Maria Mitronika

Jacopo Profili

Antoine Goullet

Nicolas Gautier

Nicolas Stephant

Luc Stafford

Agnès Granier

Mireille Richard-Plouet
email: mireille.richard@cnrs-imn.fr

TiO 2 -SiO 2 Nanocomposite thin films deposited by Direct Liquid Injection of colloidal solution in an O 2 /HMDSO low-pressure plasma

Keywords: low-pressure plasma processing, direct-liquid injection, colloidal solution, TiO2 nanoparticles, TiO2-SiO2 nanocomposite thin films

TiO2 nanoparticles (NPs), 3 nm in size, were injected inside a very-low-pressure O2 plasma reactor using a liquid injector and following an iterative injection sequence.

Simultaneously, hexamethyldisiloxane (HMDSO) vapor precursor was added to create a SiO2 matrix and a TiO2-SiO2 nanocomposite (NC) thin film. Both the liquid injection and vapor precursor parameters were established to address the main challenges observed when creating NCs. In contrast to most aerosol-assisted plasma deposition processes, Scanning/Transmission Electron Microscopy (S/TEM) indicated isolated (i.e. non-agglomerated) NPs distributed in a rather uniform way in the matrix. The fraction of the TiO2 NPs inside the SiO2 matrix was estimated by SEM, Spectroscopic Ellipsometry (SE), and X-ray Photoelectron Spectroscopy.

All techniques provided coherent values, with percentages between 12 and 19%. Despite the presence of TiO2 NPs, SE measurements confirmed that the plasma-deposited SiO2 matrix was dense with an optical quality similar to the one of thermal silica. Finally, the percentage of TiO2 NPs inside the SiO2 matrix and the effective refractive index of the NCs can be tuned through judicious control of the injection sequence.

Introduction

Nowadays, there is an ongoing demand for smaller, faster, and low-cost integrated systems. Achieving this often requires new materials and structures that combine the more conventional ones. In this context, nanocomposite (NC) thin films formed of nanoobjects embedded in a matrix have gained great attention, as the properties of the resulting composite may be more likely to resemble those of the interface zones rather than those of the original constituents [START_REF] Nelson | Dielectric polymer nanocomposites[END_REF]. The high surface-area-to-volume ratio in NCs can also lead to enhanced macroscopic properties compared to the bulk materials.

One of the most significant challenge in the development of NC thin films is the control of the growth mechanisms, the final morphology and the incorporation of the nanoobjects using reproducible, versatile and low-cost processes. Several chemical and physical methods have been used in recent years. For example, sol-gel is a well-established process based on hydrolysis and condensation reactions. [START_REF] Guldin | Self-Cleaning Antireflective Optical Coatings[END_REF][START_REF] Kermadi | Microstructure and optical dispersion characterization of nanocomposite sol-gel TiO 2 -SiO 2 thin films with different compositions Spectrochim[END_REF] The main drawback of this approach is the need of several steps to produce NC thin films. In addition, the toxicity of the nanoparticles (NPs) during the drying of the film [START_REF] Charitidis | Manufacturing nanomaterials: from research to industry[END_REF] limits the industrialization. Vacuum processes such as PECVD or CVD have also demonstrated their ability to synthesize NC thin films by controlling the plasma-or thermally-activated reaction pathways. [START_REF] Larouche | Microstructure of plasma-deposited SiO2/TiO2 optical films[END_REF][START_REF] Li | Structural and Optical Properties of PECVD TiO 2 -SiO 2 Mixed Oxide Films for Optical Applications: Structural and Optical Properties of PECVD TiO 2 -SiO 2 … Plasma Process[END_REF] In parallel, sputtering approaches have also been used to create metal, metal-oxide or metal-nitrides NC thin films. [START_REF] Charitidis | Manufacturing nanomaterials: from research to industry[END_REF][START_REF] Nedfors | Reactive sputtering of NbCx-based nanocomposite coatings: An up-scaling study Surf[END_REF] Recently, there have been some attempts using hybrid (aerosol-assisted, direct liquid injection etc.) approaches for the creation of NC thin films. Profili et al. [START_REF] Profili | Deposition of nanocomposite coatings on wood using cold discharges at atmospheric pressure Surf[END_REF] implemented a one-step approach for the deposition of ZnO-SiO2 NC coatings on wood by the atomization of a stable suspension of ZnO dispersed in hexamethyldisiloxane (HMDSO) in an atmospheric pressure dielectric barrier discharge (DBD). Ross et al. [START_REF] Ross | The CVD of Nanocomposites Fabricated via Ultrasonic Atomization[END_REF] elaborated composite films using an ultrasonic atomizer and a PECVD system using Tetraethoxysilane (TEOS) as a matrix precursor and polystyrene nanoparticles with a mean size of 96 nm. The pressure in the specific system was varied between 100 and 500 mTorr. More recently, Kahn et al. [START_REF] Carnide | Procédé de préparation de nanoparticules[END_REF] proposed a new method that allowed to synthesize NPs from organometallic precursor in a reactor-injector prior to their injection in a RF low-pressure plasma reactor at 750 mTorr. The resulted NC thin films consisted of small (6 ± 3 nm) and isolated ZnO NPs homogeneously dispersed in an amorphous hydrogenated carbon matrix. This increased interest in hybrid approaches led to the development of new direct liquid injection systems [START_REF] Vervaele | Development of a new direct liquid injection system for nanoparticle deposition by chemical vapor deposition using nanoparticle solutions[END_REF] and studies regarding the interactions of the liquid droplets with the low pressure plasma systems [START_REF] Ogawa | The direct injection of liquid droplets into low pressure plasmas[END_REF] (misty plasmas). As a general trend, most of these studies focus on the deposition of inorganic NPs-organic matrix NC film. The majority of the reposted works involve high-pressure and atmospheric-pressure plasma system. These systems have been thoroughly studied mainly because of the technically-challenging conditions existing at very low pressures (e.g. risk of contamination of the vacuum chamber, malfunction of the turbomolecular pump, fast liquid vaporization without reaching the substrate). [START_REF] Fanelli | Aerosol-Assisted Atmospheric Cold Plasma Deposition and Characterization of Superhydrophobic Organic-Inorganic Nanocomposite Thin Films[END_REF][START_REF] Bruggeman | Plasma-liquid interactions: a review and roadmap Plasma Sources[END_REF] In most cases, however, the NPs in NC thin films are agglomerated, and their size is increased compared to the one in the colloidal solution before their injection in the plasma. [START_REF] Fanelli | Aerosol-Assisted Atmospheric Cold Plasma Deposition and Characterization of Superhydrophobic Organic-Inorganic Nanocomposite Thin Films[END_REF][START_REF] Profili | Nebulization of Nanocolloidal Suspensions for the Growth of Nanocomposite Coatings in Dielectric Barrier Discharges: Nebulization of Nanocolloidal Suspensions… Plasma Process[END_REF][START_REF] Dembele | Deposition of Hybrid Organic-Inorganic Composite Coatings Using an Atmospheric Plasma Jet System[END_REF][START_REF] Bardon | Dispersion of Cerium-Based Nanoparticles in an Organosilicon Plasma Polymerized Coating: Effect on Corrosion Protection Plasma Process[END_REF][START_REF] Frari | Hybrid Layers Deposited by an Atmospheric Pressure Plasma Process for Corrosion Protection of Galvanized Steel[END_REF][START_REF] Avril | Alumina particle reinforced TiO2 composite films grown by direct liquid injection[END_REF][START_REF] Profili | Influence of the voltage waveform during nanocomposite layer deposition by aerosol-assisted atmospheric pressure Townsend discharge[END_REF][START_REF] Profili | Deposition of nanocomposite coatings on wood using cold discharges at atmospheric pressure Surf[END_REF][START_REF] Brunet | Control of composite thin film made in an Ar/isopropanol/TiO2 nanoparticles dielectric barrier discharge by the excitation frequency Plasma Process[END_REF] The main bottlenecks for producing NC thin films are incorporating a high-volume fraction, preventing the agglomeration of NPs and dispersing them homogeneously in the matrix. Using a very-low-pressure system, the objective is to deposit thin films incorporating 3-nm TiO2 NPs in a high-quality inorganic SiO2 matrix. This is accomplished by the direct liquid injection of the 3-nm TiO2 NPs colloidal solution without a carrier gas and the addition of HMDSO vapor precursor in the very-low-pressure O2 plasma. Furthermore, as several works involving atmospheric-pressure plasma systems reported challenges to deposit in one-step and acquire good quality NC films, the differences and prospects in using very-low-pressure hybrid systems is investigated. The evaluation of the NC's quality, homogeneity and distribution of the TiO2 NPs are achieved by various characterization techniques such as high-resolution Scanning Transmission Electron Microscopy (HR-STEM), Energy Dispersive X-ray spectrometry (EDX), Atomic Force microscopy (AFM) and X-ray photoelectron spectroscopy (XPS).

Finally, with this work, besides establishing this novel and versatile approach, the objective is to identify experimental conditions for acquiring thin films oriented towards optical applications. Therefore, the optical characteristics of the film, such as its effective refractive index, are being extracted with Spectroscopic Ellipsometry (SE).

Experimental Details

TiO2 NPs colloidal solution

The colloidal solution used for the injection of the TiO2 NPs in the very-low-pressure plasma has been synthesized following a specific protocol developed at the Institut des Matériaux Jean Rouxel. [START_REF] Karpinski | Anatase colloidal solutions suitable for inkjet printing: Enhancing lifetime of hybrid organic solar cells Sol[END_REF][START_REF] El Kass | TiO 2 Anatase Solutions for Electron Transporting Layers in Organic Photovoltaic Cells[END_REF] The NPs enclosed in the colloidal solution were well-defined and monodispersed with an anatase crystalline structure. Their mean hydrodynamic diameter (including species of the solvent surrounding the NPs to counterbalance surface charge) obtained by Dynamic Light Scattering (DLS) with Zetasizer NanoZS (Malvern) was centered on 3.7 nm (Figure 1). The double layer surrounding the NPs has been estimated at 1 nm. [START_REF] El Kass | TiO 2 Anatase Solutions for Electron Transporting Layers in Organic Photovoltaic Cells[END_REF] Therefore, the actual TiO2 nanocrystallite diameter lay at 2.7 nm. As synthesized, the solvent of this solution was a mixture of 4-methyl-1,3-dioxolan-2-one also known as propylene carbonate (PC) and 1,2-propanediol or propylene glycol (PG), with a molar fraction of 27%:73% respectively. From previous studies of our group [START_REF] Karpinski | Anatase colloidal solutions suitable for inkjet printing: Enhancing lifetime of hybrid organic solar cells Sol[END_REF][START_REF] Mitronika | Modification of the optical properties and nano-crystallinity of anatase TiO2nanoparticles thin film using low pressure O2 plasma treatment Thin Solid Films[END_REF], we identified that some solvent molecules remain attached to the surface of the NPs, creating an 4.7 Å hydroxylated and organic shell around TiO2 NPs. For the effective injection and evaporation of the colloidal solution in the very-low-pressure PECVD system, the dilution of the colloidal solution with a more volatile one was carried out.

Therefore, the final colloidal solution used was a mixture of PC:PG:Methanol with a volume fraction of 20.25:54.75:25 . The size distribution of TiO2 NPs in the diluted solution was verified with DLS and was not altered by either dilution or long-time storage..

Deposition of TiO2-SiO2 NC thin films using the hybrid approach

For the elaboration of the TiO2-SiO2 NC thin film, the colloidal solution was injected in a very-low-pressure PECVD reactor. The schematic representation of the hybrid experimental setup is given in Figure 2. The reactor comprises a very-low-pressure high-density Inductively Coupled Plasma (ICP) source created at 13.56 MHz (400 W) in a fused silica discharge tube using a helicon antenna which is also connected to a matching network. A static magnetic field (50 G) is applied to the source in order to confine the electrons and densify the plasma. The plasma then expands into a stainless-steel processing chamber where the substrate holder is located. The substrate holder is 17 cm in diameter located in the middle of the processing chamber with a distance of 16.8 cm from the bottom of the ICP antenna. The Si substrates used, which were positioned in the middle of the substrate holder, were 1 cm x 2 cm in dimension and their temperature was measured at 400 K through temperature-sensitive labels located at the substrate's backside. In this system, plasma gases (95% O2, 5% Ar here) are introduced at the top of the plasma source and the high-vacuum pumping system is located 11 cm from the bottom of the processing chamber. The outlet of the vacuum pumping system that includes also a cold trap, is 10 cm in diameter. In Figure 2b, its location related to the substrate holder and the liquid doser can be seen. More information about the reactor can be found in [START_REF] Nicolazo | Study of oxygen/TEOS plasmas and thin SiOx films obtained in an helicon diffusion reactor Surf[END_REF][START_REF] Granier | Diagnostics in helicon plasmas for deposition Plasma Sources[END_REF].

For the deposition of the SiO2 matrix, HMDSO vapor was introduced from a heatedcontainer (at 35.8 °C) without any carrier gas through a distribution ring (indicated with blue in Figure 2), located 8 cm above the substrate. The injection of the TiO2 colloidal solution was achieved through a separate industrial direct-liquid doser (indicated with red in Figure 2). It is an injection system from Kemstream that features a normally closed fast solenoid valve (or injector), with a small conductance (flow coefficient), commonly called Cv. This valve is controlled in pulsed regime. The orifice of the injection system includes 8 holes with a diameter of 70 µm each. This injection system was mounted on the PECVD reactor with an angle of 55° with respect to the substrate holder. The distance of the injector to the reactor was fixed at 20 cm and therefore the distance of the injector to the substrate can be calculated at 48.24 cm. The pipe connecting the injector to the reactor was heated (yellow lines indicating the heating wire in Figure 2) at 423 K to enhance the evaporation process of the produced droplets. For maintaining very-low pressure during the deposition (3 mTorr) and therefore obtaining a high-quality inorganic matrix, no additional carrier gas has been used to assist the injection of TiO2 NPs. In such conditions, the droplet creation and the formation of the aerosol were mainly driven by the difference in pressure between the reservoir of the colloidal solution (760 Torr) and the plasma reactor (3 mTorr). [START_REF] Kooij | What Determines the Drop Size in Sprays?[END_REF] For this specific pressure difference, the mean droplet size obtained by Phase Doppler Anemometry (PDA) was 65.0 µm with a standard deviation of 29 µm. The opening of the injection valve was trigged by applying a positive pulse controlled by the VapSoft software of the direct-liquid injector. It is worth highlighting that during each pulsed injection of the colloidal solution in the plasma reactor the pressure instantaneously increased from 3 to 15 mTorr due to solvent evaporation [START_REF] Ogawa | The direct injection of liquid droplets into low pressure plasmas[END_REF] and then returned back to the steady-state pressure (3 mTorr) within less than 2 s. The pressure variation was detected from the capacitance diaphragm gauge (MKS Instruments France SA) and was read on the go. Additionally, the 15 mTorr was the maximum value that was provided by the gauge reading. Upon injection and pressure increase, the intensity of the ICP O2 plasma is decreased retrieving within less than 2 seconds its usual brightness. Despite this fact, the plasma is never extinguished remaining ignited during the total duration of the process. As a matter of fact, the intensities of O and Ar lines (at 394.5 nm and 419.8 nm, respectively) measured by time resolved OES are supplied as Electronic Supporting Information (ESI). These results, indicate that during each injection (within the DLI-ON), the emission intensities of the Ar and O lines are decreased trying to retrieve their steady state value within the pulses (Figure S1). This variation, especially for the Ar emission intensity is related to the pressure variation. Both intensities never reach a zero value verifying once more that the plasma remains ignited during the total duration of the hybrid deposition process. Regarding the RF power supply, during the droplet injection, a reflected power with a mean value of 2 W is provided by the CESAR RF Power Generator.

For deposition experiments, the total sequence length was set at one minute (Figure 2). More specifically, during the first 20 s, N=10 pulses of 1 ms (TON =1 ms) were triggered each two seconds (T=2 s). Thus, after this first-time lapse of N*2 s = 20 s, the valve remained closed for 60-N*2 s = 40 s (TOFF). Such sequence was repeated for 50 minutes. The solvent mass rate was estimated at 2.33 mg per pulsed injection and therefore 23.3 mg per minute for N=10 cycles.

During this intermittent mode of injection (Figure 3), the oxygen mass flow rate of the working gas was set at 24 SCCM (standard cubic centimeters per minute) while the continuous mass flow rate of HMDSO vapor was maintained at 0.11 SCCM. The fraction between HMDSO and O2 was chosen in a way to produce a silica-like matrix with a very low carbon content [START_REF] Bousquet | Comparison of structure and mechanical properties of SiO2-like films deposited in O2/HMDSO pulsed and continuous plasmas Surf[END_REF]. O2 was also used to ensure the removal of solvent-based organic moieties surrounding TiO2 NPs [START_REF] Mitronika | Modification of the optical properties and nano-crystallinity of anatase TiO2nanoparticles thin film using low pressure O2 plasma treatment Thin Solid Films[END_REF]. for a total of 20 s) and the second part without liquid injection (for a total off time TOFF = 40s).

Characterization of the NC thin films

The cross section and distribution of the TiO2 NPs inside the SiO2 matrix were investigated (Figure 4 a, b ,c) using transmission electron microscopy (S/TEM Themis Z G3, 300kV, Thermo Fisher Scientific). A typical image of high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM, 28.5 mrad, probe corrected) is reported in Figure 4 d. The elemental mapping (Figure 4 e f g) was performed using EDX (energy dispersive X-ray spectrometer) analysis. For the effective characterization of the samples through TEM, the deposition was carried on Tin doped Indium oxide deposited on polyethylene terephthalate (ITO (50 nm) / PET (125 µm), Sheldahl). Cross-sections of the sample were prepared by using an ultramicrotome (Leica UC7/FC7) equipped with diamond knife (Diatome). The morphological analysis of the TiO2-SiO2 NC film on Si was carried out through Scanning Electron Microscopy (SEM) with a JEOL JSM 7600F, in a secondary electron mode, operating at 5 kV. The chemical analysis by EDX was conducted with a SAMx SDD system on a SEM JEOL JSM 5800LV. To increase the EDX signals, a dedicated thicker film was prepared (224 nm). To ensure that no additional Si coming from the substrate would interfere with the quantification of the silica matrix the dedicated thicker NC film was deposited on ITO/PET substrate. To probe different thicknesses, EDX spectra were recorded at three accelerating voltages: 15 kV, 10 kV and 7 kV and a magnification of X1000 (2552 µm 2 ) was chosen to smooth any local anomalies of chemical concentrations. The software Stratagem was used for EDX measurements because first the volume irradiated by the electron beam and second the X-ray collected are related to a larger depth than the thickness of the top layer. [START_REF] Pouchou | X-Ray Microanalysis of Thin Surface Films and Coatings Microchim[END_REF] The atomic percentages were determined by considering a theoretical model for the probed zone [START_REF] Christien | The Use of Electron Probe MicroAnalysis to Determine the Thickness of Thin Films in Materials Science X-Ray Spectroscopy[END_REF]. For this, three layers were considered: first, 224 nm of NC (Ti, Si, O), second, 50 Raman spectroscopy measurements were performed using a microconfocal Raman microscope (Renishaw inVia). The instrument was equipped with a double edge filter to eliminate the Rayleigh scattering and a charged couple device (CCD) camera working at a temperature of 220 K with a 1024x256 pixel array. The setup was composed of a confocal microscope that was equipped with an automated XYZ table and an X50 magnification objective. The laser excitation source used was set at 514 nm and the focused power was adjusted to avoid any degradation of the samples. Therefore, the power was fixed at 1.5mW and a recording time of 1 s with an accumulation of 10 measurements was used.

The chemical characterization of the topmost surface was investigated with X-ray photoelectron spectroscopy (XPS) on a Kratos Nova spectrometer with Al Kα radiation at 1486.6 eV. Neutralization of the surface was applied to avoid differential charging effect. Survey spectra were measured from 1200 to -2 eV (Binding Energy) using a pass energy of 160 eV whereas high-resolution spectra over specific regions of interest were recorded with a pass energy of 40 eV. Short time measurements were recorded (dwell of 100 ms per 0.1 eV stepscan for O1s, Ti 2p and 200 ms for C1s). After checking that no X-ray damage was induced, the scans were summed up to increase the signal-to-noise ratio. The decomposition and fitting of the measured XPS peaks was carried out using CasaXPS software [START_REF] Fairley | Copyright©[END_REF]. After accounting for the background using a U2-Tougaard background, Gaussian-Lorentzian (30-70%) functions were selected to model the different components. The signals were calibrated in energy according to the position of the C 1s band at the binding energy of 285 eV.

To be able to compare the results obtained from different characterization techniques, detailed calculations have been achieved (which are supplied as Electronic Supporting Information) to transform at% of elements to vol% of oxides.

Results

Structural characterization of the TiO2-SiO2 NC thin films

Cross section structural analysis of the plasma-deposited NC thin film was performed using TEM and STEM (Figure 4). Through multiple measurements on low magnification images (Fig. 4a), the thickness of the film was estimated at 85 +/-6 nm (a growth rate of 1.7 nm/min for a total plasma deposition time of 50 min). This value is comparable to the one achieved in O2/HMDSO plasmas without injection of the TiO2 colloidal solution [START_REF] Bousquet | Comparison of structure and mechanical properties of SiO2-like films deposited in O2/HMDSO pulsed and continuous plasmas Surf[END_REF]. Using highresolution images on the rectangular area (black contour) shown in Fig. 4b and based on the contrast between the amorphous matrix and the basal planes of the nanocrystals, some TiO2

NPs can be identified (enlarged NPs as an inset in Figure 4) and stressed with white-rounded contour shapes (Figure 4c). Interestingly, no agglomeration of these NPs is observed but rather an assembly of small and isolated TiO2 NPs in the SiO2 matrix. Using ImageJ software, the size of each of these NPs (14 nanoparticles identified in the specific zone probed by electron microscopy) was extracted and their mean value was found to be at 2.7 ± 0.4 nm. By comparing this value with the hydrodynamic diameter of TiO2 NPs in the colloidal solution obtained by DLS (2.7 nm + 1 nm, Figure 1), it can be confirmed that the NPs' size remains intact following the aerosol formation and plasma deposition. This is a remarkable result with respect to other experiments on hybrid plasma deposition of NC thin films for which agglomeration phenomena were deemed unavoidable [START_REF] Fanelli | Aerosol-Assisted Atmospheric Cold Plasma Deposition and Characterization of Superhydrophobic Organic-Inorganic Nanocomposite Thin Films[END_REF][START_REF] Profili | Nebulization of Nanocolloidal Suspensions for the Growth of Nanocomposite Coatings in Dielectric Barrier Discharges: Nebulization of Nanocolloidal Suspensions… Plasma Process[END_REF][START_REF] Dembele | Deposition of Hybrid Organic-Inorganic Composite Coatings Using an Atmospheric Plasma Jet System[END_REF][START_REF] Bardon | Dispersion of Cerium-Based Nanoparticles in an Organosilicon Plasma Polymerized Coating: Effect on Corrosion Protection Plasma Process[END_REF][START_REF] Frari | Hybrid Layers Deposited by an Atmospheric Pressure Plasma Process for Corrosion Protection of Galvanized Steel[END_REF][START_REF] Avril | Alumina particle reinforced TiO2 composite films grown by direct liquid injection[END_REF][START_REF] Profili | Influence of the voltage waveform during nanocomposite layer deposition by aerosol-assisted atmospheric pressure Townsend discharge[END_REF][START_REF] Profili | Deposition of nanocomposite coatings on wood using cold discharges at atmospheric pressure Surf[END_REF][START_REF] Brunet | Control of composite thin film made in an Ar/isopropanol/TiO2 nanoparticles dielectric barrier discharge by the excitation frequency Plasma Process[END_REF]. Moreover, TiO2 NPs displayed in Figure 4c seem dispersed in a rather random way in the SiO2 matrix, with no evidence of preferential incorporation sites. No cracking of the SiO2 layers is visible, showing that the insertion of the NPs does not strongly affect the growth of the silica layer.

The typical HAADF-STEM image given in the same figure confirms that no bright contrast associated to possible agglomeration of the electronically denser TiO2 NPs is detected, indicating the homogeneity of the NPs dispersion, at this scale of observation. Moreover, EDX elemental mapping also confirms that both oxygen and silicon (Fig. 4 e,f) are homogeneously distributed along the cross section of the NC thin film. On the other hand, Ti (Fig. 4g) signal, which is only linked to the TiO2 NPs, is identified with some small dots verifying the TEM images (Fig. 4c) and their distribution inside the film. Since the data are a projection of 80 nm thin sample, the reinforcement in contrast, (in some small parts of Fig. 4g) may come from particles distributed along this thickness. The crystalline phase of the TiO2 NPs was identified using Raman spectroscopy. Figure 5 presents the Raman spectra of the plasma-deposited NC thin film (black line) and the TiO2 NPs spin coated thin films (red line) using the as synthesized colloidal solution, with a focus on the Eg anatase peak. The analysis was limited to using the peak positions since extreme caution is required when using the Raman intensities of very thin films. Raman modes can be assigned to the Raman spectra of the anatase crystal: 144 (Eg), 200 (Eg), 400 (B1g), 513 (A1g), 518 (B1g) and 642 cm -1 (Eg) [START_REF] Zhang | Raman scattering study on anatase TiO 2 nanocrystals[END_REF][START_REF] Ohsaka | Raman spectrum of anatase[END_REF]. In this figure, it can be observed that the Eg peak lies at 154 cm -1 , a value shifted from the expected anatase peak at 144 cm -1 [START_REF] Ma | Investigation of structural transformations in nanophase titanium dioxide by Raman spectroscopy[END_REF][START_REF] Pighini | Low-frequency Raman characterization of size-controlled anatase TiO2 nanopowders prepared by continuous hydrothermal syntheses[END_REF]. It is well-established experimentally that Raman spectra of low-dimensional semiconductor crystals are modified when compared to the one of bulk crystals: (1) the Raman peaks are asymmetrically broadened and (2) their positions are shifted [START_REF] Gao | Size-Dependent Raman Shifts for nanocrystals[END_REF][START_REF] Rolo | Raman spectroscopy of optical phonons confined in semiconductor quantum dots and nanocrystals[END_REF]. The former is not observed mainly due to the size distribution of the specific nanocrystals. This effect, results in masking the confinement effect and averaging the Raman response [START_REF] Kelly | Raman Spectroscopy as a Morphological Probe for TiO 2 Aerogels[END_REF]. However, the results presented in Figure 5 (black line) confirm the strong shift of the Eg band that the plasma-deposited TiO2 NPs embedded in the SiO2 matrix are anatase crystallites and in nanometric size. In addition to this, the Raman spectrum of the TiO2 NPs thin film through the spin-coated technique is given with red color.

The fact that for both cases the anatase peak is found in the same position (154 cm -1 ) confirms that upon the colloidal solution dilution, the TiO2 NPs injection and the SiO2 matrix deposition, the nano crystallite is not affected. and the spin coated TiO2 NPs thin film using the as synthesized colloidal solution.

Morphological analysis of the TiO2-SiO2 NC thin films

The surface morphological analysis of the TiO2-SiO2 NC thin film was investigated by SEM and AFM as shown in Figure 6. From the SEM top view image (Fig. 6a) in high magnification, the surface of the film is homogeneous and nanostructured, revealing some rounded domains.

Since the resolution of the SEM is at 10 nm, one can understand that the exact identification and quantification of these domains is impossible. These same rounded domains are highlighted in the AFM measurements (bright yellow-white color in Fig 6b). They are located higher on the z axis and their size is higher than the one of the initial NPs in solution measured by DLS in Figure 1. From detailed analysis of multiple AFM scans at low and high magnification, the mean size of the domains was extracted and plotted as a histogram in Fig 6c . The fitting of a LogNormal distribution provided a mean size of 23 nm of these rounded domains and a standard deviation of 0.3 nm. The Rq (RMS) roughness of the NC film was also deduced. A value of 2 nm was obtained, which was far different from the measured roughness of the silica thin film (Fig. 6c, Rq (RMS) = 0.3 nm) obtained by PECVD with the same reactor in comparable experimental conditions. The AFM scan for this case is given in Figure 6c, revealing the much smoother topography having a Rq (RMS) roughness lower than 5 nm. Since the deposition on flat silicon substrate should not lead to additional roughness or nanostructuration, especially for the deposition of amorphous SiO2 (matrix), the nanometric pattern observed by AFM must be linked to the presence of TiO2 NPs inside the SiO2 matrix. It is worth highlighting, however, that the mean domain size displayed in Figure 6 (23 nm) is much larger than the mean size of TiO2 NPs obtained from both DLS (Figure 1) and TEM (Figure 4a). This is because TiO2 NPs are located at different locations across the thickness of the SiO2 matrix (85 +/-6 nm) and the PECVD process used in this study provides highly conformal plasma deposition. [START_REF] Raupp | Conformality of SiO 2 films from tetraethoxysilane-sourced remote microwave plasma-enhanced chemical vapor deposition[END_REF][START_REF] Park | High-quality conformal silicon oxide films prepared by multi-step sputtering PECVD and chemical mechanical polishing[END_REF][START_REF] De Larclause | Conformity of Silica-like Thin Films Deposited by Atmospheric Pressure Townsend Discharge and Transport Mechanisms[END_REF][START_REF] Sobel | Conformal SiO 2 coating of sub-100 nm diameter channels of polycarbonate etched ion-track channels by atomic layer deposition[END_REF][START_REF] Arts | Film Conformality and Extracted Recombination Probabilities of O Atoms during Plasma-Assisted Atomic Layer Deposition of SiO 2 , TiO 2 , Al 2 O 3 , and HfO 2[END_REF] Therefore, the deeper the particle is located inside the matrix, the larger is the size of the structure observed by SEM and AFM on the topmost surface [START_REF] Profili | Nebulization of Nanocolloidal Suspensions for the Growth of Nanocomposite Coatings in Dielectric Barrier Discharges: Nebulization of Nanocolloidal Suspensions… Plasma Process[END_REF][START_REF] Profili | Influence of the voltage waveform during nanocomposite layer deposition by aerosol-assisted atmospheric pressure Townsend discharge[END_REF]. From the high-magnification and highly-localized analysis obtained both from TEM and SEM/AFM, the plasma-deposited TiO2-SiO2 NC thin films indicated a rather homogenous spatial distribution. To have an overall view of this film, low-magnification EDX mappings of the film's surface were recorded. To extract adequate resolution for each element, a thicker film was especially deposited on the ITO/PETsubstrate (224 nm). The results are presented in Figure 7. While homogeneous distributions are observed for silicon and oxygen elements over the 2552 µm 2 surface, abnormal rounded-ring shapes with diameters of several tens of micrometers can be seen for the titanium element. Such feature suggests the well-known coffee-ring effect, happening when the solute migrate in the edge of the evaporating droplet due to the formation of preferential flow and faster evaporation at this area. [START_REF] Deegan R D, Bakajin | Capillary flow as the cause of ring stains from dried liquid drops[END_REF][START_REF] Maki | Fast Evaporation of Spreading Droplets of Colloidal Suspensions[END_REF][START_REF] Yunker | Suppression of the coffee-ring effect by shape-dependent capillary interactions[END_REF] Based on such large-scale SEM and EDX observations, it seems that TiO2 species are provided to the substrate during growth as droplet and that evaporation and removal of organic species from the solvent [START_REF] Mitronika | Modification of the optical properties and nano-crystallinity of anatase TiO2nanoparticles thin film using low pressure O2 plasma treatment Thin Solid Films[END_REF] occur, at least partially, on the substrate surface rather than in the gas phase. A quantitative EDX analysis was conducted on the large region (2552 µm 2 ) avoiding an area exhibiting droplet traces such as the ones shown in Figure 7. In order to avoid all inhomogeneity aspects (in particular those related to the Ti element), only averaged values are shown in Table 1. The results are presented only for Ti and Si due the weak energy of the carbon and oxygen EDX peaks and the strong interactions throughout the layers for these elements (see details in Sec. 2.2). From such analysis, 15% Ti and 85% Si is obtained (atomic %). Based on the calculations provided in the ESI, the volume percentage corresponding to 15 at% is estimated to reach 16 %.

Surface chemical composition investigation of the TiO2-SiO2 NC thin films

The chemical analysis of the topmost surface of the TiO2-SiO2 NC thin films was carried out with XPS. Four different positions were measured on the same sample. Figure 8 represents the intensity of the O 1s, Ti 2p, C 1s and Si peaks. From these elementary spectra, the mean atomic concentration and the standard deviation in percentage is given by considering the components included in each atomic peak. The contribution of oxygen bonded to silicon atoms, noted as O-Si appears at the binding energy of 532.6 eV whereas the one corresponding to the bond with titanium, noted as O-Ti, at 530.5 eV. [START_REF] Li | Structural and Optical Properties of PECVD TiO 2 -SiO 2 Mixed Oxide Films for Optical Applications: Structural and Optical Properties of PECVD TiO 2 -SiO 2 … Plasma Process[END_REF] A noteworthy observation in this figure is that no peak corresponding to Ti-O-Si (at 531.5 eV [START_REF] Ondračka | Unravelling local environments in mixed TiO2-SiO2 thin films by XPS and ab initio calculations[END_REF]) can be found, as expected for a NC, confirming that only two different phases are detected in the film. [START_REF] Ondračka | Unravelling local environments in mixed TiO2-SiO2 thin films by XPS and ab initio calculations[END_REF] As reported by many authors [START_REF] El Kass | TiO 2 Anatase Solutions for Electron Transporting Layers in Organic Photovoltaic Cells[END_REF][START_REF] Moulder | R C and Physical Electronics, Incorporation 1995 Handbook of X-ray photoelectron spectroscopy: a reference book of standard spectra for identification and interpretation of XPS data[END_REF], the signal of titanium element in XPS results from two separated peaks that can be attributed to Ti 2p3/2 and Ti 2p1/2. The 2p3/2 peak lies at 458.5 eV as expected for Ti 4+ cations in TiO2. Since the injected colloidal solution of TiO2 NPs was synthesized in an organic solvent, the contribution of carbon element was investigated. Therefore, three components were identified in the C 1s peak. The first and most prominent one, at 285eV, is usually assigned to C-C bonds of both adventitious elemental carbon and the solvent, while the second, at 286.0eV, and the third, 288.5eV, can be related to -C-O bonds and C=O bonds, respectively. Finally, the Si 2p3/2 peak has a binding energy of 103 eV, which is related to the fully oxidized SiO2 matrix [START_REF] Moulder | R C and Physical Electronics, Incorporation 1995 Handbook of X-ray photoelectron spectroscopy: a reference book of standard spectra for identification and interpretation of XPS data[END_REF]. Consequently, from the Ti2p and Si 2p signals only, the estimated atomic percentage in Ti is at 12 ± 0.004 % and for Si at 88 ± 0.004 %; these values are consistent with those obtained by EDX at the SEM (Table 1). The atomic percentage from XPS can be treated in the same manner as in Section 3.2 and the calculations provided in the ESI, leading to a volume percentage of 13 %.

Spectroscopic ellipsometry of the TiO2-SiO2 NC thin films

The investigation of the optical and physical characteristics was carried out with spectroscopic ellipsometry. The ellipsometric ratio ρ = tan(Psi)exp(jDelta) is analyzed from the measured Psi and Delta. To extract the structural and optical information from these data, the sample was modeled as a homogenous layer on silicon substrate, without any top-layer roughness. Due to droplet traces observed at low magnification in the Section 3.1, a more pronounced diffusion of the incident light was observed (in the measurement) at shorter wavelengths. Hence, the fitting was carried out in the limited range of 550 -1000 nm. The validity of the model and therefore the evaluation of the accuracy of the fit was estimated by the Mean Square Error (MSE) factor. [START_REF] Saha | Spectroscopic ellipsometry characterization of amorphous and crystalline TiO 2 thin films grown by atomic layer deposition at different temperatures[END_REF] Assuming the ellipsometry was not sensitive enough to detect the NC nature of the sample, the first ellipsometric model used was a one phase material with Tauc-Lorentz dispersion law (Figure 9a, Model A). [START_REF] Saha | Spectroscopic ellipsometry characterization of amorphous and crystalline TiO 2 thin films grown by atomic layer deposition at different temperatures[END_REF] However, the experimental and simulated data show discrepancies, providing a high value of the error value (MSE=22). Therefore, a two-phase material (SiO2 and TiO2) was set up to account for the ellipsometric data of the TiO2-SiO2 NC thin films. First, the dispersion laws for the SiO2 matrix and the TiO2 NPs have to be assessed. Conducting an experiment of depositing the SiO2 matrix while injecting the solvent without the NPs, using the same injection conditions as the plasma-deposited TiO2-SiO2 NC thin films, allowed us to acquire a reference SiO2 matrix film and assess its quality, in the range between 550 and 1000 nm. The model used to fit the ellipsometry data of the matrix was the one of the classical thermal SiO2 material, as expressed by Herzinger et al. [START_REF] Herzinger | Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation[END_REF] using the Sellmeier/Cauchy dispersion law.

The measured and simulated Psi and Delta signals are given in Figure 10a. In this figure, the quality of the fit appears to be high and the MSE for all the wavelength range was given at MSE=10. In Figure 10b, the refractive index (n) and the extinction coefficient (k) of the SiO2 matrix are given. For the TiO2 NPs, the Tauc-Lorentz dispersion law was used based on a spincoated 50 nm TiO2 NPs film. The effective refractive index for this spin coated film was found at 1.68 at 633 nm wavelength (as voids where evidenced in the film) and the optical bandgap (Eg) at 3.4 eV. The detailed results can be found in a very recent publication of the group. [START_REF] Mitronika | Modification of the optical properties and nano-crystallinity of anatase TiO2nanoparticles thin film using low pressure O2 plasma treatment Thin Solid Films[END_REF] The fit of the NC film using the second model (Model B), resulted in an improved error (MSE=4).

Then, based on the cross-section results obtained with TEM (Fig.

Comparison with lower and higher TiO2 NPs injection rate

To assess the agility of this hybrid process into acquiring NC thin films with different composition and optical characteristics, experiments with lower and higher injection rate were attempted. The N of the injection sequence (for deposition of TiO2 NPs) was set either at N=2

or at N=30 (instead of 10) and the flow rate of the HMDSO precursor (for deposition of the SiO2 matrix) was kept at 0.11 sccm. The total deposition time was maintained at 50 min. Figure 11a provides the stacked column (%) resulting from the XPS core level spectra fitting as explained in Section 3.3. In order to assess the reproducibility of the hybrid approach the XPS stacked column provide the mean value and the standard deviation (as error bars) for two to four different experiments in each condition (including the N=10). As expected, for lower N and therefore lower amount of TiO2 NPs injected in a minute sequence, the atomic percentage of TiO2 on the surface of the TiO2-SiO2 NC thin film was much lower than N=10 (1% for N=2 and 8% for N=10). Additionally, for the highest TiO2 NPs injection rate (N=30), we can observe a three times higher TiO2 percentage (25% for N=30) than the N=10. Furthermore, using the data obtained by spectroscopic ellipsometry, and the Model B proposed in Section 3.4, the percentage of the TiO2 NPs and the effective refractive index of the NC at 633 nm wavelength was obtained (shown in Figure 11b). Once more, to account for the reproducibility of this hybrid process, the results shown in Figure 11b

Discussion

As demonstrated above, over the range of experimental conditions investigated, the deposition of TiO2-SiO2 NC thin films using a hybrid approach leads to small and isolated (nonaggregated) TiO2 NPs well-distributed into a dense and inorganic SiO2 matrix. Compared to other processes reported in scientific literature (mostly based on atmospheric-pressure systems) [START_REF] Profili | Nebulization of Nanocolloidal Suspensions for the Growth of Nanocomposite Coatings in Dielectric Barrier Discharges: Nebulization of Nanocolloidal Suspensions… Plasma Process[END_REF][START_REF] Brunet | Control of composite thin film made in an Ar/isopropanol/TiO2 nanoparticles dielectric barrier discharge by the excitation frequency Plasma Process[END_REF][START_REF] Profili | Influence of the voltage waveform during nanocomposite layer deposition by aerosol-assisted atmospheric pressure Townsend discharge[END_REF][START_REF] Profili | Interaction of atomized colloid with an ac electric field in a dielectric barrier discharge reactor used for deposition of nanocomposite coatings[END_REF][START_REF] Denis | Effect of Titanium Oxide Nanoparticle Incorporation into nm Thick Coatings Deposited Using an Atmospheric Pressure Plasma[END_REF], the proposed method significantly limits agglomeration of the NPs and improves their dispersion in the matrix. An additional important factor leading to the limited agglomeration of the NPs could be the initially very small size of the NPs in the stable (over time and composition) colloidal solution. Changes in the agglomeration dynamics could partially be attributed to the differences affecting the aerosol formation dynamics between operating at low pressure compared to atmospheric conditions. In particular, at low pressure, the nature of the flow is mainly molecular flow whereas it is laminar or even turbulent at atmospheric pressure. [START_REF] Ogawa | The direct injection of liquid droplets into low pressure plasmas[END_REF][START_REF] Reist | Aerosol science and technology[END_REF][START_REF] Hinds | Aerosol technology: properties, behavior, and measurement of airborne particles[END_REF] It could also be related to the low injection rates of the colloidal solution. Using a direct-liquid injection in a CVD system at low pressure, Varvaele et al. [START_REF] Vervaele | Development of a new direct liquid injection system for nanoparticle deposition by chemical vapor deposition using nanoparticle solutions[END_REF] have identified that low colloidal solution injection rates (as in this study) promotes the dispersion of the NPs whereas at higher rates, they tend to coalesce such that the mean size distribution shifts towards higher values.

Because of the overall good homogeneity of the film, the optical and dielectric properties of the films could also be deduced by proper modelling of the ellipsometric signals (especially between 550 and 1000 nm). This feature qualifies the process as suitable for the preparation of TiO2-SiO2 NC thin film for optical applications in this wavelength range. Despite the solvent injection, the plasma-deposited matrix proved to be of high quality as the fit with thermal silica was significantly accurate. This is a proof that carbon is not incorporated inside the matrix confirming its inorganic nature. By tuning the injection sequence, it was further possible to tune the effective refractive index, having the possibility to reach up to 53 vol% in TiO2. To the best of our knowledge, these results are unique compared to the reported approaches. [START_REF] Ogawa | The direct injection of liquid droplets into low pressure plasmas[END_REF][START_REF] Profili | Nebulization of Nanocolloidal Suspensions for the Growth of Nanocomposite Coatings in Dielectric Barrier Discharges: Nebulization of Nanocolloidal Suspensions… Plasma Process[END_REF][START_REF] Brunet | Control of composite thin film made in an Ar/isopropanol/TiO2 nanoparticles dielectric barrier discharge by the excitation frequency Plasma Process[END_REF][START_REF] Profili | Influence of the voltage waveform during nanocomposite layer deposition by aerosol-assisted atmospheric pressure Townsend discharge[END_REF][START_REF] Profili | Interaction of atomized colloid with an ac electric field in a dielectric barrier discharge reactor used for deposition of nanocomposite coatings[END_REF][START_REF] Vietro | Aerosolassisted low pressure plasma deposition of antimicrobial hybrid organic-inorganic Cucomposite thin films for food packaging applications[END_REF] Due to the set of experimental conditions used, the combination of adjusted volatility of solvents, remarkable stability of the colloidal solution and condition of spraying in a low-pressure plasma reactor resulted in an efficient transport of the droplets containing the non-agglomerated NPs towards the substrate. The optimization of the injection sequence permitted to keep the NPs protected inside the liquid droplets down to the substrate, thus preventing them to aggregate following their interaction with the plasma species. A downside of these liquid droplets as observed by the low magnification SEM-EDX images is the coffee ring patterns of the TiO2 NPs. Several works report the depiction of droplets due to the limited volatility of the injected solution or the increased mean size (due to system limitations) of the produced droplets [START_REF] Ross | The CVD of Nanocomposites Fabricated via Ultrasonic Atomization[END_REF][START_REF] Goeckner | Progress report: Direct injection of liquids into low-pressure plasmas[END_REF][START_REF] Mauchauffé | Liquid-Assisted Plasma-Enhanced Chemical Vapor Deposition of Catechol and Quinone-Functionalized Coatings: Insights into the Surface Chemistry and Morphology: Liquid-Assisted Plasma-Enhanced Chemical Vapor Deposition of Catechol… Plasma Process[END_REF]. Given that some of the produced droplets in the specific system (by the direct-liquid injector) do not have enough time to evaporate (low volatility of the solvent), the evaporation process happens at the surface of the film. Further works are under progress to get better insight in the mechanisms involved during the process in terms of droplet formation, droplet travel speed and evaporation of droplet solvents due to the interaction with the lowpressure plasma. This will open a promising chapter for understanding the impact of lowpressure plasma on the physical and chemical state of the liquid droplet during plasma deposition of NC thin films.

Conclusion

In this work, the creation of TiO2-SiO2 NC thin films by injecting a colloidal solution in a very-low-pressure O2/HMDSO plasma was reported for the first time. The iterative injection sequence was selected in a way to control and maintain in low levels the deposition rate of the NC films. The pressure was maintained at 3 mTorr (with slight variations) leading to the deposition of good-quality inorganic SiO2 matrix. The cross section of the film revealed a fairly homogeneous distribution of the NPs inside the matrix, at the local scale. The amount of these 3-nm NPs was evaluated by multiple characterization techniques and was estimated at 16 vol.

% at the SEM scale and 19 vol. % as deduced from ellipsometry i.e. on a larger area. The surface composition revealed a slightly lower value (13 vol. %), which could be explained due to the coating effect of the SiO2 matrix (during the last second of the injection sequence). Notably, not significant agglomeration of NPs was observed. Adjusting the parameters of the injection sequence (from N=2 to to N=30), NC films with lower and higher TiO2 NPs content and thus different optical properties could be achieved. Due to the originality of this work and its novelty, large-surface area deposition has still to be improved. This requires conducting theoretical and further experimental studies to understand the behavior and evaporation of the liquid droplets inside low-pressure plasmas. Through these first results it is evidenced that potentially, an agile approach like this could give the freedom to tune and adjust the NC's properties by adjusting both the nature of the NPs and the matrix but also of the injection parameters reaching a large range of NPs content inside the matrix.

Figure 1 :

 1 Figure 1: Hydrodynamic size distribution of the TiO2 NPs inside the as-prepared (PC-PG) colloidal solution deduced from DLS. Line corresponds to a Log-Normal distribution fit, based on which the mean diameter size was extracted.

Figure 2 :

 2 Figure 2: Schematic of the hybrid experimental setup including the low pressure ICP O2 reactor, the liquid doser apparatus (with red) and the HMDSO vapor distribution system (with bleu) at a) side view and b) top view. The in situ characterization techniques are also indicated on this schematic.

Figure 3 :

 3 Figure 3: Description of the plasma deposition sequence including the first part of pulsed injection of the colloidal solution (10 cycles of TON with an off time of 2 s between each pulse

  nm of ITO (In, Sn, O), and third, PET (C, O). The different atomic percentages were fitted to reproduce the intensity of EDX signals as a function of the accelerating voltage, accounting for the thicknesses of the NC and ITO and attenuation of signals throughout the different layers.The determination of carbon and oxygen was not reliable because the corresponding signals may come from ITO and PET underneath (one must also consider that the effective thickness of PET analyzed by EDX depends on the accelerating voltage). However, the precautions taken before acquisition and the good fit between theoretical curves generated by Stratagem and experimental values make it possible to rely on the results concerning silicon and titanium atomic percentages.The optical characteristics and composition of the TiO2-SiO2 NC thin films deposited on silicon substrates were also examined through a rotating compensator spectroscopic ellipsometer (J.A. Woollam M-2000). The measurements were conducted in situ (i.e. during plasma deposition) in the wavelength range of 245 to 1000 nm at 71.7° incident angle.Atomic Force Microscopy (AFM) was used in tapping mode to investigate the nanostructured topography and roughness of the NC thin films. AFM Nanowizard II (JPK Instruments), Alcoated cantilever (PPP-NCHR-50 from NANOSENSORS) with a free resonance frequency of 330 kHz and a typical spring constant of about 42 N/m were used. The size distribution of the NPs was deduced from the AFM scans by using the ImageJ software after adjustment of the threshold and transformation in binary format.

Figure 4 :

 4 Figure 4: TEM cross section of the TiO2-SiO2 NC thin film a) in low magnification indicated with white dashed lines, b) high magnification and c) higher magnification image focusing on the distributed TiO2 NPs inside the SiO2 matrix. An inset is given with higher magnification

Figure 5 :

 5 Figure 5: Raman spectra indicating the Anatase Eg band at 154 cm -1 and the second order Si band peak at 303 cm -1 for the TiO2-SiO2 NC thin film through the hybrid approach (black color)

Figure 6 :

 6 Figure 6: a) SEM secondary electron image of the surface, b) AFM scan (1 μm x 1 μm) of the TiO2-SiO2 NC thin films, c) AFM scan (1 μm x 1 μm) of SiO2 only when deposited on Si substrate and d) size distribution of the rounded domains deduced from the AFM scans.

Figure 7 :

 7 Figure 7: SEM backscattered electrons image and EDX map of Ti superimposed (left) and corresponding EDX maps of Si and O (right).

Figure 8 :

 8 Figure 8: O 1s, Ti 2p, C 1s and Si 2p peaks of the TiO2-SiO2 NC thin film measured by XPS (mean atomic percentages and standard deviation of the components from 4 different positions in between brackets).

4 ,

 4 Section 3.1) for the nanocomposite film, the Bruggeman Effective Medium Approximation (BEMA) was used (Figure 9b, Model B) to model the TiO2 NPs dispersed in the SiO2 matrix. A much better agreement was observed between the experimental and the simulated data.

Figure 9 :

 9 Figure 9: Psi (red) and Delta (green) results of the measured (continuous) and modeled (dashed) data with ellipsometry, based on Model A (a) and Model B (b).

Figure 10 :

 10 Figure 10: a) Psi (red) and Delta (green) results of the measured (continuous) and modeled (dashed) data with ellipsometry of the matrix only fitted with SiO2 thermal model.

  are extracted from four different experiments for each condition. From these results, a clear dependency of the NC's characteristics on the injection parameters is observed. More specifically, for N=0 when no NPs are injected (SiO2 matrix only) the refractive index is found at 1.47. As we start injecting a low amount of NPs, hence N=2, the NC film contains 7 vol.% of TiO2 and reveals a refractive index of 1.50. In accordance, for higher injection (N=10), the NC film contains 16 vol.% and reveals an index of 1.54. For N=30, during the one minute sequence, pulses are applied on the injector constantly and consequently no DLI-OFF exists. Hence, this induces a strong pressure variation (from 16 to 20 mTorr) and may produce less stable plasma conductions both the NPs deposition and the silica matrix. Therefore, the significantly higher error bars than the rest conditions could be explained. Nevertheless, for the highest amount of the injected NPs (N=30), the NC film can reach up to 53 vol.% in TiO2 NPs leading to a refractive index of 1.74.

Figure 11 :

 11 Figure 11: a) Stacked columns (%) of the NC surface composition obtained from XPS for N=2, N=10 and N=30 with measurements obtained from two to four samples. b) Volume percentage of TiO2 extracted from the BEMA and refractive index at 633 nm wavelength for N=0, N=2,N=10 and N=30 extracted from four samples.

Table 1 :

 1 Quantitative results by EDX on large surface (2552 µm 2 -magnification X1000).

	Weight %	Atomic %

  "Nanomatériaux Multifonctionnels Contrôlés" is thanked for the financial support to short-term exchanges between University of Montréal and Institut des Matériaux Jean Rouxel. The CIMEN Electron Microscopy Center in Nantes is greatly acknowledged, for access to TEM funded by the French Contrat Plan État-Région and the European Regional Development Fund of Pays de la Loire. Dr Quentin Hatte is thanked for fruitful discussions regarding image processing.

Acknowledgements

Franck Petitgas is thanked for the technical assembly of the reactor and Dr Simon Bulou (LIST) is thanked for his support in conducting PDA measurements. The France-Québec International Research Network on


Michel Hirschowitz

Hirschowitz André

Tom ⋆⋆

Hirschowitz

A Theory for Game Theories

Keywords:

.

Introduction

The flavor problem

Game semantics appeared in the early 90's [START_REF] Abramsky | Full abstraction for PCF[END_REF][START_REF] Hyland | On full abstraction for PCF[END_REF] and provided convenient denotational semantics to proof theories and programming languages, including their non functional features [START_REF] Abramsky | A fully abstract game semantics for general references[END_REF][START_REF] Abramsky | Concurrent games and full completeness[END_REF][START_REF] Abramsky | Full abstraction for Idealized Algol with passive expressions[END_REF][START_REF] Laird | Full abstraction for functional languages with control[END_REF][START_REF] Harmer | Games and Full Abstraction for Nondeterministic Languages[END_REF][START_REF] Laurent | Sémantique des jeux[END_REF]. However, game semantics has roughly as many variants as it has authors. Each of these game theories starts from a notion of "arrow" game (with corresponding positions and moves), yielding the natural notion of strategy. The crucial construction is then the composition of strategies, with the crucial feature that various meaningful classes of strategies (deterministic, innocent, winning) are preserved by composition.

All these compositions clearly have a common flavor (sometimes called "compose+hide"). In the present work, we propose an explanation for this common flavor. To this effect, we define, through a single construction, a huge class of game theories where the composition of strategies preserves good properties. This class contains those among existing game theories which respect the so-called switching condition [START_REF] Danos | The structure of multiplicatives[END_REF]. This restriction is only due to the fact that we have chosen to present the simplest version of the construction. Indeed, the more general version [START_REF] Hirschowitz | Jeux abstraits et composition catégorique[END_REF] involves a serious amount of weak categorical material. Nevertheless, future game models relying on our framework will avoid the burden of re-proving the combinatorial lemmas leading to the category of games and strategies. We now proceed to give a more detailed overview.

Playing in a one-way category

In our approach, a play may take place in any one-way category, which we define to be a category where objects have a sign (1/0) and where morphisms cannot go from a 1-object to a 0-one. Equivalently, a one-way category is a category C, equipped with a functor λ : C → ✷, where ✷ is the ordered set 0 ≤ 1.

The crucial part of our construction builds a wild game W C from a one-way category C. This game is wild in the sense that the two players play without any restriction (meaningful restrictions will be considered later). Let us sketch the construction of W C . It is a directed graph, whose vertices are the morphisms of C. Thus we have one kind (01) of odd vertices and two kinds (00, 11) of even vertices. We think of these "states" as follows: at an odd vertex, Player has to play and reach an even vertex; at a 11-vertex, Opponent has to play "on the left-hand side" (and reach an odd vertex) , while at a 00-vertex Opponent has to play "on the right-hand side" (and reach an odd vertex). This yields the following diagram of states

11 ✛ ML L ✲ 01 MR ✲ ✛ R 00.
In other words we have four kinds of edges (ML and MR for Player's moves, L and R for Opponent's) which we now describe in more detail. The rule is that only one end of the vertex (a morphism in C) changes, and the slogan says that O composes while P decomposes, as pictured in Figure 1: an edge from f to g, consists of an odd morphism m respectively satisfying the following rule:

Kind of move R L MR ML Rule g = m • f g = f • m f = m • g f = g • m.
Because each move changes the signs, all the m's above and in Figure 1 have sign 0 → 1.

The wild game we have constructed so far offers essentially the complete picture which we want to show, in particular one may define strategies and their composition. On the other hand, as far as meaning is concerned, the wild game is trivial, in the sense that players can easily neutralize each other. Indeed, for instance, assume Opponent moves from the current position f to, say, m • f by composing with m. Then, Player may move back to f by decomposing m • f into m and f . Thus, in the wild game, all moves are undoable. More meaningful and sufficiently general games are obtained as subgames of the wild game simply by restricting the set of odd morphisms allowed in the process of composition/decomposition. For this reason we define a game setting to be a one-way category equipped with two sets of odd morphisms as explained in Section 2.2. In the rest of the paper, we explain the basic theory of plays and strategies in a game setting, and we show how the theory of HO games may be recovered in terms of a game setting. [START_REF] Cockett | Polarized category theory, modules, and game semantics[END_REF] offer another categorical investigation into game semantics. The relationship between their work and ours remains unclear to us. Let us also mention a recent paper [START_REF] Harmer | Categorical combinatorics for innocent strategies[END_REF] which describes a categorical reconstruction of "pointer" games and innocent strategies from "general" games and strategies. In this sense, they reduce one sophisticated (but efficient) category of games to a much simpler one. Thus they aim at a better understanding of one (very important) category of games, and of the concept of innocence, while we aim at a better understanding of what could be a category of games, and do not consider the concept of innocence.

Related work Cockett and Seely

Organization of the paper In Section 2, we provide the categorical construction which, from a so-called game setting, constructs a double category of plays, where vertical composition is sequential composition, while horizontal composition is reminiscent of the usual composition of strategies. We then instantiate our framework in Section 3: after recalling the basics of (a standard variant of) HO games, we exhibit a game setting hidden in it, for which our construction yields the usual notion of plays and arrow arenas. In Section 4, we describe strategies in our abstract framework, as well as their composition, and we show that the obtained notion of HO strategies closely corresponds to the standard one.

The abstract framework: building the double category 2.1 Game settings

In order to restrict moves in the game sketched above, we should a priori specify four sets M OL , M OR , M PL , M PR of legal odd morphisms, one for each of the four kinds of moves in Figure 1. However, these restrictions will be compatible with the composition of strategies only if we impose M OL = M PR and M OR = M PL . This leads to our definition: a game setting G (C, Σ R , Σ L ) consists of a one-way category C equipped with a pair of sets of odd morphisms: Σ R is the set of forward moves (or f-moves; those going downwards in Figure 1); Σ L is the set of backward moves or b-moves. The wild game (on C) is obtained by taking as Σ R and Σ L the whole set of odd morphisms.

In a game setting G, we view objects as positions in a two-player game, actually a signed graph. Morphisms in Σ R and Σ L are Opponent and Player moves, respectively. On 0-labeled objects, Opponent is to play, whilst on 1labeled ones, it's Player's turn. As illustrated in Figure 2, from some 0-labeled position p, Opponent plays by choosing an f-move m : p → q with domain p, thereby reaching the 1-labeled position q. Conversely, from such a q, Player plays by choosing a b-move m ′ : r → q with codomain q, thereby reaching the position r. This defines a graph whose vertices are the objects of C, which we call the 0-dimensional game (0-game for short) of G and denote by G 0 (G). We call the free category over this graph the category of 0-plays over G, and denote it by C 0 (G).

The 1-dimensional game

As in standard game semantics, this yields a natural notion of arrow game, also a graph, which we call the 1-dimensional game (1-game for short) of G and denote by G 1 (G). We describe the positions of this game first, then its moves, and finally we show how to equip it with signs, in a way that refines the above interpretation of signs in the 0-game. Positions (or vertices) in G 1 (G) are morphisms in C. Given the constraints on signs, there are just three kinds of positions: 00, 01, and 11. Then, moves from f to g in the 1-game are defined to be commutative triangles in C, of one of the four shapes in Figure 1. The interpretation of signs in 1-games, illustrated in Figure 3, entirely follows from the idea that in 0-games, Player lives on the left-hand side of the position, whilst Opponent lives on its right-hand side. For a 1-position, there is thus one agent M in the middle, and one agent on each side, which we call L and R in the obvious way. M plays Opponent in the domain 0-game, and Player in the codomain 0-game. L plays Player in the domain 0-game, whilst R plays Opponent in the codomain 0-game. This yields the following rule for the 1-game: Signs of the 1-position Who's to play?

0 ✲ 0 R 0 ✲ 1 M 1 ✲ 1 L .
We consider the free category over this graph G 1 (G): we call it the category of 1-plays over G and denote it by C 1 (G). Again each 1-play v has a predecessor Pred(v) obtained by deleting the last move (if any).

Finally, we define the (horizontal) source and target functors on 1-plays, s, t : C 1 (G) → C 0 (G), by the obvious induction (or adjunction). We thus have a pullback category C 1 (G) s × t C 1 (G)) of composable pairs of 1-plays.

The double category associated to a game setting

In this section, we derive a double-categorical structure from our game setting G. For this, we will define a notion of horizontal composition of 1-plays, yielding a category whose objects are 0-plays, and whose morphisms are 1-plays. We start by defining the graph G 2 (C) of primitive interactions as follows. As vertices, take composable pairs of morphisms in C, and as edges from the pair to the pair

f ✲ g ✲ 1 f ✲ 1 g = g ′ ✲ 1 0 m ✻ . . . . . . . f ′ ✲ 0 f ✲ 1 g = g ′ ✲ 1 1 m ❄ . . . . . . . f ′ ✲ 0 f ✲ 0 g ✲ 1 1 m ❄ . . . . . . . g ′ ✲ f ′ ✲ 0 f ✲ 1 g ✲ 1 0 m ✻ . . . . . . . g ′ ✲ f ′ ✲ 0 f = f ′ ✲ 0 g ✲ 1 0 m ✻ . . . . . . . g ′ ✲ 0 f = f ′ ✲ 0 g ✲ 0 1 m ❄ . . . . . . .
f ′ ✲ g ′
✲ , take all the commutative diagrams as in Figure 4. This gives four kinds of vertices (000, 001, 011, 111) according to the signs of objects, yielding the following state diagram:

111 L ✲ ✛ 1 L 011 ✛ M 2 L M 1 R ✲ 001 M 2 R ✲ ✛ R 000.
For G 2 (C), the intuition is that there are two players M 1 and M 2 , and two opponents L and R, who interact respectively on the left-hand side with M 1 and on the right-hand side with M 2 . Thanks to categorical composition, both players act exactly as if they were facing two opponents. For instance, M 1 interacts with L on the left-hand side, and with M 2 on the right-hand side. Because of sign rules, at most one of M 1 and M 2 may play at a given time, which prevents any conflict to arise.

Next, we let C 2 (G) denote the free category generated by G 2 (G), and we call its morphisms interactions in G. Accordingly, the edges in G 2 (G) are primitive interactions. Let us also deem the primitive interactions of the middle row internal, and the other ones external. Now a key observation is that the functor

C 2 (G) π 1 , π 2 ✲ C 1 (G) s × t C 1 (G)
which maps a path in G 2 (G) to its left and right borders is an isomorphism, which says altogether that interactions are determined by their projections, and that C 1 (G) s × t C 1 (G) is freely generated by the primitive interactions. Thanks to this statement, it is enough to define our 1-horizontal composition Y • X on primitive interactions, which is straightforward: for an internal interaction, the 1-horizontal composition is the empty 1-play. Otherwise it is the obvious move from g • f to g ′ • f ′ , for each external interaction as in Figure 4.

To construct our horizontal category, we finally define identity morphisms, by mimicking what is standardly called copycat in game semantics: let copycat be the unique functor from G 0 (G) to G 1 (G) such that f-moves m : p → p ′ and b-moves m : p ′ → p are respectively sent to plays

p id p ✲ p p id p ✲ p p ′ m ❄ id p ′ ✲ p ′ m ❄ m ✲ p ′ m ✻ id p ′ ✲ m ✲ p ′ . m ✻
By the standard adjunction between categories and directed graphs, this defines copycat uniquely: on arbitrary plays, copycat simply piles up sequences of such elementary plays.

Proposition 1. The horizontal composition of 1-plays is associative and unital.

The proof of associativity relies on a freeness result concerning 3-interoctions, completely analogous to our previous freeness result concerning 2-interactions. This all gives the data for a double category. A short definition is as follows: a double category is a category object in the category of categories. A more explicit, elementary definition may be found, e.g., in Melliès [START_REF] Melliès | Double categories: a modular model of multiplicative linear logic[END_REF]. We've already checked all the required properties, except the interchange law, which makes • into a functor from the pullback

C 1 (G) s × t C 1 (G) to C 1 (G). Explicitly: (Y 1 •X 1 )•(Y 2 •X 2 ) = (Y 2 • Y 1 ) • (X 2 • X 1 )
. It happens to be satisfied, which entails: Theorem 1. For any game setting G, the categories C 0 (G) and C 1 (G), the domain and codomain functors s, t : C 1 (G) → C 0 (G), the horizontal composition functor

• : C 1 (G) s × t C 1 (G) → C 1 (G)
, and the horizontal identity functor I : C 0 (G) → C 1 (G) form a double category.

The one-way category underlying Hyland-Ong games 3.1 A brief review of HO-arenas and HO-plays

We briefly recall some definitions of HO game theory, and refer the reader to Harmer's notes [START_REF] Harmer | Innocent game semantics[END_REF] for details.

An arena A is a triple (M A , λ A , ⊢ A ), where M A is a set of moves, λ A gives signs to moves, i.e., is a function from M A to {0, 1}, and ⊢ A represents altogether a binary relation (justification) and a predicate (initiality) on M A , such that:

1. if ⊢ A m, then λ A (m) = 0 and for all m ′ ∈ M A , m ′ ⊢ A m, 2. if m ⊢ A m ′ , then λ A (m) λ A (m ′ ).
Moves m such that ⊢ A m are called initial. When m ⊢ A m ′ , we say that m justifies m ′ .

A position in an arena A is a pair (s, ρ), where s = m 1 . . . m n is a sequence of moves of alternate signs in A, and ρ is a function from {1 . . . n} to {0 . . . n -1} such that for all i ∈ {1 . . . n} 1. (priority condition) ρ(i) < i, 2. if ρ(i) = 0, then m i is initial, 3. if ρ(i) = j 0, then m j justifies m i .

We say that n is the length of the position. Our position p also has an initial part Init p ⊂ [1, . . . , n] which is the set of indices i for which m i is initial. Since positions carry their history, they also may be seen as plays, and we freely call them either way. A position of length 0 is called initial, and a non initial position p of length n has a predecessor position Pred(p), of length n -1, obtained by deleting the last move. For simplicity, we define Pred(p) p when p is initial. A set of positions is prefix-closed when it is closed under application of Pred.

Given a sign function λ, we write λ for the opposite one. Given two arenas A and B, one constructs the arrow arena A ⊸ B by taking M A + M B as the set of moves, [λ A , λ B ] as a sign function, the (injections of) initial moves of B as initial moves, and for the binary ⊢ A⊸B , taking the union (up to injection) of ⊢ A and ⊢ B , plus the pairs (m, m ′ ) with m initial in B and m ′ initial in A. Note that a position p in an arrow arena A ⊸ B determines two projections p A and p B which are in general not positions in A and B. Intuitively, this is because Opponent may switch sides, and, when asked a question in A, ask a question in B.

Define a position p in A ⊸ B to be valid if its two projections are again positions, respectively in A and B. Combinatorially, if n A and n B are the lengths of these projections, p determines a shuffle p S = [1, . . . ,

n A + n B ] → [1, . . . , n B ] ∐ [1, . . . , n A ].
We say that such a shuffle p s satisfies the switching condition, or is even when 1) is on the B-side, -for i satisfying 1 < 2i < n A + n B , p S (2i) and p S (2i + 1) are on the same side.

-if n A + n B > 0 then p S (
It turns out that p is valid exactly when p S is even. We note that p determines a restricted justification map RJ p : Init p A → Init p B . Conversely, given the projections p A and p B , a position p is determined by an arbitrary map RJ : Init p A → Init p B and an even shuffle compatible with RJ (with respect to the priority condition).

Strategies from A to B are defined to be non-empty, prefix-closed sets of valid positions in A ⊸ B. One then shows that strategies compose and have identities, which yields a category of games and strategies Strat HO .

The one-way category C HO

Let us now describe the one-way category C HO relevant for HO games. An object (A, (s, ρ)) of C HO is merely a position (s, ρ) in a game arena A, while a morphism from p = (A, (s, ρ)) to q = (B, (s ′ , ρ ′ )) is a (valid) position f = (A ⊸ B, (t, τ)) whose projections respectively give p and q. Thus our morphisms also have predecessors. Note that f and Pred( f ) share one end, but in general not both.

We are especially concerned with two kinds of morphisms. Firstly, for each position p = (A, (s, ρ)), we have a copycat morphism copycat p : p → p, which is defined by induction on the length of p: the empty play on A ⊸ A is the copycat of the initial position on A, and for greater lengths, copycat p is determined by the requirement that its second predecessor is the copycat of Pred(p): the last two moves are determined by the given projections (p and p). Secondly, we are interested in those morphisms whose predecessor is a copycat, which we call subcopycat morphisms. Each subcopycat morphism is also the predecessor of a unique copycat morphism. Thus, for a non initial position p, define Sub p to be the predecessor of copycat p . Then, each subcopycat morphism can be written Sub p in a unique way. Furthermore, if p is even, then Sub p goes from p to Pred(p) while if p is odd, then Sub p goes from Pred(p) to p.

Next, we define the composition of our morphisms. Consider two consecutive arrows, i.e., valid positions f in some A ⊸ B and g in B ⊸ C with the same projection p B on B. We denote by p A the projection of f on A, and by p C the projection of g on C and by n A , n B , n C the corresponding lengths. We will define h g • f by its restricted justification map RJ h and its even shuffle h S . For RJ h , we take the composition RJ g • RJ f . For h S , we observe that, thanks to the switching condition, there is a unique shuffle

s : [1, . . . , n A + n B + n C ] → [1, . . . , n C ] ∐ [1, . . . , n B ] ∐ [1, . . . , n A ]
compatible with f S and g S . We view this shuffle as an order on [1, . . . ,

n C ] ∐ [1, . . . , n B ] ∐ [1, . . . , n A ] and take for h S its restriction to [1, . . . , n C ] ∐ [1, . . . , n A ].
This composition is easily seen to be associative, and it is easily checked that the identity on a position p is the copycat morphism copycat p .

This altogether gives a category C HO , whose objects may be given a sign as follows: the sign of a position is 0 if Opponent is to play, or equivalently if its length is even, and 1 otherwise. Thus, a priori, we have four kinds of morphisms, 0 → 0, 0 → 1, 1 → 0, 1 → 1. However, we easily check that there are no morphisms of type 1 → 0. This is a consequence of the switching condition, and the convention that plays always start with a move by Opponent, which furthermore, in the case of arrow arenas, has to be on the right-hand side. Our category may thus be seen as a one-way category.

Remark 1 (Relaxing the switching condition). If we relax the switching condition, and allow Opponent to switch sides in an arrow game, the main new feature is that the horizontal composition of 1-plays is no more well-defined, because interactions are no more determined by their projections. As a consequence, the double category constructed above has to be replaced by some kind of weak double category, to be defined accordingly. This approach has been pursued in the first author's PhD thesis [START_REF] Hirschowitz | Jeux abstraits et composition catégorique[END_REF], where one eventually recovers a proper category when passing to strategies.

The game setting G HO

Now we explain how HO-moves may be seen as morphisms in C HO . Playing a move in a position p in A is understood as extending p (with one move in A), yielding a new position q. To this move, we attach the morphism Sub q . Note that Sub q goes from p to q if p is even, and from q to p if p is odd. Hence in our view, the set of HO-moves is precisely the set of subcopycat morphisms, which we split into the set RHO of subcopycat morphisms where the length of the codomain exceeds the length of the domain by one, and the set LHO of subcopycat morphisms where the length of the domain exceeds the length of the codomain by one. Thus, standard HO plays are 0-plays starting on an initial position in the game setting G HO (C HO , LHO, RHO). (In the game setting, we also consider plays starting on non initial positions.)

Now let us see how our view fits with plays in an arrow game: consider a valid position f in the game A ⊸ B and its extension to a new valid position g, through a HO-move m (in A or in B). We have four kinds of extensions corresponding to who is playing and where. A careful inspection shows that

-if O plays in B, then we have g = m • f (in C HO ), -if O plays in A, then we have g = f • m, -if P plays in B, then we have f = m • g, -if P plays in A, then we have f = g • m;
which shows that, indeed, O composes the original position with her move, while P decomposes the original position with her move. Thus, standard HO arrow plays are precisely 1-plays in G HO starting on an initial position.

An abstract view of strategies

In this section, we show how some standard results on strategies may be understood abstractly in a game setting G = (C, Σ R , Σ L ). Recall that an object of C is even when its sign is 0 and odd otherwise. We say that a 1-position f : p → q is even when p and q have the same sign, and odd otherwise. We note that f is odd exactly when the middle player M is to play, and even exactly when it's L or R's turn. Let us now define strategies, writing • for concatenation. Definition 1. A 0-strategy (or strategy) σ on a 0-position p is a non empty, prefixclosed set of 0-plays of domain p such that, for any x in σ with even codomain q, and for any move m : q → r in G 0 (G), x • m is also in σ.

A 1-strategy (or strategy) Σ on a 1-position f is a non empty prefix-closed set of 1-plays of domain f , such that, for any X in Σ with even codomain g, and for any move

M : g → h in G 1 (G), X • M is also in Σ.
We use S to range over 0 or 1-strategies (or both), leaving the context to disambiguate. Given f : p → q and g : q → r, we define the horizontal composition of strategies σ and σ ′ (respectively on f and g) to be the set of all plays on g • f of the form Y • X for some (horizontally) composable X ∈ σ and Y ∈ σ ′ . We easily prove that this definition is sensible: Proposition 2. A composition of 1-strategies is again a 1-strategy. Proposition 3. The composition of 1-strategies is associative.

The proof of the latter statement is an easy consequence of the associativity of our horizontal composition of plays.

We define the copycat strategy on an identity 1-position p as the smallest strategy containing the copycat 1-plays (as defined above) starting at p. These copycat strategies are neutral for our composition. We thus have a category Strat(G) whose objects are 0-positions, and morphisms are pairs of a 1-position and a strategy for it.

In the case of our running example G HO , this new category fits with the "classical" one, up to the fact that we also consider non empty plays as objects in the new category.

Theorem 2. The map sending an arena to the corresponding initial play yields a full embedding Strat HO ✲ Strat(G HO ).

Next, we show that two crucial properties of strategies are stable under composition. A strategy is deterministic iff it does not contain two plays ending on an even position and sharing all their proper prefixes. Proposition 4. The composition of deterministic 1-strategies is again deterministic.

A play is final in a strategy S when it has no extension in S. A strategy is complete iff its final plays all end on an even position. In other words, a complete strategy is one which never gets stuck. However, this definition is a bit loose w.r.t. potential infinite plays. Indeed, a complete strategy may contain infinite plays, and the composition of two complete strategies may not be complete. Intuitively, it may get lost in infinite internal "chattering" between M 1 and M 2 . Thus, we refine the picture as follows. We deem a strategy noetherian iff it contains only finite plays, and winning iff it is noetherian and complete. This yields the following: Proposition 5. The composition of two winning 1-strategies is again winning.

The previous notion of a winning strategy is not totally satisfactory. For instance, we would like copycat strategies to be winning. This somehow forces to consider some kind of non noetherian strategies. Anyway, we also wish to handle infinite plays in the spirit of Abramsky [START_REF] Abramsky | Semantics and Logics of Computation, chapter Semantics of interaction[END_REF], but this is beyond the scope of the present work.

Conclusion

We have designed a notion of game theory. This is not one more category whose objects are new kinds of arenas. Rather we have shown how to build such a category from a very minimal set of data: a (one-way) category and two sets of morphisms therein. We have sketched how our composition of strategies has the desired stability properties (but we did not consider innocence). We hope that our framework will help in the design of new, helpful game semantics. We believe that it can be extended in various ways in order to encompass most of existing game semantics, and plan to explore some of these extensions in the near future.

Fig. 1 .

 1 Fig. 1. The four kinds of edges in W C , from f to g

Fig. 2 .

 2 Fig. 2. Example play in the 0-game

✛Fig. 3 .

 3 Fig. 3. All agents (L, M, R) act as Player on their rhs and as Opponent on their lhs

Fig. 4 .

 4 Fig. 4. The six kinds of edges in G 2 (C) (each edge top-down)

⋆ CEA-LIST michel.hirschowitz@cea.fr ⋆⋆ UMR 6621 CNRS-Université de Nice-Sophia-Antipolis ah@math.unice.fr ⋆ ⋆ ⋆ UMR 5668 CNRS-ENS Lyon-INRIA-UCBL tom.hirschowitz@ens-lyon.fr

Acknowledgements We thank Vincent Danos for having advised the first author's PhD thesis, Pierre-Louis Curien for his constant benevolence and assistance, and Martin Hyland for encouraging us to write the present work.