellar envelope of C-rich evolved stars. And then, there are clues for the needed pre-existence of FLG in space: the possible extragalactic detection of planar C24 was reported by using the Spitzer Space Telescope although it has not yet been confirmed through laboratory infrared spectroscopy at different temperatures, which is extremely difficult

Acknowledgements

This research was funded by the Ministry of Higher Education, Science and Technology of the Dominican Republic (2010-2011, 2012 and 2015 FONDOCyT programs). F.P. and M.M. greatly acknowledge MESCyT and PUCMM for strong financial, administrative and technical support; 2017 NEXT Guest scientist Program (www.next-toulouse.eu/guest-scientists) and the French Embassy in the Dominican Republic for travel support. F.P. is thankful to Richard Bormett from Renishaw for technical support with Raman spectroscope. I.C.G. acknowledges the Calcul en Midi-Pyrénées initiative, CALMIP (Project p0812), for computer time allocation. This work was also granted access to the HPC resources of CINES and IDRIS under the allocation 2018-A0040906649 made by GENCI.


Uwe Ehrenstein

François Gallaire

Global oscillations in a separating flow Global low-frequency oscillations in a separating boundarylayer flow

Keywords: Boundary-layer separation, global instability, non-normality

A separated boundary layer flow at the rear of a bump is considered and twodimensional flow states at increasing Reynolds numbers are computed using a nonlinear continuation procedure for the stationary Navier-Stokes system. The global instability analysis of the steady states is performed by computing two-dimensional temporal modes. The analysis reveals non-normal modes which are able to describe localized initial perturbations associated with large transient energy growth. At larger time a global low-frequency oscillation is found accompanied with periodic regeneration of the flow perturbation inside the bubble, as the consequence of nonnormal cancellation of modes. The initial condition provided by the optimal perturbation analysis is applied to Navier-Stokes time integration and is shown to trigger nonlinear 'flapping' typical for separation bubbles.

Introduction

There is general evidence that laminar detached boundary layers are likely to undergo two-dimensional low-frequency global oscillations [START_REF] Dovgal | Laminar boundary layer separation: instability and associated phenomena[END_REF] which have been observed both in experiments, whether the separation bubble is triggered by leading-edge geometries ( [START_REF] Cherry | Unsteady measurements in a separating and reattaching flow[END_REF]) or by adverse pressure gradients ( [START_REF] Häggmark | Experiments on a two-dimensional laminar separation bubble[END_REF], [START_REF] Pauley | The structure of two-dimensional separation[END_REF]). The physical mechanisms at the origin of this type of instability which occurs above a critical Reynolds number are only partially understood. For instance, the frequencies associated with possible transition from convective to absolute instability of local velocity profiles with a certain amount of reverse flow appear to be higher than the typical frequencies of the oscillations known as 'flapping' (cf. [START_REF] Hammond | Local and global instability properties of separation bubbles[END_REF], [START_REF] Rist | Investigations of time-growing instabilities in laminar separation bubbles[END_REF]). By providing some evidence for the appearance of secondary recirculating flow regions at the rear of the main bubble, it has been conjectured ( [START_REF] Dallmann | Flow field diagnostics: topological flow changes and spatio-temporal flow structure[END_REF], [START_REF] Theofilis | On the origins of unsteadiness and threedimensionality in a laminar separation bubble[END_REF]), that topological flow changes might be responsible for the overall instability behaviour. This possibility has been explored in [START_REF] Marquillie | On the onset of nonlinear oscillations in a separating boundary-layer flow[END_REF] for a separated flow induced by a bump mounted on a flat plate but no clear-cut confirmation of topological flow changes could be given, in the absence of a basic state above criticality. In this work we readdress the low-frequency oscillations for an elongated separation bubble induced by the bump geometry which has already been considered in [START_REF] Marquillie | On the onset of nonlinear oscillations in a separating boundary-layer flow[END_REF]. We focus on the two-dimensional global instability behaviour. The three-dimensional transverse instability characterized by a global steady and weakly growing eigenmode has recently been analyzed in [START_REF] Gallaire | Three-dimensional transverse instabilities in detached boundary-layers[END_REF]. The resulting longitudinal instability first takes place when increasing the Reynolds number, which has also been observed in backward-facing step flows ([3], [START_REF] Barkley | Three-dimensional instability in flow over a backward-facing step[END_REF]). However, the low-frequency fluctuations which lead to an overall motion of the separation bubble is the dominant instability mechanism at higher Reynolds numbers. The analysis combines a quasi-Newton approach to determine steady states for Reynolds numbers above criticality and a numerical method suitable for computing two-dimensional temporal modes. Section 2 is devoted to the description of the numerical tools. The global stability results are discussed in Section 3 and an optimal perturbation analysis is performed in Section 4. Optimal initial conditions are then used in the direct numerical simulation procedure and are shown to produce low-frequency 'flapping' described in Section 5. Some conclusions are provided in Section 6.

Numerical tools

The flow domain is 0 ≤ x ≤ L, η(x) ≤ y ≤ H, with η(x) the lower boundary containing the bump which has already been considered in [START_REF] Marquillie | On the onset of nonlinear oscillations in a separating boundary-layer flow[END_REF] (cf. figure 1 for the geometry). The height H has been chosen large enough in order to recover uniform flow, the Navier-Stokes system being made dimensionless using the displacement thickness δ * at inflow, where the Blasius profile is described, and the uniform freestream flow velocity U * ∞ . The dimensionless bump height is h = 2 and the mapping

x = 2 L x -1, ȳ = 1 H (1 -γ)y + γ with γ = H + η η -H (1)
transforms the domain into [-1, 1] × [-1, 1]. A Chebyshev-Chebyshev collocation discretization is used for the transformed system, together with a stretching in the wall-normal direction which redistributes the collocation points in order to take into account the boundary-layer structure. Here, we focus on nonlinear equilibrium states of the stationary Navier-Stokes system for increasing Reynolds number Re = δ * U * ∞ /ν and hence the flow velocity u = (u, v) and pressure p are solution of

f (u, p, Re) = -(u • ∇)u -∇p + (1/Re)∇ 2 u ; ∇ • u = 0. (2)
Homogeneous Neumann boundary condition for the flow field u are imposed at outflow x = L, whereas at y = H uniform flow u = (1, 0) is prescribed and the noslip condition applies on the wall boundary. At inflow the Blasius profile (U (y), 0) is imposed. The Chebyshev-Chebyshev discretization is efficient in terms of precision versus grid-size. The problem of spurious pressure modes associated to a Chebyshev discretization in the square is overcome using extra-conditions, by imposing the continuity of the normal derivative of the pressure at the corners and by eliminating the four modes for which the gradient vanishes due to the Chebyshev discretization (cf [START_REF] Peyret | Spectral Methods for Incompressible Flows[END_REF]). A quasi-Newton method, here the Broyden rank-one update procedure (cf [START_REF] Stoer | Introduction to Numerical Analysis[END_REF]), is considered to solve the system (2) for the steady state by adding the Reynolds number as a parameter in an arc-length continuation procedure. The linear systems to be solved during the iterations in the quasi-Newton approach involve the Jacobian matrix

A(u, Re) = D (u,p) f (u, Re) (3)
which is evaluated at an initial guess of the flow field and a QR decomposition is performed. The rank-one update of the decomposition at successive iterations may easily be performed with little extra computational cost (cf. [START_REF] Stoer | Introduction to Numerical Analysis[END_REF]). Once the solution converged to a steady state (u s , p s ), its stability is computed by considering two-dimensional temporal modes

u = û(x, y)e -iωt , p = p(x, y)e -iωt . (4)
Taking into account in the Jacobian evaluated at a steady state u s that the flow perturbation is zero at inflow, the modes are solution of the generalized eigenvalue problem

-iωBq = A(u s , Re)q (5)
with q = (û, p) and Bq = (û, 0). The resulting large eigenvalue problem is solved using large-scale Krylov subspace projections together with the Arnoldi algorithm similar to the approach used in [START_REF] Åkervik | Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes[END_REF]. In all the computations the length of the computational domain is L = 300 and the height H = 30. The length of the box proved to be large enough to minimize effects of the box-size on the stability results. The modes are discretized using 250 collocation points in x and 40 collocation points in y and a Krylov subspace with dimension m = 1600 has been considered.

Basic states and global instability analysis

The basic states at different Reynolds numbers are depicted in figure 1, the recirculation length increasing with the Reynolds number. In [START_REF] Marquillie | On the onset of nonlinear oscillations in a separating boundary-layer flow[END_REF] no steady state could be obtained using time integration of the Navier-Stokes system above a Reynolds number of 610 and it has been speculated whether the observed global oscillations were somehow connected with topological flow changes [START_REF] Theofilis | On the origins of unsteadiness and threedimensionality in a laminar separation bubble[END_REF]. With the present nonlinear continuation procedure flow states are computed for higher Reynolds numbers in the unstable range. The results to be discussed hereafter show that the flow at Re = 620 is indeed unstable. But even at Re = 670 (cf figure 1c)) there is no evidence of a change in topology.

Once a flow state obtained, its stability is computed with the method outlined in Section 1. Figure 2a) shows the spectrum at Re = 590: There are several weakly unstable modes and this Reynolds number hence appears to be slightly above the margin of instability. In figure 2b) only the unstable parts of the spectrum are depicted for comparison at Re = 590, 620, the amplification rates increasing with the Reynolds number. The modes labelled from 1 to 6 are shown in figure 3, figure 3a) depicting the streamlines of the steady state at Re = 590. One observes that the modes originate approximately at the center of the recirculation bubble. While the mode labelled (1) in figure 2a) at a lower frequency reaches the outflow boundary (cf. figure 3b), the modes (2) to [START_REF] Dovgal | Laminar boundary layer separation: instability and associated phenomena[END_REF] with equally spaced frequencies have the same type of structure, the spatial spreading of the modes decreasing with increasing frequency. The similarity of the eigenmode structure is a typical feature of nonnormal operators and the next section will describe the implication of this.

Optimal growth

When addressing the possibility of growth in a flow system, the notions of optimal initial condition and non-normality of the underlying operators are essential [START_REF] Schmid | Stability and Transition in Shear Flows[END_REF].

We are looking for initial disturbances u 0 that maximize the energy at time t

G(t) = max u 0 =0 ||u(t)|| 2 E ||u 0 || 2 E ( 6
)
and a convenient form of this expression can be obtained by expanding the solution in terms of the generalized eigenmodes u(t) = N l=1 κ l (t)û l with ûl solution of (5). Hence the flow dynamics is described by

dκ dt = Λκ, κ(0) = κ 0 , (7)
where κ is the vector of expansion coefficients and Λ is a diagonal matrix whose elements are given by Λ l = -iω l . The flow perturbation energy in this basis is

||u|| 2 E = ||F exp(Λt)κ 0 || 2 2
, where F is the Cholesky factor of the Hermitian energy measure matrix M with entries M ij = ûH i ûj dxdy. Hence, the maximum growth expressed in the basis of eigenmodes reads

G(t) = ||F exp (Λt)F -1 || 2 2 . ( 8
)
and the largest growth at time t is given by the largest singular value of F exp (Λt)F -1 and the optimal initial condition κ 0 is the corresponding right singular vector providing the optimal initial flow condition u 0 through the eigenmode expansion. The optimal energy gains G(t) are depicted in figure 4, for Re = 590 and Re = 620. The large Krylov subspace procedure (with m = 1600) gives rise to a set of eigenvalues and its convergence in terms of optimal growth dynamics has been assessed. At Re = 590 for instance the results with N = 337 modes and the gain obtained for a much lower truncation at N = 150 (starting with the most unstable modes) are very close. One observes a fast initial energy growth of a magnitude of almost 10 9 followed by a global cycle with a period close to 200. This global undulation is accompanied by a weak energy growth due to the amplification rates of the individual modes.

Inspecting figure 2, one observes that the real parts of the unstable eigenvalues in the right half of the spectrum (the modes labelled from ( 2) to ( 6)) are distant of about δ ≈ 0.03 and the structures of the corresponding eigenmodes are similar. The superposition of these modes as provided by the optimal perturbation analysis gives rise to cancelling leading to the global period of 2π/δ ≈ 200. Similar non-normal effects have also been described in a recent work on a separated flow in a cavity-like geometry [START_REF] Åkervik | Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes[END_REF].

To illustrate the dynamical behaviour, the time evolution of the perturbation in the eigenmode system is depicted in figure 5, starting with the optimal initial condition which is mainly located in the vicinity of the rear part of the bump. The perturbation evolves along the plate as a localized wavepacket accompanied with a tremendous increase of energy and at approximately t = 250 it is leaving the recirculation bubble.

At the same time it reappears in the the rear part of the bubble, which is visible at t = 320. The perturbation than evolves downstream (cf. structure at t = 470) before it reappears again in the bubble (cf. t = 520) and the cycle restarts. The modes responsible for the oscillating flow pattern are individually weakly unstable which leads to the overall growth shown in figure 4 for t > 200.

Direct numerical simulation dynamics

Focusing on the relation between optimal energy growth and the low-frequency oscillations observed in previous investigations of the separation bubble, the optimal initial condition has been considered in the direct numerical simulation (DNS) procedure of the Navier-Stokes system used in [START_REF] Marquillie | On the onset of nonlinear oscillations in a separating boundary-layer flow[END_REF]. The flow at Re = 590 is considered, which according to the present analysis is slightly supercritical. As shown in [START_REF] Marquillie | On the onset of nonlinear oscillations in a separating boundary-layer flow[END_REF], at this Reynolds number it was however possible to converge by time marching to a steady state using the DNS procedure. Weak stabilization effects might be attributed to the influence of the outflow boundary of the shorter domain considered in the DNS (with L = 200), where a classical convection condition is applied. The streamwise direction is discretized using fourthorder finite differences with 1024 equidistant grid points. In y the region up to H = 80 is covered (with 97 Chebyshev-collocation points) for the solution to be uniform in the upstream region, a simple linear coordinate transform being employed. The gain in energy being expected to be close to 10 9 , the optimal initial condition has been affected with a small maximum amplitude of 5 10 -5 (however above the residual noise of order of 10 -6 in the time-marching towards the steady basic state u b ). The energy of the perturbation u = uu b has been integrated in the whole domain and the gain is depicted in figure 6. Up to t = 200 the energy growth follows closely the curve provided by the eigenmode system before nonlinear saturation occurs.

The integration has been pursued up to t = 3000 and the flow is seen to remain in a nonlinear state. When entering the nonlinear saturation, the oscillations visible in the global energy curve are somehow reminiscent of the global oscillations in the envelope curve provided by the modes. The amplitude of the nonlinear flow perturbation, by simply subtracting the base flow, along the plate at increasing time has been computed and the results are shown in figure 7. At t = 100 the perturbation is similar to that depicted in figure 5 for the dynamical system formed with the global modes. At t = 300 a regeneration behaviour of the perturbation is visible upstream the previous wavepacket at t = 250. The perturbation is highly nonlinear and is hence far from being a superposition of eigenmodes.

In [START_REF] Marquillie | On the onset of nonlinear oscillations in a separating boundary-layer flow[END_REF] low-frequency oscillations of the bubble have been reported at the supercritical Reynolds number Re = 650. In the present analysis similar fluctuations are observed as a consequence of the initial optimal perturbation at Re = 590. Figure 8 depicts the amplitude of the Fourier transform for the streamwise velocity profile at x = 60, y = 1, that is at a position which corresponds approximately to the center of the steady bubble. The low frequency f = δ/(2π) ≈ 0.005 corresponds to the non-normal 'beating' frequency in the eigenmodes system and indeed the spectrum exhibits peaks in this frequency range, the nonlinear flow being highly aperiodic. Similar to the results reported in [START_REF] Marquillie | On the onset of nonlinear oscillations in a separating boundary-layer flow[END_REF] a second dominant range of higher frequency is visible. Figure 9 shows an instantaneous vorticity field illustrating the vortex-shedding behaviour resulting from the two-dimensional global oscillations.

Conclusion

In the aim of shedding new light on the phenomenon of global 'flapping' in separated wall-bounded flows, the two-dimensional flow over a bump has been considered, by computing nonlinear states of the stationary Navier-Stokes system. When increasing the Reynolds number, no topological changes in the flow structure are detected. The separation bubble becomes unstable with respect to two-dimensional temporal modes starting at the center of the bubble and extending more or less downstream the reattachment point. The corresponding eigenvalues cross the axis of marginal instability almost simultaneously and they exhibit equally spaced frequencies. By performing an optimal initial disturbance analysis, a periodic regeneration mechanism of the perturbation at the rear of the bubble, resulting from non-normal cancelling of eigenmode structures, has been detected. The time-integration of the Navier-Stokes system starting with the optimal initial condition leads to nonlinear aperiodic flow with however the reminiscence of a global low-frequency oscillation of the bubble. This gives some strength to the conjecture, that non-normal interaction of global modes is at the origin of the 'flapping' behaviour.

Figure 1 .

 1 Figure 1. Streamlines of flow states at a) Re = 510, b) Re = 620, c) Re = 670.

Figure 2 .

 2 Figure 2. Eigenvalue spectrum at (a) Re = 590. The modes labelled from (1) -(6) are depicted in figure 3. (b) Unstable part of the spectrum at Re = 590 (×) and Re = 620 (+).

Figure 3 .

 3 Figure 3. a) Streamlines of basic state at Re = 590. b)-g) Streamwise velocity components of eigenfunctions corresponding to eigenvalues labelled (1)-(6) in figure 2.

Figure 4 .

 4 Figure 4. Envelope of maximum energy growth from initial condition, at Re = 590 with truncation N = 337 (solid line) and N = 150 (dotted line). Broken line: envelope at Re = 620.

Figure 5 .

 5 Figure 5. Streamwise velocity component of the perturbation for increasing time from top to the bottom, starting with the optimal initial condition (Re = 590). The vertical line shows the location of the reattachment point.

Figure 6 .

 6 Figure 6. Global energy of u-u b in the time-integrated Navier-Stokes system at Re = 590, starting with optimal initial condition (solid line). Envelope of the eigenmode system is depicted as the dashed line.

Figure 7 .

 7 Figure 7. a) Amplitude ||u ||dy of perturbation flow field in the DNS, as function of at various times. b)-d) Instantaneous streamwise velocity components of the perturbation flow field.

Figure 8 .

 8 Figure 8. Power spectrum in time 500 ≤ t ≤ 3000 of streamwise velocity component at x = 40, y = 1.

Figure 9 .

 9 Figure 9. Instantaneous vorticity at t = 1500.

Acknowledgements

Parts of the computations have been performed on the NEC-SX8 of the IDRIS, France. The authors would like to thank Matthieu Marquillie for providing the DNS code.