Lolwa Samet
email: lolwa.samet@ipeiem.utm.tn

Katia March

Nathalie Brun

Faouzi Hosni

Odile Stephan

Radhouane Chtourou

Effect of gamma radiation on the photocatalytic properties of Cu doped titania nanoparticles

Keywords: Titanuim dioxide (TiO 2 ), Sol-gel, Gamma irradiation, Photocatalysis, EELS, Oxygen vacancy

The aim of this paper is to study the effect of gamma-rays on structural, physicochemical, optical and photocatalytic performance of TiO 2 nanoparticles doped with different concentrations of copper ranging from 0 to 6 at.%. The powders have been prepared by sol-gel technique and annealed at 400 °C. They were irradiated by gamma-rays with doses varying from 14 to 60 KGy. These investigations confirm the formation of anatase TiO 2 nanoparticles and Cu 2+ ion substitution for Ti 4+ sites within the TiO 2 structure. This study also shows that, once the TiO 2 structure is saturated with copper, a metallic copper segregation is formed at the crystallite surfaces. After gamma irradiation, samples present a crystalline core and a disordered shell structure as a result of the formation of oxygen vacancies. Such oxygen vacancies at the TiO 2 nanocrystal surface lead to a remarkable enhancement of the photocatalytic activity of Cu-doped TiO 2 catalysts.

Introduction

Heterogeneous photocatalysts based on titanium dioxide (TiO 2 ) have been widely investigated in academic research and extensively involved in industrial applications. However, their photo-conversion efficiency under solar illumination is very limited owing to a too wide intrinsic band gap energy (3.06 eV for rutile and 3.23 eV for anatase). Using sunlight more efficiently to provide energy or to initiate chemical reactions still remains a challenge. Therefore, much effort has been made to extend the light absorption from UV down to the visible region and improve the photoelectrochemical properties of TiO 2 . The most widely adopted strategies are chemical modification by doping (including single metal or non-metal elements, two-element co-doping or self-doping) leading to the extent of the visible light absorption capability in addition to the introduction of cooperative catalysts, that facilitate the separation of photogenerated carriers. Several investigations on doped TiO 2 [START_REF] Long | Band gap engineering of double-cation-impurity-doped anatase titania for visible-light photocatalysts: a hybrid density functional theory approach[END_REF][START_REF] Long | Electronic properties of F/Zr co-doped anatase TiO 2 photocatalysts from GGA + U calculations[END_REF][START_REF] Long | Electronic structure of cation-codoped TiO 2 for visiblelight photocatalysts applications from hybrid density functional theory calculations[END_REF] have succeeded in providing some indication about the atomic level details of doping such as the dopant location (e.g. substitutional or interstitial) and discussed the effects of doping on the physical properties and electronic structure of TiO 2 .

Impurity-doped TiO 2 has been attracting a great deal of interest owing to its universal applications especially in different processes for degradation of pollutants and water splitting technologies for photoelectrochemical hydrogen production, see e.g. [START_REF] Yue | Characterization of metal doped titanium dioxide and behaviors on photocatalytic oxidation of nitrogen oxides[END_REF][START_REF] Hu | Effective water splitting using CuOx/TiO 2 composite films: Role of Cu species and content in hydrogen generation[END_REF][START_REF] Xia | Origin of significant visible-light absorption properties of Mn-doped TiO 2 thin films[END_REF][START_REF] Hsieh | Absorption and visible-light-derived photocatalytic kinetics of organic dye on Co-doped titania nanotubes prepared by hydrothermal synthesis[END_REF][START_REF] Rauf | An overview on the photocatalytic degradation of azo dyes in the presence of TiO 2 doped with selective transition metals[END_REF][START_REF] Castro | Doped titanium dioxide nanocrystalline powders with high photocatalytic activity[END_REF][START_REF] Ni | A review and recent developments in photocatalytic water-splitting using TiO 2 for hydrogen production[END_REF][START_REF] Pan | Defective TiO 2 with oxygen vacancies: synthesis, properties and photocatalytic applications[END_REF][START_REF] Wang | One-step synthesis of size-controlled Br-doped TiO 2 nanoparticles with enhanced visible-light photocatalytic activity[END_REF][START_REF] Tang | Construction of N-doped TiO 2 /MoS 2 heterojunction with synergistic effect for enhanced visible photodegradation activity[END_REF][START_REF] Sun | The photocatalytic performance of Ta and Rh codoped TiO 2 tuned by the average dopant valence[END_REF]. The presence of non-metal and transition-metal ion dopants in the TiO 2 crystalline matrix significantly influences photoreactivity, charge carrier recombination and interfacial electron-transfer rates.

However, the photoelectrochemical efficiency of doped TiO 2 is still limited by a relative high electron-hole recombination rate. Optimum conditions should be fixed in order to enhance absorption in the visible range and to reduce the charge recombination for an improvement of the photo-response. Indeed, a compromise must be achieved between the light-absorption capability and surface charge transfer rate of the TiO 2 photocatalyst. To avoid these problems, one approach is to exploit doping or co-doping that involves oxygen vacancies. Previous investigations have revealed that the band gap energy, the electron-hole recombination process as well as the photoelectrical properties of TiO 2 can be effectively tuned by oxygen vacancies [START_REF] Pan | Defective TiO 2 with oxygen vacancies: synthesis, properties and photocatalytic applications[END_REF][START_REF] Sun | Novel method to directly prepare high-surfacearea anatase titania nanoparticles with trapped electrons on oxygen vacancies[END_REF][START_REF] Bharti | Formation of oxygen vacancies and Ti 3+ state in TiO 2 thin film and enhanced optical properties by air plasma treatment[END_REF].

Besides doping, several treatment methods are proposed to achieve visible light absorption for titania catalysts. Some authors studied the effect of plasma treatment on the photocatalytic activity of anatase [START_REF] Bharti | Formation of oxygen vacancies and Ti 3+ state in TiO 2 thin film and enhanced optical properties by air plasma treatment[END_REF]. They reveal that oxygen vacancies produced by plasma treatment play an important role in the performance activity of treated TiO 2 .

Recently, reduction methods (e.g., by high-energy particle bombardment) showed interesting results in producing catalysts with great photocatalytic performance [START_REF] Tan | A facile and versatile method for preparation of colored TiO 2 with enhanced solar-driven photocatalytic activity[END_REF][START_REF] Bello Lamo | Study of gamma irradiation effect on commercial TiO 2 photocatalyst[END_REF][START_REF] Kralchevska | Influence of gamma-irradiation on the photocatalytic activity of Degussa P25 TiO 2[END_REF]. Those methods lead to generate disordered TiO 2 nanophases with simultaneously incorporated oxygen vacancies self-dopants over a few atomic layers at the surface of TiO 2 nanocrystals. In this context, some results, obtained by the application of gamma irradiation treatments on P25 (Degussa) TiO 2 photocatalyst commercial products, showed a surface effect with no significant change on morphology and crystal structure. Optical property investigations revealed an extension of the light absorption from UV down to the visible region, leading to an improvement of the photo-response [START_REF] Bello Lamo | Study of gamma irradiation effect on commercial TiO 2 photocatalyst[END_REF][START_REF] Kralchevska | Influence of gamma-irradiation on the photocatalytic activity of Degussa P25 TiO 2[END_REF]. These findings have been assigned to trapped electrons in oxygen vacant sites or to Ti 3+ states formed in the few atomic layers at the surfaces of TiO 2 crystallites. We also note that, according to these articles, both optical band gap value and photocatalytic performance do not vary linearly with the applied gamma doses.

However, this research area is relatively unexplored and many questions deserve to be clarified. In fact, there has been a lack of systematic nanoscale investigation on the nature of the surface modification (structure and composition) in connection with the applied gamma dose. Based on this, we investigated the changes caused by gamma irradiation on anatase undoped and transition metal incorporated to TiO 2 by a combination of local and global scale characterization techniques. Doping with copper was chosen. In fact, Cu doped TiO 2 catalysts have recently attracted enormous attention due to their visible light absorption high efficiency. Moreover, the low cost of preparation together with improved catalytic performance, make copper-doped TiO 2 a promising photocatalyst for wide environmental applications. It has been found that the incorporation of copper Cu 2+ ions into the TiO 2 matrix could be at the origin of such high photocatalytic activity, see e.g. [START_REF] Ganesh | Preparation and characterization of Cu-doped TiO 2 materials for electrochemical, and photocatalytic applications[END_REF][START_REF] Hudson | TiO 2 -Cu photocatalysts: a study on the long-and short-range chemical environment of the dopant[END_REF][START_REF] Xin | Effect of surface species on Cu-TiO 2 photocatalytic activity[END_REF][START_REF] Colon | Cu-doped TiO 2 systems with improved photocatalytic activity[END_REF][START_REF] Choudhury | Defect generation, d-d transition, and band gap reduction in Cu-doped TiO 2 nanoparticles[END_REF][START_REF] Aguilar | A route for the synthesis of Cu-doped TiO 2 nanoparticles with a very low band gap[END_REF][START_REF] Dashora | Formation of an intermediate band in the energy gap of TiO 2 by Cu-N-codoping: first principles study and experimental evidence[END_REF][START_REF] Navas | Experimental and theoretical study of the electronic properties of Cu-doped anatase TiO 2[END_REF], by reducing the band gap value down to 1.7 eV as predicted theoretically for a 2 at.% Cu amount [START_REF] Navas | Experimental and theoretical study of the electronic properties of Cu-doped anatase TiO 2[END_REF]. However, the nature of the active copper species (CuO, Cu 2 O, metallic Cu, or Cu 2+ ions in the TiO 2 lattice) that govern photo-reactivity processes is still under debate.

In addition, to further improve photocatalytic performance, it is important to reduce the electron-hole pair recombination and assure a sufficiently long lifetime for this pair which diffuses to the catalyst's surface and initiate a redox reaction. Here, we propose to use Gamma irradiation treatment as a viable way to improve the charge separation. We therefore present a simple and economical method to synthesize efficient TiO 2 photocatalysts reactive to visiblelight.

The effect of sunlight illumination on the photocatalytic activity of prepared catalysts was studied by removal of methylene blue (MB) as model of pollutant. To elucidate the factors responsible for the observed effects, initial and gamma irradiated undoped and Cu doped sol-gel TiO 2 products were examined. Changes in absorption properties were evaluated by ultraviolet-visible spectroscopy (UV-vis). Electron paramagnetic resonance (EPR) was used, as a powerful tool to characterize paramagnetic metal ions and defects in the oxyde catalysts. The particle size, morphology and crystal structure for each sample type were determined by transmission electron microscopy (TEM). Microanalysis studies were conducted by electron energy-loss spectroscopy (EELS).

Experimental details

Catalysts Preparation

Pure and Cu 2+ -doped TiO 2 (Cu-TiO 2 ) powders were prepared via a sol-gel technique [START_REF] Samet | Heat treatment effect on the physical properties of cobalt doped TiO 2 sol-gel materials[END_REF][START_REF] Ben Naceur | Annealing effects on microstructural and optical properties of nanostructured-TiO 2 thin films prepared by sol-gel technique[END_REF]. Titanium isopropoxide (Ti(OC 3 H 7 ) was used as TiO 2 precursor, while isopropanol (CH 3 CH(OH)-CH 3 ) and methanol (CH 3 OH) were used as solvent and acetic acid (CH 3 COOH) as catalyst. Copper chloride (CuCl 2 •2H 2 O) was used as precursor for Cu 2+ dopant. The atomic ratio of Cu to Ti were 3 and 6 at.%. Note that undoped TiO 2 samples were prepared as a control in the same way. The obtained millimetric solids were ground in an agate mortar to get fine nanopowders. As-made powders are amorphous. To obtain crystallized samples, these powders were calcined in air at 400 °C during 6 hours. This method allows to obtain powders of high chemical purity.

The synthesized sol-gel powders were exposed to gamma irradiation at ambient temperature of 25 °C (±0.5 ºC) using the Tunisian Cobalt-60 irradiator installed in the Tunisian National Center for Nuclear Science and Technology (CNSTN, Technopole of Sidi Thabet, Tunisia). The gamma source consists of eight 60 Co pencils like-sources arranged and encapsulated in axial symmetry in order to have a better isotropy of gamma photon emission. This source emits gamma photons of 1.2 MeV average energy. In a typical irradiation experiment, 100 mg of each type of TiO 2 powder, undoped and doped with copper, were inserted in eppendorf tubes placed at 15 cm from the gamma source. The applied irradiation doses were 14, 28, and 60 kGy with a dose rate of 80 Gy/min, corresponding respectively to 2h55mn, 5h50mn, 12h30mn time of irradiation. This dose rate was determined by alanine dosimeters irradiated by the facility and returned to aerial for dose rate assessment [START_REF] Hosni | Effect of gamma-irradiation on the Colorimetric properties of epoxy-resin films: potential use in dosimetric application[END_REF][START_REF] Hosni | A study of the fractionation dose on the radiation response of Harwell Red-Perspex PMMA dosimeter[END_REF].

Catalysts characterization

A test of the photocatalytic activity of the synthesized samples were carried outdoors under direct sunlight illumination, during the period of May-June and under clear skies and ambient temperature (25-30 °C) at the university of Tunis el manar (Tunisia, local latitude 36°49'8.29'' N, longitude 10°9'56.84" E). The photocatalytic activity was evaluated on the degradation of methylene blue (MB) in aqueous solution. In a typical experiment, the initial concentration of MB aqueous solution was 10 mg/l. The solutions were prepared with deionized water. Prior to irradiation, 60 mg of catalyst was dispersed in 200 ml of prepared MB solution (the amount of catalyst was 0.3 g/L). The solution was magnetically stirred for one hour in darkness to reach adsorption equilibrium. Then, it has been exposed to sunlight illumination by maintaining a continuous magnetic stirring. MB concentration was estimated with a UV-vis spectrophotometer (IC6400). The intensity of the MB absorption peak at about 664 nm was considered to estimate the MB concentration in the solution at the term of the photocatalysis reaction. The efficiency of the degradation process was estimated with the following relation: Degradation % = (C 0 -C)/C 0 , where C 0 is the initial MB concentration (after one hour in the dark) and C is the MB concentration after certain solar irradiation time.

The optical measurements were performed by diffuse reflection over the 250 to 800 nm wavelength range. The reflection spectrum was measured using a UV-vis-near infrared spectrometer (Lambda-950) equipped with an integrating sphere.

For EPR analysis, the sample was inserted into a standard rectangular cavity in an EMX 71 X-band EPR spectrometer equipped with a 100 kHz magnetic field modulator operating at room temperature.

Samples were investigated in a conventional TEM (TOPCON 002B at 100 keV) for general microstructure characterization. Chemical and electronic properties were investigated by means of EELS performed in two dedicated scanning transmission electron microscopes (STEM): a probe C s -corrected Nion Ultra-STEM microscope and a STEM-VG HB501, both equipped with a field emission source operated at 100 keV and coupled to a Gatan EELS spectrometer optically coupled to a high sensitivity (either nitrogen cooled or electron amplifying) CCD camera. Chemical and electronic structure analyses were performed by acquiring collections of EELS spectra in the spectrum-imaging Mode [START_REF] Jeanguillaume | Spectrum-image: the next step in EELS digital acquisition and processing[END_REF]. In such a mode, the focused beam is scanned over a region of interest and a whole spectrum is acquired at each spatial position of the scan. Such a spectrum-image contains typically 10000 spectra.

These spectra can be processed individually, usually by removing the background and summing the intensity corresponding to a characteristic edge and thus building elemental maps. However, as this data set contains a lot of redundant information, it is often very useful to process it as a whole and to use multivariate statistical techniques. We used principal component analysis (PCA) as a filtering method for separating meaningful signal components from noise [START_REF] De La Peña | Mapping titanium and tin oxide phases using EELS: An application of independent component analysis[END_REF].

Results and discussion

Photocatalytic activity results: Outdoor experiments

The photocatalytic efficiency of the TiO 2 catalyst depends on many factors (crystal structure, specific surface area, bulk and surface defects...). Controlling this performance requires the control of the catalyst structural properties. Doping with transition-metal elements affects the lifetime of the charge carriers. In fact, metal doping controls the competition between the electron-hole pair recombination and the transfer rate of the charge carriers from the bulk to the surface, favorable to the increase of photoctalytic activity of TiO 2 catalyst. Therefore the optimization of the doping element concentration is necessary.

The photocatalytic activity evolution of undoped and Cu-TiO 2 initial sol-gel powders and irradiated powders at 60 KGy gamma dose are presented in Fig. 1. For non-gamma irradiated samples, undoped TiO 2 showed the better photocatalytic performance under sunlight illumination as compared to Cu-TiO 2 sol-gel powders. The insertion of Cu at rates of 3 and 6 at.%, leads to a decrease from 40 to 30% (after 5 hours of exposure to sunlight) of the photocatalytic efficiency. Gamma irradiation treatment seems to contribute in favor of enhancing the photocatalytic activity of all prepared TiO 2 catalysts and especially of those at 3 at.% of Cu. The photocatalytic activity of those samples after 60 KGy gamma irradiation is 80% higher than that of the non-irradiated samples.

Optical and structural investigations

UV-visible diffuse reflectance spectroscopy

The optical properties were determined using diffuse reflectance spectra. The diffuse reflectance data R% was converted to the Kubelka-Munk function F(R) by the equation:

F(R)
which is proportional to the absorption coefficient [START_REF] Samet | Heat treatment effect on the physical properties of cobalt doped TiO 2 sol-gel materials[END_REF].The optical gap value is

= (1 -R) 2 2R
estimated relying on the Kubelka-Munk method combined with the Tauc relation [START_REF] Tandon | Measurement of forbidden energy gap of semiconductor by diffuse reflectance technique[END_REF]. The absorption spectra of non-irradiated and gamma irradiated undoped TiO 2 and Cu-TiO 2 powders are represented in Fig. 2. For non-irradiated samples, undoped TiO 2 exhibits strong absorption in the UV region and high transparency in the visible one.

Copper incorporation in the TiO 2 lattice leads to a general shift in the UV-visible absorption spectra and an increase of the absorption broad band in the visible range. This typical behavior and this type of electronic transitions is consistent with those obtained in the literature [START_REF] Colon | Cu-doped TiO 2 systems with improved photocatalytic activity[END_REF][START_REF] Choudhury | Defect generation, d-d transition, and band gap reduction in Cu-doped TiO 2 nanoparticles[END_REF][START_REF] Aguilar | A route for the synthesis of Cu-doped TiO 2 nanoparticles with a very low band gap[END_REF] and argue in favor of the substitution of Cu 2+ for Ti 4+ ions that remain in octahedral or pseudo octahedral configuration in the TiO 6 framework. In our case, the most reduced band gap (1.5 eV) was observed for samples synthesized with 3 at.% of Cu. This finding is related to the appearance of new O2p-Cu3d hybridized states at the top of the valence band and the bottom of the conduction band upon Cu 2+ doping [START_REF] Samet | Radiocatalytic Cu-incorporated TiO 2 nano-particles for the degradation of organic species under gamma irradiation[END_REF]. According to experimental and theoretical investigations, the substitution of Cu 2+ for Ti 4+ in anatase (2 at.% Cu-doped TiO 2 ) should lead to a decrease in the band gap, down to 1.7 eV, and an increase in the visible light absorption [START_REF] Dashora | Formation of an intermediate band in the energy gap of TiO 2 by Cu-N-codoping: first principles study and experimental evidence[END_REF][START_REF] Navas | Experimental and theoretical study of the electronic properties of Cu-doped anatase TiO 2[END_REF]. Indeed, The Cu 2+ substitution within the crystalline environment of TiO 2 , also leads to an absorption hump extending from 600 to 900 nm that is attributed to d-d transitions [START_REF] Choudhury | Defect generation, d-d transition, and band gap reduction in Cu-doped TiO 2 nanoparticles[END_REF].

When passing from a copper content of 3 to 6 at.%, the intensity of the absorption band between 400 and 500 nm increases while the broad 600 -900 nm absorption band decreases (Fig. 2). This suggests that the effective doping rate of Cu in the 6 at.% Cu-TiO 2 samples is lower than in 3 at.% Cu-TiO 2 . At a higher incorporation rate of Cu, segregated copper species are observed on the surfaces of the crystallites. More details concerning the effect of doping on structural and optical properties of Cu-TiO 2 sol-gel powders annealed at 400°C have been presented in our previous work [START_REF] Samet | Radiocatalytic Cu-incorporated TiO 2 nano-particles for the degradation of organic species under gamma irradiation[END_REF].

Gamma irradiation treatment significantly affects the absorbance spectra of all studied samples (Fig. 2). The absorbance of irradiated samples is always found higher than for nonirradiated ones and an extra peak appears within the wavelength range between 465 nm and 500 nm. No such peak is seen for as prepared sol-gel powders. Also, it is noted that the value of the applied gamma dose did not have much detectable effect on the absorption properties, except for samples prepared with a high Cu level. However, it is known that gamma irradiation leads to the desorption of oxygen from the crystallites surfaces. The extra oxygen vacancies thus formed, may form donor levels in the forbidden band gap of TiO 2 [START_REF] Pan | Defective TiO 2 with oxygen vacancies: synthesis, properties and photocatalytic applications[END_REF][START_REF] Sun | Novel method to directly prepare high-surfacearea anatase titania nanoparticles with trapped electrons on oxygen vacancies[END_REF] and also affect the electron-hole recombination therefore strongly affecting the photoelectrochemical properties of these catalysts. We will come back to this statement in the nanoscale analysis section.

EPR spectroscopy analysis

An EPR spectroscopy analysis has been performed in order to monitor the involved paramagnetic centers with respect to the Cu incorporation rate and to gamma irradiation dose.

This investigation also allows for measuring the oxygen vacancy distribution that occurred during internal incorporation of Cu 2+ ions in the TiO 2 lattice and gamma treatment. The evolution, as a function of the gamma radiation dose, of the EPR signal of undoped and Cu-TiO 2 samples (3 at.% and 6 at.%) are shown in Fig. 3(a,b,d) respectively. The sharp peak observed at g = 2.001 is close to the free electron g value (g = 2.0003) [START_REF] Choudhury | Defect generation, d-d transition, and band gap reduction in Cu-doped TiO 2 nanoparticles[END_REF]. In TiO 2 , this value is reported to be due to a single electron trapped in an oxygen vacancy ( ) [START_REF] Sun | Novel method to directly prepare high-surfacearea anatase titania nanoparticles with trapped electrons on oxygen vacancies[END_REF]. 𝑉 -

𝑂

Undoped samples display only this paramagnetic signal whose intensity slightly increases with the dose of gamma irradiation, whereas the line width hardly changes (Fig. 3(a)).

After Cu insertion (Fig. 3(b,d)), an asymmetric EPR signal shape appears. This EPR signal is present in all Cu-TiO 2 initial and gamma treated samples and corresponds to the presence of Cu 2+ (3d9) ions in the distorted octahedral coordination of O 2-ions [START_REF] Choudhury | Defect generation, d-d transition, and band gap reduction in Cu-doped TiO 2 nanoparticles[END_REF], which confirms the substitution of Cu 2+ for Ti 4+ ions in the TiO 2 structure. These EPR peaks are also broad suggesting that a dipolar interaction is taking place among neighboring Cu For non-irradiated samples, we note that the intensity of the EPR peak assigned to 𝑉 - 𝑂 decreases notably after addition of Cu in the TiO 2 samples [START_REF] Samet | Radiocatalytic Cu-incorporated TiO 2 nano-particles for the degradation of organic species under gamma irradiation[END_REF]. After gamma irradiation such EPR peak increases notably and especially for 3 at.% Cu-TiO 2 , while the 6 at.% Cu-TiO 2 EPR spectrum changes radically. We speculate that a new Cu-rich phase is formed. It has been disclosed that the substitution of Cu 2+ for Ti 4+ ions in the TiO 2 lattice leads to the formation of oxygen vacancies inhibiting the formation of Ti 3+ ions [START_REF] Choudhury | Defect generation, d-d transition, and band gap reduction in Cu-doped TiO 2 nanoparticles[END_REF][START_REF] Navas | Experimental and theoretical study of the electronic properties of Cu-doped anatase TiO 2[END_REF]. Indeed, within the detection limits of EPR spectroscopy, no signal corresponding to Ti 3+ ions was detected. We also note that these oxygen vacancy sites are electron traps. In fact, following the Cu 2+ doping, the place occupied by the O 2-anion in the regular TiO 2 lattice is taken by one or two free electrons, thus forming a donor level below the conduction band. It is demonstrated that the number of such donor states in both anatase and rutile titania increases with increasing number of oxygen vacancies and can overlap the conduction band [START_REF] Pan | Defective TiO 2 with oxygen vacancies: synthesis, properties and photocatalytic applications[END_REF]. This may explain the great decrease of the TiO 2 band gap upon Cu 2+ doping. But, in our case the incorporation of Cu 2+ ions in TiO 2 crystallites is accompanied by a significant decrease in oxygen vacancies or rather in free electrons trapped in these vacant sites. In order to explain these findings, we have performed a nanoscale investigation via EELS analysis.

Nanoscale analysis

Examples of bright Field (BF) images of 3 at.% Cu-TiO 2 annealed at 400°C, obtained during spectro-microscopy measurements, are shown in Fig. 4(a). It can be seen that the average diameter of the crystallites is between 10 and 15 nm and that these crystallites exhibit high crystallinity and well-resolved lattice features throughout the whole polyhedral particles.

The application of the PCA technique to EELS performed at sub-nm resolution shows the presence of metallic copper nanoparticles (1-2 nm in diameter) at the surface of the TiO 2 crystallites (see Fig. 4(b,c)). From this doping level, the TiO 2 structure begins to be saturated with Cu [START_REF] Samet | Radiocatalytic Cu-incorporated TiO 2 nano-particles for the degradation of organic species under gamma irradiation[END_REF]. The electrons trapped in oxygen vacancies can be transferred to the Cu 2+ ions segregated on the crystallite surface resulting in metallic Cu formation. This charge transfer can explain the decrease of the EPR signal PPH in comparison with undoped TiO 2 𝑉 - 𝑂 samples. This decrease in the density of electrons trapped in oxygen vacancies close to the surface may also explain the photocatalytic reactivity reduction of copper doped TiO 2 catalysts.

In addition, in a previous work [START_REF] Samet | Radiocatalytic Cu-incorporated TiO 2 nano-particles for the degradation of organic species under gamma irradiation[END_REF], we have shown by EELS elemental mapping that the insertion of copper in the TiO 2 structure is homogenous within a grain. The amount of Cu inserted (in substitution) in the TiO 2 structure does not exceed 1 at.%. When the copper solubility limitis exceeded, Cu 2+ ions are segregated to the grain surfaces in metallic or oxidized form. The total (inserted or segregated) Cu amount varies from a fraction of a percent to about 10% from one grain to another. The same behavior was observed when doping the TiO 2 structure with Co 2+ ions as reported in a previous article [START_REF] Samet | Heat treatment effect on the physical properties of cobalt doped TiO 2 sol-gel materials[END_REF].

We now discuss the use of gamma irradiation for the purpose of extra creation. As 𝑉 -

𝑂

shown from the TEM observations (Fig. 4(d)) one advantage (at controlled doses) of this method is the little damage at the surface of the crystallites.

For undoped TiO 2 samples, a 27% increase in was estimated from EPR 𝑉 - 𝑂 measurements, which leads to an increase in absorbance in the visible range and an improvement of 10% in photocatalytic efficiency. We have mentioned that 3 at.% Cu-TiO 2 is saturated with less than 1 at.% Cu in substitutional site. In non-gamma irradiated samples, the formation of the metallic copper nanoparticles, which neutralize the free electrons trapped in the sites created by Cu 2+ doping, inhibits the photocatalytic reactivity. As shown in 𝑉 - 𝑂 Fig. 3(a,b,c), in such relatively less stable structures with larger specific surface areas (25% larger than the undoped TiO 2 samples [START_REF] Samet | Heat treatment effect on the physical properties of cobalt doped TiO 2 sol-gel materials[END_REF]), the creation of by gamma irradiation is 𝑉 - 𝑂 facilitated. In fact, we note that the Cu 2+ EPR signal intensity slightly decreases with gamma irradiation dose, whereas the EPR peak assigned to increases notably. A 400% increase of 𝑉 - 𝑂 is accompanied by a 80% improvement in photocatalytic efficiency. The oxygen vacancy 𝑉 - 𝑂 density increase implies a charge carrier density increase on the surface of the crystallites.

Moreover, the transfer of the photogenerated electron to the catalyst surfaces can be further facilitated via the Schottky barrier created by the metal-semiconductor interaction at the metallic copper/TiO 2 interface. Also the excellent electric conductivity of metallic copper has been shown to play an important role in improving photocatalytic activity of Cu-TiO 2 samples.

In order to get some insight about the effect of gamma irradiation at a nanometer scale, anatase 3 at.% Cu-TiO 2 powders irradiated at 60 KGy gamma dose were analysed by high spatially-resolved EELS. For these samples, the typical surface effect (amorphization) is shown on the bright field image of Fig. 4(d). The corresponding EELS spectra for different crystallites in the sample are shown in the energy regions of the TiL 2,3 (Fig. 5(a)) and OK (Fig. 5(b)) edges and compared to non-gamma irradiated samples. Both TiL 2,3 Energy Loss Near Edge Structures (ELNES) in non-gamma irradiated and irradiated samples, are in agreement with Ti 4+ oxidation state in anatase. They exhibit a spin-orbit splitting into 2p 3/2 (L 3 ) and 2p 1/2 (L 2 ) levels, with a separation of 5 eV. The degree of crystallinity is reflected by the further splitting of the L 2 and L 3 peaks into two peaks due to crystal-field effect: the octahedral coordination of titanium atoms with oxygen splits the Ti3d states into the t 2g and e g symmetries. The right shoulder in the L 3-eg peak separated by about 1 eV from the maximum is typical for anatase and has been interpreted as the signature of a structural long-range effect within the crystal [START_REF] Krüger | Multichannel multiple scattering calculation of L 2,3 -edge spectra of TiO 2 and SrTiO 3 : Importance of multiplet coupling and band structure[END_REF]. A similar spitting can be observed in the OK edge, which results from the O2p-Ti3d hybridized states of t 2g and e g symmetry [START_REF] Gloter | Electronic structure of titaniabased nanotubes investigated by EELS spectroscopy[END_REF]. The increase of the oxygen vacancy concentration with applied gamma doses leads to the formation of a defective layer at the TiO 2 crystallite surface. Tan et al. reported similar observations on colored TiO 2 [START_REF] Tan | A facile and versatile method for preparation of colored TiO 2 with enhanced solar-driven photocatalytic activity[END_REF]. Indeed, after gamma treatment,TiL 2,3 and OK ELNES show significant changes. The shape of the TiL 3 edge is very sensitive to local disorder but also to long range order modification induced by the desorption of oxygen. A broadening of the different features in both edges is observed. Such broadening is the result of the superimposition of various crystal field splitting effects and is an indication of the amorphisation of the structure (in the case of TiL 2,3 , the right shoulder on the L 3-eg peak also vanishes as an indication of long range order disappearance). We note that these spectral variations are also compatible with the appearance of a small amount of Ti 3+ .

At high Cu incorporation level (6 at.% Cu-TiO 2 ) the solubility limits of Cu in the TiO 2 structure is exceeded and the titanium crystallites are mainly encapsulated by a Cu thin layer (Fig. 4(e)). EPR investigations confirm that the 6 at.% Cu-TiO 2 non-gamma irradiated crystallites are mainly undoped and that copper species are distributed over the surface of the crystallites, in metallic or Cu 2 O forms [START_REF] Samet | Radiocatalytic Cu-incorporated TiO 2 nano-particles for the degradation of organic species under gamma irradiation[END_REF]. Those copper species, by covering the surface of the crystallites, may be at the origin of the photocatalytical efficiency drop by limiting the interaction between the sunlight and the catalyst. According to EPR investigation (Fig. 3(d))

and TEM imaging (Fig. 4(f)), these copper species are unstable under gamma radiation and transform into Cu-rich nanoparticles, which agglomerate on the surface of the TiO 2 crystallites. TEM images (Fig. 4(f)), also show the destructive effect of gamma rays that release the bare surfaces of crystallites. A photocatalytic efficiency improvement of 20% was observed after gamma radiation treatment at 60 KGy.

We now come back to the previous discussion on the peak observed close to 500 nm in the absorption spectra of gamma irradiated samples (Fig. 2). Our TEM investigations show a gamma irradiation induced amorphization over thicknesses of a few atomic planes at the surface of the TiO 2 crystallite (section 3. 2. 3). Gamma irradiation also leads to the desorption of oxygen from the TiO 2 crystallite surface. A theoretical study [START_REF] Pham | Oxygen Vacancy and hole Conduction in amorphous TiO 2[END_REF] has shown that the formation energy of oxygen vacancies reduces significantly (by a few eV) upon amorphization. Formed oxygen vacancies introduce localized states in the band gap [START_REF] Chen | A facile approch to prepare black TiO 2 with oxygen vacancy for enhancing photocatalytic activity[END_REF] and can be related to the resolution of the extra induced visible band with gamma irradiation.

Other studies [START_REF] Abdelghany | Effect of TiO 2 doping and gamma ray irradiation on the properties of SrO-B 2 O 3 glasses[END_REF] showed that the gamma irradiation of borate glasses containing a high TiO 2 content (10%) is responsible for the appearance of small peaks or a shoulder at 540 nm. This specific induced visible peak is ascribed to octahedrally coordinated trivalent titanium (Ti 3+ ) ions. Surface Ti 4+ ions can capture released electrons during the irradiation process and are transformed to Ti 3+ ions leading to the generation of the low intensity visible peak.

Our EPR investigation does not provide any evidence for the presence of Ti 3+ ions.

However, the local EELS analysis suggests the possible presence of such ions. Therefore, the peak observed close to 500 nm for irradiated samples could also be related to trace Ti 3+ in the crystallite surface (shell) amorphous layers.

Our photocatalytic experiments show that Cu-TiO 2 with higher oxygen vacancies is more efficient as compared to pure TiO 2 , and we confirm that oxygen vacancies play a significant role in both absorption of visible light and photocatalytic efficiency due to the high mobility of their induced charge carriers. Several studies reported a similar synergetic effect in TiO 2 when co-doped with and nitrogen: photogenerated carrier recombination was 𝑉 - 𝑂 shown to be reduced and visible light absorption improved in comparison to N-doped or undoped TiO 2 [START_REF] Chen | A facile approach to synthesize N-doped and oxygen-deficient TiO 2 with high visible-light activity for benzene decomposition[END_REF][START_REF] Chen | Origin of the visible-light photoactivity of NH 3treated TiO 2 : Effect of nitrogen doping and oxygen vacancies[END_REF].

Conclusion

This study presented a simple and economical method to synthesize efficient visiblelight-responsive TiO 2 photocatalysts. Defective pure and Cu doped TiO 2 powders were successfully prepared by combining sol-gel synthesis and gamma treatment. Such samples were characterized by EPR, TEM, EELS and UV spectroscopies. The photocatalytic activity of the obtained powders was evaluated by the photocatalytic degradation of MB. Compared to the non-gamma irradiated TiO 2 powders, it was found that the introduction of defect sites at the TiO 2 catalyst surfaces results in a significant improvement in the photocatalytic efficiency due to the generation of oxygen vacancy electron trapping states. The enhancement of the absorbance of visible light was attributed to both Cu 2+ substitution and to the presence of extra oxygen vacancies induced by irradiation, which introduce isolated states in the band gap of the TiO 2 catalyst. This study also showed a close relationship between the gamma irradiation dose and the concentration of defect sites which are related to the destiny of photogenerated electrons and holes. Indeed, the defective sites inhibit the recombination of these electron-hole pairs and thereby increase the concentration of photogenerated carriers at the crystallite surface, leading to an enhancement of the activities of the TiO 2 powders under sunlight irradiation. These results demonstrate the relevance of a strategy exploiting (Cu 2+ , ) cooperative effects for 𝑉 - 𝑂 promoting the photoelectrochemical properties of TiO 2 powders. This preparation method by gamma treatment could be potentially used for large-scale production of catalysts with remarkable visible light absorption and photocatalytic activity for wide applications in environmental remediation.

  2+ . The peakto-peak height (PPH) of Cu 2+ ion and oxygen vacancy ( ) signals as a function of the 𝑉 - 𝑂 applied gamma dose are shown in Fig. 3(c) for undoped and 3 at.% Cu-TiO 2 samples.

Fig. 1 :

 1 Figures captions: Fig. 1: Photodegradation efficiency results of MB, under natural sunlight: without catalyst, with initial undoped and Cu-TiO 2 catalyst powders and with undoped and Cu-TiO 2 catalyst powders treated by 60 KGy gamma irradiation dose.

Fig. 2 :

 2 Fig. 2: Diffuse reflectance spectra as a function of gamma irradiation doses of (a) undoped TiO 2 , (b) 3 at.% Cu-TiO 2 and (c) 6 at.% Cu-TiO 2 .

Fig. 3 :

 3 Fig. 3: EPR spectra of Cu-TiO 2 catalysts annealed at 400°C as a function of gamma irradiation doses: (a) undoped TiO 2 , (b) 3 at.% Cu-TiO 2 and (d) 6 at.% Cu-TiO 2 .(c) Evolution of the peak-to-peak height of both EPR signatures for divalent copper ions (Cu 2+ ) and oxygen vacancies (Vo) as a function of gamma doses for undoped TiO 2 and 3 at.% Cu-TiO 2 catalyst.

Fig. 4 :

 4 Fig. 4: (a) High-resolution Bright Field (BF) image of a 3 at.% Cu-TiO 2 powder annealed at 400°C, (b) Cu elemental map and associated CuL 2,3 EELS signal extracted after PCA (c) from the reduced area indicated in (a). (d) Bright Field (BF) image of a 3 at.% Cu-TiO 2 powder annealed at 400°C and treated by 28 KGy gamma dose. (e) TEM micrograph of a 6 at.%. Cu-TiO 2 powder annealed at 400 °C and (f) after treatment at a 60 KGy gamma dose.

Fig. 5 :

 5 Fig. 5: (a) TiL 2.3 and (b) OK edge fine structures of 3 at.% Cu-TiO 2 powders annealed at 400°C, before and after 60 KGy gamma treatment.

  Fig.1

Fig. 4

 4 Fig.4


Marc Boyer

Hugo Daigmorte

Impact on credit freeze before gate closing in CBS and GCL integration into TSN

The Time Sensitive Networking (TSN) task group has added a set of mechanisms to Ethernet in order to provide a real-time network. In particular, the output port scheduling based on a Credit-Based Shaper (CBS) algorithm, that was introduced formerly by the Audio-Video Bridging (AVB) task group, has been enhanced with a time driven Gate Control List (GCL). This implies some update in the credit evolution rules, and several solutions may exist. In this paper, we compare the solution used in the standard with another one used in most papers, and also with a third one, designed as a tradeoff between the two others. The comparison is first done on some hand-made examples, showing some credit overflow and unfairness potential problems. Then, simulations are done on a single switch with 3 CBS queues.

INTRODUCTION

In order to offer real-time guarantees to audio and video flows over Ethernet, the Audio-Video Bridging (AVB) task group has defined a set of extensions of the IEEE 802.1Q standard [START_REF]IEEE Standard for Local and Metropolitan Area Networks -Bridges and Bridged Networks[END_REF], related to the forwarding policy in bridges (aka. switches). Among the 8 queues, the two with higher priority (called A and B) are scheduled at the output port of each switch with a specific policy, the Credit-Based Shaper (CBS). At configuration, each queue X ∈ {A, B} receives a part of the output port rate R, called the "idle slope". Then, a scheduler based on a credit variable implements this constraint (this algorithm will be described in Section 2.2).

Thereafter the Time Sensitive Networking (TSN) task group has added new extensions. One of them, named "Enhancement for scheduled traffic", introduced per queue gates: each queue has access to the output port only when its gate is open, and the opening/closing of the gates is controlled by a cyclic time based schedule, called Gate Control List (GCL). The introduction of this mechanism is interpreted by some authors as the introduction of Scheduled Traffic or Time-Triggered communications in TSN [START_REF] Alderisi | Introducing support for scheduled traffic over IEEE audio video bridging networks[END_REF]. The CBS algorithm has been adapted to take into account the GCL mechanism. There are several ways to do such an adaptation and, surprisingly, the choice made in the standard is different from what is considered in the literature. This paper will first present in Section 2 the sub-part of the TSN standard that is considered in this paper. Section 3 gives an overview of the assumptions done in the literature. Then, for some examples, the qualitative impacts of these new rules will be discussed in Section 4: the transformation of the operator slope into the credit idle slope, the possible credit overflow and some unfairness between flows w.r.t. AVB behaviour. After these preliminary considerations, Section 5 will formally define in an unified framework the standard evolution rule, the one assumed by most research papers, and a new one, a trade-off inspired by AVB. These three rules are compared by simulating a single output port with three CBS queues in Section 6. For sake of simplicity, this paper does not address preemption.

OUTPUT PORT MODEL

This section presents the sub-part of the 802.1Q standard [START_REF]IEEE Standard for Local and Metropolitan Area Networks -Bridges and Bridged Networks[END_REF] relevant for this study. Subsection 2.1 presents the global standard architecture, subsection 2.2 presents the credit based shaper rules defined before the introduction of gates, subsection 2.3 presents the evolution of those rules due to the introduction of gates, and subsection 2.4 presents some hypotheses limiting the scope of this study.

Output port architecture

In the current 802.1Q standard [START_REF]IEEE Standard for Local and Metropolitan Area Networks -Bridges and Bridged Networks[END_REF], a TSN output port is made of (up to) 8 queues, numbered from 0 to 7. To each queue is associated an optional transmission selection algorithm, and a gate that can be either open (o) or closed (c) as illustrated in Figure 1. A frame in a queue can be selected for transmission only if the transmission selection algorithm allows it (subsection 2.2 presents the credit based shaper, CBS, selection algorithm) and if the gate is open. A static global cyclic table, the GCL, schedules the gate openings and closings. If several frames are ready for transmission, the higher priority queue is selected (the higher the queue id, the higher the priority). Each frame is mapped into a class, and each class into a queue. In this paper, we will assume one queue per class, and use queue and class as synonyms.

Credit evolution rules without GCL

The Credit-Based Shaper (CBS, [1, §8.6.8.2]) is one possible transmission selection algorithm. This section presents the selection rules of CBS, when no GCL is implemented in the output port. These selection rules rely, for each CBS queue X , on the value of an integer variable, called credit and a parameter, id X , the idle slope. The rules, when no GCL is used, are the following: R1: The head of queue frame can be selected for transmission only if the credit of this queue is non negative, and no higher priority frame is ready for transmission. R2: During frame transmission, the credit decreases as a function of time with "send slope" rate sd X = id X -R where R is the transmission rate of the port. R3: The credit increases with rate id X when either the credit is negative, or it is positive and there is some frame waiting in the queue. R4: If the credit is positive and the queue is empty, the credit is reset to null.

This behaviour is illustrated on an example with three active classes: A, B, and C, with A having the highest priority and C the lowest. Figure 2 shows the arrivals of frames (the frame X-n being the n-th frame entering queue X), the port transmission schedule and the credit of class B (with null initial value). When a message B-1 is set in the queue, the credit is null, and from rule R1, it can be selected for transmission. Then, rule R2 implies that the credit decreases. When message B-2 is received, it can not be selected since the credit is negative, and it must wait until the credit of class B reaches 0 to be selected. At the end of B-2 transmission, the output port is idle, and a lower priority message C-1 can start its transmission. Since we do not consider preemption, when the credit of class B reaches 0, the message can not be send, and the credit still increases, reaching positive values. However at the end of C-1 transmission, a higher priority frame A-1 is waiting, and B-3 has to wait. Thus its credit still increases. Finally message B-3 is selected for transmission, rule R2 implies that the credit decreases. At the end of B-3 transmission, the credit is positive and the queue is empty so, from rule R4, the credit is reset to null.

Credit evolution rules with GCL

The introduction of gates in [1, §8.6.8.4] comes with a modification of credit evolution rules. The main ideas are:

time time Class B credit Transmission Arrivals C-1 C-1 B-1 B-2 B-3 B-1 B-2 B-3 A-1 A-1
Figure 2: CBS credit evolution rules (without GCL) R5: The credit is frozen when the gate is closed (overriding rules R2 and R3). R6: A frame can start its transmission if it can be sent up to completion before the next gate-close event.

Moreover, the idle slope is scaled proportionally to the accumulated closing duration (more details will be given in Section 4). This new behaviour is illustrated in Figure 3. Moreover the GCL list is in this example such that, when the gate of a queue is open, all other gates are open, and when the gate of a queue is closed, all other gates are closed. When message B-1 is set in the queue, the credit is null, and from rule R1, it can be selected for transmission. Then, rule R2 implies that the credit decreases. When message B-2 is received, it can not be selected since the credit is negative, and it must wait until the credit of class B reaches 0 to be selected. When message A-1 is received, the output port is idle, it can be selected for transmission. When the credit of class B reaches 0, the message can not be sent (the output port is used by frame A-1), the credit still increases, reaching positive values. The gate closing influence starts at the end of A-1 transmission (time instant s): rule R6 implies that message B-2 has to wait. However a lower priority message C-1 can start its transmission. When the gate is closed, from rule R5, the credit of class B is frozen. When the gate opens, message B-2 can start its transmission. At the end of B-2 transmission, the credit is positive and the queue is empty so, from rule R4, the credit is reset to null.

One point deserves some attention: what happens just before gate closing? The rule R6 prevents a frame to encroach on a closed gate period: a frame can not start its transmission if it does not have enough time to finish before the next gate-close event 1 . A time interval when a frame can not start its transmission due to rule R6 will be called a "pre-closing" time interval. But what is the evolution rule of the credit during this time? In previous studies [START_REF] Maxim | Delay Analysis of AVB traffic in Time-Sensitive Networks (TSN)[END_REF][START_REF] Zhao | Timing Analysis of AVB Traffic in TSN Networks using Network Calculus[END_REF], this period has been considered as an "extension" of the gate closing period, and it was assumed that the credit was also frozen. But in the current standard [START_REF]IEEE Standard for Local and Metropolitan Area Networks -Bridges and Bridged Networks[END_REF], the credit increases during this waiting time 2 . This paper focuses on the effect of these rules on the port traffic scheduling. To ease comparison with AVB, CBS queues will be named A,B,C,D,E,F, as illustrated on Figure 4.

Port with CBS and exclusive gating

STATE OF THE ART

Scheduling synchronous and asynchronous workload on a given resource is an old issue. For example, the "deferrable server" [START_REF] Strosnider | The deferrable server algorithm for enhanced aperiodic responsiveness in Hard Real-Time Environnments[END_REF] considers a set of strictly periodic tasks, plus a deferrable server, designed to execute the asynchronous workload. Like CBS, this server receives periodically a capacity, a credit, of execution time. But a deferrable server can use all its credit to execute several pending asynchronous tasks, whereas the replenishment rules of CBS are designed to limit such bursts (as illustrated by the delaying of frame B-2 in Figure 2). Moreover, deferrable server assumes preemptive scheduling, whereas the core is this paper is the impact that non-preemption and encroaching avoidance have on credit.

More recently, another Ethernet-based solution mixing synchronous (Time-Triggered) and asynchronous (Rate Constrained) flows has been developed: TTEthernet [START_REF] Steiner | TTEthernet Dataflow Concept[END_REF]. Three integration policies 2 A detailed analysis of this sub-part of the standard devoted to CBS credit evolution rules and GCL interactions can be found in [START_REF] Daigmorte | Does the integration of CBS and GCL behaves as you expect? And can it be enhanced[END_REF]. have been defined in [START_REF] Steiner | TTEthernet Dataflow Concept[END_REF] to solve (or limit) the encroaching of asynchronous flow on synchronous time windows. But this technology has no credit-based shaping mechanism.

There are, to the best of our knowledge, few studies discussing this specific point of credit evolution rule before gate-close event in the TSN context.

The first proposal on adding some time scheduled traffic to AVB seems to be done in [START_REF] Alderisi | Introducing support for scheduled traffic over IEEE audio video bridging networks[END_REF], calling the proposal "AVB ST". It shows that to avoid encroaching on reserved time windows, called "ST slots", some "guard bands" have to be considered. The credit evolution rule is the one of AVB: when a CBS queue waits, its credit increases, whether this waiting is due to any guard band or a ST slot. A worst case response time AVB ST is done in [START_REF] Ashjaei | Schedulability analysis of Ethernet Audio Video Bridging networks with scheduled traffic support[END_REF], making clear distinction between the standard behaviour of TSN and the AVB ST proposal.

All other papers, analysing TSN configurations with CBS and GCL mechanisms, either by simulation or evaluating the worst case latency, assume that there is a "guard band" before the gateclose event, and that during this guard band, the credit is frozen [START_REF] He | Impact Analysis of Flow Shaping in Ethernet-AVB/TSN and AFDX from Network Calculus and Simulation Perspective[END_REF][START_REF] Maxim | Delay Analysis of AVB traffic in Time-Sensitive Networks (TSN)[END_REF][START_REF] Maxim | Delay Analysis of AVB traffic in Time-Sensitive Networks (TSN)[END_REF][START_REF] Zhao | Timing Analysis of AVB Traffic in TSN Networks using Network Calculus[END_REF].

Last, in [START_REF] Daigmorte | Modelling in network calculus a TSN architecture mixing Time-Triggered, Credit Based Shaper and Best-Effort queues[END_REF], when looking for a worst case latency bound, several integration policies are considered, and a generic model that generalises all these policies is defined. It might be generic enough to be used with the standard rules.

IMPACT OF CREDIT EVOLUTION RULES

This evolution of credit before a gate-close event can have unexpected consequences. The reason is that, for a given class, there are two sources of bandwidth loss due to gate closing: some is lost while the gate is closed, and during these intervals, the credit is frozen, whereas some is lost while the gate is open, and during these intervals, the credit increases.

In this section, we will first show that the pre-closing time (this time when a frame can not be sent since there is not enough time to send a frame before the next gate-close event) can not be "used" by CBS flows: if this time is statically allocated to CBS flows, it can lead to buffer overflow (Section 4.2.1), and if it is not, it will lead to CBS idling time, whereas there is backlog and idle port The duration of the GCL GateOpenTime(n)

The total amount of time during the GCL duration that the gate state for queue n is Open PreCloseTime(n, s, t) Accumulated time of pre-closing between s and t MaxPreCloseTime(n) Maximum value of pre-closing during a GCL (Section 4.2.2). Second, we will show in Section 4.3 that these rules can be unfair: it allows the CBS queues with the highest priority to gain more credit, and send larger bursts than in a pure AVB system, creating larger jitter for low priority CBS queues. But before discussing the differences and interactions between these elements, some vocabulary has to be introduced.

Vocabulary

Let portTransmitRate be the transmission rate of the port of interest and let OperCycleTime be the duration of the GCL table (the period of the behaviour). Let n ∈ [0, 7] be a queue number, GateOpenTime(n) is the total amount of time during the GCL duration that the gate state for queue n is open, GateCloseTime(n) def = OperCycleTime-GateOpenTime(n), the total amount of time during the GCL duration that the gate state for queue n is Closed. The network designer has to set the operIdleSlope(n) of each queue n using CBS, the bandwidth allocated to this queue [1, § 34.3]. All these values are statically defined. Now, given an interval [s, t], PreCloseTime(n, s, t) is the accumulated time where some frame has been delayed because of rule "there is insufficient time available to transmit the entirety of that frame before the next gate-close event" 3 . This duration is by nature dynamic: it depends on the presence of frame in the queue before some gate-close event, on the credit value, and on the frame size. It can be upper bounded: each close-gate event may generate only one pre-close interval, and its length is at most the duration of one frame of maximal length. Let MaxPreCloseTime be its maximal value on a GCL cycle 4 .

Then, the idleSlope parameter is computed as in [1, §8.6.8.2] 1 summarises all of this information.

idleSlope(n) = operIdleSlope(n) × OperCycleTime GateOpenTime(n) (1)

The pre-closing time can not be used by CBS flows

This section presents how the evolution rules of credit prevent the use of this pre-closing time by CBS flows: first, in Section 4.2.1, how the static configuration of parameters prevents from statically reserving it for CBS flows, second, in Section 4.2.2, how the transmission rules prevents CBS flows from dynamically using it.

4.2.1 Pre-closing time can not be allocated to CBS flows. During the configuration, the pre-closing time can not be allocated to CBS. Otherwise, the value of the credit is unbounded and its implementation variable can overflow (in the remainder of this paper we will write "credit overflow" to speak about the credit implementation overflow).

Consider a TSN output port with CBS at queues 1 to 6 and exclusive gating, as presented in Section 2.4. For any priority n ∈ [START_REF]IEEE Standard for Local and Metropolitan Area Networks -Bridges and Bridged Networks[END_REF][START_REF] Daigmorte | Modelling in network calculus a TSN architecture mixing Time-Triggered, Credit Based Shaper and Best-Effort queues[END_REF], the condition

i ∈hp(n) operIdleSlope(i) R + GateCloseTime(n) OperCycleTime + MaxPreCloseTime(n) OperCycleTime ≤ 1 (2)
must hold, where hp(n) denotes the set of CBS queues with priority not smaller than the nth queue 5 . Otherwise, the credit of the priority level n can overflow. This condition just states that the global load must be less than 1, considering equal to 1000ms, the portTransmitRate is also 1b/ms, all messages are of constant sizes 100b and the GateCloseTime represents 20% of the OperCycleTime divided into two windows of duration 100ms (One may object that it is not possible to have Ethernet frames of size 100b, and that 1b/ms is unrealistic. But this example has been built to illustrate some behaviour with a small number of frames, human-friendly constants and can be scaled to realistic values). Both examples assume that the queues are never empty during the considered interval. In this example the sum of allocated bandwidth plus the GateCloseTime is equal to portTransmitRate (class A and class B operIdleSlope equal 400b/s). But since a part of the bandwidth allocated is in fact lost by pre-closing time, it leads to credit overflow.

ρ i = operIdleSlope(i) R

4.2.2

Un-allocated time can not be used by CBS flows. The previous example has shown that the pre-closing time, which is by nature dynamic, can not be allocated to CBS flows. One may wonder if it can be dynamically used by these flows. Consider a second example (see Figure 6) with the same parameters as the previous one, except that the sum of allocated bandwidth plus the GateCloseTime plus the MaxPreCloseTime is equal to portTransmitRate (class A and class B operIdleSlope equal 300b/s). In this case, there is no more overflow; however there is some unused bandwidth. The reason is that the idle slope of a CBS queues acts both as a minimal reserved capacity, but also as a maximal bandwidth usage (it is both the slope of the minimal service curve and the shaping curve [START_REF] De | Complete modelling of AVB in Network Calculus Framework[END_REF]). Then, since the MaxPreCloseTime has not been allocated to CBS flows (to avoid possible credit overflow), it can not be used by these CBS flow. This behaviour is not surprising: whereas GPS-like policies (WFQ, WRR, DRR [START_REF] Katevenis | Weighted round-robin cell multiplexing in a general-purpose ATM switch chip. Selected Areas in Communications[END_REF][START_REF] Abhay | A generalized processor sharing approach to flow control in integrated services networks: the single-node case[END_REF][START_REF] Shreedhar | Efficient fair queueing using deficit round robin[END_REF]) are designed to dynamically share the bandwidth in a fair way (with configurable weights between flows), credit based shaping reserves a constant capacity. In TSN, to avoid encroaching on static gate closing, some dynamic loss of bandwidth is created, and credit based shaping can not share it between flows (but it may be used by some Best-Effort queue, i.e. queue with low priority flows and without shaper).

The pre-closing time credit evolution rule can be unfair

A more unexpected consequence of the credit evolution rule is that it can be unfair: it can allow high priority classes to accumulate more credit, and create larger bursts than what would happen In order to visualise this unfair behaviour we will use the original situation described in Figure 7. In these examples there are three CBS flows : class A, class B and class C. In this example the GateCloseTime is null and the portTransmitRate is 1b/ms. 30% of the OperCycleTime is allocated to class A (operIdleSlope equal 300b/s), 20% is allocated to class B (operIdleSlope equal 200b/s) and 20% is allocated to class C (operIdleSlope equal 200b/s). Messages of each class are of constant sizes : 150b for A and 100b for B and C. Figure 7 shows that there is an alternation in the transmission of messages between these three classes. Now we will consider a different situation. The traffic of the class A is split: two thirds remains in the class A, and one third is set to the highest priority queue. The highest priority queue gets an exclusive access to the medium through GCL mechanism, as described in Section 2.4. The operIdleSlope of the class A is decreased to 200b/s, its message size is decreased to 100b. The part sent to the higher priority queue is cut into smaller packets of size 50b, with a schedule 6 given at the bottom of Figure 8. Then, the behaviour of the system is represented in Figure 8: the first frame of the C queue is delayed by two frames from A and two frames from B. The long term service is the same (in both Figures 7 and8, 12 frames of C are served), but the traffic is more bursty, due to gate closing, but also to the credit evolution rule. Now we will consider that during the pre-closing time, the credit is frozen (like when the gate is closed). The Figure 9 represents the same system than in Figure 8, but with credit frozen during preclosing time. In this situation there is a perfect alternation in the transmission of messages between the three classes. This situation is more fair. However the part of unused bandwidth increases: since the credits no longer increase during PreCloseTime, only 10 frames per class are served during this schedule. The reason is in the conversion between the operIdleSlope and the idleSlope: the eq. 1 does not consider the effect of pre-closing. This point will be discussed in Section 5.

DIFFERENT EVOLUTION RULES

In this section, we will formally define in an unified framework the standard evolution rule, the one assumed by most research papers, and a new one, a trade-off inspired by AVB. We will first take a look at some important CBS properties (Section 5.1) before discussing several possible credit evolution rules (Section 5.2).

Two framing remarks

Before discussing and comparing different rules for the credit evolution, let us make two framing remarks.

Keeping credit value bounded. First, one must avoid any credit overflow. We have shown in Section 4.2 that the pre-closing time can not be allocated to CBS flows. In other words, one can interpret the pre-closing time as the time where the link is used by a virtual higher priority flow. Since the operate idle slope operIdleSlope(i) of a class i can be interpreted as a part of the bandwidth reserved for this class, one must take into account the bandwidth used by the virtual flow, and the total bandwidth allocation must be less than the link capacity. That is to say, the equation 2 must hold. Note that, like for other CBS flows, the bandwidth that was reserved for this virtual "pre-closing" flow and that is not used is not lost since it may be used by best-effort flows.

AVB credit and bandwidth. Second, one should consider how the credit-based rule can ensure a throughput of idleSlope(X ) to a class X .

In AVB, when a class X starts at some time t to send a frame of length L with null credit, it has to wait up to t + L id X before emitting a new frame, letting other flows send their frames. The credit goes up to positive values only when others AVB flows are 6 Like in example of Figure 5, this example has been built to illustrate some behaviour with human-friendly constants, leading to unrealistic Ethernet frames of size 50b, and a very non efficient schedule with very small intervals. Link Class X credit time sending frames (or due to the non-preemption of one best-effort frame). That is to say, the credit can always increase and return back to zero when it has negative value, but it can increase to positive values only if another flow is using the link. Moreover, the amount of data sent in an interval [u, v] (where the credit ceases to be null at u and comes back to 0 at v) is id X (v -u), as illustrated in Figure 10. More generally, the credit mechanism offers a bandwidth id X to class X (up to bounded credit variations). Let us now discuss several possible credit evolution rules.

X-1 L 1 L 1 id X u v X-2 L 2 L 2 id X u ′ v ′

Discussing new rules

Operator slope vs. idle slope. As presented in Section 5.1, the CBS mechanisms in AVB ensure that a class X with idle slope id X (expressed in bit per second), on any backlogged interval receives at least id X , and can not use more than id X (up to the -boundedcredit variation).

The integration of CBS and GCL has introduced some intervals where the credit is frozen. Then, considering two instants u, v such that the credit ceases to be null at u and comes back to 0 at v, as illustrated in Figure 11. The quantity of data send is only id X × (v -u -F (u, v)), where F (u, v) denote the credit freezing time between u and v (in Figure 11, this is a single interval, but in the general case, it may be the sum of several).

Considering the example of Figure 11, the relation between

L, u, v, F , id X is L = id X × (v -u -F ).
Then, if the network operator aims to offer to class X a bandwidth op X , one must set

id X = op X × v -u v -u -F . ( 3
)
In the current standard rule, this time F (u, v) is statically known (this is the GateCloseTime). Then, as presented in eq. 1, in order to offer a long-term bandwidth op X , on all a GCL schedule, starting at s and ending at e, the idle slope is set as

id X = op X × e -s e -s -F (s, e) = op X × OperCycleTime GateOpenTime(X) . (4)
But when considering a rule where this freezing time is dynamic, one may only know F , F such that F ≤ F (s, e) ≤ F , leading to two possible definitions of the idle slope,

id X = op X × OperCycleTime OperCycleTime -F (5)
or

id X = op X × OperCycleTime OperCycleTime -F . (6)
Credit evolution rule. Consider first the situation where a frame X is received while the credit of its class is negative (Figure 12, left part) and this frame is too large to be sent before the next gate-close event. The standard behaviour, labelled (S) considers that the credit increases up to gate-close event, and is frozen while the gate is closed. Previous studies assume that the credit is frozen is such case, leading to the dotted credit curve (F). But one may consider, by symmetry with AVB behaviour, that a negative credit can always increase and return to zero, but that a positive credit must be linked to a use of the link by another flow, leading to the dashed credit curve (R).

Considering several flows, more options appear, as presented in right part of Figure 12. While the frame X is blocked, it can exist another frame Y using the link. Moreover, it introduces another variation in the possible credit evolution rule: while Y is being sent, does the credit of X evolve or not? We chose in this study to let it increase, since it reduces the frozen time, which is a kind of "waste" of credit opportunity. Moreover, another class may have a head-of-queue frame Z small enough to be sent between Y and the gate-close event. In the current standard rules, this frame Z will be sent, and this appear to be a reasonable behaviour. But this highlights the fact that there is no "guard band" but a per-class behaviour, that we named "pre-closing".

To formally define the rules (R) and (F), let us introduce two new boolean variables pre-closing, and gate-closed. The boolean gateclosed is true when the gate is closed, false otherwise. Definition of pre-closing depends on the policy and is postponed. In the cases of AVB behaviour (R) and of frozen behaviour (F) the credit evolution rules are the following:

• the credit decreases during a frame transmission (which implies that the gate is open, i.e. gate-closed=False); • the credit increases when gate-closed=False and pre-closing=False and either the credit is negative, -or there is some frame waiting in the queue; • the credit is frozen when gate-closed=True or pre-closing=True.

It remains to define when pre-closing turns True and False. First pre-closing turns False when the gate closes. Second pre-closing turns True when the gate is open, the link is available (no message is being sent, and no high priority message can start its transmission) and Return to zero (R): the credit is positive and the frame could not be sent up to completion before the next gate-close event. Frozen behaviour (F): the frame could not be sent up to completion before the next gate-close event.

Combining remarks on evolution rules and idle slope leads to 5 behaviours of interest: the standard one, denoted S, with the idle slope computed using eq. ( 4), the one freezing the credit when a frame can not be fully sent, denoted F, using either the idle slope id or id (resp. defined in eq. ( 5) and eq. ( 6)), and the negative values return to zero, denoted R, also with either id or id. But keep in mind that using numerical values, id = id since the minimal freezing time is the GateCloseTime.

EXPERIMENTS

This section evaluates the effects that credit evolution rules have on the traversal time of CBS messages. In order to evaluate these impacts, we first define an experimental configuration (Section 6.1). The results obtained are presented in Section 6.2 before being interpreted in the Section 6.3.

Simulation model and assumptions

In this paper, we decide to focus on a single node and a single transmit port. Then our configuration pattern is the following:

• The portTransmitRate is equal to 100 Mb/s.

• The architecture conforms to the hypotheses of Section 2.4: no transmission algorithm for higher and lower priority queues, CBS for all others, exclusive-gating between the higher priority queue and all others. • Only 3 CBS queues are active (queue A, queue B and queue C):

-Their operIdleSlope are equal and set to 20 Mb/s.

-The OperCycleTime is equal to 1s and the GateOpenTime is equal to 0.8s. -Each queue is shared by 20 stricly periodic flows with offsets and constant packet size. -The period and size of each flow was uniformly chosen from the set {(1ms, 125bytes), (2ms, 250bytes), (4ms, 500bytes), (8ms, 100bytes)} so that the load of each flow is equal to 1 Mb/s. • There is 1 BE flow in the lower priority queue, with messages of size randomly chosen between 125 bytes and 1250 bytes.

We consider two different configurations for the closing events of CBS and BE queues:

-Random: 1000 closed intervals, with duration of 0.2ms, are randomly set (without overlaping). -Uniform: Every ms starts a closed interval (leading to 1000 closed interval), with a duration of 0.1ms, 0.2ms or 0.4ms randomly chosen, but such that there are 400 intervals of 0.1ms, 400 intervals of 0.2ms and 200 intervals of 0.4ms per OperCycleTime.

One simulation is done using this configuration pattern and by choosing randomly an offset for each flow. One simulation lasts 5 second and provides the delay for each message. In order to obtain representative results, statistics are done over 50 simulations, leading to more than 5 × 10 6 messages. Now that we have our configuration it is possible to compute the idle slopes defined in equations (4) ( 5) and ( 6): id = 25Mb/s, id = 25 Mb/s and id = 27.78 Mb/s.

Results

As presented in Section 5, our aim is to evaluate 3 possible credit evolution rules, with different values of idle slope. In each case, 50 simulations have been run, each one with a different set of offsets between the (periodic) flows. The delays of each frame have been recorded during 5s of simulated time, and then grouped per class to get per class delay distribution. Since the delays depend strongly on the set of offsets, all the 50 delays distributions have been merged into a single one.

A first framing observation is that, for each experiment, the shapes of the three classes are similar: they may have different average or median values, but the shapes of the distributions curves look very similar, as shown on Figure 13. Then, next results plotting delay distribution will aggregate all classes into a single one, and the per class results will be given using box plots. The ends of the box are the lower and upper quartiles (Q1 and Q3) and the band inside the box is the median (Q2). The whiskers (the two lines outside the box) extend to the highest and lowest observations.

We then can compare the three policies, with the same value of idle slope (id = id = 25Mb/s), first on the experiments with the uniform gate closing (cf. Figure 14, and mind that the scales are different). Clearly, the frozen policy gives the worst results, and in fact, such a policy will lead to buffer overflow (but our simulator assumes infinite memory). The reason is that this policy, with this value of idle slope, does not offer enough throughput to the flows. In fact, it also is the case for the policy R considering the worst possible behaviours. But in this simulation, the credit has enough opportunities to get back to 0 before gate closing and avoid buffer overflow.

When considering a random allocation, (cf. Figure 15), the box plots are globally equivalent to the one of the uniform allocation, but the distributions of delay are different, as illustrated on Figure 16. This shows that the distribution of the gate closing instants has a significant impact on the delays, but this influence is out of the scope of this study.

The previous experiments have been done with the small value of the idle slope, id (25Mb/s in the experiment), which is insufficient for the policy R to serve all input flows, in theory, and insufficient in practice for the policy F. We may now have a look on the effect of a larger idle slope, id (27.78Mb/s in the experiment). In this set of experiments, we are going to set the idle slope of the S policy to 27.78Mb/s. This is not the value of the idle slope given by the standard, but comparing policy S with R and F, while giving S a smaller idle slope would have been an unfair comparison.

The delay distributions of all classes are plotted in Figure 17 and the per class box plots are given in Figure 18. When comparing the policies, it also appears that F is worse than the others, and that S and R have very similar results. But there is another important fact: while the idle slopes have been increased by only 11%, the average delays have been decreased by a factor of 8.

Discussing results

What can be learned from all these results? Mainly two things. First, freezing the credit of a class when it can not send any frame due to a gate closing (policy F) is not a very good solution with regard to the other solutions (policies S and R). Second, the delays do not increase linearly with the load with the standard mechanisms: when considering average delays, it is better to let the load be slightly smaller than the bandwidth allocated through the operator slope parameter. Last, the policy R, inspired by AVB, performs comparably to the standard mechanism.

CONCLUSION

This article pays attention to a small part of the TSN behaviour: the instants before a gate-close event, and their interactions with the CBS selection algorithm, and in particular the credit evolution rules before a gate-close event.

The starting point of this study is that several behaviours exist in the literature, without (up to our knowledge) any qualitative or quantitative comparison. In this paper, we have considered in a unified framework three policies: the one of the standard (denoted S), one that, as assumed in most of papers, freezes the credit before the gate closing (denoted F), and a third one, inspired by AVB, that allows the credit to return up to zero before a gate closing (denoted R). Whereas the differences can be considered as minor (just a few words to change in the standard), it has an impact. We have shown on some specific hand-made examples, presented in Section 4, that small modifications can lead to buffer overflow or unfairness between classes. The impact on these variations have been evaluated by running 200 simulations for each of the three policies. It shows that it can have very significant impact on the performances. In particular, the policy F that is mainly considered in the literature, gives on simulation worse results than the standard behaviour S. This study must now be completed by some analysis on the worst case delays. The policy R has been designed to avoid too large values of the credit, which is a root of unfairness. The simulations have shown that it has mean performances equivalent to the standard, and one must investigate what are the differences on the worst case delays.

However, the preemption (presented in [START_REF]IEEE Standard for Local and metropolitan area networks -Bridges and Bridged Networks -Amendment 26: Frame Preemption. IEEE Standard[END_REF][START_REF]IEEE Standard for Ethernet Amendment 5: Specification and Management Parameters for Interspersing Express Traffic[END_REF]) will reduce the impact of gate closing on CBS flow, but it will not disappear. Without preemption, the maximum pre-closing is a full Ethernet frame (15KB), and with preemption, it will be reduced to a full fragment (1.6KB).

Figure 1 :

 1 Figure 1: TSN outputput port architecture

Figure 3 :

 3 Figure 3: CBS credit evolution rules (with GCL)

Figure 4 :

 4 Figure 4: TSN output port with High Priority / CBS / Best Effort and exclusive gating between High Priority queue and other queues. {x, y} = {o, c}

Figure 5 :

 5 Figure 5: Credit overflow

Figure 6 :

 6 Figure 6: Unused bandwidth

  Figure 7: Three CBS flows

Figure 8 :

 8 Figure 8: Credits increase during the pre-closing time

Figure 9 :

 9 Figure 9: Credits are frozen during the pre-closing time

Figure 10 :Figure 11 :

 1011 Figure 10: AVB credit evolution w.r.t. packet size; L i is the size of packet X-i, id X the credit of class X.

Figure 12 :

 12 Figure 12: Possible credit evolution scenarios

Figure 13 :

 13 Figure 13: Per class delay distribution for policy S, idle slope id, random gate closing.

Figure 14 :Figure 15 :

 1415 Figure 14: Box plot of delays (in ms), for policies S, R, and F, idle slope id = id, uniform gate closing.

Figure 16 :

 16 Figure 16: Delay distribution of all classes for policy S, idle slope id, uniform and random gate closing.

Figure 17 :Figure 18 :

 1718 Figure 17: Delay distribution of all classes for policies S, R and F, idle slope id, uniform gate closing.

Table 1 :

 1 Glossary

	portTransmitRate	The transmission rate of the port of in-
		terest
	R	Shorthand for portTransmitRate
	CBS	Credit-Based Shaper
	operIdleSlope(n)	Bandwidth (in b/s) reserved for queue n
	GCL	Gate Control List
	OperCycleTime

The use of preemption may reduce this "forbidden" time, but it can not completely avoid it.

We chose the term "pre-closing" time for these intervals, instead of "guard band" which is more commonly used, because depending on the authors, the "guard band" is this dynamic pre-closing time, whereas for others, the "guard band" is a static interval before the gate-close event, whose length must be large enough to prevent any encroaching. In the standard itself, the term "guard band" is used only in an informative appendix [1, Appendix Q], and without any exact definition.

The terms portTransmitRate, OperCycleTime, GateOpenTime operIdleSlope, are from the standard, the terms GateCloseTime, PreCloseTime and MaxPreCloseTime have been introduced to ease the discussion.

Whereas it is common in real-time community to consider that the index 0 has the highest priority, in the 802.1Q standard, the queue with index 0 has the lowest priority, and so, hp(n) = {n, n + 1, . . . ,

6}.

ACKNOWLEDGMENTS

We would like to thank Luxi Zhao from Technical University of Denmark for the valuables comments on a preliminary version of this paper.