Christian Chatillon
email: christian.chatillon0979@orange.fr

F Teyssandier
email: teyssandier@univ-perp.fr

Thermodynamic assessment of the different steps observed during SiC oxidation

Keywords: oxidation, SiC, high temperature, active oxidation, passive oxidation

Oxidation studies of SiC are reviewed and analyzed in terms of thermodynamics by crossing the high temperature behavior of the SiC material -mainly the activities of its components during the vaporization processes -with the gas flows supplied to its surface. Additional information is gathered from the abundant literature dealing with surface characterizations using specific instruments in order to elucidate the main chemical surface and interface mechanisms and the associated reactions involved in active and passive oxidation as well as for active-passive or passive-active transitions. Experimental data -erosion rate, growth rate of silica layers and active-passive transition -are discussed in relation with the simulation of equilibrium processes based on thermodynamics.

Introduction

The SiC compound is manufactured by several processes in order to obtain different qualities of materials, which correspond to the physical properties desired in service. It has been mainly used as a refractory compound, in particular in processes carried out under atmospheric conditions, while more recent applications are devoted to high power electronic devices.

Refractory ceramics are mainly made by powder technology, while silicon carbide used in electronic devices is made by a combination of different processes such as chemical vapor deposition (CVD) or crystal pulling.

All of these processes are performed at high temperature: from 1100 K for CVD, over 2300 K for Acheson manufacturing and even up to 3000 K for crystal pulling. The presence of oxygen as a pollutant or possibly added voluntarily can inhibit the process, although in some cases it allows it to be carried out more efficiently as has been observed in the crystal pulling. This is why the influence of oxygen, which depends on the different oxidation modes, is critical. These processes have been studied at length by different research groups. The present study is a review enriched by a complete thermodynamic justification of the different oxidation behaviors of the SiC compound.

Before any thermodynamic analysis, the main characteristics related to the behavior of SiC at high temperature are summarized as follows:

• When heating SiC at high temperature, either under vacuum or in a non-reactive atmosphere, it vaporizes and incongruently forms a vapor phase rich in silicon. A direct consequence is the formation (or precipitation) of carbon on the surface of SiC crystals during heating or its presence in commercial silicon carbide as a residual pollutant.

• The oxidation behavior of SiC under an oxidizing atmosphere, which is also observed for many other carbides, reveals two main oxidation regimes: an active oxidation regime and a passive oxidation regime. The active oxidation regime is observed for a low flow rate of incident oxygen, that is to say for low partial pressures of oxygen. The main reaction occurring under these conditions at the surface of SiC can be summarized by the following main balance equation: SiC (s) + O2 (g) = SiO (g) + CO (g) (1a) Due to the balance of gas flows during this regime, no accumulation of carbon or oxygen is observed on the surface of the silicium carbide, which remains free of silica layer. During the active oxidation regime, the SiC surface is assumed to remain free of any other compound. An increase in the incident oxygen flow rate promotes the formation of a silica layer according to the main reaction, SiC(s) + 3/2 O2(g) = SiO2(s) + CO(g) (1b).

• The transition between active and passive oxidation regimes (A-to-P) has been studied a lot because the two oxidation regimes lead to very different corrosion rates. Additionally, the mechanisms that occur during the reverse, passive to active (P-to-A) transition have a critical influence on high temperature applications involving SiC.

The present work was undertaken to establish the state of our thermodynamic knowledge applied to the oxidation behavior of SiC as well as to identify the kinetic limitations.

Experimental and theoretical background

Information on the main studies concerning the oxidation or corrosion of SiC at high temperature -mainly by oxygen but also including some work under other atmospheres -that are quoted in this article are presented in Table 1. Various experimental methods have been used such as:

• Coupled methods: for example, oxidation step in a circulating gas reactor with postmeasurement of mass gain or thickness of the oxide layer.

• Microbalance under vacuum or Thermo-Gravimetric Analysis (TGA) under different atmospheres in order to follow the loss or gain of weight and thus reveal the oxidation regime.

• In situ measurements of the thickness of the oxide layer during growth by Ellipsometry and / or Interferometry.

• Profilometry applied after various spectrometric measurements such as Auger electron spectroscopy (AES), X-ray photoelectron spectroscopy (XPS), scanning electron spectroscopy (SEM), electron probe microanalysis (EPMA)….

• Experimental techniques used to characterize gaseous species: mass spectrometry with Knudsen cells (KCMS), gas analysis (quadrupole spectrometer), or emission / adsorption spectroscopy. The identification of these species can also be coupled with the characterization of the solid phases by different methods.

Table 1 includes articles relating to the chemical interactions between SiC and SiO2 since these interactions produce gaseous species SiO(g) and CO(g), whose partial pressures are controlled by the same thermodynamic equilibria as those involved in the oxidation of SiC by SiO2 through the gas phase. This table 1 is completed by theoretical studies, which are mainly the modeling or simulation of the erosion rate of SiC or the growth rate of the silica layer, as well as various compilations of data related to oxidation. Samples were heated via conventional resistance heated furnaces, laser heating, solar furnace heating, resistance heating of the sample itself, controlled fuel burner or arc-jet as quoted in Table 1.

Table 1. Summary of literature data concerning the oxidation of SiC, the high temperature behavior of SiC-SiO2 mixtures and the thermodynamic modeling of oxidation.

Authors / Date and references

Sample nature

Experimental technique T range / K

Main results and observations

Erwin (1958) [START_REF] Ervin | Oxidation Behavior of Silicon Carbide[END_REF] Green Turkdogan [START_REF] Turkdogan | Enhancement of Diffusion-Limited Rates of Vaporization of Metals[END_REF]. These authors propose different steps depending on the increase in the partial pressure of oxygen starting from a gaseous flow of neutral gas: -(i) increase in the vaporization rate for a low incident oxygen flow rate corresponding to the oxidation that is currently called active, -(ii) then a transition characterized by a sudden decrease in the vaporization rate, today qualified as active-passive transition, -(iii) growth of an oxide layer at a parabolic rate and therefore assumed to be limited by the diffusion of species in the thickness of the oxide layer formed, -(iv) sometimes formation of gas-phase nucleation, which can generate particles in front of the surface under study due to supersaturation, -(v) and vice versa when the incident oxygen flow is reduced, a transition observed or predicted for much lower oxygen pressures. In this article, all these phenomena are reviewed and discussed on the basis of thermodynamic arguments with the exception of gas-phase nucleation, which is not directly related to the formation of the oxide layer. The present study mainly addresses the role of thermodynamics applied to the understanding of processes involved in oxidation or erosion of SiC at high temperature and its ability to represent experimental observations.

3. Approach to the oxidation of SiC by coupling thermodynamics and mass transfer

Phase diagram and equilibrium vaporization

Oxidation of SiC occurs during high temperature service and its simultaneous vaporization is of prime importance. In the Si-C binary system, SiC is the only defined compound identified at equilibrium. Although identified as a defined compound, SiC exhibits an extremely narrow compositional domain, which can be written SixC(1-x). Within this compositional domain the activities of Si and C vary from the two-phase equilibrium Si-SiC to the C-SiC one.

Vaporization of SiC is not congruent, neither under neutral gas nor under vacuum. Equilibrium calculations performed by Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. I. SiC and SiC-SiO2 mixtures under neutral conditions[END_REF] to evaluate the evaporation rates of SiC as a function of temperature show a large excess of Si loss compared to C, and the subsequent precipitation of C. These authors have shown that carbon precipitates very quickly on the surface of a SiC virgin as soon as temperature rises: depending on the experimental ultra-high vacuum conditions, this carbon at the surface of SiC as observed by AES begins to be detected in the 900 to 1100K range. Numerous experiments also confirm the rapid graphitization of the SiC surface as quoted by Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. I. SiC and SiC-SiO2 mixtures under neutral conditions[END_REF].

If we consider the oxidation of SiC, it is the ternary Si-C-O phase diagram that must be looked at. Apart from SiC and SiO2, no other condensed stable phase compound is known in this phase diagram, especially in the pseudo-binary SiC-SiO2 section, which is the relevant one in the oxidation process when a silica layer forms at the SiC surface. As a result, the SiC-SiO2 interface must be considered as thermodynamically stable and therefore these two compounds can co-exist.

It should be noted that an abundant literature is devoted to SixCyOz amorphous materials usually called Blackglasses. However, and although these compounds resist to oxidation up to high temperatures, they are exclusively obtained from polymeric ceramic precursors and should be considered as metastable compounds. This is why they will not be taken into account in this article. When SiC is heated under a neutral atmosphere or under vacuum, any composition of the pseudo-binary section first vaporizes according to a congruent behavior as explained by Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. I. SiC and SiC-SiO2 mixtures under neutral conditions[END_REF]. The corresponding main equilibrium equation is as follows (equation ( 2)):

SiC + 2 SiO2 (s) = 3 SiO (g) + CO (g) (2)
Above ≈ 2000 K, thermodynamics predicts the incongruent vaporization of any composition belonging to the pseudo-binary section, which produces liquid Si. This behavior is the starting point for the production of Si in metallurgical arc furnaces. In metallurgical plants, the starting temperature is higher, around 2200 to 2300K, probably due to the hindered (or retarded) vaporization of silica as explained by Chatillon and Nuta [START_REF] Chatillon | Mass Spectrometric Studies of Non-Equilibrium Vaporizations in Knudsen Cells: A Way to the Determination of the Nature of Kinetic Processes[END_REF]. Authors who have studied the interaction of oxygen with SiC have very often tried to explain their experimental results, either with the above-mentioned reference to the congruent vaporization equation ( 2), or from each of the three-phase domains in equilibrium with SiC, which produce invariant oxygen pressures:

SiC (s) + O2 (g) = SiO2 (s) + C (s) (3)
2 SiC + 2 O2 (g) = SiO2 (s) + Si (s or l) + 2 CO (g) ( 4)

Note that the oxygen potential of equation ( 4) is the same as for the two-phase domain:

SiO2 (s) = Si (s, or l) + O2 (g) (5).

Non-equilibrium vaporization

Considering the out-of-equilibrium vaporization conditions -that is, when the evaporation flow is not completely returned to the surface -SiC or SiO2 surfaces exhibit hindered vaporization (also called "retarded "). Under such conditions, the evaporation rate becomes lower than the equilibrium rate, which means that the vaporization is kinetically limited: in the case of SiO2, the difference between the two values can reach a factor of 10 -2 to 10 -3 as compiled by Chatillon

and Nuta [START_REF] Chatillon | Mass Spectrometric Studies of Non-Equilibrium Vaporizations in Knudsen Cells: A Way to the Determination of the Nature of Kinetic Processes[END_REF] and 5.10 -2 for the two-phase SiC-C binary domain as measured by Rocabois et al.

[79] and confirmed by Pisch et al. [START_REF] Pisch | Evaporation behavior of SiC powder for single crystal growth-an experimental study on thermodynamics and kinetics[END_REF]. The influence of these evaporation coefficients on the vaporization of SiC-SiO2 mixtures, which are also important in explaining the behavior of the SiC / SiO2 interface, was estimated by Jacobson et al. [START_REF] Jacobson | Reactions of Silicon Carbide and Silicon (IV) Oxide at Elevated Temperatures[END_REF] and Honstein et al. [START_REF] Honstein | High temperature mass spectrometric study of the vaporization behaviour of SiC-SiO2 system[END_REF][START_REF] Honstein | Mass spectrometric and thermodynamic analyses of (SiC-SiO2) powders vaporization behavior[END_REF] using mass spectrometry coupled to Knudsen cells. These studies show that -(i) very low evaporation coefficients (around 10 -4 ) imply a drastic reduction in the partial pressures of vaporized gaseous species (more than a hundred in Knudsen cells which have small effusion rates compared to evaporation of the sample), -(ii) the ratio between SiO (g) and CO(g), which are by far the two main gaseous species observed by mass spectrometry, is within the 1-2 range and therefore significantly different from the congruent value 3 (equation ( 2)) or 3.76 which is the vacuum value deduced from the Knudsen flows. This result will be used to explain the mechanisms observed during the oxidation of SiC later in this article. Note that the evaporation coefficients are the same as the condensation coefficients when it comes to the same molecule or the same atom rising to the surface (principle of micro-reversibility in atomic physics). The evaporation coefficients as determined by free vaporization or Knudsen method measurements (performed under vacuum) are strictly related to the last step in the vaporization process occurring at the surface of the sample and are not representative of any global kinetic process in a reactor.

Conventional thermodynamic calculations to explain oxidation behavior

Heuer and Lou [START_REF] Heuer | Volatility Diagrams for Silica, Silicon Nitride, and Silicon Carbide and Their Application to High-Temperature Decomposition and Oxidation[END_REF] build a "volatility diagram" (also called Kellogg diagram [START_REF] Kellogg | Vaporization Chemistry in Extraction Metallurgy[END_REF]) in which the stability domains of the condensed phases in the ternary Si-C-O diagram (Si, SiC, C and SiO2) are presented according to two partial pressures used as axis: -(i) on the abscissa the partial pressure of oxygen (or its activity) in equilibrium with the various condensed phases present in the system (and not the gas phase pressure which is introduced into the experiment), -(ii) on the ordinate other partial pressures such as CO(g) or SiO(g). The transition from A to P can initially be defined in the Si-O system as the critical pressure of oxygen at the surface of the two-phase Si-SiO2 domain, which corresponds to the appearance of a silica layer (equation ( 5)). This pressure is very low and corresponds to the highest partial pressure of SiO(g). Any increase in the partial pressure of oxygen at the surface of Si above this critical value induces the formation of SiO2 (s). Since the equilibrium oxygen partial pressure at the surface of SiC is very low, the equilibrium of the atomic O flux at the transition can be given by the relation p(O2, applied) = ½ p (SiO, g, product), which corresponds to equation ( 5). This approach, first developed for pure Si, was further extended to SiC by Heuer and Lou [START_REF] Heuer | Volatility Diagrams for Silica, Silicon Nitride, and Silicon Carbide and Their Application to High-Temperature Decomposition and Oxidation[END_REF] (see figure 17 of their article). However, this article does not provide any interpretation for the production of CO(g), which must be added to SiO(g) in the vaporization balance at the surface of SiC. Indeed, in this case it is equation (1a), which is to be considered. Note that the approach of Heuer and Lou [START_REF] Heuer | Volatility Diagrams for Silica, Silicon Nitride, and Silicon Carbide and Their Application to High-Temperature Decomposition and Oxidation[END_REF] is applied to partial pressures and not to flows of species to and from the surface of SiC, while the latter are the key parameters to interpret the oxidation process.

A recent and more conventional approach to the oxidation of SiC is the use of a thermodynamic calculation software to minimize the Gibbs free energy of a known mixture at given temperature and total pressure or volume. For example, Narushima et al. [START_REF] Narushima | High-Temperature Active Oxidation of Chemically-Deposited Silicon Carbide in an Ar-O2 Atmosphere[END_REF] calculated the partial equilibrium pressures of gas species above SiC when increasing the amount of oxygen added to a given amount of Ar and SiC: for a small amount of oxygen, their calculations show its total consumption to form SiO (g) + CO (g). SiO2 is formed when the amount of oxygen reaches a threshold value. The same type of calculation was performed by Eck et al. [START_REF] Eck | Behavior of SiC at high temperature under helium with low oxygen partial pressure[END_REF] but for different total applied pressures, and the same observation was made. Gibbs free energy minimization has also been used by Wang et al. [START_REF] Wang | Theoretical Investigation for the Active-to -passive Transition in the Oxidation of Silicon Carbide[END_REF] to compare the influence of different gas mixtures (Ar + O2) on SiC under a total pressure of 8.10 -3 bar, in the temperature range from 600 to 2200K.

They identified the different areas of phase formation as a function of the partial pressures of the gaseous species. They show that in the absence of a silica layer (active oxidation) the partial pressures of CO(g) and SiO(g) are in a ratio given by equation (1a). These calculations being based on the global thermodynamic equilibrium at constant applied pressure, they are able to show the major trends, but are obviously not able to identify the basic reaction mechanisms, which control kinetics.

Nickel et al. [START_REF] Nickel | High Temperature Corrosion of Sic in Hydrogen-Oxygen Environments[END_REF] considered a stable SiO(liq) phase instead of SiO2 (solid or liquid) in equilibrium with a two-phase mixture: either SiC-C or Si-SiC. He varied the activity of C (or correlatively of Si via the formation of SiC) in order to obtain a pressure of CO(g) close to 1 bar, which is the total pressure applied to the system. Under these conditions, an oxide should form but is assumed to be eliminated by spallation caused by the CO bubbles generated at the SiC / SiO2 interface. Continuous repetition of this process generates oxidation equivalent to active oxidation. This process is named by the author as limiting secondary active / passive transition. Under these conditions, Nickel et al. [START_REF] Nickel | High Temperature Corrosion of Sic in Hydrogen-Oxygen Environments[END_REF] assume that the transition occurs (for p (CO) <1 bar) according to the conditions of the three-phase equilibrium SiC-SiO(l) -Si(l). Two arguments allow this hypothesis to be rejected: -(i) the SiO phase (s, or l) is not a stable phase as shown by Rocabois et al. [START_REF] Rocabois | Vapour pressure and evaporation coefficients of SiO(Amorphous) and SiO2(s)+Si(s) mixtures by the multiple Knudsen cell mass spectrometric method[END_REF] thanks to mass spectrometry measurements, which also explains the low pressures measured above SiO2 or Si-SiO2 due to low evaporation coefficients, -(ii) the equilibrium proposed between the three phases SiC-SiO (l) -Si (l) (invariant equilibrium) does not satisfy the equality of the pressures (or flow rates) of SiO(g) and CO(g)

for the active oxidation according to equation ( 1) as indicated later in the text.

Thermodynamic analysis of Song and Smith [START_REF] Song | Effects of Low-Pressure Oxidation on the Surface Composition of Single Crystal Silicon Carbide[END_REF] takes into account two main reactions to determine the active / passive transition:

SiC(s) + O2 (g) = SiO2 (s) + C(s) (equation (3))
and the congruent reaction

SiC(s) +2 SiO2(s) = 3 SiO(g) + CO(g) (equation (2))
Each of these reactions limits two different domains, one of them being dominated by equation (1a), i.e. the active oxidation reaction. The A-to-P transition is assumed to be close to the congruent vaporization of the pseudo-binary SiC-SiO2 (equation ( 2)).

Thermodynamic modeling approach of the active oxidation and active-to-passive transition

A purely thermodynamic approach is based on the chemical potentials of the species and therefore on the partial pressures of the gaseous species. The steady-state oxygen partial pressure at the surface of SiC (active oxidation) or at the SiC / SiO2 interface (passive oxidation) has, so far, not been directly measured. One way to identify this value is to perform thermodynamic calculations to relate the oxygen flow imposed, and therefore known, at the input of the experimental device to the effective partial pressure of oxygen at the surface of SiC or to the activity of oxygen at the SiC / SiO2 interface. These calculations involve the gaseous species formed on the surface of SiC, that is to say mainly the two species SiO(g) and CO(g).

Under 1 bar total pressure, the basic relations are written according to a theory, which was first developed by Wagner [START_REF] Wagner | Passivity during the Oxidation of Silicon at Elevated Temperatures[END_REF] for the oxidation of silicon taking into account the concentration boundary layer between the Si surface and atmosphere when a viscous laminar flow is established. For this purpose, both the diffusion coefficients of the gas phase species and the thickness of the diffusion layer are required. The first complete solution was proposed by Balat et al. [START_REF] Balat | Active to Passive Transition in the Oxidation of Silicon Carbide at high Temperature and Low pressure in Molecular and Atomic Oxygen[END_REF] and later by Eck et al. [START_REF] Eck | Behavior of SiC at high temperature under helium with low oxygen partial pressure[END_REF] taking into account reaction (1a) for the active oxidation associated with the mass balance of all the atoms at the SiC / gas interface, that is to say with the stoichiometric vaporization of SiC without accumulation of C or Si on the surface. The Ato-P transition is introduced in the model by the formation of silica, that is to say by use of reaction [START_REF] Rosner | High Temperature Kinetics of the Oxidation and Nitridation of Pyrolytic Silicon Carbide in Dissociated Gases[END_REF]. The value of the partial pressure of oxygen entering the device as identified by Balat et al. [START_REF] Balat | Active to Passive Transition in the Oxidation of Silicon Carbide at high Temperature and Low pressure in Molecular and Atomic Oxygen[END_REF] for the A-to-P transition is only affected by the uncertainty on the diffusion coefficients since in their model the thickness of the diffusion layer is eliminated. Partial pressures of SiO(g) and CO(g) in the gas phase as well as of O2(g) at the surface of SiC are assumed to be negligible in the flow balance (zero value) and, therefore, the steady-state oxygen potential at the SiC surface is not known. Similarly, Jacobson et al. [START_REF] Jacobson | Oxidation Transitions for SiC Part I. Active-to-Passive Transitions[END_REF] have calculated the Ato-P transition using the diffusion boundary layer associated with different interface reactions such as the congruent vaporization equation (equation (1a)), or the equations relating to the two three-phase mixtures (equations ( 3) and ( 4)) in order to compare it with their experimental data obtained with Si-SiC, SiC and SiC-C samples.

More recently, Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF] performed thermodynamic calculations using the Knudsen flux balance at the SiC surface, i.e. for the molecular regime (pressure in the device ≤ 0.1 mbar).

In this regime, there is no diffusion boundary layer, which means that the molecular fluxes of evaporation and condensation are the highest possible. The system is in equilibrium, that is to say that the evaporation and condensation coefficients of the gaseous species are equal to 1. In these calculations, -unlike the previous case including a diffusion layer -the partial pressures in steady-state of all gaseous species are calculated at the SiC surface when the incident oxygen flux is taken into account through the mass balance of each component. In this type of calculation, the partial pressure of oxygen at the surface of SiC is not removed and for active oxidation and the A-to-P transition these surface oxygen potentials are calculated based on a partial pressure of oxygen given at the input (or at wall for vacuum systems) of the experimental set-up using Knudsen's relation. Partial pressures of all the many gaseous species on the surface of SiC are calculated and taken into account in the mass balance of the components. These calculations confirm that the two predominant species are by far SiO(g) and CO(g) and that the partial pressure of oxygen calculated at the surface of SiC in steady-state conditions is very low.

The Si(g) species is the third species contributing to the vaporization of SiC.

Calculations by Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF] define the two windows of thermodynamic activity of oxygen and silicon corresponding to oxidation in its active regime. At constant temperature and under a very low flow of incoming oxygen, C precipitates at the surface of SiC due to its incongruent vaporization, because the flow of oxygen reaching the surface of SiC does not convert all of the C formed into CO(g). The activity of C in SiC is equal to 1, while the activity of silicon reaches its lowest value leading to minimal erosion of SiC. As the incident oxygen flux increases, the activity of Si in SiC increases rapidly due to its very narrow composition range. The A-to-P transition occurs as soon as the silica layer begins to form according to equation [START_REF] Rosner | High Temperature Kinetics of the Oxidation and Nitridation of Pyrolytic Silicon Carbide in Dissociated Gases[END_REF]. By increasing the flow of oxygen, the activity of silicon varies from its minimum value in the two-phase equilibrium SiC-C (precipitation of C) to a determined composition SixC(1-x) rich in Si when the layer of silica is formed. This situation as calculated is quite close to the congruent vaporization of the pseudo-binary SiC-SiO2 system (equation ( 2)). However, this Si-rich composition of SixC(1-x) at the transition never reaches the Si-SiC phase limit (aSi always remains below 1 at silica formation). Thus, Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF] delimit the field of activity values of silicon for active oxidation in SixC(1-x) by a window of activity of silicon for SiC in relation to the window of oxygen pressure on the surface of SiC.

3.5. Review of passive oxidation using thermodynamic analysis.

Luthra [START_REF] Luthra | Some New Perspectives on Oxidation of Silicon Carbide and Silicon Nitride[END_REF] carried out thermodynamic calculations to identify possible limiting steps encountered during the passive oxidation of SiC in the parabolic regime, i.e. when the flows of oxygen or other oxidizing species are limited by their diffusion through the SiO2 layer at the SiC surface. Parabolic regime is written as a function of partial pressures of oxygen at the SiC / SiO2 interface and at the SiO2 / gas interface. Two types of SiC / SiO2 interfacial reactions can occur:

SiC(s) + O2(g) = SiO2(s) + C(s) (3)
with one mole of oxygen consumed per mole of SiC and carbon precipitation, which corresponds to the three-phase SiC-SiO2-C equilibrium and, SiC(s) +3/2 O2(g) = SiO2(s) + CO(g) [START_REF] Antill | Active-to-passive transition in the oxidation of SiC[END_REF] with 1.5 moles of oxygen consumed per mole of SiC. In this latter equation the excess carbon is removed as CO(g). Limiting steps identified by Luthra [START_REF] Luthra | Some New Perspectives on Oxidation of Silicon Carbide and Silicon Nitride[END_REF] are: -(i) diffusion of oxygen (transport of O 2-on the SiO2 network) or permeation of O2 through the silica layer towards the SiC / SiO2 interface, -(ii) interfacial reaction at the SiC / SiO2 interface, or -(iii) diffusion or permeation of the reaction products, that is to say C(s) or CO(g) from the SiC / SiO2 interface through the silica layer. Luthra opposes three arguments to the hypothesis of a limitation by the diffusion / permeation of oxygen: the observation of CO bubbles, the fact that the rate of permeation of CO is not greater than that of oxygen, and the fact that the oxidation rate is lower on SiC than on pure silicon. A limitation by the permeation of CO(g) escaping from the SiC / SiO2 interface through the silica layer would be associated with a precipitation of C at the SiC / SiO2 interface and, depending on temperature, with the formation of CO bubbles. It should also be noted that a carbon layer may already be present under the SiO2 layer when it is formed before oxidation: either because the sample of SiC chosen was manufactured under experimental conditions corresponding to the two-phase SiC-C domain, or due to incongruent vaporization of SiC during the temperature ramp before the oxidation step. Growth of the SiO2 layer limited by CO diffusion should follow the same law as for the limiting step by the oxygen diffusion, an assumption which is invalidated by the growth observed on different faces of a single crystal of SiC. By comparing different experimental oxidation regimes, Luthra [START_REF] Luthra | Some New Perspectives on Oxidation of Silicon Carbide and Silicon Nitride[END_REF] proposes a mechanism controlled by mixed interface reaction / diffusion for SiO2 growth, with outward CO diffusion controlling the rate instead of inward oxygen diffusion. He also suggests the formation of a SiCxOy compound at the SiC / SiO2 interface by analogy with silicon oxynitride. Finally Jacobson [START_REF] Jacobson | Corrosion of Silicon-Based Ceramics in Combustion Environments[END_REF] asserts that the absence of a gradient of carbonaceous species in the thickness of the silica layer as well as the absence of proof of precipitation of C at the SiC / SiO2 interface are the proofs of a mixed interface / diffusion control reaction.

Active oxidation and erosion rates

At low incident oxygen mass flow, oxygen is completely consumed to form gaseous species the predominant species being SiO(g) and CO(g) in the ratio =1. The SiC surface remains free of silica layer as observed post-mortem (but rarely in situ) using various characterization tools.

Ogura and Morimoto [START_REF] Ogura | Mass Spectrometric Study of Oxidation of SiC in Low-Pressure Oxygen[END_REF] attempted to observe these gaseous species in this oxidation regime by sampling the gas phase using a supersonic molecular beam device (sampling at atmospheric pressure) turned towards a quadrupole mass spectrometer under high vacuum. They observed only CO(g) from the multiple reflections (as written by the authors) on the skimmer and the surrounding walls of the device, while SiO(g) quickly condensed on these walls before reaching the mass spectrometer because of these multiple reflections. So the supersonic sampling device did not work properly. Under low total pressures (≤ 10 -4 bar), Honstein et al. [START_REF] Honstein | Vaporization of Mixed SiC Powders. Partial Pressures and Grain Morphology Changes under Vacuum Conditions[END_REF] (p. 3854 and fig. 3 of their paper) recently confirmed by mass spectrometry measurements on Knudsen cells the main contribution of reaction (1a). To do this, they measured the flow of species at the very start of the active oxidation of a SiC surface at the instant corresponding to the disappearance of the native silica layer during the de-oxidation step of various SiC powders.

In previous studies, the slope of weight loss plotted against time as determined from thermogravimetric analysis (TGA) was found to be 1 according to Goto et al. [START_REF] Goto | Effect of oxygen partial pressure on the high-temperature oxidation of CVD SiC[END_REF], which corresponds to the active regime under p(O2) = 0.1 MPa, while this value becomes ½ in the passive regime (parabolic law).

Modeling of the active oxidation of SiC in steady-state at low or atmospheric total pressure can be easily performed because reaction (1a) is completely shifted towards the formation of products as evidenced by the very low partial pressure of oxygen at the surface of SiC in active oxidation steady-state. As a result, the whole incident oxygen stream is converted into SiO(g) + CO(g) species. The equilibrium (maximum) vaporization flow rates are thus calculated from literature data (figures or tables) as a function of the partial pressure of incoming oxygen at the inlet of the device and compared with the experimental values in figure 1. Four remarks can be made on these calculations: -(i) the erosion rate does not depend on the value of the equilibrium constant that is a function of temperature since reaction (1a) is totally shifted to the right side, -(ii) the erosion rate nevertheless depends on the partial pressure of the input oxygen pressure, which generally is imposed by a flow rate regulated at the inlet of the experimental device, -(iii) the influence of temperature and flowrate remains significant but difficult to evaluate, -(

the partial pressures of the two predominant gaseous species CO(g) and SiO(g) are generally much higher than that of all the other species such as CO2(g), Si(g), Si2(g), Si2C(g)…This is why we prefer to plot the erosion rate as a function of the only partial pressure of oxygen imposed at room temperature at the inlet of the device in order to relate in figure 1 the different erosion regimes reported in or from literature under atmospheric pressure or vacuum. However, in very high temperature ranges and for low oxygen partial pressures, some other gas species can competitively evaporate, as shown by Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF]. Furthermore, under constant oxygen pressure, the composition of the SixC(1-x) sample tends to move towards its SiC-C limit when temperature increases. According to equation (1a), which is the equilibrium equation for active oxidation, and with decreasing silicon activity, the erosion rate of SiC decreases as shown in figure 1 by experimental data of Jacobson et al. [START_REF] Jacobson | Oxidation Transitions for SiC Part I. Active-to-Passive Transitions[END_REF] carried out with a sample of SiC-C. The experimental erosion rate of pure SiC corresponds to that calculated with a diffusion boundary layer, which is in fact the thermodynamic (equilibrium) erosion rate. For the Si-SiC sample -a limiting SiC two phase composition, which cannot be obtained under active oxidation as proven by Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF] -the erosion rate observed by Jacobson et al. [START_REF] Jacobson | Oxidation Transitions for SiC Part I. Active-to-Passive Transitions[END_REF] is higher than for the pure SiC due to the oxidation contribution of Si producing an excess of SiO(g). From A-to-P experiments Jacobson et al. [START_REF] Jacobson | Oxidation Transitions for SiC Part I. Active-to-Passive Transitions[END_REF] observed post-mortem some "silica rods" formed at the surface of SiC due to excess SiO(g) flow.

Plotting the rate of erosion as a function of the reciprocal of temperature should be relevant but has never been done correctly because the composition of the SixC(1-x) sample moves in its nonstoichiometric range towards a composition at the limit of the two-phase SiC-C domain. In addition to the preponderant contribution of the gaseous species CO(g) and SiO(g), the contribution of all other gaseous species was again also taken into account in the calculations of Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF]. We first observe in Figure 1 that the two allotropic forms α-SiC and β-SiC (respectively the black and red marks) do not show significant differences in erosion rate for the same incident oxygen flow, despite the difference of 7.0 kJ in their standard enthalpy of formation (ref.

JANAF [START_REF] Chase | NIST-JANAF Thermochemical Tables[END_REF] (β → α allotropic variety). This is explained by the total shift of reaction (1a) towards the formation of gaseous species with consumption of all the incident oxygen flow.

The authors of the experimental data verified in different ways that the experimental erosion rates presented in Figure 1 correspond to a true active oxidation regime. The vertically aligned experimental data correspond to the same partial pressure of oxygen and were obtained for different temperatures or for different flow regimes and/or rates. Several authors mention that the erosion rate increases sharply with temperature: Dawi et al. [START_REF] Dawi | High Temperature oxidation of SiC under helium with low-pressure oxygen. Part 3: beta-SiC-SiC/PyC/SiC[END_REF] between 1650 and 2250 K, Gulbransen et al. [START_REF] Gulbransen | The Oxidation of Silicon Carbide at 1150° to 1400°C and at 9x10 -3 to 5x10 -1 Torr Oxygen Pressure[END_REF] between 1523 and 1623 K, Charpentier et al. [START_REF] Charpentier | High temperature oxidation of SiC under helium with low-pressure oxygen. Part 2: CVD bete-SiC[END_REF] between 1550 and 2050 K, Hald [START_REF] Hald | Operational limits for reusable space transportation systems due to physical boundaries of C/SiC materials[END_REF] between 2250 and 2380 K.

The temperature dependence of the oxidation rate can be assessed:

• for devices under atmospheric total pressure, from the variation of the diffusion coefficients in the concentration boundary layer: according to Jacobson and Myers [START_REF] Jacobson | Active Oxidation of SiC[END_REF] between 1573 and 1773K, the variations of the diffusion coefficients increase the erosion flux by +1, 7%. This increase is not sufficient to explain the variations observed. It is probable that the variations in flow regime in the reactor should also modify the thickness of the diffusion boundary layer and contribute to the measured values.

• for devices under low total pressure from the Hertz-Knudsen flow rate equilibrium ratio: it increases by + 17% between 1500 and 2200K. Trends observed for some experiments are larger than those provided by the only temperature increase.

In conclusion, the observed increase in erosion cannot come solely from variations in the flow regime with temperature. For high pressures, there is probably also a large variation in the thickness of the boundary layer. In fact, the notion of concentration (or diffusion) boundary layer is rather crude and consists in assimilating the momentum boundary layer to the concentration boundary layer of all gaseous species. However, the momentum boundary layer does not correctly represent the concentration boundary layer because the definition of the concentration boundary layer means that its thickness may be different for different species.

The whole flow regime in the reactor can also impose a decrease of oxygen partial pressure at the frontier of the boundary layer for diffusion.

Dawi et al. [START_REF] Dawi | High Temperature oxidation of SiC under helium with low-pressure oxygen. Part 3: beta-SiC-SiC/PyC/SiC[END_REF] observed the erosion of a CVD coated β-SiC layer on a SiC / SiC based composite with a pyrocarbon interphase. Three successive heat treatments of 10 min were applied at three different temperatures: two temperatures close to the A-to-P transition and a third at a higher temperature. It is observed that the erosion rate increases due to the state of the sample reached after each successive heat treatment. By analogy with the variations observed by Rosner and Allendorf [START_REF] Rosner | High Temperature Kinetics of the Oxidation and Nitridation of Pyrolytic Silicon Carbide in Dissociated Gases[END_REF] as presented in Figure 2, a kind of plateau is reached as soon as the transition A-to-P is exceeded. For temperatures above about 2000-2100K, -(i) erosion probably reveals the underlying composite structure, which is more actively oxidized than the SiC coating (Figure 2A), or -(ii) in the case of pure SiC, its vaporization must also be taken into account (increase observed in Figure 2B) while the composition of SixC(1-x) tends towards that of the two-phase SiC-C domain.

In figure 1, we observe that by varying the parameters (temperature, flow rate…) some authors measured the same erosion rate for different oxygen partial pressures. It should also be noted that these values never reach the thermodynamic limit calculated under low pressure. Such behavior must be attributed to the kinetic limitations in reaction (1a) (elementary stages in the adsorption and desorption processes). Such a kinetic limitation is confirmed by Berton et al.

[23] who observe a sharp increase in the rate of erosion when using the same atomic flow of incident oxygen but in the form of atomic oxygen produced by microwave radiation applied to the same incident flow of regulated molecular oxygen.

The two erosion lines calculated by Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF] define the active erosion domain at equilibrium under vacuum -that is to say in molecular flow regime (total pressure <10 -4 bar)between the SiC-C phase limit and the non-stoichiometric Si richer composition of SixC(1-x) at the A-to-P transition. Experimental results obtained under a low total pressure never reach thermodynamic equilibrium, which is the maximum erosion flux, thus suggesting a kind of kinetic limitation for reaction (1a). Conversely, the experimental results obtained under 1 bar total pressure (neutral gas mixed with oxygen) are generally close to current calculations based on diffusion in the boundary layer. For intermediate pressures corresponding to a flow regime between molecular and viscous, the experimental results agree with the expected trend between the two sets of theoretical values represented by the lines in Figure 1. Another interesting feature is that the rate of erosion under 1 bar total pressure increases when the oxygen concentration decreases, and the rate of erosion tends to reach the maximum value calculated under vacuum:

the system behaves as if the boundary layer vanished at lower oxygen partial pressures in favor of another exchange mechanism.

Recent measurements obtained using C / C or C / SiC composites -as is or covered with a CVD layer of β-SiC (gray marks in Figure 1) -show erosion values in the same range as pure SiC,

i.e. between the calculated values of high pressure and those under vacuum.

In conclusion, the residual difference between the maximum steady-state erosion rate (Hertz-Knudsen) as calculated by Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF] under the assumption of thermodynamic equilibrium and the determinations under reduced pressures in molecular regime can be explained by an activation barrier when O2(g) adsorbs to the SiC surface. O2 (g) is probably adsorbed as O species (ads) -with a probability <1 -on different atomic sites before reacting in the surface layer to produce the adsorbed species SiOads and COads precursors of the gaseous SiO(g) and CO(g) species. In Figure 2B, the erosion ratio (evaporated molar flow rate of SiC relative to oxygen taken for incident atomic flow rate) shows a plateau for the ratio = 0.17 as Experimental data at the A-P oxidation transition of β-SiC (red marks) are scattered but a certain agreement is observable between the values of Hinze and Graham [START_REF] Hinze | The Active Oxidation of Si and SiC in the Viscous Gas-Flow Regime[END_REF], Narushima et al.. [START_REF] Narushima | High-Temperature Active Oxidation of Chemically-Deposited Silicon Carbide in an Ar-O2 Atmosphere[END_REF],

Balat [START_REF] Balat | Determination of the Active-to-Passive Transition in the Oxidation of Silicon Carbide in Standard and Microwave-Excited Air[END_REF], Charpentier et al.. [START_REF] Charpentier | High temperature oxidation of SiC under helium with low-pressure oxygen -Part 1:: Sintered alpha-SiC[END_REF][START_REF] Charpentier | High temperature oxidation of SiC under helium with low-pressure oxygen. Part 2: CVD bete-SiC[END_REF], Jacobson and Myers [START_REF] Jacobson | Active Oxidation of SiC[END_REF] and Brisebourg et al.. [START_REF] Brisebourg | Oxidation of β-SiC at high temperature in Ar/O2, Ar/CO2, Ar/H2O gas mixtures: Active/passive transition[END_REF].

The experimental data of the oxygen pressures at the A-to-P transition of SiC (blue marks) seem a little higher than for β-SiC. The difference between the partial pressures of oxygen at the transition can be estimated mainly from the formation of the Gibbs free energies of the two allotropic varieties of SiC using reaction (1a):

∆ 𝐺 °= -𝑅𝑇 ln 𝐾 (1) = -𝑅𝑇𝑙𝑛 𝑝 • 𝑝 𝑎 (= 1) • 𝑝
with the active oxidation balance for reaction (1a)

1 2 𝑝 = 𝑝 = 𝑝 (∆ 𝐺 °) -(∆ 𝐺 °) = 7000 𝑗𝑜𝑢𝑙𝑒𝑠 = 𝑅𝑇𝑙𝑛 .
Gibbs free energy difference between the two α and β allotropic varieties of SiC -based on JANAF [START_REF] Chase | NIST-JANAF Thermochemical Tables[END_REF] values -gives 𝑙𝑜𝑔 = 0.56 at 1500K and 0.42 at 2000K. The difference between the input oxygen partial pressures corresponding to the two SiC allotropic varieties are within this range in figure 3.

Although least squares smoothing for the full set of vacuum results is close to that under atmospheric pressure, some of these data approach the limit calculated in vacuum by Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF]. On the contrary, the experimental results under 1 bar never approach the values calculated by Balat et al. [START_REF] Balat | Active to Passive Transition in the Oxidation of Silicon Carbide at high Temperature and Low pressure in Molecular and Atomic Oxygen[END_REF] or Eck et al. [START_REF] Eck | Behavior of SiC at high temperature under helium with low oxygen partial pressure[END_REF] under the assumption of thermodynamic equilibrium, which takes into account the diffusion of species in the boundary layer, with the exception of the latest data from Jacobson et al. [START_REF] Jacobson | Oxidation Transitions for SiC Part I. Active-to-Passive Transitions[END_REF].

Rosner and Allendorf [START_REF] Rosner | High Temperature Kinetics of the Oxidation and Nitridation of Pyrolytic Silicon Carbide in Dissociated Gases[END_REF] studied the oxidation of β-SiC by pure atomic oxygen under a total pressure of 1.3 • 10 -3 bar. For the same temperature, the transition A-to-P occurs at an inlet pressure of O(g) (or atomic flow) lower than that of O2(g). These conclusions assume that the kinetic barrier associated with the adsorption of oxygen in reaction (1a) disappears (or decreases soundly) for atomic oxygen, thus allowing it to adsorb directly to Si-C surface sites for incorporation into the surface adsorbed layer. Balat et al. [START_REF] Balat | Active to Passive Transition in the Oxidation of Silicon Carbide at high Temperature and Low pressure in Molecular and Atomic Oxygen[END_REF] confirms these results results by studying the oxidation of α-SiC or β-SiC under molecular or dissociated O2(g) in a pressure range from 2 • 10 -3 to 21 • 10 -3 bar. More recently, using air or dissociated air flows Balat [START_REF] Balat | Determination of the Active-to-Passive Transition in the Oxidation of Silicon Carbide in Standard and Microwave-Excited Air[END_REF] observes an opposite effect: a higher activation barrier for O(g) than for O2(g). The excess of N2(g) in this case probably has an influence on the discharge (production of N?), which favors the formation of a certain amount of Si3N4 or oxynitride due to the N atoms competing with the formation and growth of silica.

Passive-to-active transition

When performing experiments to study the passive-to-active (P-to-A) transition regime, Rosner

and Allendorf [START_REF] Rosner | High Temperature Kinetics of the Oxidation and Nitridation of Pyrolytic Silicon Carbide in Dissociated Gases[END_REF] measured experimental partial pressures of O2(g) at the P-to-A transition far below -approximately 10 -3 to 10 -4 -the theoretical equilibrium A-to-P value calculated by Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF] in the same low range of total pressure. Indeed, switching from the passive mode to the active mode requires a prior vaporization of the silica layer, theoretically a congruent vaporization of pure silica. Such a pressure difference can also come from the influence of an evaporation coefficient, which delays the mass loss of silica. There may also be some uncertainty in the temperature measurement of the Joule heated filament used by Rosner and Allendorf [START_REF] Rosner | High Temperature Kinetics of the Oxidation and Nitridation of Pyrolytic Silicon Carbide in Dissociated Gases[END_REF] when coated with silica. Schneider et al. [START_REF] Schneider | A theoretical and experimental approach to the active-to-passive transition in the oxidation of silicon carbide[END_REF] have also proposed a variation of the partial pressure of oxygen encompassing four orders of magnitude (large hysteresis) depending on the direction of the transition path: A-to-P or P-to-A. Therefore, they define the transition as a large domain (see their fig. 10). Recently Jacobson and Myers [START_REF] Jacobson | Active Oxidation of SiC[END_REF] carried out a detailed experimental study on the influence of the direction of variation of the partial pressure of oxygen -increasing or decreasing -in the determination of the transition temperature: the authors observed a small difference, which was, however, of the order of magnitude of their experimental uncertainties. In the work of Jacobson and Myers [START_REF] Jacobson | Active Oxidation of SiC[END_REF], the oxygen partial pressure difference (or hysteresis) varies by a factor of 1.5 to 3 in the temperature range 1660 -1760K

under Ar + O2 at 1 bar, and a factor 3 at 1760K under reduced pressure. However, these observations were carried out on very thin silica layers, previously formed in the same experiment, the vaporization of which may be influenced by the proximity of the underlying SiC / SiO2 interface. With the same team, and in order to confirm the influence of the experimental delay in the observation of the P-to-A transition, Harder et al. [START_REF] Harder | Oxidation Trasitions for SiC Part II. Passive-to-Active Transitions[END_REF] studied the time necessary to obtain the complete vaporization of a 1 µm thick layer of silica (independently previously formed) under an oxygen imposed pressure whose starting value is lower than that of the transition, that is to say in the active oxidation domain. The onset of P-to-A oxidation is determined by weight changes measured by TGA while the inlet oxygen pressure is gradually decreased. Thus, the marks (+) relating to the P-to-A transition in figure 3 correspond to a low limit compared to the marks (red full square) for the A-to-P transition obtained by the same team with the same equipment. Erosion of silica is proposed to occur by vaporization of CO(g) + SiO(g) species, which are assumed to be formed at the SiC / SiO2 interface. These species further diffuse through the cracks inside the silica layer, which have been post-mortem observed on the sample by detachment and thinning of this layer. Three different samples -pure SiC and mixtures of SiC-0.5% C and 12-19% Si-SiC -show different silica erosion times, the last sample being rapidly eroded during the time required for the heating ramp up to 1773K.

Harder et al. [START_REF] Harder | Oxidation Trasitions for SiC Part II. Passive-to-Active Transitions[END_REF] calculated the pressure of oxygen at equilibrium at the surface of a layer of pure SiO2 vaporizing congruently at equilibrium. Using the published evaporation coefficients for pure silica, they found an oxygen pressure much lower than that corresponding to the P-to-A transition. In addition, the pressure value at thermodynamic equilibrium, i.e. corresponding to a pseudo-binary SiC-SiO2 system vaporizing congruently, remains much greater than the observed P-to-A transition. The conclusion of these measurements and calculations is that even when the interfacial reaction between SiC and silica is decisive for the production of gaseous species, their diffusion through the silica layer remains an important limitation for the flow exchanged with the atmosphere. This coupling between interface reaction and diffusion is that which was proposed by Honstein et al. [START_REF] Honstein | Mass spectrometric and thermodynamic analyses of (SiC-SiO2) powders vaporization behavior[END_REF] to explain the equality of the partial pressures CO(g) and SiO(g) during the vaporization of mixtures of SiC-SiO2 powders, as well as their very low partial pressure values measured by mass spectrometry compared to theoretical equilibrium values: the SiC powder grains remain covered with SiO2 (passive oxidation) due to the excess oxygen supplied by the vaporization of the silica grains in SiC + SiO2 mixtures. Thus, the P-to-A experiments described above cannot measure with precision the residual oxygen pressures, which correspond to the transition, since they depend on the thinning of the silica layer necessarily obtained after a certain time and under the only oxygen partial pressures lower than the value corresponding to the transition. Conversely using vacuum cannot lead to any oxygen pressure value.

Main interface chemical reactions at the active to passive oxidation transition

Many authors, as summarized in Table 1 of the article by Jacobson and Myers [START_REF] Jacobson | Active Oxidation of SiC[END_REF], attempted to determine the main vaporization reactions occurring at the solid / gas interface for the A-to-P transition above various domains of the phase diagram: congruent vaporization in the SiO2-SiC pseudo-binary domain, three-phase Si-SiC-SiO2 or C-SiC-SiO2 domains for example.

According to Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF] the so-called "active oxidation" regime covers a SixC(1-x) composition field ranging from Six1C(1-x1) + C to Six2C(1-x2) + SiO2 depending on the oxygen partial pressure. In this composition range, the partial pressure of oxygen at the surface (under steady-state) varies by a factor ≈ 1000. At the upper limit of this composition range, the partial pressure of O2(g) at the SiC surface corresponding to the oxidation transition A-to-P remains slightly greater than the value corresponding to the congruent vaporization of the pseudo-binary SiC-SiO2 (reaction (2)). In the oxygen pressure range of active oxidation, there is a correspondence between the partial pressure and/or the flow of oxygen at the surface of SiC and the non-stoichiometric composition SixC(1-x) insofar as the adsorption of Si and C atoms at the surface does not impose any kinetic limitation: any bulk component or surface diffusion for example during its transfer into the adsorbed layer. However, the main reaction for any active regime remains reaction (1a) until the transition. The transition is then characterized by the value of the activity of Si in SixC(1-x), which evolves towards the Si-SiC limit without ever reaching it.

Thermodynamics of the transition: hysteresis and interfacial reaction

Is the A-to-P transition "reversible"? That is to say, does it correspond to the same variable pair (aSi, p(O2)) at the surface of SiC or interface SiC/SiO2 when the incident oxygen flow increases or decreases for a given temperature? From a thermodynamic point of view -due to the variance of the system including the two reactions (1) and ( 5) -there is only one pair (aSi, p(O2))

corresponding to the existence of a layer of silica on the surface of SiC, but do the experiments work "reversibly" once a layer of silica is formed? As analyzed by Honstein et al. [START_REF] Honstein | Mass spectrometric and thermodynamic analyses of (SiC-SiO2) powders vaporization behavior[END_REF], once a layer of silica has formed, and the imposed incident oxygen pressure has been removed by pumping under vacuum or removed by a flow of extra pure neutral gas, vaporization of the silica layer should take place in the pseudo-binary SiC-SiO2 domain, that is to say by pseudobinary congruent vaporization, either under a perfectly neutral atmosphere or under high vacuum. Figure 4 shows the non-stoichiometric composition range of SixC(1-x), which is not known but represented arbitrarily large for the purposes of discussion. Figure 4 also shows the two compositions in the two regimes, A-to-P and pseudo-binary congruent vaporization in the non-stoichiometric domain of SixC(1-x), (the relative positions of the activities are based on the calculations in Table 1). Under active oxidation conditions, the vaporization ratio p(SiO) / p(CO) is equal to 1 until the transition, while in the de-oxidation step this ratio should be 3 under an atmospheric pressure of neutral gas (reaction (2)) or 3.76 under vacuum. Assuming equilibrium conditions, Table 1 gives the partial pressures of the main gaseous species (SiO(g), CO(g)) and the corresponding O2(g) surface or interface pressure) calculated for the two situations: A-to-P transition and pseudo-binary congruent vaporization (thermodynamically similar to the passive to active transition). Table 1 gives the corresponding Si activity in SixC(1-

x),. Note that according to table 1, a similar figure for oxygen equilibrium partial pressures can be drawn as figure 4, the relative positions of p(O2) for transition and congruent vaporization being inverted (relative to Si). In figure 4, p(O2) at P-to-A transition is somewhat lower than for A-to-P, a conclusion that differs from the pure thermodynamic reasoning on variance (beginning of this part) because we take into account chemical reactions coupled with mass transfer. 1.4 10 -2 5.9 10 -3 6.1 10 -18 3.9 10 -18 0.34 0.59

These pressure values are calculated from Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF] at temperatures where the various quoted authors observed the "vacuum" or P-to-A transition. In figure 16 of the article by Jacobson and Myers [START_REF] Jacobson | Active Oxidation of SiC[END_REF] (1490 ° C (1763 K)), the A-to-P transition is observed for a value of 6.5 10 -4 atm (pressure of O2 at the inlet of their device). Conversely in their figure 18, a beginning of "congruent" vaporization corresponding to the first weight loss is observed at 6.6 10 -4 atm and the beginning of the active regime is observed at 2.9 10 -4 atm. Note that the O2(g) partial pressure measured at the inlet of their device is totally converted into CO(g)+SiO(g) partial pressures at the surface of SiC. This is why the incoming experimental pressure of O2 must be compared with the calculated sum of the partial pressures CO(g)+SiO(g) and not with the partial pressure of oxygen calculated in Table 1, which corresponds to the steady-state partial pressure of O2 at the SiC / gas interface. This comparison in Table 1 shows that:

• The equivalent oxygen pressure (sum (SiO(g)+CO(g))/2) calculated under the equilibrium assumption for the congruent vaporization or the transition pressure P-to-A is much higher than those measured since p(SiO)+p(CO) = 1.6. 10 -2 bar, which is much higher (two order of magnitude) than the value of 2.9 10 -4 atm applied to the inlet of their device by Jacobson and Myers [START_REF] Jacobson | Active Oxidation of SiC[END_REF]. This confirms the presence of a kinetic barrier for the reaction (1a) or for the vaporization of SiO(g) and CO(g) compared to the pseudo-binary SiC-SiO2 system at equilibrium. Therefore the evaporation coefficients must be lower than 1.

• The input partial pressure of oxygen corresponding to the observed disappearance of the silica layer is approximately 2.24/2=1.12 times lower than the value corresponding to the formation of silica at the experimental A-to-P transition. This ratio is greater than the one calculated (0.9) from the equivalent oxygen pressures in Table 1. Conclusion by Jacobson and Myers [START_REF] Jacobson | Active Oxidation of SiC[END_REF] of a "small and no significant difference" between A-to-P and P-to-A does not agree with the present predicted and calculated difference, which remains significant but of the same order of magnitude:oxygen pressure at the A-to-P transition remains a little bit higher than at the P-to-A transition associated to the congruent vaporization. The larger experimental ratio of 1.12 may result from imposed oxygen partial pressures lower than the one needed to observe the disappearance of the silica layer during a time interval compatible with the duration of the experiment.

Furthermore, observations by mass spectrometry of the vaporization of SiC-SiO2 mixtures carried out by Jacobson et al. [START_REF] Jacobson | Reactions of Silicon Carbide and Silicon (IV) Oxide at Elevated Temperatures[END_REF] or Honstein et al. [START_REF] Honstein | High temperature mass spectrometric study of the vaporization behaviour of SiC-SiO2 system[END_REF][START_REF] Honstein | Mass spectrometric and thermodynamic analyses of (SiC-SiO2) powders vaporization behavior[END_REF] shows that the ratio of the partial pressures p(SiO) / p(CO) is always within the 1-2 range, which differs significantly from the expected value: 3.76 for a congruent behavior under vacuum. The ratio in the range 1 to 2observed in the case of a mixture of SiC (with silica layer)+SiO2 powders -along with the lower measured partial pressures are associated by Honstein et al. [START_REF] Honstein | Mass spectrometric and thermodynamic analyses of (SiC-SiO2) powders vaporization behavior[END_REF] with the reactions taking place at the SiC-SiO2 interface below the studied silica layer. It corresponds to the steady-state release of the gaseous species from the SiC / SiO2 interface by diffusion or permeation through the silica layer and not to the congruent vaporization in a "theoretical" quasi-binary pure SiC-SiO2 system as calculated when assuming thermodynamic equilibrium. Mass spectrometric observations of this decreased ratio let suppose a loss of SiO(g) during the counter-diffusion of this species with oxygen in the silica layer thickness: indeed, Jacobson and Myers [START_REF] Jacobson | Active Oxidation of SiC[END_REF] observe the formation of silica outgrows at the surface of the silica layer (formation of some kind of whiskers). Note that in the low pressure regime (case of mass spectrometric measurements using effusion cells) there exists also a counter oxygen flow provided by the silica grains mixed with SiC grains as proposed by Honstein et al. [START_REF] Honstein | Mass spectrometric and thermodynamic analyses of (SiC-SiO2) powders vaporization behavior[END_REF]. Thus, the experimental "global" SiO/CO vaporized ratio ≈ 1 does not invalidate the congruent vaporization reaction at the SiC / SiO2 interface and results from the coupling of the interfacial reaction with probably a reaction of formation of silica in the silica layer (provided by the reaction of SiO(g) with O2(g) in the voids and channels along the permeation/diffusion process). Likewise, Harder et al. [START_REF] Harder | Oxidation Trasitions for SiC Part II. Passive-to-Active Transitions[END_REF] propose quite the same de-oxidation process by vaporization at the SiC / SiO2 interface followed by gaseous diffusion of SiO(g) + CO(g) conversely to O2(g) through the cracks previously formed in the preparation of the silica layer. Such a phenomenon takes place at the only P-to-A transition. Among the different reactions proposed to occur at the SiC / SiO2 interface, the most probable is from our point of view the congruent vaporization since no silicon nor carbon residues are ever observed at the SiC surface after the P-to-A transition.

Conclusion on the oxidation transitions

A-to-P and P-to-A transitions occur for very close values of the oxygen partial pressure at the interface of SiC with either the gas phase or SiO2. The oxygen partial pressure at the formation of the first silica layer (A-to-P) remains slightly higher than the one corresponding to the congruent pseudo-binary (P-to-A) vaporization that drives the de-oxidation process from the interface.

The oxygen partial pressure at the SiC surface and the silicon activity are inter-related at the oxidation transitions, and their values depend on the nature of the allotropic variety of SiC formed. Therefore, the A-to-P transition can be observed for different oxygen incident flows.

For the active oxidation regime under low total pressure, there exist some significant kinetic barrier for di-oxygen reaction that decreases slightly the erosion rates of SiC calculated at equilibrium. There is probably also a loss of oxygen atoms at the first adsorption step : reflexion of O2 molecules during collision on different sites. The influence of this step decreases under atmospheric pressure since the gas flow towards the surface sites is limited by gaseous diffusion.

From a thermodynamic point of view, the SiC-SiO2 interface remains at equilibrium and this is why no carbon nor silicon is formed there.

Passive oxidation

Looking at the growth rate of the silica layer, two domains are generally observed: a first domain where SiO2 grows linearly with time (linear regime) followed by a second domain occurring beyond a specific thickness of the silica layer where SiO2 grows linearly with the square root of time (parabolic regime). The Deal and Grove (DG) [START_REF] Deal | General Relationship for the Thermal Oxidation of Silicon[END_REF] model based on a linearparabolic law is a practical way to combine these two behaviors. The model was originally developed for the oxidation of silicon and has been successfully applied to the case of SiC oxidation.

Presser and Nickel [START_REF] Presser | Silica on Silicon Carbide[END_REF] have extensively studied the oxidation of α-SiC and β-SiC and provided an exhaustive review of this subject in 2008. In general, studies dealing with polycrystalline β-SiC are mainly devoted to thermomechanical applications at high or even very high temperatures, while monocrystalline α-SiC (4H or 6H) is mainly involved in metal-oxidesemiconductor (MOS) devices such as SiC-MOS devices. In the latter case, the oxide layer is generated at 1373K under dry or wet oxygen. The high corrosion rate and the thick layers of silica encountered in the oxidation of β-SiC have been widely studied by thermogravimetric analysis (TGA) and mainly in the parabolic regime, whereas for α-SiC the thickness of silica layer does not usually exceed 100 to 200 nm, which corresponds to the linear growth regime.

TGA measurements are not very accurate when the silica scale is thin, which is the case for the linear regime, while the spectroscopic ellipsometry used to determine the thickness of the thin silica layers in the studies related to α-SiC is very precise but cannot be applied beyond 50 nm thickness.

6.1. Linear regime

Summary of measurements and interpretations

In the linear regime, the silica layer is thin and is therefore not supposed to limit the diffusion of oxidizing species towards the SiC / SiO2 interface or the diffusion of CO towards the SiO2 / gas interface. The DG model assumes a limiting first order oxidation reaction at the SiC / SiO2

interface. The oxidation process can be summarized by the following equilibrium equations at the interface:

either: SiC + O2 (g) = SiO2 + C (3)
or: SiC + 3/2 O2 (g) = SiO2 + CO (g). ( 6)

However, there is no direct evidence that these main reactions are slow enough to control the rate of oxidation, especially at high temperature. Indeed, the presence of an additional layer at the interface between SiC and SiO2 can also be at the origin of the linear regime. On condition that such a layer has a constant thickness and that it limits the diffusion of certain species involved in the oxidation process, it can account for the linear regime. In the case of the oxidation of Si, Watanabe et al. [START_REF] Watanabe | New Linear-Parabolic Rate Equation for Thermal Oxidation of Silicon[END_REF] proposed a modified DG equation in which the oxidation reaction is no longer kinetically limiting. These latter authors suggest that a compression-strained oxide layer of constant thickness located at the Si / SiO2 interface imposes the limiting step for the diffusion of oxygen. When characterizing at the atomic scale 6H-SiC materials oxidized at 973K, Hoshino et al. [START_REF] Hoshino | Atomic scale characterization of oxidized 6H-SiC(11-20) surfaces[END_REF] observed such a layer in compression just below the oxide layer during the oxidation of α-SiC. The compressive stress starts from the border of the thin SiO2 film and develops through the interlayer with very little penetration into the SiC substrate (2%). The thickness of the layer affected by the compressive stress was about 1 nm. A phase of constant thickness generated at the SiC / SiO2 interface by the oxidation reaction could also play the role of a layer limiting diffusion in the linear regime.

Interface layer or interphase

Several authors have shown that SiC oxidation process can generate a domain composed of Si, C and O atoms at the SiC / SiO2 interface. In the case of hexagonal SiC polytypes, Virojanadara

and Johansson [START_REF] Virojanadara | Studies of oxidized hexagonal SiC surfaces and the SiC/SiO2 interface using photoemission and synchrotron radiation[END_REF] reviewed the experimental studies dealing with the properties of the interface between SiO2 and a SiC single crystal: either on polar surfaces with Si termination, or C termination. They concluded that no carbon clusters or carbon-containing by-products are detected when the oxide layer thickness is greater than about 1 nm on the surface of a Siterminated polar surface, while a mixture of Si-C-O bonds is detected during the oxidation of the polar surface with the C termination. Ramberg et al. [START_REF] Ramberg | Passive-Oxidation Kinetics of High-Purity Silicon Carbide from 800° to 1100°C[END_REF] also observed a layer of unknown composition at the interface between the carbon polar surface and the SiO2 layer. The authors assumed that this layer would be more resistant to oxygen diffusion than pure SiO2, which would cause the parabolic rate constant to be much lower on the C-terminated face than on the Si-terminated face. However, it should be mentioned that Song et al. [START_REF] Song | Modified Deal Grove model for the thermal oxidation of silicon carbide[END_REF] observed the opposite behavior. Zheleva et al. [START_REF] Zheleva | Transition layers at the SiO2/SiC interface[END_REF] analyzed the structure and chemistry of the interface between a 4H-SiC crystal doped N on the Si-terminated face and a SiO2 layer obtained by oxidation at 1473K in pure oxygen. They observed the presence of distinct layers several nanometers thick:

3 to 5 nm on each side of the interface which they attributed to the formation of a ternary Si -C -O phase, and a top layer of partially amorphous SiC. More recently, Kouda et al. [START_REF] Kouda | Oxygen partial pressure dependence of the SiC oxidation process studied by in-situ spectroscopic ellipsometry[END_REF] found that regardless of the surface polarity of 4H-SiC or the partial pressure of oxygen, an interface layer of about 1.5 nm is formed when the thickness of SiO2 reaches about 7 nm. This SiC-type layer is assumed -(i) to result from the accumulation of Si and C interstitials in the silica layer near the interface followed by diffusion into the silica layer, a process similar to that called "emission model" initially proposed by Hijikata et al. [START_REF] Hijikata | A Kinetic Model of Silicon Carbide Oxidation Based on the Interfacial Silicon and Carbon Emission Phenomenon[END_REF] and -(ii) be responsible for the decrease in the growth rate of the oxide just after the onset of oxidation. The thickness and structure of this interface layer would remain unchanged as the layer continues to grow.

After oxidation of a β-SiC layer produced by CVD, Pampuch et al. [START_REF] Pampuch | New aspects of the oxidation of SiC[END_REF][START_REF] Pampuch | Formation of ternary Si-O-C Phase(s) During Oxidation of SiC[END_REF] observed by IR spectroscopy the existence of Si-C-O absorption bands. Berjoan et al. [START_REF] Berjoan | AES study if the SiO2/SiC interface in the oxidation of CVD beta-SiC[END_REF] oxidized polycrystalline β-SiC at 1573-1633K for 10 minutes and followed the erosion profile by Auger and SIMS electron spectroscopy: they observed a 40 nm thick layer containing a mixture of SiO2, SiC and carbon. Hornetz et al. [START_REF] Hornetz | ARXPS studies of SiO2-SiC interfaces and oxidation of 6H SiC single cxrystal Si-(001) and C-(00-1) surfaces[END_REF] observed by angle-resolved XPS a 1 nm thick interface of Si4C(4-x)O2 composition on the oxidized Si-terminated polar face of an α-SiC wafer. Using the same characterization technique with low energy ion scattering (LEIS), Radtke et al. [START_REF] Radtke | Initial stages of SiC oxidation investigated by ion scattering and angleresolved x-ray photoelectron spectroscopies[END_REF] also observed a transition domain containing a SiCxOy phase, between a layer of SiO2 and a 6H-SiC substrate (0001). Liu et al. [START_REF] Liu | High temperature oxidation behavior of SiC coating in TRISO coated particles[END_REF] also revealed by X-ray photoelectron spectroscopy (XPS) the presence of silicon oxycarbide at a β-SiC / SiO2 interface after oxidation in air in the temperature range between 1073 and 1873K. The binding energies observed were interpreted as a mixture of silicon oxycarbons with the following proposed compositions: SiO3C, SiO2C2, SiOC3. By XPS analysis of oxidized SiC used to coat TRISO fuel particles for nuclear applications, Cho and Lu [START_REF] Cho | High temperature oxidation behavior of bulk SiC with low partial pressures of air and water vapor in argon[END_REF] did not observe any free C at the SiC / SiO2 interface, instead some mixed Si-C and Si-O bonds. By comparison between the diffusion activation energies calculated for the O2 or CO species through the crystallized phases of SiCO4, Si2CO6 or SiO2 (α-Quartz, ß-Quartz, cristobalite or tridymite), they showed that the outward diffusion of CO through the silica layer is the step controlling the oxidation of both the Si-and C-terminated faces of α-SiC. Since the molecular size of the O2 and CO species is comparable, this difference in behavior has been attributed to a greater Lewis base character of the CO molecule. layer at the SiC / SiO2 interface when β-SiC obtained by CVD is oxidized above 1783K. Since such a layer is not detected below 1783K, the authors suggested that carbon formation corresponds to a growth rate of SiO2 limited by the outward diffusion of CO. In addition, they observed the formation of bubbles generated by excess CO above 1873K. Such a carbon layer has also been detected by means of Raman spectroscopy by Lee et al. [105] during the oxidation at 1673K of 6H polycrystalline SiC using a gas mixture composed of either 0.9% O2 in Ar, or 50% H2O in air. They observed that the quantity of carbon trapped at the interface increases with the partial pressure of the oxidizing species. However, we can notice that such a carbon layer might as well have been formed during the heating ramp of SiC in the absence of oxidizing species as a consequence of the non-congruent vaporization of SiC. In this regard, the heating sequence of the experimental procedure influences the nature of the surface subjected to oxidation, at least for the first oxidation steps. However, Liu et al. [START_REF] Liu | High temperature oxidation behavior of SiC coating in TRISO coated particles[END_REF] observed by RAMAN spectroscopy the formation of graphite at 1673K or 1873K for long oxidation times. They interpreted this result as the accumulation of C at the SiC / SiO2 interface when the oxygen diffusion flux through the silica layer decreases: this means that silicon would be more easily oxidized than carbon (see surface analysis in the next part). [114] proposed that an abrupt interface cannot be obtained and that a transition layer must be present in all cases.

Thermodynamic aspects

From a thermodynamic point of view, the vaporization of pure SiC produces Si(g) as the main species in the gas phase with a partial pressure at least one hundred times greater than the species containing carbon. Due to this non-congruent vaporization as the final step in heating a SiC sample before it is subjected to oxidation by oxygen, a carbon-rich surface is first obtained with Si ad-atoms, which can turn into a graphite structure stable with time and temperature.

Considering the above-mentioned surface reconstructions with Si ad-atoms (which are those producing Si(g) in the last desorption step of the vaporization process), the Si concentration in the SiC surface bilayer decreases from 1/2 for the stoichiometric compound down to 1/3. The surface of SiC thus becomes unsaturated with Si ad-atoms and the vaporization process at this time is no longer in equilibrium as evidenced by the existence of an evaporation coefficient above the two-phase SiC-C domain (see part 3.2). The evaporation flux of Si(g) decreases by a factor of 5.10 -2 and the remaining steady-state concentration of Si ad-atoms is provided by the diffusion of Si from bulk SiC to the surface layer, a phenomenon generally called surface adsorption under chemical forces induced by surface tension. As observed by Powers and Somorjai [106], these Si ad-atoms preferentially bind to the oxygen atoms arriving at the surface, rather than to the C atoms. Indeed, the concentration of C ad-atoms probably remains very low due to the surface tension, which is the result of vaporization enthalpies and / or partial pressures favoring the presence of silicon. There is therefore a lack of C ad-atoms for the reaction with oxygen to produce CO (g). The Si ad-atoms are trapped by oxygen and then give either seeds of silica or SiO(g), which can first evaporate or diffuse outward through the formed upper layer of silica (diffusion or permeation process similar to that of CO(g)).

The SiC surface at the beginning of the formation of a silica layer or any SiC / SiO2 interfaceincluding connected channels and cavities due to the presence of amorphous materials during the formation of the silica like layer -can be simulated by SiC + SiO2 powder mixtures. Mass spectrometry applied to such mixtures reveals that the pressure of the SiO species (g) is of the same order of magnitude as that of CO(g). In addition, these pressures are very low compared to the values calculated under the assumption of thermodynamic equilibrium. In the thickness of the reconstructed SiC surface (about one SiC bilayer) the excess of C atoms in the upper lattice react with oxygen to form CO ad-atoms in the adsorption layer but this reaction seems slower than the reaction of oxygen with silicon. These CO ad-atoms further form cavities and / or diffuses through the layer of silica already formed (permeation and / or interstitial diffusion).

Due to the diffusion in the opposite direction of oxygen through the silica layer, as well as the preferential reaction of oxygen with the Si ad-atoms during the first step of the oxidation process, the silica layer forming can thus trap carbon atoms or reorganize itself with those that

have not yet reacted with oxygen. This situation can continue as long as the silica layer grows This mechanism proposed by Kouda et al. [START_REF] Kouda | Oxygen partial pressure dependence of the SiC oxidation process studied by in-situ spectroscopic ellipsometry[END_REF] is called "emission of atoms". It is related to vaporization assisted diffusion processes in cavities leading to permeation and interstitial diffusion. In addition to the reactions (3) and ( 6) already mentioned, several other reactions can take place in the intermediate layer:

Si (ad-atom) + ½ O2(g) = SiO(g) ( 7)

C (from SiC) + ½ O2(g) = CO(ad-atom) ( 8)

CO(ad-atom) = CO(g) ( 9)

Si(ad-atom) = Si(g) [START_REF] Harris | Oxidation of 6H-alpha Silicon Carbide Platelets[END_REF] ... Gas molecules formed in the cavities at the SiC / SiO2 interface diffuse in the silica layer (interstitials and / or permeation). All the other molecules mentioned by Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF] in the active oxidation of SiC -namely at the active to passive transition -probably also form in the same cavities of the interface layer, but thermodynamics indicate that their relative importance remains very low: the amount of Si(g) formed is already three order of magnitde magnitude smaller than that of SiO(g) or CO(g). In such a general scheme, the limiting step at the interface can arise from: -(i) the diffusion of silicon atoms from bulk SiC to provide Si adatoms bonded to the C sites at the cavities of the interface or, -(ii) the reaction of carbon with oxygen entering the carbon-rich interface to produce CO(g), which decreases the amount of C sites available for Si adsorption and hence the number Si-O-Si bonds that allow the nucleation of silica. Recently, using the substrate bending technique Li et al. [START_REF] Liu | High temperature oxidation behavior of SiC coating in TRISO coated particles[END_REF]115] measured the stress resulting from the growth of a thin layer of silica on 4H-SiC substrates (Si and C side). The measured stress, which is low at high temperature, is significantly increased on return to ambient temperature. The thickness of the silica layer, which is stressed, is about 40 nm except on the C face when it is oxidized under wet oxygen. In the latter case, no stress is observed beyond about 20 nm. In addition, the measured stress is at least 3 times lower on the C face. The main effect of this compressive stress is to decrease the molecular diffusion (or permeation) of oxygen due to the modified interconnection between the cavities in the amorphous lattice of the silica layer and increase the chemical potential of oxygen in the interface layer compared to the pure silica layer. Indeed, as calculated by Shen et al. [116] for the same type of tetragonal deformation in covalent epitaxial III-V compounds taking into account the elastic energy of the crystal, the chemical potential of the more volatile component -i.e. oxygen in the present case -is that which increases mainly under compressive stress. Correspondingly, the chemical potential gradient of oxygen, which causes diffusion through the upper silica layer must decrease. In conclusion, there are three reasons in favor of the hypothesis that the reactions in the interface layer are the stage limiting the growth of SiO2: concentration of active species, topographical changes and chemical potentials. The present thermodynamic analysis remains compatible with the hypothesis of intermediate layer responsible for the linear regime (see part 6.1.1.) if we consider that this layer is very thin and has high defects concentration: this layer would then be the one in which all the interface reactions take place.

6.1.8. Preparation of surfaces prior to the silica growth process

Looking at the preparation steps of SiC substrates before oxidation, standard chemical cleaning with HF leaves a layer of native silica on the surface. This layer can be removed before oxidation by vaporization at 1123K under ultra-high vacuum or under very pure neutral gas.

Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. I. SiC and SiC-SiO2 mixtures under neutral conditions[END_REF][START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF] have calculated that a first exposure to a low flow of oxygen, which can for example correspond to a level of impurity encountered in a flow of neutral gas, can consume -within a certain temperature range -the excess carbon and thus rapidly change the composition of the surface by reaching the active oxidation regime. Under such experimental conditions of active oxidation (the so-called oxygen and silicon windows as defined by Honstein et al. [START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF]), the SiC surface becomes richer in silicon when one gradually increases the incident oxygen flux at constant temperature. This is due to the steady-state which continuously increases the activity of silicon before reaching the transition A-to-P. The surface reconstruction in this case becomes probably very different (and unknown to date) and mainly enriched in adsorbed Si atoms.

Brisebourg et al. [START_REF] Brisebourg | Oxidation of β-SiC at high temperature in Ar/O2, Ar/CO2, Ar/H2O gas mixtures: Kinetic study of the silica growth in the passive regime[END_REF][START_REF] Brisebourg | Oxidation of β-SiC at high temperature in Ar/O2, Ar/CO2, Ar/H2O gas mixtures: Active/passive transition[END_REF] used such a preliminary surface preparation by keeping the SiC surface for a short time under active oxidizing conditions before increasing the incident oxygen flux to start the growth of the silica layer. Such a process removes carbon, native silica as well as surface impurities and considerably improves the reproducibility of the passive oxidation process. The Si-rich SiC surface is then favorable to the direct growth of SiO2 and no intermediate layer is observed with such a procedure. The stress at the SiC / SiO2 interface can in this case be high, but has not been measured by the authors. However such an interface remains stable as shown by thermodynamics in the Si-C-O ternary system.

Parabolic regime

If a growth rate limited by the diffusion of species through the silica layer is a good interpretation of the parabolic growth rate, the nature and direction of the limiting species is debated. It can be:

 either a diffusion of molecular species by permeation and interstitial diffusion along micro-channels and cavities interconnected by thin walls as described by [START_REF] Singhal | Oxidation of silicon-based structural ceramics[END_REF]134], Mieskowsky [START_REF] Mieskowski | Bubble Formation in Oxide Scales on SiC[END_REF]), which showed on the basis of the activation energy values that the outward diffusion of CO could limit the growth rate of SiO2 in the temperature range 1473-1723K. Even if the debate on the species limiting the growth rate is not settled, it is today recognized that below 673K the transport through the silica layer of dissolved O2 or CO molecules (permeation and / or interstitial diffusion) is dominant. In this transport regime, it is easy to check which molecule controls the growth rate of SiO2: if the O2 diffusion controls this rate, we must observe a proportionality between the growth rate of SiO2 and the O2 permeation rate and therefore the partial pressure of O2 in the gas phase (Williams [135]). Such a proportionality is not observed if CO permeation controls this rate. At higher temperatures and using isotopic tracers, Zheng et al. [START_REF] Zheng | Oxidation of Single Crystal of Silicon Carbide. II. Kinetic Model[END_REF] have suggested that diffusion of ionic oxygen (O 2-) via the lattice (lattice transport) and presumably by means of a vacancy mechanism becomes dominant at low partial pressures of oxygen. This latter diffusion process has a high activation energy relative to molecular permeation and is independent of the partial pressure of oxygen in the gas phase because electron transport limits the rate of oxidation. These two transport processes take place simultaneously, but one prevails over the other depending on the temperature domain: transport by molecular species is dominant at low temperature and is characterized by low activation energies for the constants kl and kp (typically around 100 kJ / mole), although network transport is dominant at high temperature (typically above 673K), and its growth rate constant is independent of gas phase composition and has a high activation measured a continuity of the activation energy for the oxidation rate in the parabolic regime between amorphous SiO2 at low temperature and crystallized cristobalite at high temperature, suggesting that interstitial diffusion of molecular O2 in cristobalite is also expected to be the dominant diffusion process at high temperature.

Silica growth rate in the presence of H2O

Of all the molecules that can transport oxygen, the water molecule is believed to have the highest rate of diffusion through the silica layer (Doremus [141]). As for O2, the oxidation rate of SiC is controlled by the diffusion of molecular H2O in the low temperature range and the ionic diffusion of O 2-in the reticular network at high temperature (Narushima et al. [142]).

However, unlike O2, H2O can form gaseous species of silicon hydroxide or silicon oxyhydroxide (Brady [143]) when it reacts with SiO2 at high temperature. Among these species, Si(OH)4 and SiO(OH)2 -also called "silanols" -are those, which must be formed preferentially in the H2O concentration and temperature ranges encountered in most thermomechanical applications of SiC. Vaporization of the SiO2 layer is in this case in competition with its growth rate. A recession in the thickness of the SiO2 layer can be even observed (Mouche and Terrani

[144]) under a strong flow of water vapor.

When the amount of silicon hydroxide or silicon oxyhydroxide species formed is small, the growth of silica can be described by the DG model, but the linear regime is in this case much more extensive (Opila and Hann [START_REF] Opila | Paralinear Oxidation of CVD SiC in Water Vapor[END_REF], Narushima et al. [142]). The diffusion of oxidizing species in silica is indeed so rapid in the presence of atmospheres containing H2O that the transition to a parabolic regime (where diffusion is limiting) is observed for silica layers thicker than in the case of oxidation by O2. According to Opila and Hann [START_REF] Opila | Paralinear Oxidation of CVD SiC in Water Vapor[END_REF] kp is increased by an order of magnitude in the presence of water vapor due to the increased solubility of H2O molecules in silica. Note that the H2O solubility is directly related to the adsorption in voids and channels of the amorphous silica. On the other hand, when the quantity of gaseous silicon hydroxide or silicon oxyhydroxide species formed is high, the linear vaporization rate of SiO2 by the hydroxide species must be taken into account in the coupled growth-vaporization modeling. For this purpose, Opila and Hann [START_REF] Opila | Paralinear Oxidation of CVD SiC in Water Vapor[END_REF] [147], Singhal [START_REF] Singhal | Oxidation of silicon-based structural ceramics[END_REF]134,148]). Deal and Grove [START_REF] Deal | General Relationship for the Thermal Oxidation of Silicon[END_REF] reported that although they observed a diffusivity of H2O in SiO2 lower than the diffusivity of O2, the increased oxidation rate observed in the presence of H2O must be due to the equilibrium concentration of H2O in SiO2 (solubility) which is about three orders of magnitude greater than that of O2 under the experimental conditions encountered. More recent measurements by Brisebourg et al. [START_REF] Brisebourg | Oxidation of β-SiC at high temperature in Ar/O2, Ar/CO2, Ar/H2O gas mixtures: Active/passive transition[END_REF] confirm the effect of H2O on the speed of linear oxidation in the low temperature range where both O2 and H2O diffuse in their molecular state. The oxidation rate of SiC in this case is in fact proportional to the partial pressure of H2O. On the other hand, the presence of H2O does not increase the parabolic growth rate in the whole temperature range. The parabolic growth rate increases with the partial pressure of H2O in the low temperature domain while in the high temperature range it decreases as the partial pressure of H2O increases. Such a reduction, which results from the reactive volatilization of SiO2, is however not observed in a linear regime, that is to say for small thicknesses of the SiO2 layer. This suggests that oxide volatilization does not fully explain this behavior and that H2O may somewhat limit the diffusion of oxygen vacancies through the oxide layer at high temperature. This is because the influence of H2O is complex and OH -ions can also participate in the process. These ions can either contribute to the diffusion of oxidizing species with molecular H2O as suggested by Opila et al. [START_REF] Opila | SiC Recession Caused by SiO2 Scale Volatility under Combustion Conditions: II, Thermodynamics and Gaseous-Diffusion Model[END_REF], or modify the structure of SiO2 during its dissolution: the reaction of the silica network to hydroxyl breaks an oxygen bridge However, it should be noted that the promotion of the crystallization of silica into cristobalite by water vapor is still debated.

Silica growth rate in the presence of CO2

Use of CO2 as an oxidizing species of SiC has been less studied. Opila et al. [START_REF] Opila | Oxidation of Chemically-Vapor-Deposited Silicon Carbide in Carbon Dioxide[END_REF] have shown that the oxidation of SiC in the presence of CO2 in the temperature range 1473 to 1673K no longer follows the Deal-Grove model. In contrast, it is characterized by an initial weight gain according to a parabolic law followed a few hours later by a linear decrease. This kinetics seems independent of temperature in the field studied, which suggests that diffusion of the oxidizing species inside silica is no longer the limiting rate when CO2 is the oxidizing species.

Dissociation of the CO2 molecule at the surface of SiO2 can in this case be limiting. When using CO2 as an oxidizing species, oxidation rates measured are always much lower than those measured when using O2 or H2O. Goto et al. [START_REF] Goto | Effect of oxygen partial pressure on the high-temperature oxidation of CVD SiC[END_REF] revealed in the range 1873 to 1973K that kp is strongly temperature dependent and varies linearly with the chemical activity of oxygen, such behavior being consistent with a network oxygen diffusion inside the silica layer. More recently, Chokawa et al. [156] determined by ab-initio calculations that CO(g) diffuses through the silica layer and reacts at the SiC / SiO2 interface to form CO2(g) according to the following equilibrium equation: SiO2 (stoichiometric) + CO(g) = SiO2 (including oxygen vacancies) + CO2(g). [START_REF] Hinze | The Active Oxidation of Si and SiC in the Viscous Gas-Flow Regime[END_REF] This process is responsible for the generation of oxygen vacancy defects at the SiC / SiO2 interface.

Conclusion

Thermodynamic analysis of the oxidation of SiC was carried out taking into account the vaporization phenomena occurring in binary Si-C, Si-O and ternary Si-C-O systems including the pseudo-binary SiC-SiO2 system. The important point is that the vaporization of SiC is not congruent. Thus, largely preferential vaporization of Si(g) leads to the formation of a precipitate of C at the surface, which can lead rather quickly to the formation of a graphitic phase, as has been demonstrated by various techniques of surface analysis. This behavior influences the state of the SiC surface at the beginning of oxidation.

As the solubility of oxygen in SiC is almost zero and there are no stable ternary compounds in the Si-C-O ternary equilibrium phase diagram, the behavior of SiC under oxygen flow must be investigated by combining thermodynamics with mass transfer at the surface. The oxidation process then corresponds to a steady state vaporization of the SiC elements coupled to surface reactions involving oxygen and producing the two predominant species CO(g) and SiO(g).

These species have almost identical vapor pressures. According to thermodynamics -that is to say, under the assumption of rapid surface reactions -the behavior of SiC under oxidation can be classified into three domains: -(i) at low partial pressures of oxygen vaporization of SiC is predominant and produces carbon at the surface, -(ii) at higher oxygen partial pressures, this carbon in excess is totally converted into CO(g). These conditions characterized by a CO / SiO vaporization ratio equal to 1 and a "clean" quasi-stoichiometric SiC surface correspond to the active oxidation regime -(iii) at even higher oxygen partial pressures, the formation of a silica layer corresponds to the passive oxidation regime. CO(g) and SiO(g) fluxes are maximum at thermodynamic equilibrium, while the vaporization experiments show a significant reduction in these fluxes (a factor of 100 to 10,000) as soon as a silica layer is formed at the surface of SiC.

The active oxidation regime has been the subject of numerous experiments measuring the erosion rate of SiC as a function of the partial pressure of oxygen. This partial pressure is that imposed at the inlet of the reactor when the total pressure is high (pressure > 10 mbar) or that inside the reactor or device when the total pressure is low under molecular vacuum conditions (pressure <1 mbar). The earlier modellings of the mass transfer at the interface between the SiC surface and the gas phase under total atmospheric pressure involve a concentration boundary layer in which the variation in the concentration of the gaseous species is assumed to occur by pure diffusion process. The modelling correctly reproduces the erosion rates in an active oxidation regime at atmospheric pressure. Under reduced pressure, this modeling is based on Knudsen fluxes (molecular flows) and exhibits erosion rates at equilibrium slightly higher than those observed experimentally. Such a difference can be attributed to the kinetics of the surface Initial SiC surface is all the more modified as the heating period is long and the temperature is high. This has a direct influence on the structure of amorphous silica at the beginning of growth as well as on its carbon content in the phase at the SiC / SiO2 interface. It is therefore not surprising that growth rates and activation energies are widely dispersed. The only way to obtain reproducible and stable silica layer growth rates is to start with a "clean" SiC surface, which can be obtained by a first treatment in an active oxidation regime as observed by some authors using optical interferometry.

Oxidation by water vapor is faster than by oxygen not only because the diffusion of H2O in silica is faster, but also because the solubility of water is greater, which highlights the role of channels voids and cavities in amorphous silica. At high temperature, vaporization of silanols causes the growth of the layer to stop due to the equilibrium state established between its growth and its vaporization. It can even lead to recession of the silica layer.

Oxidation by CO2 is of the parabolic type but the thickness of the silica layer rapidly reaches a very low limit value due to the very low permeability of CO2 in the silica. Beyond this stage, the decrease of the thickness of the silica layer is associated with the decomposition of CO2 on its surface, which imposes a very low oxygen potential favoring the vaporization of silica mainly through the SiO molecule (g).

This article shows that the oxidation of SiC involves mechanisms that are well documented today. However, its modeling depends mainly on the knowledge of the diffusion mechanisms in SiC and in SiO2 and therefore on the choice of the diffusion coefficients. Besides, the structure of the amorphous silica plays a large role in the corrosion of SiC and would need further characterization as a function of growth conditions.

Figure 1 .

 1 Figure 1. Active oxidation erosion rates as deduced from experiments compared to calculations corresponding to vacuum conditions (SiC-C and A-to-P lines) by Honstein et al. [73] and present calculations with the same thermodynamic references and using the diffusion boundary layer assumption (Ar+O2 and He+O2 lines), as a function of the input oxygen pressures. Experiments under 1 bar total pressure: black marks are for α-SiC, red marks for β-SiC. Blue marks are for low pressures. Dot and dash lines correspond respectively to the transition under viscous regime (p total > 10 mbar) and molecular regime (p total < 10 -4 bar). A-to-P is for active to passive transition, SiC-C for C precipitation, other for silica passivation. Experimental data: Jacobson (2011) × β-SiC 1 bar Ar+O2; ○, Δ β-SiC low pressure; Schneider (1998) ◊ β-SiC O2 0.1 bar; Berton (1992) ○ α-SiC O2 0.01 bar, + O2+O 0.01 bar; Charpentier (2010) 1 bar Ar-O2 ■ α-SiC, + β-SiC; Hinze (1976) □ Polyβ-SiC 0.197 bar; Gulbransen (1966, 1972) □ α-SiC vacuum; Keys (1977) • Sint. β-SiC 1 bar O2; Narushima (1991) ∆ α-SiC 1 bar Ar+O2; Hald (2003) ■ C/C SiC Plasma arc jet; Hatta (1999) ▬ C/C coated β-SiC Natural Convection; Rosner (1970) ♦ 0.03 Torr O2 , ▲ 0.01 Torr O(g); Dawi (2012) ▲ β-SiC/PyC He 1-7 bar; Jacobson (2013) β-SiC 1 bar Ar-O2, ■ β-SiC, ○ β-SiC -C, ∆ Si-SiC.

Figure 2 .

 2 Figure 2. (A): Dawi et al. [62] mass variation after 10 min treatments at 1 bar and post-mortem characterization of the SiC composite surface. (B): Rosner and Allendorf [5] from surface recession of a filament exposed to a vacuum flow. The authors determined the A-to-P transition (vertical lines). At the beginning of the active oxidation, the weight loss or erosion ratio (SiC evaporated molar flow versus O incident atomic flow ratio) is constant and directly related to a constant incident oxygen flow, but when temperature increases the SiC vaporization contributes to the weight loss of the sample.

2 Torr ( 1 .

 21 the temperature increases from 2042 to 2267K at an input pressure of di-oxygen constant of 3.10 -2 Torr (4.0 Pa). Using atomic oxygen, the ratio is 0.26 for a pressure O(g) equal to 1.10 -33 Pa). These plateaus confirm the main consumption of the oxygen flow to form SiO(g) + CO(g), which justifies the choice of our abscissa in figure1. The erosion rates corresponding to these plateaus are shown in figure1. The values measured by Rosner and Allendorf[START_REF] Rosner | High Temperature Kinetics of the Oxidation and Nitridation of Pyrolytic Silicon Carbide in Dissociated Gases[END_REF] (the full blue marks in figure1correspond to figure 2 of their article) are just below the maximum theoretical value predicted by Honstein et al.[START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF]: the difference observed comes from the erosion rate lower than 1 in relation to the existence of an oxygen condensation coefficient lower than 1. The equation proposed by Honstein et al.[START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF] at equilibrium overestimates the observed erosion rate by a factor ≈ 3 compared to the data of Rosner and Allendorf[START_REF] Rosner | High Temperature Kinetics of the Oxidation and Nitridation of Pyrolytic Silicon Carbide in Dissociated Gases[END_REF].Conversely, under 1 bar total pressure, the experimental results are in agreement with the calculations based on diffusion fluxes in a boundary layer for partial pressures of oxygen in the range 10 -4 to 1 bar. Because the oxygen flux at the surface of SiC is lower when controlled by diffusion at high pressure than under Knudsen flux (a factor of 10 4 in Figure1), the difference between experimental results and calculations is not large enough to reveal a kinetic limitation from reaction (1a). When the partial pressure of oxygen in the inlet gas stream under atmospheric pressure decreases below 10 -4 bar, the rate of erosion increases to the value measured under vacuum. This feature also suggests that diffusion in the boundary layer is no longer a limitation of the oxidation flux for low oxygen content. The calculated lines presented in figure1correspond to the following equations: He-O2 , 1 bar: 𝑙𝑜𝑔 𝐸𝑟𝑜𝑠𝑖𝑜𝑛(/𝑚𝑔 • 𝑐𝑚 • ℎ ) = 0.9426 𝑙𝑜𝑔 (p /bar) + 3.4088 Ar-O2 , 1 bar: 𝑙𝑜𝑔 𝐸𝑟𝑜𝑠𝑖𝑜𝑛(/𝑚𝑔 • 𝑐𝑚 • ℎ ) = 0.9319 𝑙𝑜𝑔 (p /bar) + 3.7299 SiC-C , vacuum: 𝑙𝑜𝑔 𝐸𝑟𝑜𝑠𝑖𝑜𝑛(/𝑚𝑔 • 𝑐𝑚 • ℎ ) = 1.0072 𝑙𝑜𝑔 (p /bar) + 8.3733 SiC, A-to-P, vacuum 𝑙𝑜𝑔 𝐸𝑟𝑜𝑠𝑖𝑜𝑛(/𝑚𝑔 • 𝑐𝑚 • ℎ ) = 1.0003 𝑙𝑜𝑔 (p /bar) + 7.7658 5. Active-to-passive oxidation transition5.1. Experimental and theoretical determinationsThe oxidation transition from A-to-P is related to the incident oxygen flux, which is a key parameter for the onset of growth of a silica layer. The Arrhenius plot of the experimental values is shown in Figure3.

Figure 3 .

 3 Figure 3. Arrhenius plot of the experimental literature values corresponding to A-to-P transition for SiC oxidation compared to theoretical calculations. The oxygen partial pressure is measured at the entrance of the device. Red marks are for β-SiC (or CVD-SiC), blue for α-SiC and gray for composites coated with CVD-β-SiC. Experimental data: Gulbransen (1966, 1972) × α-SiC vacuum; Rosner (1970) ◊ 1 Torr Ar-O2, ♦1 Torr Ar-O(g); Antill (1971)coated β-SiC vacuum; Hinze (1976) + Poly-β-SiC 0.2 bar; Keys (1977) × Sint. β-SiC 1 bar O2; Vaughn (1990) □ Sintered α-SiC 1 bar Ar+O2; Narushima (1991) □ α-SiC 1 bar Ar+O2; Balat(1992) Eck (2008) Wagner model ▬▬▬; Berton (1992) × α-SiC O2 0.01 bar;

Figure 4 .

 4 Figure 4. Schematics of A-to-P and P-to-A transitions. When O2(g) pressure increases up tothe formation of the silica scale (red arrow), silicon activity increases from the SiC-C limit up to the A-to-P transition. Note that according to Honstein et al.[START_REF] Honstein | Thermodynamic approach to the vaporization and growth phenomena of SiC ceramics. II. The SiC surface under oxidative conditions[END_REF] activity of Si at transition is lower than for congruent vaporization. When the oxygen pressure is no longer applied, the de-oxidation step should follow congruent vaporization at constant silicon activity till the disappearance of the silica scale i.e. P-to-A. Then pure SiC loses silicon (blue arrow) till the SiC-C phase limit and finally C precipitates at the surface.

6. 1 . 3 .

 13 Interphase stabilityThe formation of silicon oxycarbons has also been predicted by atomic scale modeling. Using ab-initio molecular dynamics, Da Silva et al. [100] found that for low carbon concentrations, a quartz-like structure Si2CO6 can be stabilized as an accommodation layer between SiO2 and SiC. More recently, using first-principle density function calculations, Wang et al. [101] have suggested that a dense interface layer composed of Si(1-x)CxO2 (e.g. cristobalite SiCO4 or α-Quartz Si2CO6-like structures) can play a critical role in the thermal oxidation process of SiC.

6. 1 . 4 .

 14 Carbon at the SiC / SiO2 interface Some authors have also observed the presence of a layer of pure carbon at the SiC / SiO2 interface. Chang et al. [102] observed by TEM a 1-1.5 nm thick layer of C at the 6H-SiC interface oxidized up to 40 nm of silica. By using AFM after HF etching of the silica layer Afanas'ev et al. [103] observed carbon clusters at the interface between SiO2 and 4H-SiC (Si terminated surface). However, these clusters were mainly attributed to the original crystal surface. Katsui et al. [104] observed by Raman and TEM spectroscopy the formation of a carbon

6. 1 . 5 . 5 x 5 )

 1555 photoemission techniques coupled to a synchrotron to clarify the role of carbon at the SiC / SiO2 interface. For this purpose, they started from well-defined original SiC surface quality analyzed before oxidation. Using AES, XPS and LEED, Powers and Somorjai [106] compared original ultra-vacuum (UHV) treated SiC surfaces in the 300 to 1373 K range. They observed competition between oxidation by the residual gases (the residual pressure was 10 -8 mbar in their apparatus), which produces O and C contaminations as well as a sublimation of the silicon until a graphitization of the SiC surface above 1273K. More importantly, they noticed that oxygen is bound to two Si sites and not to C on the SiC surface, and that the rate of adsorption of Si (diffusion from bulk SiC) to the surface (probably at the C sites) is rapid up to the formation of the first monolayer of silica.The nature of the interface constructed at the first moment of the oxidation process depends on the nature of the initial SiC surface before the beginning of the growth of the oxide. Indeed, throughout the heating ramp and whatever the atmosphere, the SiC surface degasses the absorbed species (low temperature desorption ≈ 373-423K), then adsorbs the species (up to ≈ 923-1123K) and begins to evaporate significantly. This can modify or eliminate the native silica layer and change the structure and composition of the extreme surface.Martensson et al. [107] published a review of the surface reconstruction of 6H-SiC (0001) (the Si-terminated polar face) when heated under ultra-vacuum (UHV). At room temperature, a SiC surface cleaned by the standard RCA chemical process exhibits a surface reconstruction (1 x 1) covered with a thin layer of native silica. When heated to 1123K under UHV, the silica layer is evaporated and at 1223K a reconstruction (√3 × √3) takes place, which corresponds to 1/3 of the adsorbed Si atoms bonded in the SiC surface bilayer. This bilayer is depleted in Si as evidenced by the valence band of silicon different from that corresponding to solid SiC. Such a surface structure composed of Si ad-atoms on a C-rich SiC surface bilayer has been confirmed by Coati et al. [108] and Simon et al. [109]. Heated to 1423K under UHV, a new surface reconstruction (6√3 × 6√ (3)) R30 ° (R30 ° for its rotation with respect to the underlying SiC layer) appears without graphitic bond, but still depleted in silicon. Slow heating to 1623K causes the formation of (5 x 5) and (6 x 6) surface structures with graphitic bonds. The two primary reconstructions above are entirely compatible with the preferential adsorption of oxygen on two atoms of Si proposed by Powers and Somorjai [106] and Hoshino et al. [91]. The oxidation treated on these reconstructed surfaces forms a very fine mixed intermediate layer and an upper layer of SiO2 without any carbonaceous contamination according to Virojanadara and Johansson [110, 111] and Watanabe et al. [112]. Using a molecular beam source, Amy et al. [113] carried out a preliminary deposit of Si in order to saturate the carbon-rich surface with Si ad-atoms. They obtained a SiC / SiO2 interface without C and a transition layer a few nm thick. Using ab-initio

  at the expense of SiC and oxygen diffuses through both the silica layer and Si-C-O intermediate layer where interfacial exchange reactions occur. Consequently, any Si-C-O intermediate layer observed must include numerous defects due to the absence of Si atoms adsorbed creating either vacancies or cavities generally associated with amorphous silica. On the SiC side of this thin interface a permanent diffusion of Si atoms towards the interface from SiC preferably adsorb on the C sites preventing oxygen from doing so. Therefore, the silica layer grows towards the SiC substrate.

(

  Sucov [149],Williams [135],Cappelen et al. [147]) thus causing a decrease in the density of SiO2. The ability of H2O to modify the silica network may be responsible -(i) for a lower ability of the silica layer to protect SiC and thus for its greater oxidation and -(ii) for the formation of an increased amount of bubbles due to the increased production of gas species by the oxidation reaction (Opila et al.[START_REF] Opila | SiC Recession Caused by SiO2 Scale Volatility under Combustion Conditions: II, Thermodynamics and Gaseous-Diffusion Model[END_REF]).Using a moderate partial pressure of H2O, several authors observed the promoting effect of H2O on the crystallization rate of SiO2 (Jorgensen et al. [3], Ainger [150], Ogbuji [151], Tressler et al. [152], Maeda et al. [153], Opila et al. [70]. When the water vapor pressure is high (> 0.15 MPa), the formation of two SiO2 sublayers is observed: a porous cristobalite layer above a thin and dense amorphous SiO2 surface layer (More et al. [154], Tortorelli and More [155], Cheng and Tortorelli [102]). Such high water pressures lead to the formation of very thick cristobalite layers above 1273K whose thickness increases with the partial pressure of water vapor.

  reactions, and in part to the adsorption step of molecular oxygen (sticking or condensation coefficient <1) since the use of an equivalent oxygen flux but composed of atomic oxygen increases the rate of erosion. Partial pressure of oxygen corresponding to the transition from the active oxidation regime to the passive regime -that is to say the formation of silica on the surface of SiC -has been The different parabolic growth regimes of silica have been discussed in relation to the diffusing species but without very convincing results. Other causes have therefore been sought, and in particular that of the influence of an interface layer between SiC and SiO2. It should be noted here that little work has been devoted to amorphous silica, the structure of which can be very different from one growth to another, which modifies the value of the diffusion or permeability coefficients. Similarly, the possibility of crystallization of amorphous silica in the form of cristobalite during growth is also source of dispersion in the diffusion of species in the thickness of the silica layer and therefore in the activation energy of its growth. A comparative and exhaustive study of the experimental diffusion coefficients of O2, O 2-, CO, SiO and Si could help to explain these differences (this work is ongoing).Studies on the nature of the interface between SiC and SiO2 have shown the presence of carbon or mixed C-Si-O bonds in some atomic layers between silica and SiC. Compressive mechanical stresses were also observed in these interface layers. Studies on the surface of SiC -in terms of composition or structure -show the rapid depletion of silicon as the temperature increases, followed by the formation of C-C (graphitic) bonds, while the incident oxygen reacts preferentially with the Si atoms adsorbed on the remaining carbon sites. These observations are to be put in perspective with the preferential vaporization of Si (the SiC does not vaporize congruently) coupled with diffusion of Si in SiC. Oxygen therefore binds to silicon to create the Si-O bonds precursors of silica. Two growth possibilities then exist: -(i) the interface propagates on the surface of SiC by trapping the C remaining ad-atoms, -(ii) the Si(ads) or the SiO(ads) formed vaporize and diffuse through the silica layer in the growth course. Given the amorphous nature of the silica formed, the existence of channels, voids or cavities favors, when the SiO2 layer is thin, the arrival of oxygen to the SiC / SiO2 interface and the departure of CO(g) (as well as other species such as SiO(g) and Si(g)) thanks to the permeability of the SiO2 layer. The interface is modified by the departure of Si, SiO and CO through the silica layer.

Table 2 .

 2 Equilibrium partial pressures and Si activity of the SiC compound for the A-to-P

	transition under vacuum calculated by Honstein et al. [73] under thermodynamic
	equilibrium assumption and comparison with the congruent vaporization in the SiC-SiO2
	pseudo-binary field during the scale disappearance, i.e. P-to-A transition. Temperatures used
	for calculations are those quoted by the authors.
	p / bar	SiO(g)	CO(g)	O2(g)		a(Si)
	Authors	Trans. Cong.	Trans.	Cong.	Trans.	Cong.	Trans. Cong.

   or the diffusion of oxygen ions O 2-on the silica network. Indeed, experimental measurements of the transport properties of these species are difficult to carry out and are very sensitive to the presence of impurities, in particular traces of H2O(Doremus [119]) or Na(Zheng et al. [120], Opila[START_REF] Opila | Oxidation Kinetics of Chemically Vapor-Deposited Silicon Carbide in Wet Oxygen[END_REF] 121]). In addition, in the range of parameters where O2 or CO species are expected to diffuse in their molecular state through the silica

	layer, its amorphous nature and microstructure (with varying amount of "channels or
	cavities") should have a significant influence on the transport processes. The
	microstructural properties of the silica layer probably also vary depending on the
	experimental parameters used to oxidize SiC (and in particular the surface preparation),
	resulting in a wide variety of activation energy values experimentally observed as listed
	by Jacobson et al. [69]. In addition, some authors mentioned the permeation and
	interstitial or reticular diffusion of species such as Si [122-127] or SiO [128-132], which
	can be formed at the interface as discussed above.

Lamkin and Ryley [117]

or

Molyneux [118]

. These species could be O2 diffusing from the SiO2 / gas interface towards the SiC / SiO2 interface, or CO diffusing in the opposite direction

6.2.1. Permeation and lattice diffusion

Zheng et al.

[START_REF] Zheng | Oxidation of Single Crystal Silicon Carbide. I. Experimental Studies[END_REF]

carried out the first isotopic study devoted to understanding the oxidation of SiC. They build Si

13

C epitaxial CVD films on Si 12 C substrates. After oxidation at 1473 or 1673K, SIMS depth profiling showed no gradient of carbonaceous species in the silica layer, indicating that these species were not responsible for the kinetic control of the oxidation reaction.

Vickridge et al. [132]

confirmed this result by measuring the

18

O enrichment of a 100 nm thick SiO2 layer annealed in C 18 O (1373K, 350 mbar, 1 h). The excitation curve revealed an almost constant CO concentration profile after the initial surface exchange, thus indicating a very high CO diffusion coefficient. These two results disagree with several other propositions

(Pultz [133]

, Singhal

  proposed a modified version of the DG model, which they called the para-linear model. A consequence of the competition between the growth rate and the vaporization rate of SiO2 is observed at long oxidation times: the appearance of a limiting thickness of the silica layer as a result of a steady state equilibrium between the two competing phenomena (Hui et al. [145], Pham et al. [146]). Most authors agree that when H2O is added to O2 the oxidation rate of SiC increases (Opila [70] , Deal and Grove [88], Cappelen

Acknowlegements : The present work has been performed from scientific demands in the framework of the GDR TherMatHT (Groupe de Recherche Thermodynamique sur les Matériaux à Haute Temperature), a Research Group supported by CNRS-France during the 2013 -2021 period.

energy (typically about 590 kJ / mole). Brisebourg et al. [START_REF] Brisebourg | Oxidation of β-SiC at high temperature in Ar/O2, Ar/CO2, Ar/H2O gas mixtures: Kinetic study of the silica growth in the passive regime[END_REF][START_REF] Brisebourg | Oxidation of β-SiC at high temperature in Ar/O2, Ar/CO2, Ar/H2O gas mixtures: Active/passive transition[END_REF] observed for the linear and parabolic regimes that the growth rate of SiO2 is as expected roughly proportional to the partial pressure of O2 at low temperature and becomes independent of the composition of the gas phase above 1923K. 6.2.2. Exchange between oxygen network and oxygen permeating inside the silica layer Some authors have also proposed the possible exchanges between the oxygen of the network and the transported oxygen in the form of O2 or CO by the permeation mode through the SiO2 layer. Using double oxidation experiments, Zheng et al. [START_REF] Zheng | Oxidation of Single Crystal Silicon Carbide. I. Experimental Studies[END_REF] observed the incorporation of 18 O in the initially oxidized sample both in the superficial and interfacial regions but also in the intermediate region, unlike the case of Si oxidation. However, this point was not confirmed by Vickridge et al. [132] who observed that O2 species have almost no exchange with the silica matrix. On the other hand, these last authors showed that a significant exchange between the oxygen of the matrix and the oxygen of CO is observed during its permeation.

Amorphous silica and crystallization

Depending on both temperature and composition of the gas phase, the silica layer formed can remain amorphous or crystallize, which can affect the diffusion properties of the species through the SiO2 layer. ß-cristobalite is the stable form of crystallized SiO2 above 1743K but it undergoes a β-to-α transition at about 543K according to Opila [121] or at 846K according to Liu et al. [START_REF] Liu | High temperature oxidation behavior of SiC coating in TRISO coated particles[END_REF] upon cooling. This allotropic transformation is accompanied by a volume contraction, which generates cracks in the silica layer (Opila [121]). Presser et al. [136] studied the size and growth rate of cristobalite crystals during the dry thermal oxidation of single crystal silicon carbide (6H-SiC) at 1673K. They showed that the process starts at the SiO2 / atmosphere interface from the beginning of the oxidation. They describe the progressive formation of cristobalite "radialites" embedded in the entire amorphous layer. By measuring the thickness of these "radialites" and that of the remaining amorphous part, they deduce a ratio ≈ 4.2 between the parabolic rate constants of vitreous silica and cristobalite. This value is close to the ratio of the effective oxygen diffusion coefficients in the two phases ≈ 2.8 as measured by Rodriguez-Viejo et al. [137]. Differences in oxygen solubility in the two phases can also explain the difference between the two values. This ratio is much lower than the decrease in growth rate, which can reach the value of 30 times as reported by Ogbuji and Opila [138]. However, the different temperature regimes used by these authors may partly explain this difference as well as the real nature of the silica layer -in particular the number of voids and channels and the geometry -. Presser et al. [139] also underlined the influence of impurities on the formation of cristobalite. In particular, they have shown that the presence of alkaline or alkaline-earth elements causes a structural transition of cristobalite from disc-shaped "radialites" to small crystalline spheres called "globulites". This transformation is catalyzed by a melt in relationship to the impurities formed above the devitrified zones. As a result, a vitreous network forms a backbone in the silica layer, which acts as an accelerated diffusion pathway for O2. Ramberg and Worrell [140] also observed that impurities stabilize a small amount of interconnected amorphous domains, which is responsible for increasing the parabolic constant leading to a change in activation energy at high temperature. In the absence of impurity, these last authors observed in various experiments by imposing an increased partial pressure of oxygen. These experiments show some dispersion, which can be attributed mainly to the flow regime in the reactors used, because these flows control the oxygen amount arriving at the surface of SiC.

The choice of the partial pressure of oxygen at the inlet of the reactor (or in the vacuum chamber) is practical but not relevant because the transition oxidation regime depends on the stationary partial pressure of oxygen near the surface of SiC, which is imposed by the surface reaction. Among these models, those calculating the partial pressure of oxygen in the stationary state at the surface of SiC, as well as the state of the SiC compound characterized by the activity of silicon (or carbon), are the most relevant. Indeed, the active oxidation regime preceding the transition imposes the stoichiometry of SiC via the silicon activity at the time of formation of the silica layer. It should be noted that the composition of SiC varies very little when going from the SiC+Si to the SiC+C two-phased domains, while the corresponding activities of Si and C vary greatly, which is the feature of a defined compound.

Using partial pressures of oxygen at the inlet of the reactors, the passive-to-active transition, i.e. the disappearance of the silica layer, shows large differences in experimental determinations: over 4 orders of magnitude. This strong dispersion does not arise from some chemical "hysteresis" behavior but from the experimental delay for observation mainly related to the thickness of the silica layer and therefore to the vaporization time of this layer. From a thermodynamic reasoning concerning the oxygen potential at the surface of oxidized SiC, and by considering the vaporization phenomena of silica and those of the congruent vaporization of SiC-SiO2 mixtures studied elsewhere, the oxygen potential at the interface of oxidized SiC appears very close to the congruent vaporization of the pseudo-binary SiC / SiO2. This thermodynamic reasoning does not take into account the vaporization kinetics, which produce partial pressures much lower than the equilibrium pressures. As the SiC/SiO2 interface is thermodynamically stable, observations of low vapor pressures for CO(g) and SiO(g) as well as their ratio close to 1 let suppose that these species originate from the voids and channels that are interconnected to the SiC surface probably at the triple line SiC-SiO2-gas. Nevertheless, the relative values of the oxygen flows between A-to-P and P-to-A remain of the same order of magnitude as shown by the few experiments carried out in the same run when the silica layer first obtained has a limited thickness before the de-oxidation step. The small difference is confirmed by thermodynamic reasoning on vaporization / diffusion coupling flows in the silica layer.

Passive oxidation of SiC corresponds to the growth of the silica layer under different oxygen pressures higher than those of the active to passive transition. Two regimes are observed: a first one so-called linear regime associated with the oxidation reaction at the SiC / SiO2 interface, and a second one so-called parabolic regime governed by the diffusion of oxygen in the silica layer, the diffusion of carbonaceous species (mainly CO) being much faster. These two regimes have very different activation energies. The linear regime corresponds to small thicknesses of the silica layer and can therefore only be measured using specific instruments. The experimental activation energies obtained for the parabolic regime appear very dispersed and depend on the temperature range. In fact, at high temperature, the amorphous silica crystallizes in the form of cristobalite. This latter variety has a lower oxygen diffusion coefficient than amorphous silica.