Fabrice Piazza
email: fpiazza@pucmm.edu

Marc Monthioux

Pascal Puech

Iann C Gerber

Towards a better understanding of the structure of diamanoïds and diamanoïd/graphene hybrids

Hot-filament process was recently employed to convert, totally or partially, few-layer graphene (FLG) with Bernal stacking into crystalline sp 3 -C layers at low pressure. The result reported earlier relies on Raman spectroscopy and Fourier transform infrared microscopy. As soon as the number of graphene layers in the starting FLG is higher than 2-3, the sp 2 -C to sp 3 -C conversion is partial only, due to the prevalent Bernal stacking sequence. We report new evidences confirming the sp 2 -C to sp 3 -C conversion from electron diffraction at low energy, Raman spectroscopy and Density Functional Theory (DFT) calculations. Partial sp 2 -C to sp 3 -C conversion generates couples of twisted, superimposed coherent domains (TCD), supposedly because of stress relaxation, which are evidenced by electron diffraction and Raman spectroscopy. TCDs come with the occurrence of a twisted bilayer graphene feature located at the interface between the upper diamanoïd domain and the non-converted graphenic domain underneath, as evidenced by a specific Raman signature consistent with the literature. DFT calculations show that Raman T peak originates from a combination of the sp 3 -C stretching mode of a sp 3 -C layer with the optical out-of-plane mode of a graphene layer; both layers being sandwiched between a highly hydrogenated sp 3 -C surface and the underneath unconverted graphene layer(s).

Introduction

Since 2009, diamane has been a hypothetical nanocarbon material [1]. Interest in genuine diamane (two sp 3 -C layers only, both hydrogenated) and more generally in diamanoïds (sp 3 -C layers with more than two layers, where inner layer(s) is(are) not hydrogenated) [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF], which are wide band-gap 2D materials, derives from its potential use in technologies such as transistors for nanoelectronics, host system for single photon emission in quantum computing devices and building material for miniaturized biomedical devices [1,[START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[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] Chernozatonskii | Determination of ultrathin diamond films by Raman spectroscopy[END_REF][START_REF] Ruoff | Personal perspectives on graphene: New graphene-related materials on the horizon[END_REF][START_REF] Gupta | Two-Level Quantum Systems in Two-Dimensional Materials for Single Photon Emission[END_REF]. Currently, the ability to create graphene transistors is limited by the lack of band-gap in that material [START_REF] Fiori | Electronics based on two-dimensional materials[END_REF]. Unlike graphene, according to computational studies, diamanoïds display the electronic structure of semiconductors, with a direct band-gap depending on the film thickness and crystalline structure (stacking order) [1,[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]. Another interest lies in the possibility of using nitrogen-vacancy (N-V) centers in diamane through substitution by nitrogen atoms in the C lattice to configure qubits for use in quantum computing [START_REF] Ruoff | Personal perspectives on graphene: New graphene-related materials on the horizon[END_REF][START_REF] Gupta | Two-Level Quantum Systems in Two-Dimensional Materials for Single Photon Emission[END_REF]. Diamane was recently expected to be superior to diamond as host system for single photon emission [START_REF] Gupta | Two-Level Quantum Systems in Two-Dimensional Materials for Single Photon Emission[END_REF]. Finally, the expected strength, low coefficient of friction and bio-compatibility could render diamane and diamanoïds very competitive as building materials to make lower power and miniaturized electronics and biomedical devices [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF].

In 2019, the original synthesis of few-layer and crystalline sp 3 -carbon (lonsdaleite or diamond or a combination of both) was demonstrated over surface areas up to ~1,700 µm 2 [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF]. This breakthrough was achieved from the chemisorption of H radicals produced by the hot-filament process at low temperature (below 325 °C) and pressure (50 Torr) on the basal planes of few-layer graphene and the subsequent interlayer bonding between sp 3 -C. Until then, even if various studies based on calculations hypothesized that diamanoïds could form when non-supported few-layer graphene (FLG) are placed in hydrogen cold plasma [1], there was no report of any experimental evidence of such scenario. A structure transformation of FLG into a diamond-like layer was reported in [START_REF] Rajasekaran | Interlayer carbon bond formation induced by hydrogen adsorption in few-layer supported graphene[END_REF]. However, crystal structure was not evidenced. No Raman spectroscopy nor electron diffraction results were reported. The material obtained in [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 [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]. Temporary interlayer bonding between graphene layers were reported but this was shown to happen only when graphene layers are exposed to very high pressure [START_REF] Barboza | Room temperature compression-induced diamondization of few-layer graphene[END_REF][START_REF] Martins | Raman evidence for pressure-induced formation of diamondene[END_REF][START_REF] Gao | Ultrahard carbon film from epitaxial two-layer graphene[END_REF]. In contrast, in [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF], it was unambiguously shown, for the first time, the preparation at low pressure of stable nanometer-thick crystalline sp 3 -bonded carbon materials. sp 3 -C related peaks from diamond and/or lonsdaleite and/or hybrids of both were detected in visible and UV Raman spectra. C-H bonding was directly detected by Fourier Transform Infrared (FTIR) microscopy over an area of ~150 µm 2 and one single component attributed to sp 3 -C-H mode was detected in the C-H stretching band showing that carbon atoms were bonded to a single hydrogen and strongly suggesting that the sp 3 -C materials obtained were nanometer-thick films with basal planes hydrogenated [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF]. Carbon atoms from the graphene edges got hydrogenated as well, but (i) their related IR cross-section is expected to be higher than that of the sp 3 -C-H mode, as in free molecules [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF], and (ii) the number of hydrogenated carbon atoms from the graphene faces is much larger, which makes that they prevail in the FTIR spectrum.

However, the sp 2 -C to sp 3 -C conversion was not confirmed by transmission electron microscopy (TEM) techniques, because the converted material was found to be electron sensitive, even for electron energy as low as 80 keV, supposedly because of the high hydrogenation degree of the surface, which was evidenced by FTIR microscopy [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF]. Following the publication of [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF], another possibility for the pressureless sp 2 -C to sp 3 -C transformation of nanometer-thick and crystalline films, from fluorine chemisorption, was submitted for publication [START_REF] Bakharev | Chemically Induced Transformation of CVD-Grown Bilayer Graphene into Single Layer Diamond[END_REF]. In [START_REF] Bakharev | Chemically Induced Transformation of CVD-Grown Bilayer Graphene into Single Layer Diamond[END_REF], the disappearance of sp 2 -C features in Raman spectra following fluorine chemisorption was shown although sp 3 -C features were not evidenced in Raman spectra. However, some evidences of the sp 2 -C-to-sp 3 -C conversion were provided by optical transmission and transmission electron microscopy techniques.

The most convincing evidence is the disappearance of the 1s-2p (π * ) energy loss peak at 285 eV in the high energy loss spectra of fluorinated bi-layer graphene obtained at 80 keV. The electron sensitivity of the film was shown at 80 kV by varying the size of the parallel electron beam and hence the electron dose rate. Notably, there is a lack of overlap between the experimental results published in the so far only two reports on nanometer-thick and crystalline sp 3 -C sheets [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF][START_REF] Bakharev | Chemically Induced Transformation of CVD-Grown Bilayer Graphene into Single Layer Diamond[END_REF].

Further structure analysis is required.

No Raman activity was detected in Raman spectra below 900 cm -1 , showing that the nanosized and crystalline sp 3 -C sheets evidenced in [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF] differ from Rh6-II structure [START_REF] Wang | A New Carbon Allotrope with Six-Fold Helical Chains in all-sp 2 Bonding Networks[END_REF][START_REF] Calderon | HRTEM low dose: the unfold of the morphed graphene, from amorphous carbon to morphed graphenes[END_REF].

In this work, we studied the structure of nanometer-thick and crystalline sp 3 -C sheets obtained by the exposure of FLG to H radicals in a hot-filament chemical vapor deposition (HFCVD) reactor, before and after hydrogenation, by TEM and electron diffraction at 5 keV, Density Functional Theory (DFT) calculations, and further interpretation of Raman spectra.

Experimental

Graphene pristine materials

As-received FLG films deposited on 3 mm copper transmission electron microscopy grids (2000 Mesh) from Graphene Supermarket (SKU # SKU-TEM-CU-2000-025) were used as graphene materials. The FLG films were 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 %.

Hydrogenation process

A commercial hot filament reactor 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 nanometer-thick and crystalline sp 3 -C sheets. The details of the process were previously disclosed [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF]. The samples studied here were prepared at a pressure of 10 Torr and using a H2 flux of 1 sccm. The duration of the hydrogenation process was of 9 minutes and 50 seconds. The maximum FLG temperature was ~325 °C [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF].

Material characterization

The sp 2 -C to sp 3 -C conversion was confirmed by UV Raman spectroscopy following the work published in [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF]. For that purpose, UV Raman spectroscopy was employed to examine the material structure before and after hydrogenation. Raman spectra were recorded with a Renishaw InVia Reflex Spectrometer System equipped with a stigmatic single pass spectrograph including holographic grating of 3600 grooves.mm -1 and using the 244 nm lines of Ar ion laser (85 Lexel Second Harmonic Generation laser). The scattered light was collected in the 180° backscattering geometry. A x40 objective was employed. The detection system was a Peltier-cooled UV coated Deep Depletion CCD array detector and the entrance slit was set to 50 µm. The resolution was of 3.5 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. Highly-oriented pyrolytic graphite (HOPG) was used for peak position calibration. 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 µm 2 . Peaks were fitted using Lorentzian function using Wire 4.4 software from Renishaw.

TEM and electron diffraction were employed to further examine the material structure before and after hydrogenation. 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 [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF]. This is consistent with the photon sensitivity observed under certain conditions [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF] and the reported electron sensitivity of F-diamane [START_REF] Bakharev | Chemically Induced Transformation of CVD-Grown Bilayer Graphene into Single Layer Diamond[END_REF]. This is the reason why a low-voltage benchtop transmission electron microscope at 5keV from America Delong was used. The instrument includes a 5kV Schottky type field emission gun and a 2560×2160 pixel Front Illuminated Scientific CMOS (6.5 µm 2 pixel size). Maximum magnification and resolution were of ×700,000 and 1.2 nm, respectively. Electron diffraction patterns were obtained from 100 nm-large areas, and they were analyzed using ImageJ 1.52a software.

Electron diffraction patterns have been calculated using the XaNSoNS open source software considering a wavelength of 0.01nm [START_REF] Neverov | XaNSoNS: GPU-accelerated simulator of diffraction patterns of nanoparticles[END_REF]. Typically 10 6 atoms have been considered for a bilayer domain and 2x10 6 atoms for a 4-layer domain. The results have been plotted using Matlab® software.

The signal intensity in XRD spectra is proportional to the amount of atoms. Because displayed on the TEM grid with electron transparent thickness, the amount of material is not enough to obtain an exploitable XRD signal from it. An access to synchrotron could be considered but, from the calculated diffracted patterns, it seems useless as a clear discrimination between the patterns from each phase does not seem possible.

Computational Details

The electronic structures were obtained from DFT calculations with 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] which uses the plane-augmented 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 of 0.05 eV width for partial occupancies. During geometry optimization step all the atoms were allowed to relax with a force convergence criterion below 0.005 eV/Å, using the van der Waals corrected scheme of Grimme et al. [START_REF] Grimme | A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu[END_REF]. A vacuum height of 20 Å was used to avoid spurious interaction between periodic images of the different slabs. To estimate phonon dispersion (7x7) supercells were used with a 7x7x1 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]. To validate the present computational setting, in Figure SI.1, the phonon dispersion curve of a pristine graphene monolayer is given, in which the characteristic G band is located at 1572 cm -1 .

To mimic our experimental FLG samples, which are made of a various number of layers, we have modelled partially hydrogenated systems consisting of four distinct carbon layers (FIG. 1a). The first layer (L1) corresponds to the fully hydrogenated layer typical of diamane and diamanoïds (all C atoms are sp 3 -hybridized, half of them bonded with one H atom, the other half are bonded with carbon atoms from layer L2). The second layer (L2) is not hydrogenated, hence half of the carbon atoms are truly sp 3 -C and bonded to carbon atoms from L1, while the other half are still sp 3 -hybridized but exhibit a free orbital with almost one unpaired electron mainly of pz character (FIG. 1b). The two bottom layers L3 and L4 remain pure sp 2 -C, however L3 is interacting more strongly with L2 than with L4 due to the existence of the unsatisfied valence bonds in the former.

As a consequence, the L2-L3 interlayer distance is shorter (in the range 0.3037-0.3308 nm) than the L3-L4 interlayer distance (0.3512 nm). For thicker systems, other layers similar to L4 can be added underneath while respecting the stacking sequence in graphite (ABAB), until the other side of the flake is reached where the occurrence of the L1-L2-L3 combination may repeat. Despite the stacking sequence of graphene layers in our pristine FLG is likely, it is important to figure out that the sp 3 -C to sp 2 -C conversion generates a huge stress in the system, which may induce layer decoupling upon relaxation, more preferably between layers where the interaction is weaker (typically between L3 and L4). Therefore, different L1-…-L4 stacking orders have been tested, namely ABBB, ABAB, ABCA, and ABBA, the latter being more stable by at least 15 meV/carbon atom than the other stacking sequences. As a consequence, only the ABBA stacking sequence is discussed below. However, it is worth noting that the structure changes described below would be the same for ABAB stacking.

Therefore, the model considered in FIG. 1 is based on the idea that, when starting from multilayer graphene, the hydrogenation process generates the formation of an upper diamanoïd domain (involving L1 and L2 in the model, possibly added with L3, because of the stronger interaction with L2 than with L4), lying upon a lower, untransformed FLG domain (involving L4 in the model and subsequent graphene layers underneath).

Results and discussion

FIG. 2 shows a representative UV Raman spectrum of FLG exposed to H radicals as already reported in [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF], which was demonstrated as typical of nanometer-thick and crystalline sp 3 -C sheets [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF]. It is different from the spectra obtained before the hydrogenation process, which are typical of perfect FLG [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF]. The spectrum shown in FIG. 2 displays two sharp peaks centered at ~1062.8 cm -1 and 1321.9 cm -1 , respectively, which the FWHMs are of ~14.0 and 29.6 cm -1 , respectively. The first peak was interpreted as the T peak, somehow related to bonding between sp 3 -C [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF]. This will be further discussed below. The peak centered at ~1321.9 cm -1 was interpreted as corresponding to the diamond/lonsdaleite stretching mode [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF]. Areas of the specimen for which such peaks could be detected in UV Raman spectra were considered for further TEM examination.

No change was observed in the converted materials over more than five years as checked by Raman spectroscopy. It is therefore assumed that converted crystalline sp 3 In the pristine FLG from the starting specimen (i.e., before hydrogenation), hence multilayer domains, the Bernal stacking sequence ABA is likely, as the most frequent and the most stable (as compared to AAA and ABCA stacking). Hence, the diamond structure (face centered cubic, FCC) should be favored, however, it cannot be ruled out that the huge stresses induced by the sp 2 -Csp 3 -C conversion (see below) may enable atom displacements and generate the lonsdaleite structure (hexagonal) anyway. Therefore, whether it would be possible to discriminate between remaining graphene domain, a converted film with the diamond structure, and the lonsdaleite structure from the diffraction patterns is an important issue to look at. Diffraction pattern may offer three ways for this: interplanar distances, symmetries (spot distribution), and peak intensities.

In order to check whether measuring d-spacing distances might be a reliable discrimination method, the variation in the d-spacing of the (100) planes was obtained from 298 measurements taken from 53 flakes (before hydrogenation). The same magnification (camera length) and the same illumination area were used to record each pattern. In all cases, the distance between opposite spots was considered to minimize measurement errors. The variation in the d-spacing measurement of the (100) planes (theoretical: 0.2130 nm) was found to reach 5.25 %. This value is higher than the difference between the d-spacing of the (111) planes in diamond (0.205 nm [START_REF]International Centre for Diffraction Data[END_REF]) and of the (100) planes in graphite, which is of 3.75 %. Likewise, the d-spacing of the (110) planes in diamond (0.126 nm [START_REF]International Centre for Diffraction Data[END_REF]) is very close of that of the (110) planes in graphite (0.123 nm), which makes a difference of ~2.4%. Similarly, the difference between the d-spacing of the (100) planes in lonsdaleite (0.219 nm [START_REF]International Centre for Diffraction Data[END_REF]) and in graphite is only 2.8 %. Therefore, under the present experimental conditions, it is not possible to unambiguously discriminate the diffraction patterns of pure sp 3 -C crystalline phase (diamond or diamane or lonsdaleite) from that of few-layer graphene by estimating d-spacing from the diffraction spots.

On the other hand, considering the spot distribution on the diffraction patterns should, in principle, be able to discriminate between the FCC structure of diamond on the one hand, and the hexagonal structures of lonsdaleite and graphene on the other hand. Indeed, as the diamond structure in a converted film is seen according to the [111] axis (perpendicular to the film plane, see FIG. 1c However, in particular because the FCC structure corresponds to a multiple cell, it is questionable whether the aspect of the diffraction pattern remains the same when the system involves a limited number of layers. This was simulated for 2 and 4 layers respectively (FIG. 4b and4c). It is worth noting that 6 spots with the hexagonal symmetry now appear in the first ring, as it was expected for the hexagonal structure only (FIG. 4d to 4f). The intensity of those spots is seen to decrease from the 2-layer to the 4-layer structure, which is consistent with the fact that those spots will not show up at some point as the number of layers increases. Therefore, for diamane and diamanoïd films made of few layers only, it is again not possible to unambiguously discriminate the diffraction patterns of pure sp 3 -C crystalline phase (diamond or lonsdaleite) from that of few-layer graphene by considering the spot distribution and symmetry.

Finally, we explored the possibility to discriminate between the various structures by measuring spot intensities. Indeed, in [START_REF] Bakharev | Chemically Induced Transformation of CVD-Grown Bilayer Graphene into Single Layer Diamond[END_REF], the intensity distribution over the diffraction peaks in the electron diffraction patterns of F-diamane and AB-stacked bi-layer graphene was studied and compared to data obtained from simulated diffractograms acquired from DFT-optimized structures.

Therefore, we did the same, and analyzed the intensity distribution over the diffraction peaks in the electron diffraction patterns of a single FLG domain after the exposure to the hot-filament-promoted hydrogenation process (FIG. 5). FIG. 5a shows typical electron diffraction pattern of a single FLG domain after the exposure to the hot-filament-promoted hydrogenation process. FIG. 5b-d display the profile plots of the diffraction peak intensities obtained from pattern regions highlighted by the rectangular boxes in FIG. 5a. FIG. 5b-d only show three of the six profiles for clarity purpose. For half of the profile lines, the intensity of the spots of the 1 st family planes (closest to the center of the pattern) is higher than the intensity of the spots of the 2 nd family planes (FIG. 5b and5c). This is similar to what was reported for F-diamane in [START_REF] Bakharev | Chemically Induced Transformation of CVD-Grown Bilayer Graphene into Single Layer Diamond[END_REF]. However, for the other half of the intensity profiles, there is no clear trend and the situation differs from the ideal case (FIG. 5d). This shows that such a measurement has a poor significance, because the Ewald sphere intercepts the elongated reciprocal rods at various height with respect to the local film distortion. Indeed, the fact that the coherent domains are ultrathin films makes that they distort easily. Even though the distortions are slight, because of the very small electron wavelength, Bragg angles are in the range of a fraction of degree, hence distortions in a similar range are enough for some family planes be no longer under the Bragg angle. On the other hand, the ultrathin nature of the films makes that the reciprocal nodes are elongated in the z* direction making them reciprocal lines. Therefore, although some scattering events should normally not occur because of the film distortion, the reciprocal line of the family plane involved in the distortion could be elongated enough for touching the Ewald sphere surface. The result is that the scattering occurs anyway, a spot appears in the diffraction pattern which actually is a cross-section of the reciprocal line by the Ewald sphere, but the intensity of the spot cannot be the nominal one because the Ewald sphere does not cross the reciprocal line at its maximum intensity but at various height depending on the distortion. This corresponds to the "interference error" effect well-known by transmission electron microscopists. Therefore, one could not expect discriminate between the various possible structures based on the analysis of spot intensities either.

In some regions of the FLG after exposure to the hot-filament-promoted hydrogenation process, electron diffraction patterns differ from the typical patterns of pristine FLG and/or of pure ultrathin and crystalline sp 3 -C sheets. FIG. 6 displays some examples of such patterns taken from different flakes. Over 53 flakes analyzed, such patterns were not observed in pristine FLG. The complex patterns shown in FIG. 6 including satellite peaks are characteristic of patterns for twisted superimposed coherent domains (TCD) with similar periodicities of very small twisted angle, and with secondary periodicity [START_REF] Zhang | Structural and electron diffraction scaling of twisted graphene bilayers[END_REF][START_REF] Yoo | Atomic and electronic reconstruction at the van der Waals interface in twisted bilayer graphene[END_REF]. Referring to the model in FIG. 1, such TCDs could be the upper diamanoïd domain (represented by L1 and L2, completed with L3) and the lower graphenic domain (represented by L4 and other graphenes underneath) -see the introduction to the model in Section 2.4. Therefore, at the interface between the two twisted domains are two graphene layers (L3 and L4) which are in twisted configuration. Twisted bilayer (TBL) with very small twisted angle is a system recently studied [START_REF] Zhang | Structural and electron diffraction scaling of twisted graphene bilayers[END_REF][START_REF] Yoo | Atomic and electronic reconstruction at the van der Waals interface in twisted bilayer graphene[END_REF] which generates specific Raman signatures [30]. Therefore, FIG. 6 reveals such an occurrence of TCDs upon hydrogenation.

16

~1669 cm -1 in UV Raman spectra (TBL1 and TBL2 in FIG. 7), which were previously reported for TBL [30].

It is supposed that the slight rotation of the superimposed domains from the TCDs is induced by the relaxation of huge (several GPa) local constraints which develop between superimposed layers as a result of the sp 2 -C-to-sp 3 -C conversion. In diamond-like carbon, a material that contains a variable fraction of sp 3 -C and sp 2 -C, internal stress can reach several GPa [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][START_REF] Weiler | Preparation and properties of highly tetrahedral hydrogenated amorphous carbon[END_REF]]. This will be further discussed below. Together with the sharp stretching mode at 1319-1337 cm -1 and assigned to lonsdaleite or diamond, UV Raman spectra of hydrogenated samples display a sharp T peak at 1055-1071 cm -1 assigned to bonding between sp 3 -C (FIG. 2 and7) [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF]. We further investigated the physical origin of this T peak by using DFT calculations. For pure sp 2 -C and sp 3 -C materials, no T peak is present. We have already reported [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF] a spatial correlation between the occurrence of the diamond/lonsdaleite stretching mode peak and that of the T peak while there is no spatial correlation between the location of the G peak and that of the T peak. Thus, we have hypothesised that the T peak could be related to the interface between a sp 2 -C layer and a sp 3 -C layer, and consequently we have investigated the phonon dispersion of the ABBA-stacking model proposed in FIG. 1. Upon full atomic relaxation, all frequencies remain positive, proving the stability of the proposed structure. Three distinct Raman active modes in the range of interest are present, with wavenumbers from 1050 to 1100 cm -1 (FIG. 8). Normal mode analysis shows that two of them, which correspond to a large rotation of hydrogen atoms (see SI.3 for atomic vector representation), are degenerated with a wavenumber of 1099 cm - 1 at the center of the Brillouin zone (Γ point) and are visible in Raman spectra. Interestingly, the mode at 1078 cm -1 , which involves atoms from L1 and L2, can be associated to a combination of the stretching of the sp 3 -C bonds with an optical out-of-plane (ZO) mode of graphene membrane.

Due to the periodic repetition of the motif in DFT calculations, delamination is not possible and the full cell relaxation leads to compressive strain in the sp 3 -C part (L1-L2) and to extensive strain in graphene layers (L3-L4, and further layers underneath), explaining the position of the G band which is downshifted by almost 150 cm -1 in the calculation compared to the pristine graphene (see

FIG. SI.1).

Using the conversion factor of 4.5 cm -1 /GPa [30], the stress brought to the graphene domain (L4 and further layers underneath) can be estimated at -33 GPa. Obviously, this is a huge stress value which is by far enough to generate the delamination and twisting events which occur experimentally in the material, although the periodic conditions in the calculation cannot account for them. The system exhibits a perfect, periodic interface contrary to amorphous materials. As a consequence, the experimental linewidth of the T band is as low as 14.0 cm -1 .

In summary, the model presented is FIG. 1 together with our DFT calculations are consistent with characterization results obtained here and in Reference [START_REF] Piazza | Low Temperature, Pressureless sp 2 to sp 3 Transformation of Ultrathin, Crystalline Carbon Films[END_REF]. In particular, the model is able to account for the five peaks observed in Raman spectra in the 1000-1800 cm -1 wavenumber range.

Besides, the results show that it is not possible to form pure diamanoïds starting from FLG with ABA… stacking as soon as the number of layers is higher than two. In that case, only the first two layers will be sp 3 -C. Also, only the first surface layer will be sp 3 -C if the starting FLG is turbostratic.

In that case, for some rotation angles, AA and AB configurations may locally occur, allowing local only sp 2 -C to sp 3 -C conversion, that could generate the destruction of the structure or a huge topological deformation along with huge inner constraints. The formation of diamanoïds with a number of layers higher than two requires starting from FLG with AAA… or ABC… stacking, which both are energetically unfavourable, and therefore barely preparable.

Conclusion

We reported on new additional evidences, from electron diffraction at 5 keV and Raman spectroscopy, supported by calculations, confirming the successful conversion of few-layer graphene into nanometer-thick and crystalline sp 3 -C sheets from the exposure of FLG to H radicals produced in a hot-filament process at low temperature and pressure. The results confirmed previous multi-wavelength Raman spectroscopy and FTIR microscopy analysis. Complex electron diffraction patterns differing from pristine FLG and including satellite peaks reveal that twisted coherent domains involving a minimum of one diamanoïd domain and one unconverted graphenic domain underneath, are formed, presumably due to the relaxation of stress resulting from the partial sp 2 -C to sp 3 -C conversion. A configuration of twisted bilayer graphene (TBL) is created at the interface between the twisted, superimposed domains. Peaks ascribed to TBL are also evidenced in Raman

FIG. 1 .

 1 FIG. 1. (a) and (b) two side view and (c) top view of the partially hydrogenated FLG used in DFT calculations with ABBA stacking. The projected structure is that of the face centered cubic of diamond. In (b) is shown how the pz orbitals are preferably oriented towards the layer underneath (L3). The black lines delimite the primitive cell.

  FIG. 2.Typical UV Raman spectrum (at 244 nm) of FLG after exposure to the hot-filamentpromoted hydrogenation process.

  ), there should be no spot on the first ring, and the first family of spots showing the hexagonal symmetry should be that of the (110) planes, i.e. the 220 spots, corresponding to a d-spacing of 0.126 nm, which locate on the second ring. This is what is shown on the simulated diffraction pattern of a diamond nanoparticle in FIG. 4a.

FIG. 4 .

 4 FIG. 4. (a) to (c), simulated electron diffraction patterns for the FCC diamond structure oriented with the [111] axis parallel to the electron beam. (a) Diamond nanoparticle (FCC structure) ; (b) 2layer film ; (c) 4-layer film. (d) Model of a 2-layer hexagonal structure oriented with the [001] axis parallel to the electron beam ; (e) and (f) simulated electron diffraction patterns for a 2-layer and a 4-layer film with the hexagonal structure oriented as in (d), respectively. For (a), (b), and (e) atoms were considered, for (c) and (f) 2x10 6 atoms were considered.

Piazza et al. 13 FIG. 5 .

 135 FIG. 5. (a) Typical electron diffraction pattern of FLG after the exposure to the hot-filamentpromoted hydrogenation process and (b-d) profile plots of the diffraction peak intensities obtained from pattern region highlighted by the rectangular boxes in (a). The letters in (a) are used to label the profiles shown in (b-d).

FIG. 6 .

 6 FIG. 6. (a to f) Typical diffraction patterns of specimen flakes after the exposure to the hotfilament-promoted hydrogenation process.

FIG. 7 .

 7 FIG. 7. Typical UV Raman spectrum of a specimen flake after the exposure to the hot-filamentpromoted hydrogenation process showing peaks typical of twisted bilayer graphene, labeled TBL1 and TBL2. Those peaks are also present in the Raman spectrum in FIG. 2. The G band reveals the partial conversion only.

FIG. 8 .

 8 FIG. 8. (a) Phonon dispersion curve of the partially hydrogenated ABBA stacked FLG model proposed in FIG. 1 based on DFPT calculations. (b) Corresponding vectors associated to the atomic displacements along the normal mode at 1078 cm -1 . Vector lengths are not at the scale of the model, but are proportional to each other.

FIG. SI. 2 .

 2 FIG. SI.2. (a) and (c) are the TEM images of the specimen areas having provided the diffraction patterns shown in FIGS. 6e and 6f, respectively. (b) and (d) are magnified views of the squared areas in (a) and (c), to better reveal the moiré periodicities.

Piazza et al. 25 FIG. SI. 3 .

 253 FIG. SI.3.Vectors associated to the atomic displacements along the two degenerated normal modes at 1099 cm -1 . Only one is depicted here, since the other is simply anti-symmetric in the x-y exchange.

Acknowledgements

This research was funded by the Ministry of Higher Education, Science and Technology (MESCyT) of the Dominican Republic (2010-2011, 2012, 2015 and 2016-2018 FONDOCyT programs). F.P. and M.M. greatly acknowledge MESCyT and PUCMM for strong financial, administrative and technical support; 2017 NEXT Guest scientist Program and the French Embassy in the Dominican Republic for travel support. I.C.G. acknowledges the Calcul en Midi-Pyrénées initiative CALMIP (Project p0812) for allocations of computer time, as well as GENCI-CINES and GENCI-IDRIS for Grant No. 2019-A006096649.

spectra at 1385 cm -1 and 1669 cm -1 , consistently with the recent literature. Also, by using DFT calculations, the exact origin of the T peak was revealed. It originates from a combination of the sp 3 -C stretching mode of a C-sp 3 layer with the optical out-of-plane mode of a graphene layer; both layers being sandwiched between a highly hydrogenated sp 3 -C surface layer and the underneath unconverted graphene layer (s), and interacting strongly because of the presence of free pz orbitals in the sp 3 -C layer. The frequent evidence of sp 3 -C to sp 2 -C partial conversion confirms that starting from FLG with both a number of layers higher than two and the Bernal (ABA…) stacking sequences, which is the most frequent configuration by far, is not favourable for producing diamanoïds.

However, this may be used for the fabrication of graphene/diamanoïd heterostructures. The formation of diamanoïds with a number of layer higher than two requires specific stacking sequences other than Bernal which are energetically unfavourable, and therefore difficult to synthesize. It is expected to elaborate large area diamane layer by using Bernal-stacked bi-layer graphene as pristine material.

Supplemental Information

FIG. SI.1.

Phonon dispersion curve of a pure graphene monolayer.


Pierre Auger

Ali Moussaoui

P Ierre Auger

Ali M Oussaoui

ON THE THRESHOLD OF RELEASE OF CONFINEMENT IN AN EPIDEMIC SEIR MODEL TAKING INTO ACCOUNT THE PROTECTIVE EFFECT OF MASK

Keywords: SEIR model, urban mobility, confinement, mask protection, covid-19

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

Introduction

The covid-19 epidemic has led the vast majority of states around the world to take very strict confinement measures ranging from lockdown to partial confinement. The economic 1 consequences of this confinement over more than two months are considerable and alarming. At a time when most states begin to lift confinement, sometimes with great reluctance, it seems very useful to develop reliable models as much as possible to estimate the risks of an excessive level of un-confinement which could give rise to a resurgence of the epidemic.

Many models have been developed very quickly to describe the evolution of the pandemic, D. [START_REF] Adam | Modelling the pandemic. The simulations driving the world's response to COVID-19[END_REF]. Various computer or mathematical models have been produced and show sometimes very alarming scenarios with successive phases of confinement followed by phases of restarting the epidemic, S.M. Kissler et al. (2020). While most countries are in the process of release of the confinement, it is important to develop models to assess whether the protective measures taken by the government are sufficient to prevent a restart of the epidemic.

The goal of this work is to propose a simple model allowing to estimate a threshold of release of confinement and of protection by masks below which the epidemic cannot restart. For this, we combine two approaches, our previous work, A. Moussaoui and P. [START_REF] Moussaoui | Prediction of confinement effects on the number of covid-19 outbreak in Algeria[END_REF], where we had defined a threshold for the release of confinement with that of J. Howard et al. (2020) where they had proposed a formula for reducing the basic reproduction rate of an epidemic in the case where the population uses masks to protect themselves from infection.

2. The SEIR model without intervention Y. [START_REF] Liu | Understanding Unreported Cases in the COVID-19 Epidemic Outbreak in Wuhan, China, and the Importance of Major Public Health Interventions[END_REF] considered four representative places of social contacts : (H) individual households where families are living; (S) schools, college and universities; (W) various workplaces indoor and outdoor; (P) large density public places, such as stadiums, shops, markets, including public transportation. In our model, we will group two compartments (S) and (W) into one that we continue to call (W). In this way, schools, colleges and universities are included in the workplace. We therefore only consider for simplicity 3 compartments (H), (W) and (P).

The population is structured into sub-populations associated with different places where individuals move every day. At home (H), individuals are in contact with their family members, parents, grandparents and children. At work (W), they are in contact with office colleagues inside the building or outdoors. In high density public places (P), people can come into contact with many others and catch the disease there. Figure 1 shows the possible transitions between the different states.

dS H dτ = ( γ T W S W + γ T P S P -2 γ T H S H ) -β H S H I H (1) dS W dτ = ( γ T H S H - γ T W S W ) -β W S W I W (2) dS P dτ = ( γ T H S H - γ T P S P ) -β P S P I P (3) dE H dτ = ( γ E W E W + γ T P E P -2 γ T H E H ) + (β H S H I H -kE H ) (4) dE W dτ = ( γ T H E H - γ T W E W ) + (β W S W I W -kE W ) (5) dE P dτ = ( γ T H E H - γ T P E P ) + (β P S P I P -kE P ) (6) dI H dτ = ( γ T W I W + γ T P I P -2 γ T H I H ) + (kE H -αI H ) (7) dI W dτ = ( γ T H I H - γ T W I W ) + (kE W -αI W ) (8) dI P dτ = ( γ T H I H - γ T P I P ) + (kE P -αI P ) (9) dR H dτ = ( γ T W R W + γ T P R P -2 γ T H R H ) + αI H (10) dR W dτ = ( γ T H R H - γ T W R W ) + αI W ( 11
)
dR P dτ = ( γ T H R H - γ T P R P ) + αI P (12)
We define the total proportion of susceptible individuals

S(t) = S H (t) + S W (t) + S P (t), exposed E(t) = E H (t)+E W (t)+E P (t), infectious I(t) = I H (t)+I W (t)+I P (t) and removed R(t) = R H (t) + R W (t) + R P (t) at time t.
Our variables are proportions and verify for any t the following expression:

S(t) + E(t) + I(t) + R(t) = 1. ( 13
)
τ is the fast time, t = ετ is the slow time and is a small dimensionless parameter. For instance, we can choose ε = 1 24 . In this way, the time unit of the fast model is the hour and the time unit of the epidemic model would be the day. and P. van den Driessche (2003), J. [START_REF] Arino | Disease spread in metapopulations, Non Linear and Evolution Equations[END_REF]. In our work, we assume that displacements are carried out quickly with regard to the dynamics of the epidemic. The fast part of the model is obtained when = 0 and corresponds to the daily movement model. We define time slots for average occupancy per day of the different compartments, respectively denoted T H , T W and T P for each compartment. The time slots verify that their sum is equal to one day:

T H + T W + T P = D. ( 14
)
Where D = 24. γ is a positive parameter. m AB = γ T B is the rate for change from place B to place A with B ∈ [H, W, P ] and A = B. The rate is assumed to be inversely proportional to the time slot of the departure place. Therefore, individuals are more likely to remain on places with larger time slots. The slow part of the model corresponds to all the terms of order of and is a classic SEIR model taking into account different places.

β A is the transmission rate in location A ∈ [H, W, P ]. 1 k is the average time spent in the exposed compartment before becoming infectious. 1 α is the average time spent in the infected compartment to go to the removed compartment in which the cured individuals as well as the deceased ones are found.

Let's look for the fast equilibrium of this model. A simple calculation shows that the fast equilibrium verify the following relationships for susceptible:

u * H = S * H S = T H D , u * W = S * W S = T W D , u * P = S * P S = T P D . ( 15
)
We denote the fast equilibrium with an upper star. At the fast equilibrium, the proportion of susceptible individuals in the different places is proportional to the total proportion of susceptible. We get the same fast equilibrium formulas for the exposed, the infectious and the removed because they are supposed to move in the same way. u * A is the constant proportion of individuals in place A at equilibrium with A ∈ [H, W, P ] which is propor-tional to the time slot associated with this place. We assume that exposed individuals have no symptoms and move as before they were infected. For infectious individuals, we assume that symptomatic individuals are very quickly isolated and go into compartment R. The asymptomatic infected do not know that they are infectious and continue to move as before. Such equilibrium proportions in the different places are reported in S. Xia et al. (2020) for the 2009 Hong Kong H1N1 Influenza Epidemic also reported in Y. [START_REF] Liu | Understanding Unreported Cases in the COVID-19 Epidemic Outbreak in Wuhan, China, and the Importance of Major Public Health Interventions[END_REF] for the Covid-19 epidemic.

Using the methods of aggregation of variables, Auger et al. (2008) and Auger et al. (2008b), we derive a reduced model governing global proportions in the different epidemic compartments at the slow time. For this, we replace the fast equilibrium of daily displacement in the complete model. Then, we add the 3 equations of susceptible, exposed, infectious and removed and we go back in slow time which leads us to the following reduced model which is a classic SEIR model:

dS dt = -βS(t)I(t), (16)
dE dt = βS(t)I(t) -kE(t), (17)
dI dt = kE(t) -αI(t), (18)
dR dt = αI(t), (19)
S(t) + E(t) + I(t) + R(t) = 1, ( 20
)
where we find an overall transmission rate β which is expressed as a function of local transmission rates and the proportions of individuals in the different places at the fast equilibrium as follows:

β = β H (u * H ) 2 + β W (u * W ) 2 + β P (u * P ) 2 . ( 21
)
For the example of covid-19 epidemic, the value of the transmission rate β = 2.1 was estimated using data for Algeria A. [START_REF] Moussaoui | Prediction of confinement effects on the number of covid-19 outbreak in Algeria[END_REF] and France N. Bacaër

Consideration of intervention strategies

After the start of the epidemic, most countries have put in place a set of measures to protect the population against the epidemic. The main measures were almost total confinement and protective measures, notably the compulsory wearing of masks and social distancing. Therefore, we will now consider three phases, a first phase corresponding to the start of the epidemic at t = 0 without any protective measures, followed by a second phase of severe confinement starting at time T and a third phase of partial release of confinement at time T * by taking additional protective measures such as wearing masks and social distancing.

The model with confinement

In A. [START_REF] Moussaoui | Prediction of confinement effects on the number of covid-19 outbreak in Algeria[END_REF], the first part proposed a classic SEIR epidemiological model to describe the dynamics of the Covid-19 epidemic in Algeria. The second part proposed a SEIR model taking into account confinement but without taking into account different groups of places. In this work, we consider daily movements of people in various groups of places. Therefore, we will now adapt the confinement model to the case with groups of places. After the start of the epidemic in China, in France and in most countries in the world, a very severe period of confinement was imposed on populations, especially in large cities. In the model, we will assume that a constant proportion v of the population is strictly confined. We assume that the confined people are completely removed from the chain of transmission of the epidemic. Thus, we consider total confinement as in the big cities in China where individuals stayed at home all day with a ban on going out and with deliveries of basic necessities every day at home. Confined people can no longer contaminate other people except for family members confined with them. In the case of covid-19, symptomatic people from the same family who are confined and infected are quickly taken into hospital care and can no longer infect another family. Asymptomatic people are confined to their house, cannot infect other persons outside and stop being infectious after a period of around ten days on average. u = 1 -v represents the proportion of unconfined people. In France, as in most of the countries concerned by confinement, part of the population has not been confined.

These are doctors and health personnel in hospitals, police and security personnel, persons providing minimum transport and city maintenance, etc. These people continued to move about as before. To simplify, in our model, we are going to assume that non-confined individuals move in exactly the same way as before the epidemic with the same average durations of time spent in different places. We also assume that the transmission rates remain unchanged in the various places visited. These are strong assumptions. However, if we accept these hypotheses, we get the following expressions of the fast equilibrium for susceptible as well as for exposed and infected:

S * H S = uu * H , S * W S = uu * W , S * P S = uu * P , (22)
Thus, S uc = S * H + S * W + S * P = uS corresponds to the proportion of un-confined persons. S c = vS = (1 -u)S is the proportion of totally confined individuals removed from the contamination chain. Then, the reduced model taking into account confinement is the following, similar to the one found in A. [START_REF] Moussaoui | Prediction of confinement effects on the number of covid-19 outbreak in Algeria[END_REF]:

dS dt = -β 1 S(t)I(t), (23)
dE dt = β 1 S(t)I(t) -kE(t), (24)
dI dt = kE(t) -αI(t), (25)
dR dt = αI(t), (26) (27)
However, we get a different overall transmission rate β 1 :

β 1 = u 2 β = u 2 (β H (u * H ) 2 + β W (u * W ) 2 + β P (u * P ) 2 ), (28)
u = 1 corresponds to the situation before the epidemic, no confinement at all. u = 0 corresponds to lockdown, total confinement of all the population. 0 < u < 1 corresponds to partial confinement situations.

Following the method in A. Moussaoui and P. [START_REF] Moussaoui | Prediction of confinement effects on the number of covid-19 outbreak in Algeria[END_REF], it is possible to calculate a threshold for the release of confinement making it possible to avoid a second epidemic peak. A simple way to find this threshold is to consider the effective reproduction rate of the epidemic R 0 (T * ) at time T * of the release of confinement of the third phase and which is given by the following expression:

R 0 (T * ) = (u * ) 2 R 0 S(T * ). ( 29
)
Where R 0 is the basic reproduction number of the epidemic and S(T * ) the proportion of susceptible individuals at the time of the release of confinement T * . Here we note u * = u(T * ) which is the constant proportion of un-confined persons fixed at t = T * . Those persons are free to move exactly as they did before the epidemic started. In order to avoid a secondary peak, we need an effective R 0 (T * ) smaller than 1 which leads to the next condition:

u * = 1 R 0 S(T * ) .
This formula means that it is necessary to end the confinement with a proportion below this threshold to avoid a secondary peak of the epidemic. In A. Moussaoui and P. Auger (2020), the authors show numerical simulations for successful and unsuccessful release of confinement.

The model with confinement and masks

Protective measures have also been taken such as masks including social distancing.

We will now consider the additional effect of wearing masks based on the article by J. Howard et al. (2020). The authors claim that wearing masks reduces R 0 by a factor

(1 -pq) 2 , where q is the efficacy of trapping viral particles inside the mask, and p is the percentage of the population that wears masks. Now we assume that in addition to partial release of confinement at time T * , the government decides to make the wearing of the mask compulsory in certain risky places. The SEIR model with partial confinement and using mask is given by the system:

dS dt = -β 2 S(t)I(t), (30)
dE dt = β 2 S(t)I(t) -kE(t), (31)
dI dt = kE -αI, (32)
dR dt = αI, (33)
where

β 2 = (u * ) 2 (1 -pq) 2 β.
As a consequence, combining the two models, the one for confinement, A. Moussaoui and P. [START_REF] Moussaoui | Prediction of confinement effects on the number of covid-19 outbreak in Algeria[END_REF], and the one for use of masks, J. Howard et al. (2020), we get the following formula for the effective basic reproduction number of the epidemic at the time of the release of confinement t = T * with use of masks:

The two strategies, total confinement (v * → 1) and, 100% efficient mask worn by everybody (pq → 1) allow to stop definitively the epidemic. Furthermore, both lead to the same final size of the epidemic, R 0 R(T * ). However, it seems clear that to avoid a slowdown in economic activity, it seems preferable to use protective measures with masks rather than a total confinement of the population resulting in a major economic slowdown and a significant increase in unemployment.

The model with confinement, masks and the effect of the proportion of asymptomatic infected individuals

Let f be the constant proportion of asymptomatic infected. In P. Magal ( 2020), the proportion of asymptomatic people was estimated to be f = 0.2. On the Charles de Gaulle

French aircraft carrier, around 60% of the sailors on board were infected with the virus and around half of them were asymptomatic. Last july, the French national public health agency was able to determine that around 24.3% of positive cases did not show symptoms.

Consequently, the value of the parameter f is not well known and it appears useful to study the sensitivity of the model with respect to it.

We assume that the rate of transition from the infected compartment to the removed compartment is written as follows:

α(f ) = f α 1 + (1 -f )α 2 , ( 40
)
Where 1 α 1 represents the average infection time for an asymptomatic individual which is about 10 days while 1 α 2 represents the average infection time before isolation at hospital or quarantine for a symptomatic individual which can be estimated about 1 or 2 days because symptomatic people are in principle quickly isolated. In our model, isolated individuals go into the removed compartment because they no longer contribute to the infection cycle at all.

The travel behavior, particularly in urban areas, of people infected with a disease is generally modified. In T. A. [START_REF] Perkins | Calling in sick: impacts of fever on intra-urban human mobility[END_REF], the authors study these changes in urban mobility behavior in the case of dengue. While it seems acceptable to think that asymptomatic infected people who are not in quarantine continue to move around as usual, the movement behavior of symptomatic infected individuals may be altered by the effects of the disease. In our model, we will therefore assume that asymptomatic infected move with the same rate as uninfected individuals, i.e. γ T A for a move from site A = [H, W, P ] to any other place. The proportion at the fast equilibrium of asymptomatic individuals in the different places is therefore always given by the following formula:

u * H = T H D , u * W = T W D , u * P = T P D . ( 41
)
However, it seems acceptable to assume that symptomatic individuals with the disease move less during the day and stay at home longer. In our model, we can represent this effect by increasing the time slot corresponding to the home and reducing the times slots corresponding to the places visited daily, work and public locations.

T S H = θT H , T S W = φT W , T S P = ψT P . ( 42
)
with θ > 1, φ < 1 and ψ < 1 and still checking:

D = θT H + φT W + ψT P . ( 43
)
With these assumptions, symptomatic infected individuals spend more time at home and less time at work and in public places. As a result, the fast equilibrium frequencies of symptomatic infected individuals become:

u S H * = θT H D , u S W * = φT W D , u S P * = ψT P D . (44)
For example, if we assume that asymptomatic infected people spend 14 hours per day at home, 8 hours at work and 2 hours in public places while symptomatic spend 19 hours a day at home, 4 hours at work and 1 hours in public places. The previous parameters are equal to θ = 1.36 and φ = ψ = 0.5. This only adds two parameters as they are related by relation 14θ + 8φ + 2ψ = 24.

Furthermore, the overall rate of infection is modified and will depend on the proportion f of asymptomatic infected persons. We recall that we assume that the proportion of asymptomatic infected persons remains constant for all t. Under these conditions, the global contact rate becomes β 1 (f ) :

β 1 (f ) = (u * ) 2 β(f ), (45)
Where β(f ) is the global rate of contact in the absence of confinement with a proportion f of asymptomatic infected persons:

β(f ) = f (β H (u * H ) 2 + β W (u * W ) 2 + β P (u * P ) 2 ) + (1 -f )(β H u * H u S H * + β W u * W u S W * + β W u * P u S P * ), (46)
This expression takes into account infections between sane individuals and asymptomatic infected individuals as well as those with symptomatic infected. Finally, the global rate of contact with confinement and with a proportion f of asymptomatic is written more simply as follows:

β 1 (f ) = (u * ) 2 D 2 β H T 2 H (f + θ(1 -f )) + β W T 2 W (f + φ(1 -f )) + β P T 2 P (f + ψ(1 -f )) , (47)
Finally, taking into account confinement and the masks we obtain an overall contact rate:

β 2 (f ) = (1 -pq) 2 β 1 (f ), (48)
u α 1 α 2 f β H β W β p T H T W T P θ φ ψ p q k -0.004 -0.003 -0.002 -0.001 0.000 0.001 S u α 1 α 2 f β H β W β p T H T W T P θ φ ψ p q k -2e-04 0e+00 2e-04 4e-04 6e-04 E u α 1 α 2 f β H β W β p T H T W T P θ φ ψ p q k -1e-04 0e+00 1e-04 2e-04 I u α 1 α 2 f β H β W β p T H T W T P θ φ ψ p q k -0.

Discussion of the results

Sensitivity analysis with respect to parameters

Figure 4 shows the sensitivity analysis with respect to the various parameters of the model taking into account the proportion of unconfined people, the masks as well as the effect of change in urban displacement behavior of the symptomatic. As we might expect, the effect on the population of susceptible individuals is contrary to the effect on the population of exposed, infected and removed ones. For example, a small increase in the proportion of unconfined persons, u, significantly increases the number of infections but decreases the number of susceptible persons. It is in fact normal that by reducing the number of people confined, we increase contacts and facilitate infection.

The parameters having the most effect on the infection by promoting the epidemic are the proportion of people outside confinement as well as the parameters which favor infection at work, T W , β W and φ where the contact rate is supposed to be much higher than in the other places. The parameters having the most effect in slowing the epidemic are the control parameters on the masks p and q and parameter α 2 . A small increase in α 2 corresponds to a decrease in isolation time for an asymptomatic person. It is understandable that by isolating asymptomatic people more quickly by practicing a policy of rapid and numerous tests, this will slow down the epidemic.

Threshold in case of partial release of confinement

While most countries are reluctant to lift severe confinement in fear of seeing a secondary/third peak in the covid-19 epidemic start again, it is useful to use our model for evaluating a threshold for release of confinement and protective measures to avoid a restart of the epidemic.

We now present numerical simulations in different cases. We still mention the covid-19 epidemic with parameters values mentioned above for the simulation of the complete and aggregated models. Under these conditions, N. [START_REF] Bacaër | Un modèle mathématique des débuts de l'épidémie de coronavirus en France[END_REF] found R 0 = 2.33 while A.

Moussaoui and P. [START_REF] Moussaoui | Prediction of confinement effects on the number of covid-19 outbreak in Algeria[END_REF] found R 0 = 2.1 for Algeria. However, if infectious persons are not tested and isolated rapidly, the infectivity period can last much more than one day and can be as large as ten days or more, leading to much larger values for R 0 , up to 3 to 3.5. It is for this reason that we have tested larger values of R 0 .

Assuming that the lifting of confinement occurs after a severe confinement phase and that Table 1 shows that it is necessary to keep confined a large part of the population, almost 27% for an R 0 of 2, and up to 45% for an R 0 of 3.5. These values remain high and show that it is necessary to couple partial release of confinement with protective measures such as the wearing of masks.

Threshold in case of partial release of confinement and use of masks

Figure 5 illustrates the threshold for the release of confinement which allows to avoid a second peak of the epidemic in the case R 0 = 3. In that example, the proportion of the population who is allowed to end confinement and to move again as before the start of the epidemic is 0.592, see table 1. It shows that below this confinement threshold u * = 0.59, there is no need to wear masks, Figure 5 Figure 6 illustrates the threshold for a total release of confinement u * = 1 for different R 0 ranging from 2 to 3.5. In this case all individuals are allowed to break out of confinement and behave exactly as before the epidemic. The figure 6 shows that the higher Indeed, without mask p = q = 0, R 0 (T * ) < 1. Lower figure : Above the threshold u * = 0.60, without mask p = q = 0, R 0 (T * ) > 1, there is a second peak after the release of the confinement. In order to avoid it, it is needed to use mask in order to be above the level curve R 0 (T * ) = 1. 20 the R 0 , the more the level curve corresponding to R 0 (T * ) = 1 is shifted at the top right of each of the four sub-figures, which means that to reach the un-confinement threshold making it possible to avoid a second peak, it is necessary to ask a larger proportion of the population to wear increasingly effective masks.

Figure 6 shows that it is possible to lift totally confinement with success for an epidemic of R 0 = 2, with 50% efficacy masks worn by only 50% of the population, whereas for R 0 = 2.5, we need on average 60% efficacy masks worn by 60% of the population, for R 0 = 3, 65% efficacy masks worn by 65%, and for R 0 = 3.5, 70% efficacy masks worn by at least 70% of the population.

Figure 7 up right shows that for an epidemic with an R 0 of 2.5, corresponding to the value usually reported for Covid-19, the release of confinement for 90% of the population is possible without risk, if 80% of the population wears masks in risky areas with efficacy 40% which more or less corresponds to the efficacy of masks not reserved for hospital staff but in sold in local shops for the public which seems realistic.

In addition, with teleworking, we can consider that if around 10% of people on average stay at home and are therefore not very prone to infection, this ensures successful release of confinement without a second peak with an R 0 of up to 2.5. It is also obviously very important to couple protection with masks with the use of tests to trace the chains of contamination and quickly isolate infected people in areas where the epidemic starts again after lifting of confinement.

Conclusion and perspectives

These results show that the coupling of a release of confinement at a almost total level 90% and even more, coupled to the use of masks by a reasonable fraction of the population in risky areas, is possible in good safety conditions. We may achieve this goal by keeping a rather small proportion of the population (10%) in a state of total confinement. In addition, the situation could be even more favorable because of the protective measures taken in establishments for the elderly to avoid contacts as well as teleworking allowing for many people to avoid risky places such as workplaces and public transportations. Consequently, this makes it possible to reach this goal and to obtain this target rate of partial confinement more easily. An even better efficiency of the masks as well as a higher proportion of people wearing masks, which seems to be the case especially in France where the masks are compulsory in many public places, should make it possible to carry out the end of confinement still in better conditions.

In the future, we plan to develop mathematical models taking into account the daily movements of individuals in different places, schools and universities, workplaces, transport and public places, shops where the risks of infection are different, Y. [START_REF] Liu | Understanding Unreported Cases in the COVID-19 Epidemic Outbreak in Wuhan, China, and the Importance of Major Public Health Interventions[END_REF] and human behavior P. Manfredi and A. d'Onofrio (2013). It also appears important to take into account the age structure of the populations insofar as the intensities of social contacts strongly depend on the age groups, Y. [START_REF] Liu | Understanding Unreported Cases in the COVID-19 Epidemic Outbreak in Wuhan, China, and the Importance of Major Public Health Interventions[END_REF]. Because the processes take place at different time scales, daily for travel and several months for the epidemic, it remains possible to use methods of aggregation of variables to reduce the dynamical system for which it still will be possible to obtain analytical results such as expression for confinement threshold.

Figure 1 :

 1 Figure 1: Transition graph of the SEIR model with different places.

(Figure 2 :

 2 Figure 2: Evolution of the proportions in the different places in each epidemic compartment over time. The parameters values used in the simulation are: γ = 10; β H = 2.87; β W = 10.03; β P = 1.43; T H = 14, T W = 8; T P = 2, α = 1, = 0.04; k = 0.2.

Figure 2 Figure 3 :

 23 Figure 2 shows numerical simulations of the complete model. It shows the evolution over time of the proportions of individuals in the different places for various epidemiologi-

Figure 2

 2 Figure 2 shows that the proportions in the different places for each epidemic compartment change over time while remaining proportional to the proportions at the fast equilibrium, i.e. u * H = T H D = 0.583, u * W = T W D = 0.333 and u * P = T P D = 0.083. The fast equilibrium is established quickly and is verified throughout the slow trajectory. It is therefore sufficient to simulate the reduced (aggregated) model because the fast equilibrium makes it possible to know the distribution in the different places at any time.

Figure 3

 3 Figure 3 compares the trajectories of the complete system taking into account the epidemic compartments by location with the reduced or aggregated system considering only 4 global epidemic compartments. The figure shows that the solutions of the two systems are extremely close. We can therefore use the aggregated model to understand the dynamics of the complete model.

Figure 4 :

 4 Figure 4: Sensitivity analysis for system (30)-(33) taking into account the proportion of unconfined people, the masks as well as the effect of change in urban displacement behavior of the symptomatic.

  upper figure. Above this threshold u * = 0.60, it is necessary to ask the population to wear masks, Figure 5 lower figure. For example, it is needed to use masks worn by 15% of the population and with efficacy almost equal to 15%.

Figure 5 :

 5 Figure 5: Illustration of the threshold for the release of confinement, u * = 0.592, for R 0 = 3. Upper figure : Below the threshold u * = 0.59, it is not even needed to use mask.

Figure 6 :

 6 Figure 6: Illustration of the threshold for a total release of confinement, u * = 1, for different values of R 0 . Up Left : R 0 = 2. Up Right : R 0 = 2.5. Down Left : R 0 = 3. Down Right : R 0 = 3.5.

Figure 7 :

 7 Figure 7: Illustration of the threshold for an almost total release of confinement at 90%, i.e. u * = 0.9, for different values of R 0 . Up Left : R 0 = 2. Up Right : R 0 = 2.5. Down Left : R 0 = 3. Down Right : R 0 = 3.5.

Table 1 :

 1 there are still approximately 95% of susceptible persons not yet infected, we obtain the following threshold values to avoid a secondary peak, with S(T * ) = 0.95, for different values of R 0 , see Table1. Threshold for the release of confinement for different R 0

	R 0	2	2.5	3	3.5
	u * 0.725 0.649 0.592 0.545

Acknowledgements

The authors would like to thank the editor and the anonymous reviewer for their valuable comments and suggestions, which have led to a significant improvement of the whole manuscript. Ali Moussaoui is supported by the DGRSDT, Algeria.

Consequently, we find a new threshold for the release of confinement with masks in order to avoid a second epidemic peak:

Threshold formulas were presented in A. [START_REF] Moussaoui | Prediction of confinement effects on the number of covid-19 outbreak in Algeria[END_REF] but here we generalize to the case of protection with masks.

Comparison of the final sizes of the epidemic for pure confinement strategy and pure mask strategy

In A. [START_REF] Moussaoui | Prediction of confinement effects on the number of covid-19 outbreak in Algeria[END_REF], the authors established the following final size relation in the case of pure confinement:

Where v * = 1 -u * which is the constant proportion of confined individuals imposed by the government at time t > T * . When v * → 1, (corresponding to lockdown measures), the final size of the epidemic will be:

A similar argument may be used to establish the final size of the epidemic in the case of pure mask strategy without confinement:

When pq → 1, corresponding to all the population uses masks of total efficacy, the final size of the epidemic will be: R(∞) R 0 R(T * ).

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