J Magnant

R Pailler

Y Le Petitcorps

L Maillé

A Guette

J Marthe

E Philippe

Fiber-reinforced ceramic matrix composites processed by a hybrid technique based on chemical vapor infiltration, slurry impregnation and spark plasma sintering

Keywords: A, Sintering, B, Composites

Fabrication of multidirectional continuous carbon and silicon carbide fiber reinforced Ceramic Matrix Composites (CMC) by a new short time hybrid process was studied. This process is based, first, on the deposition of fiber interphase and coating by chemical vapor infiltration, next, on the introduction of silicon nitride powders into the fibrous preform by slurry impregnation and, finally, on the densification of the composite by Liquid-Phase Spark Plasma Sintering (LP-SPS). The homogeneous introduction of the ceramic charges into the multidirectional fiber preforms was realized by slurry impregnation from highly concentrated and well dispersed aqueous colloid suspensions. The chemical degradation of the carbon fibers during the fabrication was prevented by adapting the sintering pressure cycle. The composites manufactured are dense. Microstructural analyses were conducted to explain the mechanical properties achieved. One main important result of this study is that LP-SPS can be used in some hybrid processes to densify fiber reinforced CMC.

Introduction

Non-oxide Ceramic Matrix Composites (CMC), i.e. mainly those consisting of a silicon carbide (SiC) or silicon nitride (Si 3 N 4 ) based matrix reinforced with either carbon or SiCfibers, have the potential for being used up to about 1500°C, as structural materials, in different fields including advanced engines, gas turbines for power/steam co-generation, heat exchangers, as well as nuclear reactors of the future [START_REF] Naslain | SiC-matrix composites: Nonbrittle ceramics for thermo-structural application[END_REF][START_REF] Jones | Radiation resistant ceramic matrix composites[END_REF].

CMC are processed according to (i) gas phase routes (CVI: Chemical Vapor Infiltration) [START_REF] Naslain | The CVI-process: State of the art and perspective[END_REF], (ii) liquid phase routes either from polymers (PIP: Polymer Impregnation and Pyrolysis) [START_REF] Interrante | Fabrication of SiC matrix composites using a liquid polycarbosilane as the matrix source[END_REF][START_REF] Takeda | Strength of a Hi-Nicalon/silicon-carbide-matrix composite fabricated by the multiple polymer infiltration-pyrolysis process[END_REF][START_REF] Rak | Novel CMC material based on a Si 3 N 4 matrix[END_REF], or molten elements reacting with mineral charges in the preforms or with the atmosphere (RMI: Reactive Melt Infiltration) [START_REF] Hillig | Making ceramic composites by melt infiltration[END_REF], or finally (iii) ceramic routes (SI-HP: Slurry Impregnation and Hot Processing) [START_REF] Katoh | A novel processing technique of silicon carbide-based ceramic composites for high temperature applications[END_REF][START_REF] Grenet | Carbon fibre-reinforced silicon nitride composites by slurry infiltration[END_REF][START_REF] Veyret | Hi-Nicalon reinforced silicon nitride matrix composites[END_REF]. Nevertheless, the time and cost production of CMC are some large obstacles that hamper their wider acceptance. Some hybrid processes, based on the combination of gas, liquid and ceramic routes, have been developed to optimize the densification of the fiber preforms, the microstructure of the composites and to reduce the time and cost production: CVI + (SI-MI or RMI) [START_REF] Xu | Carbon/silicon carbide composites prepared by chemical vapour infiltration combined with silicon melt infiltration[END_REF][START_REF] Hillig | Melt Infiltration Approach to Ceramic Matrix Composites[END_REF][START_REF] Luthra | Toughened Silcomp composites, Process and preliminary properties[END_REF][START_REF] Kochendörfer | Low Cost Processing for C/C-SiC Composites by Means of Liquid Silicon Infiltration[END_REF][START_REF] Krenkel | Cost Effective Processing of CMC Composites by Melt Infiltration (Lsi-Process)[END_REF], (PIP) n + CVI [START_REF] Suzuki | Fabrication and Characterization of 3D C/SiC Composites via Slurry and PCVI Joint Process[END_REF],

CVI + (SI+HP) as the NITE process (for Nano Infiltration of Transient Eutectoid) [START_REF] Kohyama | Development of SiC/SiC composites by nanoinfiltration and transient eutectoid (NITE) process[END_REF][START_REF] Katoh | Microstructure and properties of liquid phase sintered SiC/SiC composites[END_REF][START_REF] Shaoming | Preparation of SiC/SiC Composites by Hot Pressing Using Tyranno-SA Fiber as Reinforcement[END_REF][START_REF] Shimoda | Influence of pyrolytic carbon interface thickness on microstructure and mechanical properties of SiC/SiC composites by NITE process[END_REF][START_REF] Park | Efforts on large scale production of NITE-SiC/SiC composites[END_REF][START_REF] Dong | Preparation of SiC/SiC Composites by Hot Pressing, Using Tyranno-SA Fiber as Reinforcement[END_REF][START_REF] Suzuki | Preparation and properties of dense SiC/SiC composites[END_REF].

This last process was oriented to overcome the weakness of composites through other process routes for the improved matrix cracking stress and thermal conductivity [START_REF] Katoh | Microstructure and properties of liquid phase sintered SiC/SiC composites[END_REF]. The severe sintering conditions employed in NITE process require quasi-stoichiometric composition SiC fibers of third generation like Hi-Nicalon S, Sylramic or Tyranno SA fibers and thick PyC interphase. SiC f /PyC i /SiC m composites with very low porosity, high mechanical and thermal properties as well as an excellent hermiticity with respect to gaseous cooling fluids such as helium were successfully produced at 1780°C / 20 MPa by NITE process.

In this work, a new short time hybrid process was developed to (i) reduce the time and cost production, (ii) increase fiber preform densification and (iii) optimise the microstructure of the composite. This process is based on CVI fiber coating followed by Slurry Impregnation of ceramic powders into multidirectional continuous fiber preforms and then densification of the composite by Liquid Phase Spark Plasma Sintering (CVI + SI + LP-SPS).

Ex-PAN carbon fiber and SiC (Hi-Nicalon) fiber-reinforced Si 3 N 4 matrix composites (C f /Si 3 N 4m , SiC f /Si 3 N 4m ) were manufactured. Carbon fibers were used because of their unique high thermomechanical properties and their low cost, despite the fact that carbon undergoes oxidation at very low temperature (400-500°C). SiC (Hi-Nicalon) fibers of second generation are made of a mixture of SiC-nanocrystals (≈ 5 nm in mean size) and free carbon [C/Si (at) ratio ≈ 1.39] and were chosen for their excellent oxidation resistance and their cost which is about three times cheaper than quasi-stoichiometric SiC fibers of third generation. Hi-Nicalon SiC fibers creep at moderate temperature (1240°C) [START_REF] Bodet | Tensile creep behaviour of a silicon carbide-based fibre with a low oxygen content[END_REF][START_REF] Chollon | Structure, composition and mechanical behavior at high temperature of the oxygen-free hi-nicalon fiber[END_REF] but the soft sintering conditions achieve by SPS may limit their creep. Silicon nitride was chosen as constituent for the matrix because of its excellent thermomechanical properties, such as high strength, hardness, toughness, good shock resistance and good oxidation resistance [START_REF] Riley | Silicon Nitride and Related Materials[END_REF]. First, the CVI technique was used in this process for the deposition of pyrocarbon interphase (PyC) and SiC coating because it yields deposits of relatively uniform composition, structure and thickness even with preforms of complex architecture as those employed in this study. Next, slurry impregnation was used to introduce the ceramic powders into the fiber preforms since this method is cost effective and can be easily employed with fiber preforms of complex shape. Then, Liquid Phase Spark Plasma Sintering (LP-SPS) was used to densify the composites since SPS technique allows shorter densification time than traditional techniques which, firstly, is cost beneficial and, secondly, may limit the chemical reaction of fibers with matrix phases during the fabrication. Furthermore, to our knowledge, no study has successfully reported the fabrication of continuous fiber reinforced ceramic matrix by Spark Plasma Sintering.

In this paper, the different steps of our hybrid process and the first mechanical characterisations of these materials will be presented and discussed.

Experimental method

In order to identify the experimental parameters (pressure, temperature, compounds) of the sintering of composite matrix, first studies are developed on monolithic materials.

Monolithic matrix process

Ceramic powders

Sub-micronic silicon nitride powders + 7 wt% ytria (Y 2 O 3 ) + 3 wt% alumina (Al 2 O 3 ) sintering-aid powders were used (Table 1). Yttria and alumina sintering-aids react in temperature with the silica layer covering the surface of silicon nitride powders to form a liquid phase that firstly promotes the densification of Si 3 N 4 [START_REF] Riley | Silicon Nitride and Related Materials[END_REF] , and secondly, may accommodate the stress in the material due to external pressure and, as a result, limit possible fiber damage.

Sintering conditions for the monolithic matrix

Silicon nitride+yttria+alumina powders were mixed by wet ball-milling in polyethylene pots with ethanol and Al 2 O 3 balls during 24 hours. Then, mixed powders were dried and milled by dry ball-milling in polyethylene pots during 5 hours to eliminate the agglomerates. The powders were then fitted in a cylindrical carbon die with an inner diameter of 20 mm for Spark Plasma Sintering (Dr Sinter 2080, SPS Syntex Inc.). The sintering temperature varied from 1500°C to 1700°C, the heating-rate was set to 200°C/min and the holding period varied from 1 to 20 minutes. First, the powders were degassed under vacuum at a temperature of 700°C, next a mechanical pressure of 30 MPa was applied and finally a nitrogen atmosphere (P N2 = 0.02 MPa) was introduced in the chamber to avoid Si 3 N 4 decomposition at high temperature. The heating process was controlled using an optical pyrometer focused on the surface of the carbon die. It is important to notice that the temperature given is the temperature of the die and this one can be different from the temperature of the sample. Authors claim that the effective temperature is about 50-100°C higher than the one measured on the die surface [27][28]. The linear shrinkage of the monolithic matrix during sintering was recorded by an extensometer.

Composite materials process

Fiber preforms and interphases

The main characteristics of Ex-PAN carbon fibers and SiC Hi-Nicalon fibers are given in Table 2. It is well known that fiber/matrix bonding is a key factor controlling performances of ceramic matrix composites, a pyrocarbon interphase (PyC i ) was deposited on the fiber surface from gaseous precursor in order to arrest and deflect the matrix microcracks and to insure a load transfer function [START_REF] Naslain | The design of the fibre-matrix interfacial zone in ceramic matrix composites[END_REF][START_REF] Després | Mechanical properties of C/SiC composites as explained from their interfacial features[END_REF].

To minimize possible chemical reactions of carbon fibers with the matrix constituents during the fabrication, a SiC (c) coating was deposited by CVI on the PyC interphase (PyC i ).

The different combinations of PyC interphase, SiC coating and fibers used in this study are: {C f / PyC i }; {C f / PyC i + SiC c }; {SiC f (Hi-Nicalon) / PyC i + SiC c }.

Ceramic suspensions and its introduction into the fiber performs

In order to infiltrate the pwder into the preforms, aqueous ceramic suspensions have been realized.

Electrokinetic behaviour of Si 3 N 4 powders in aqueous suspension is governed by the ionisation of the surface groups according to the pH and has been extensively studied [START_REF] Mp | Processing of concentrated aqueous silicon nitride slips by slip casting[END_REF][START_REF] Bergstrom | Interfacial characterization of silicon nitride powders[END_REF][START_REF] Zhang | Dispersion of Si 3 N 4 powders in aqueous media, Colloids and Surfaces A: Physicochem[END_REF][START_REF] Ys | Consolidation of aqueous concentrated silicon nitride suspension by direct coagulation casting[END_REF][START_REF] Hruschka | Processing of βsilicon nitride from water-based α-silicon nitride, alumina, and yttria powder suspensions[END_REF]. being aware that above pH=11 the decomposition of Y 2 O 3 particles becomes important [START_REF] Hruschka | Processing of βsilicon nitride from water-based α-silicon nitride, alumina, and yttria powder suspensions[END_REF].To adjust the pH, tetramethylammonium hydroxide (TEAH, 35 wt% in water, Sigma-Aldrich Chemical Co.) was employed.

The rheological behaviour of the suspensions was investigated with a cone-plate rheometer (Rotovisco RT 20, Haake) by ascending shear rates ramp from 0 to 400 s -1 in 5 min to determine the exact pH value for which the viscosity of the suspension is the lowest. To avoid undesired influence from mechanical histories, fresh samples were homogenized by shearing at an identical shear rate of 100 s -1 for 1 min and left standing for an additional 1 min prior to measurement.

Different techniques can be used to introduce ceramic particles inside fiber preforms from colloidal suspensions: Electrophorese Impregnation (EPI) [36], submicronic powder aspiration [START_REF] Bernard | Method of densification of a porous substrate by a matrix containing carbon[END_REF] and slurry impregnation under vacuum. This last method was chosen in this work for its ability to be easily transposed at plant-level to impregnate preforms with complex shape. The impregnated preforms were then dried during 48h at 25°C. Some samples were cut, polished and characterised by optical microscopy.

Fiber preforms densification by spark plasma sintering

Silicon nitride + sintering-aid powder charged preforms were stacked in a graphite die for SPS. The Spark Plasma Sintering conditions are a compromise between minimizing fibers degradation and achieving full densification. The selection of the sintering conditions was based on the densification of monolithic Si 3 N 4 . The sintering temperature was 1650°C and hold 2 min at this temperature, while the pressure was varied from 30 to 150 MPa. Fibers degradation can be the result of:

(i) chemical reactions of the carbon fibers or pyrocarbon interphase with the matrix phases.

Luthra and Park have shown that Al 2 O 3 and Y 2 O 3 are stable with carbon provided there are no cracks for CO gas to escape [START_REF] Luthra | Chemical Considerations in Carbon Fiber/Oxide-Matrix Composites[END_REF]. However, silica undergoes carbothermic reduction at 1650°C with p CO < 300 kPa (equation 1). The silicon nitride can react as well with the carbon according to the equation 2.

SiO 2 (s) + 3 C (s) = SiC (s) + 2 CO (g) (1) Δ r G(1950°K) ≈ -53 kJ.mol -1 Si 3 N 4 (s) + 3 C (s) = 3 SiC (s) + 2 N 2 (g) (2) Δ r G(1950°K) ≈ -117 kJ.mol -1
Silicon carbide can react in a same way with silica or alumina to form CO (g) .

The fast kinetic of densification achieved by Spark Plasma Sintering may limit these reactions. Furthermore the sintering was conducted with the sintering chamber under N 2 atmosphere to limit both possible decomposition of Si 3 N 4 and reaction between carbon and silicon nitride (reaction 2).

(ii) physical degradations due to the external pressure applied.

Then, to minimize fiber degradation the pressure was applied only when the grain boundary phases began to soften but soon enough (1300°C) to limit gas evolution and thus to inhibit any possible reaction of Si 3 N 4 or secondary grain boundary phases with the pyrocarbon interphase or the fibers [START_REF] Luthra | Chemical Considerations in Carbon Fiber/Oxide-Matrix Composites[END_REF]. For comparison, some preforms were sintered with the pressure applied at 1500°C instead of 1300°C. Furthermore, to obtain a system as closed as possible to inhibit gas evolution, the preforms were embedded in a powder-bed before loading in the die. To determine the influence of the pressure media on the fiber degradation, a powder-bed of (boron nitride) BN+Si 3 N 4 and a powder-bed of graphite were used.

CMC characterization

The bulk density of the samples was measured by Archimedes method. The microstructures were characterised by scanning electron microscopy (Quanta 400 FEG, FEI) and energydispersive X-ray spectroscopy (EDAX) on polished surfaces. Crystalline phases were determined from X-ray diffraction patterns obtained in a θ-θ Bragg-Brentano configuration (D8 advance, Bruker).

The sintered composites were subsequently cut and ground into 45x12x2.5 mm 3 specimens for three-point-bending tests, with a crosshead speed of 0.5 mm/min and a span of 30 mm (Model 5569, Instrom Corp). The polished cross sections of the composites before bending tests were characterized by scanning electron microscopy, energy-dispersive X-ray spectroscopy on polished surfaces and transmission electron microscopy. A sintering temperature of 1650°C during 1 min allows a relative density of 99 % whereas a sintering temperature bellow 1650°C does not permit to achieve relative density upper to 68 % even for long sintering period (figure 4). The XRD phase composition of samples sintered 5 min at 1650°C and 1700°C are presented figure 2. Sample sintered at 1650°C is constituted of α and β-Si 3 N 4 grains whereas sample sintered at 1700°C is only constituted of β grains. As mechanical properties are changing according to the α/β phase ratio [START_REF] Kawaoka | Precise property control in silicon nitride ceramics by α/β phase ratio control[END_REF],

Results and discussion

Monolithic materials

one has to choose the couple of parameters (sintering temperature -sintering time) which corresponds to the required properties. In this study, as fiber-degradation needs to be avoided, a sintering temperature of 1650°C applied during 2 minutes was chosen.

Composite materials

The minimum in the viscosity of the ceramic powders suspension, required for infiltration, occurs for pH=10.5+/-0.1 (figure 5). As a consequence, the sintering-aids, i.e.

Y 2 O 3 and Al 2 O 3 powders, were added to the 33 vol.% Si 3 N 4 suspensions prepared at pH=10.5. The sintering-aid powders in the Si 3 N 4 suspension do not destabilize it (figure 6).

After the slurry impregnation step, the inter-tows porosity of the fiber preforms is filled homogenously by the mineral charges (figure 7).

The state of densification by SPS of the composites as a function of the applied pressure is presented in the Table 3.

When a SiC coating is deposited around the carbon fibers, the pressure required to obtain a full dense matrix is more important than without SiC coating. This lies in the fact that the SiC coating withstands the transmission of the pressure to the matrix. The composite with the SiC coating is fully dense at 150 MPa, but a maximum pressure of 75 MPa is available for large mechanical testing samples due to SPS machine limitations. Hence mechanical tests were carried out on samples sintered at this pressure.

Microstructural caracterizations

The microstructures of the composites were studied in term of fiber degradation, densification of the inter and intra-tows matrix, microcracks inside the matrix, decohesion between the fibers and the matrix and will be successively discussed.

SEM micrographs and EDX analyses (figures 8, 9 and 10) reveal the influence of the pressure cycle and of the pressure media on the fiber degradation. When pressure is applied at 1500°C, the carbon fibers embedded in a powder-bed of BN+Si 3 N 4 are degraded which is not the case when the pressure is applied as soon as 1300°C. Furthemore, when a powder bed of graphite is used instead of a powder bed of BN+Si 3 N 4 and the pressure applied at 1300°C, the fibers are not degraded. As a result, this confirms that fiber degradation can be avoided by inhibiting gas evolution. This is in agreement with the results obtained by Luthra who observed that Al 2 O 3 and Y 2 O 3 are chemically compatible with carbon if there is no cracks for CO (g) to escape [START_REF] Luthra | Chemical Considerations in Carbon Fiber/Oxide-Matrix Composites[END_REF]. Furthemore, the chemical composition of the powderbed have no influence on this degradation. The degradation of the carbon fibers is the result of the chemical reactions between the carbon and the matrix phases, what is confirmed by EDX analysis (figure 9). Indeed, the reaction zone between the fiber and the matrix is constituted of silicon carbide which should mainly be formed according to reaction 2.

The SiC (Hi-Nicalon) fibers are chemically degraded after the sintering of the composite (figure 11 and12). Indeed, the elements Al, Y and O were detected at the periphery of the SiC+C fibers (figure 12). These elements have diffused inside the fiber from the sintering aids liquid phase or from its vapor. No chemical degradation was observed in the carbon fiber composites.

The inter-tows matrix of all composites is fully dense except in some areas located between crossing tows for C-fibers composites (figures 10, 11 and 13). The fiber architecture of the carbon fiber preform which is not the same than the SiC fiber preform is probably at the origin of this partial densification.

In C f /Si 3 N 4 composites, the fibers are bonded inside the tows by the pyrocarbon and eventually by the SiC coating whereas in SiC f /Si 3 N 4m composite the thickness of the pyrocarbon interphase and of the SiC coating is too low to bond correctly the fibers. Carbon fiber composites without SiC coating have some intra-tow areas densified by the Si 3 N 4based matrix. For carbon fiber composites with the SiC coating, the accessibility to the intra-tow porosity is impeded by the SiC layer prior to powder impregnation.

The matrix of the composite is constitued of β-Si 3 N 4 grains and α-Si 3 N 4 grains embedded in an amorphous intergranular phase (figure 14).

All C f /Si 3 N 4m composites exhibit microcracks after fabrication but not SiC f /Si 3 N 4m composites (figures 10, 11 and 13). These microcracks are the result of the mismatch between the Coefficients of Thermal Expansion (CTE) of carbon-fibers and Si 3 N 4matrix. Sabouret et al. have computed around 450 MPa the internal stress inside the matrix in unidirectional C fiber-reinforced Si 3 N 4 matrix composite (with E C-fiber =250 GPa) [START_REF] Sabouret | Composites à matrice Nitrure de Silicium et fibres de carbone : Elaboration, comportement mécanique[END_REF]. In the multidirectional C-fiber arrangement composites studied here, the repartition of the thermal stress inside the matrix is more complicated than in unidirectional composites but is higher than the intrinsic stress at failure of the Si 3 N 4 matrix.

No important decohesion between the fibers and the interphase/coating is observed in carbon fiber composite. In contrast, SiC fiber composites show important heterogeneous decohesions (figure 11). SiC (Hi-Nicalon) fibers are known to creep at moderate temperature (around 1240°C) [START_REF] Chollon | Structure, composition and mechanical behavior at high temperature of the oxygen-free hi-nicalon fiber[END_REF]. Hence, these decohesions can be attributed to strain differences between the SiC fiber and the SiC coating during sintering under pressure at 1650°C. As a consequence, the soft sintering conditions in SPS do not permit to limit enough the fiber strain. In carbon fiber composite the large thickness of PyC interphase can accommodate some strain differences between the constituents of the composite which is not the case in SiC fiber composite due to lower PyC thickness. Furthermore carbon fibers

show no important strain. Perspectives to limit the decohesions between the SiC fibers and the SiC coating are (i) to stabilize the fiber microstructure by a heat treatment prior to the SiC deposit [START_REF] Chollon | Structure, composition and mechanical behavior at high temperature of the oxygen-free hi-nicalon fiber[END_REF], (ii) to deposit a thicker PyC interphase between the fiber and the SiC coating, (iii) to formulate a Si 3 N 4 based matrix sinterable at lower temperature. These perspectives will be addressed in a further paper.

Some decohesions or cracks between the PyC interphase or the SiC coating and the matrix are observed only in C f /Si 3 N 4m composite. These decohesions are the result of thermal stresses produced by the mismatch between the CTE of the C-fiber and the matrix or by differences between the strain of the fiber preform and the one of the matrix.

Mechanical Behavior

The stress-displacement curves obtain during three-point bending mechanical tests are presented in figure 15. SiC-fibers composites have a brittle behavior with a stress at failure relatively low (≈ 150 MPa). This relatively low stress at failure can be attributed to the chemical degradation of the SiC-fibers during the fabrication, the absence of intra-tows matrix and to some decohesions between interphases and fiber. The brittle behavior is more probably due to the chemical degradation of the SiC-fibers. Others possibilities to explain the brittle behavior are a too thin PyC interphase and/or inhomogeneous bonding areas between the fibers and the SiC coating. C-fibers composites exhibit damageable behavior with the maximum stress at failure around 250 MPa. The presence of the SiC coating around the fibers enhances the mechanical properties. Hypotheses formulated to explain this point are that (i) the strength of the interface bond between the SiC coating and the matrix phases (more precisely the intergranular SiO 2 -Y 2 O 3 -Al 2 O 3 glassy phase) is stronger than the bond between the PyC interphase and the amorphous matrix phase, (ii) as the strain at failure of the composites with and without SiC coating are more or less the same, the increase in ultimate stress for the composite with SiC coating is due to the increase of the elastic modulus of the composite by the SiC coating (iii) the SiC coating prevented the fibers from chemical degradation during the fabrication. However, as fiber degradations were not observed, the last hypothesis is improbable. Characterisations are in progress to better understand the influence of the SiC coating on the mechanical properties of carbon fiber composites. For comparison, a raw preform (i.e. without matrix and not sintered), with a PyC + SiC fiber coating, was mechanically tested (figure 15). Even if the SiC coating contribute to consolidate the fiber preform, the influence of the matrix on the mechanical properties is obvious.

The mechanical behavior of the C f /Si 3 N 4m composites manufactured by our hybrid process is promising even if their stress at failure around 250 MPa is lower than the 20-years underdevelopment C f /SiC m composites fabricated by CVI (≈ 350 MPa) [START_REF] Eberling-Fux | Matériaux composites à matrice nanostructurée[END_REF]. Experiments should be conducted with different fiber architectures and different thickness of PyC interphases and SiC coating to find the more appropriate ones to this hybrid process.

Conclusion

The hybrid process developed here allows short-time fabrication of ceramic matrix composites. Firstly, a mechanical fuse pyrocarbon layer and a SiC coating were deposited around the fiber in the preform by Chemical Vapor Infiltration, secondly, silicon nitride and sintering-aid powders were introduced in the preform by slurry impregnation and, thirdly, ceramic-powder-charged-fiber preforms were consolidated by Spark Plasma Sintering. SiC Hi-Nicalon fiber-reinforced Si 3 N 4 matrix composites have a brittle-type mechanical behavior with a relatively low bending stress at failure (≈ 150 MPa) due to fiber chemical degradation and the lack of bond between fibers inside the tows or to some decohesions between interphases and fiber. C f /Si 3 N 4 composites can be produced with no chemical degradation of the fibers if they are embedded in a powder-bed during sintering and if the external sintering pressure is applied soon enough in temperature to inhibit gas evolution and thus fiber degradation. The value of the pressure required to fully sinter the matrix depends on the presence of a SiC coating because this one withstands the application of the pressure inside the preform. The carbon fiber composites show a mechanical damageable behavior and a maximum bending stress at failure around 250 MPa. As a consequence Spark Plasma Sintering can be used in some hybrid processes to densify multidirectional continuous fiber reinforced ceramic matrix composites.

List of figure and table captions

  By acoustophorometric measurement, Jung et al. shown that Si 3 N 4 M11 Starck powder has a maximum absolute zeta potential value in the pH range [10-11.5] [34]. Hruschka et al. studied the evolution of zeta potential versus pH for Al 2 O 3 and Y 2 O 3 aqueous suspensions and showed that in the pH range [10.5-11.5], Al 2 O 3 and Y 2 O 3 particles are respectively charged with the same sign as Si 3 N 4 particles[35]. As a result, mixed aqueous suspensions of Si 3 N 4 + 7 wt.% Y 2 O 3 + 3wt.% Al 2 O 3 particles were prepared in the pH range [10.5-11.5],

Figure 1

 1 Figure 1 shows the shrinkage and shrinkage rate curves of Si 3 N 4 + 7 wt% Y 2 O 3 + 3 wt%

Figures 2 and 3

 3 Figures 2 and 3 reveal the phases composition and the microstructures of Si 3 N 4 + 7 wt%

Figure 1 :

 1 Figure 1: Applied temperature and pressure, shrinkage and shrinkage rate of the sample

Figure 2 :

 2 Figure 2: XRD patterns of the Si 3 N 4 + sintering-aids samples as a function of the sintering

Figure 3 :

 3 Figure 3:. Si 3 N 4 + 7 wt% Y 2 O 3 + 3 wt% Al 2 O 3 monolithic matrix (a) secondary electron image of sample sintered at 1260°C with no holding period and (b) back-scattered image of fully dense sample sintered at 1700°C during 5 min.

Figure 4 :

 4 Figure 4: Evolution of shrinkage of Si 3 N 4 + 7 wt% Y 2 O 3 + 3 wt% Al 2 O 3 samples during

Figure 5 :

 5 Figure 5:. Viscosity vs. pH for the 33 vol.% Si 3 N 4 suspension at a shear rate of 150 s -1 .

Figure 6 :

 6 Figure 6: Influence of the sintering-aids on the rheological behaviour of the 33 vol.% Si 3 N 4 -based suspension.

Figure 7 :

 7 Figure 7: Optical micrographs of fiber preforms (a) before impregnation by the mineral

Figure 8 :

 8 Figure 8: SEM micrographs of C f / PyC i / Si 3 N 4m Spark Plasma Sintered composites

Figure 9 :

 9 Figure 9: EDX analysis of zones marked 1 (unreacted matrix zone) and 2 (reacted

Figure 10 :

 10 Figure 10: SEM micrographs of C f / PyC i / Si 3 N 4m Spark Plasma Sintered composites

Figure 11 :

 11 Figure 11: SEM micrographs of SiC f Hi-Nicalon / PyC i2 + SiC c2 / Si 3 N 4m Spark Plasma

Figure 12 :

 12 Figure 12: EDX analysis of SiC-Hi-Nicalon fibers in the SiC f / PyC i + SiC c / Si 3 N 4m Spark

Figure 13 :

 13 Figure 13: SEM micrographs of C f / PyC i + SiC c / Si 3 N 4m Spark Plasma Sintered

Figure 14 :

 14 Figure 14: TEM images of C f / PyC+SiC i / Si 3 N 4m composites showing (a) the different

Figure 15 :

 15 Figure 15: Stress/displacement curves from bending test for composites embedded in a

  Figure 2

3 Si 3 N 4 +

 334 Figure 5

Figure 7

 7 Figure 7

  , zone (1) reacted matrix and fiber zone, zone[START_REF] Jones | Radiation resistant ceramic matrix composites[END_REF]

  Figure 13

Table 1 :

 1 Main characteristics of the fibers used in this study and of Si 3 N 4 .

a At 25°C; b Temperature at which m = 0.5 in BSR-test; c Depending on processing conditions

Table 2 :

 2 Some characteristics of the powders used.

Table 3 :

 3 State of densification of the matrix inside the composites as a function of the mechanical pressure applied during Spark Plasma Sintering.

Table 1 Si 3

 13

		300
		250
	(MPa)	150 200
	stress	50 100
		0
		0	1	2	3	4	5	6
	C / PyC+SiC / without matrix	displacement (mm)
	C / PyC / Si3N4
	C / PyC+SiC / Si3N4
	SiC (Hi-Nicalon) / PyC+SiC / Si3N4
				N 4	Y 2 O 3		Al 2 O 3
		provider	HC Starck HC Starck Alfa Aesar
		grade		M11	C		-
		Specific		12-15	10-16		10
		surface area

Table 2

 2

		Density	Tensile	Young's	Strain to	CTE	Thermal	Creep
			strength	modulus	failure	(10 -6 K -1 ) a	conduct.	parameter
			(MPa)	(GPa)	(%)		(W.m -1 K -1 ) a	(T 0.5 )(°C) b
	Ex-PAN C-	1.77	2100-2400	230-297	1.05-0.71	-0.1 -10	17	/
	fibers
	SiC Hi-	2.74	2800	270	1.0	3.3-3.5	5.0-7.8	≈ 1240
	Nicalon
	fibers
	Si 3 N 4	3.44	800-1600 c	210	≈ 0.004	2.8-3.5	15-40

c

Table 3

 3 The thicknesses of PyC i interphase and SiC c coating are different in C f and in SiC f composites. PyC thicknesses of C f / [PyC i1 ] and of C f / [PyC i1 +SiC c1 ] are the same.

		30 MPa	50 MPa	75 MPa	150 MPa
	C f / [PyC i1 ] *	partially	partially	full densified	full densified
	C f / [PyC i1 +SiC c1 ] *	partially	partially	partially	full densified
	SiC f / [PyC i2 +SiC c2 ] *	partially	full densified	full densified	full densified
	*

Acknowledgements

This work was supported by DGA (French Direction Générale pour l'Armement) and SAFRAN-Snecma Propulsion Solide through a grant. The authors acknowledge S. Bertrand (SAFRAN-Snecma Propulsion Solide) for material supply; G. Chevallier, C. Estournes and G. Raimbeaux (PNF 2 , University of Toulouse, France) for Spark Plasma Sintering tests and P. Weisbecker (LCTS) for TEM analysis.


E Jacques

Y Le Petitcorps

L Maillé

C Lorrette

L Chaffron

Joining Silicon Carbide Plates by Titanium Disilicide-Based Compound

Keywords: Brazing silicon carbide, Ceramic metal interface, Ceramic Matrix Composites, Intermetallic and disilicide compounds

The present work is part of the Fourth Generation reactors program where is described the methodology and the results for joining SiC substrates by metallic silicides with SiC powder reinforcements. The more severe temperature in service are in the range of 1000°C but short time incursions at 1600 or 2000°C have to be anticipated. One of the key issues is the joining of the SiC f /SiC m composites to seal the combustible cladding. We describe the results for joining SiC substrates at liquid state using TiSi 2 . Joint integrity and joint strength can be improved by adding small SiC particles to the silicides powders. The assemblies are performed in an inductive furnace. Cross sections of the assembly, wettability tests, thermo-mechanical properties and 4-bending point mechanical tests are presented.

INTRODUCTION

Joining ceramic/ceramic composites (CMC's) is of interest for numerous applications in the aeronautic or nuclear fields. This joining may concern the assembly of two CMCs parts or the assembly of one CMC part against a metallic alloy. This study deals with the assembly of two silicon carbide/silicon carbide composite plates designed for a fuel cell component within the framework of the Fourth Generation Reactor Program [START_REF] Chaffron | Innovative SiC f /SiC composite Materials for Fast Reactor Application[END_REF]. In order to achieve this goal, many requirements have to be followed: (i) the joint must be tight enough to avoid the gas leaks in service and, (ii) has to withstand temperatures such as 1000°C for very long durations (service conditions) and 1500°C for short durations (accidental conditions) without damaging the CMC properties, (iii) the process has to be compatible with the nuclear constraints and must not damage the composite properties which are very sensible to fibre degradations.

The composite material for this application is made of a woven silicon carbide fibre perform (Hi-Nicalon Type S TM or Tyranno SA3 TM fibre) infiltrated by a silicon and carbon based precursor (CVI process).

Both the matrix and the fibre are nearly pure silicon carbide which gives to the composite a good thermal stability up to 1600°C. The CMCs assembly research is not a new but this requirement in temperature stability (1500°C) makes the answer not trivial [START_REF] Riccardi | High temperatures brazing for SiC and SiC f /SiC ceramic matrix composites[END_REF][START_REF] Riccardi | Issues of low activation brazing of SiC f /SiC composites by using alloys without free silicon[END_REF][START_REF] Riccardi | Low activation brazing materials and techniques for SiCf/SiC composites[END_REF][START_REF] Jacques | Preliminar results on joining of thin SiC/SiC composites by silicides coumpounds and local heating[END_REF]. Conventional silicon based compositions cannot be used since the melting point of Si is too low. Thus a material selection study has been done in order to select compounds able to withstand these requirements. The selection of the right composition for the joint has been done from the physical properties, the thermodynamic data of the different candidates in order to avoid a too important mechanical and chemical mismatch with the silicon carbide. Various assembly processes have been evaluated (solid state bonding, liquid state bonding) and several ways of short heating have been tested (microwave, High Frequency heating). The quality of the assembly has been evaluated through metallographic observations on polished sections or with tomography images on the whole assembly.

MATERIALS AND METHODS

Material selections

The silicon carbide material is a covalent crystal characterised by a low coefficient of thermal expansion CTE, a high modulus, a high melting point and a intermediate toughness which makes the ceramic sensible to flaws (Table 1). Unfortunately the silicon carbide reacts at medium temperature (above 800°C) with most of the refractory metals. Thus these metals cannot be chosen which purpose. However the analysis of the ternary phase diagram involving Si, C and Metallic atoms indicates that some compositions can be in a thermodynamical equilibrium with SiC (Figure 1). The carbide (here TiC x ) and the disilicide (here TiSi 2 ) are thermodynamically stable with SiC in the solid state whereas the other silicides react with SiC. In addition to a poor thermal expansion mismatch with SiC, TiC has not been chosen due its too high melting point which could be a problem if the joint has to be melted for the assembly. The remaining choice is the metallic disilicide. This disilicide is related to a ionic-covalent crystals which exhibits an intermediate Young's modulus and a higher CTE than SiC. Due to this CTE mismatch, the joint will be in tension and the silicon carbide in compression on cooling. One way to reduce the CTE of the joint is to incorporate SiC particles within the silicide. By controlling the volume fraction of SiC added, the CTE and the Young's modulus can be tailored. Due to the relatively high thermal stability of TiSi 2 , this disilicide has been selected. The TiSi 2 compound will be reinforced by SiC particles (less than 1m in size) and with different volume fraction (from 0 to 30%).

Material heating

Since the disilicide do not react with the silicon carbide substrate in the solid state, the bonding between the joint and the SiC substrate should not be strong enough. In order to get this bonding, the melting of the joint should be done very rapidly in order to promote and to limit the chemical reaction with the silicon carbide substrate. A microwave furnace (2450MHz) can be used for melting ceramics which have an effective electrical conductivity between 1 and 10 Sm -1 such as semi-conductors. It is supposed to be a more economical process than the other conventional processes like the high frequency induction furnace.

The principle was to selectively heat the joint and not the silicon carbide substrate. Thus the joint could be melted without heating the composite itself. Unfortunately the silicon carbide absorbs at the same wave than the titanium disilicide. Finally, a more conventional high frequency furnace has been chosen (60% of power -Five Celes HF MP3, 12kW). With that equipment, the joint between two silicon carbide plates is melted in less than 1 minute and cooled down for few seconds later after the melting. The assembly is maintained by a graphite susceptor inside a chamber under a purified argon atmosphere. The wettability is of great importance in the brazing technology and has been studied for the different compositions by recording the value of the contact angle during the spreading of the drop.

Material characterisations

In order to characterize the physical properties of the joint (CTE and Young's modulus) model materials of TiSi 2 + x% of SiC p were prepared by spark plasma sintering. After sintering, the materials were nearly fully dense (95% of the theoretical density). The CTE has been measured by a TMA device (dilatometer Netzsch DIL402C) and the Young's modulus measured from the room temperature to 1000°C was determined by a Grindosonic apparatus (LEMMENS Inc.).

The quality of joining has been studied on polished cross sections by SEM and WDS analysis (Hitachi S4500). A particular attention has been paid to the occurrence of cracks in the joint, the thickness of the reaction zone between the melted joint and the SiC substrate, as well as its chemical composition. The macroscopic characterisation of the joint has been done by X ray tomography (Phoenix X Ray nanotom S).

RESULTS AND DISCUSSIONS

Decreasing the residual stresses in TiSi 2 by the addition of submicronic SiC particles (SiC p )

Although the values are below the data expected from a rule of mixture (ROM), the SiC p have a positive effect on the Young's modulus from room temperature to 1000°C (Figure 2). This difference from the ROM can be explained by the submicronic size of the particles, the aspect ratio of them which is closed to one and the presence of 5% of porosities. A 10% decrease in the CTE is also measured on TiSi 2 reinforced with 20% in Vo. of SiC p .

The consequences on the network of cracks in the joint can also be seen on figure 3. In order to avoid a chemical reaction between the TiSi 2 and the SiC from both the particles and the substrate, the material was also prepared by spark plasma sintering at 1200°C. The presence of the SiC particles induces a decrease in the residual stresses within the joint which exhibits also a clear interface with the silicon carbide substrate.

Wettability of TiSi 2 + SiC p

For pure TiSi 2 , the contact angle decreases from 80° to 12°, 4 seconds after the melting (Figure 4) TiSi 2 has a very low contact angle which is very useful for its infiltration in small cavities of the materials to join. The reason of this kinetic of wettability can be explained by the reactivity of the liquid Ti-Si alloy with the SiC substrate [START_REF] Gasse | [END_REF]. When SiC particles are added, the contact angle is increased to 30° for 30%Vo.

of SiC p after 4 seconds in the liquid state. The particles induce an increase in the viscosity as well as they react with the titanium disilicide matrix. However, the contact angle measured with 30% of SiC is very closed to the contact angle measured with pure silicon. The contact angle and the kinetic of spreading can be taylored by the amount of SiC p . It is important to control the time during the melting step in order to limit the reaction zone thickness between the joint and the SiC substrate.

Reactivity of liquid (Ti,Si) + SiC p with the SiC substrate

In the liquid state, the liquid containing Ti and Si atoms should react with SiC. From the SiC to the TiSi 2 , the interface is composed of Ti 3 SiC 2 and TiC (Figure 5). If the joint contains SiC p they also should react with the liquid and the liquid should be less aggressive against the SiC substrate. Consequently, the reaction zone thickness at the Joint/SiC interface decreases with an increase in SiC p amount (Tableau 2).

Characterization of a real assembly between two plates of SiC

Two plates of SiC have been joined by melting the TiSi 2 mixture and followed by a rapid cooling. It can be seen that the brazing is nearly uniformly present even though some depletion of matter can be seen. No pressure was applied on the surface of the sample, a possible explanation of this lack of matter could be the presence of gas confined during the melting. However these first results are encouraging.

CONCLUSIONS

The aim of this work was to find an adequate brazing composition which is stable with SiC between 1000°C to 1500°C for long durations. Titanium disilicide can reach such a goal but due to the mismatch in the physical properties between SiC and TiSi 2 , the joint is drastically cracked. The addition of submicronic particles of SiC inside the joint reduces the tensile residual stresses. The SiC p have also an other beneficial effect on the chemical stability of the joint when the disilicide is melted during the brazing step. The interest of this type of brazing has now to be confirmed by the mechanical testing of the assembly.

LIST OF FIGURE AND TABLE CAPTIONS

TABLES

Table 1: Physical Properties of the potential candidates for joining SiC [START_REF]Materials Handbook, A concise Desktop Reference[END_REF] Table 2: Influence of the amount of SiC p on the Reaction Zone thickness at the TiSi 2 /SiC interface

FIGURESFigure 1 :

 1 FIGURES

Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 2345 Figure 2: Young's modulus versus the temperature for TiSi 2 /SiC systems with different content of additions

2 TiSi 2 +

 22 FIGURE 1

TABLE 1

 1

			Si	SiC	TiSi 2	TiC
	Melting (°C)		1414	2700	1540	3140
	CTE (10 -6 /K)		5.7	4.5	10.4	7.6
	Young's Modulus (GPa)	130	400	260	450
			TABLE 2
	Pure TiSi 2	TiSi 2 +		TiSi 2 +	TiSi 2 +
			10%SiC Vo.	20%SiC Vo.	30%SiC Vo.
	Reaction zone	20	6-7		4-5	4-5
	thickness (m)

ACKNOWLEDGMENTS

The authors wish to thank M.Lahaye from Cecama University of Bordeaux and O. Cathy from LCTS respectively for their contributions in the chemical analysis and in the tomography characterization. The microwave experiments were done by E. Savary at Crismat (University of Caen, France).


Jerome Magnant

Laurence Maillé

René Pailler

Alain Guette

Spark Plasma Sintering of Ceramic Matrix Composites with self healing matrix

Keywords:

The processing of dense Ceramic Matrix Composites (CMC) by a short time and so by low cost process was studied. This process is based on (i) the chemical vapor deposition of carbon interphase on the fiber surface, (ii) the introduction of mineral powders inside the multidirectional continuous fiber preform and (iii) the densification of the matrix by Spark Plasma Sintering (SPS). To prevent carbon fibers and interphase from oxidation in service, a self-healing matrix made of silicon nitride and titanium diboride was processed. A thermal treatment of 3 minutes at 1500°C allows to fully consolidate by SPS the composite without fiber degradation. The ceramic matrix composites obtained have an ultimate bending stress at room temperature around 300 MPa and show a self-healing behaviour in oxidizing conditions.

INTRODUCTION

The strengthening and toughening of silicon nitride by reinforcement with continuous fibers has received considerable attention in recent years. Si 3 N 4 reinforced with carbon fibers can be fabricated by high temperature sintering under pressure: HP-HIP [START_REF] Grenet | Proceeding HTCMC II, Manufacturing and materials Development[END_REF] at 1800°C or by Spark Plasma Sintering (SPS) [START_REF] Magnant | Fiberreinforced ceramic matrix composites processed by a hybrid technique based on chemical vapor infiltration, slurry impregnation and spark plasma sintering[END_REF] at 1650°C. The coefficient of thermal expansion (CTE) of carbon fibers is anisotropic (it is very low and even negative along the axis but large and positive radially: CTE C// =-0.1x10 -6 K -1 ; CTE C┴ =10x10 -6 K -1 ) and different from that of Si 3 N 4 (CTE Si3N4 =2.8-3.5x10 -6 K -1 ). As a result, CMC as C/Si 3 N 4 composites already exhibit a microcracked matrix in the as-prepared state, these microcracks facilitating the in-depth oxygen diffusion toward the oxidation prone carbon fibers when exposed to oxidizing atmosphere [3]. Indeed carbon fibers undergo active oxidation at very low temperature (≈450°C). Hence, to obtain C-fiber reinforced CMC, which have a long lifetime (thousands of hours and even more) under load at high temperature and in oxidizing atmospheres, extensive researches have been carried out during the last decades. The solutions proposed are (i) the utilization of SiC fibers instead of C fibers [4] (but SiC fibers with good mechanical properties at high temperature are far more expensive than C fiber), (ii) the development of innovative self-healing interphases and matrices [5] and (iii) the development of specific coatings (environmental barriers coating) [6,[START_REF] Kn | High Temperature Ceramic Matrix Composites[END_REF].

The concept of self-healing interphases and matrices is based on the presence of elements as boron compounds (B 4 C, BN, TiB 2 ,…) or silicon compounds (SiC, Si 3 N 4 ,…) which form fluid oxide phases at high temperature when exposed to oxidizing atmosphere. This liquid oxide could fill the cracks rendering the interphase or the matrix self-healing. CMC with self-healing matrix were successfully fabricated by Chemical Vapor Infiltration (CVI) [START_REF] Lamouroux | [END_REF], but display long time and high cost production due to the intrinsic characteristics of the gaseous process and are not fully dense. In this work, carbon fiber composite with a dense self-healing matrix fabricated by a new short time process was studied. The hybrid process used for the fabrication of the composite is based (i) on CVI fiber coating, (ii) on slurry infiltration of ceramic powders into the multidirectional-continuous fiber preforms and (iii) on densification by Spark Plasma Sintering of the composite. This process was developed in our laboratory and has been reported somewhere else [START_REF] Magnant | Fiberreinforced ceramic matrix composites processed by a hybrid technique based on chemical vapor infiltration, slurry impregnation and spark plasma sintering[END_REF]. The matrix is a ceramic constituted of Si 3 N 4 and TiB 2 phases. Si 3 N 4 is the main constituent and was choosen for its excellent thermomechanical properties such as high strength, hardness, toughness, good thermal shock and oxidation resistance as well as for its lower sintering temperature in comparison of SiC and consequently lower manufacturing cost [9]. Titanium diboride exhibits a high melting point (~2900°C), a high hardness and has the faculty to form fluid healing B 2 O 3 phase as soon as 470°C under oxidizing conditions [9][10][11][12]. Furthemore, the high elastic modulus and coefficient of thermal expansion of TiB 2 with respect to Si 3 N 4 contribute to potential toughening mechanisms [13]. Si 3 N 4 + TiB 2 particulate composites with good mechanical properties were successfully processed by HP and HIP [13,14]. The aims of this paper are first, to determine the sintering conditions of the unreinforced Si 3 N 4 / TiB 2 ceramic matrix by Spark Plasma Sintering, second, to process carbon-fibers composites with Si 3 N 4 / TiB 2 matrix, densified by SPS process and, third, to evaluate their mechanical and self-healing properties.

2. EXPERIMENTAL METHOD

Elaboration of monolithic matrix

In order to perform the ceramic matrix, sub-micronic powders of Si 3 N 4 , TiB 2 , Al 2 O 3 , and Y 2 O 3 were used (Table I). The proportion of the different constituent was 66.5 wt% Si 3 N 4 + 20 wt% TiB 2 + 4.5 wt% Al 2 O 3 + 9 wt% Y 2 O 3 . Yttria and alumina are sintering-aids and react in temperature with the silica layer covering the surface of silicon nitride powders to form a liquid phase that firstly promotes the densification of Si 3 N 4 . Titanium diboride is the self-healing phase and was also reported to promote the densification of Si 3 N 4 at low temperatures [12].

Powders were mixed by wet ball-milling in polyethylene pots with ethanol and Al 2 O 3 balls during 24 hours. Then, mixed powders were dried and milled by dry ball-milling in polyethylene pots during 5 hours to eliminate the agglomerates. The powders were then fitted in a cylindrical carbon die with an inner diameter of 20 mm for Spark Plasma Sintering (Dr Sinter 2080, SPS Syntex Inc.). The sintering temperature varied between 1450°C and 1750°C, the heating-rate was set to 200°C/min and the holding period varied from 1 to 20 min at the selected temperature. First, the powders were degassed under vacuum at 700°C, next a mechanical pressure of 30 MPa was applied and finally a nitrogen or argon atmosphere (P gas = 0.02 MPa) was introduced in the sintering chamber. The heating process was controlled using an optical pyrometer focused on a hole in the carbon die. The axial shrinkage of the unreinforced matrix during sintering was recorded by an extensometer.

The bulk density of the samples was measured by Archimede's method. Crystalline phases were determined from X-ray diffraction patterns in Bragg-Brentano geometry (D8 advance, Bruker), using a copper anode (CuK).

Elaboration of CMC

Multidirectional fiber preforms, fabricated from ex-PAN carbon fibers were used. Their main characteristics were given in a previous paper [START_REF] Magnant | Fiberreinforced ceramic matrix composites processed by a hybrid technique based on chemical vapor infiltration, slurry impregnation and spark plasma sintering[END_REF].

As it is well known, the fiber/matrix bonding is a key factor controlling performances of ceramic matrix composites [START_REF] Després | Mechanical properties of C/SiC composites as explained from their interfacial features[END_REF], so a pyrocarbon interphase (PyC i ) was deposited on the fiber surface by CVI in order to arrest and deflect the matrix microcracks and to insure a load transfer function.

To minimize possible chemical reactions of carbon (fibers or interphases) with the matrix constituents during the fabrication, a SiC coating (SiC c ) was deposited by CVI on the PyC interphase. The different combinations of fibers, PyC interphase and SiC coating used in this study are: {C f / PyC i } and {C f / PyC i + SiC c }.

The preforms were infiltrated by a slurry impregnation under vacuum, using a suspension containing 30 vol.% of ceramic powders (Si 3 N 4 +Al 2 O 3 +Y 2 O 3 +TiB 2 ), as reported in a previous paper [START_REF] Magnant | Fiberreinforced ceramic matrix composites processed by a hybrid technique based on chemical vapor infiltration, slurry impregnation and spark plasma sintering[END_REF].

The impregnated preforms were then dried during 48h at 25°C. The dried carbon preforms charged with the silicon nitride + titanium diboride + sintering-aid powders were then put into a graphite die with an inner diameter of 50 mm for SPS. The SPS conditions are a compromise between minimizing fiber degradation and achieving full densification of the matrix. The choice of the sintering conditions resulted of the densification conditions of the monolithic matrix (Si 3 N 4 +TiB 2 + sintering-aid powders). The sintering temperature was fixed at 1500°C and hold 3 min under N 2 atmosphere, while the pressure was fixed at 75 MPa.

Yttria and alumina sintering-aids react in temperature with the silica layer covering the surface of silicon nitride powders to form a liquid phase that may accommodate the stress in the material due to external pressure during preparation of the CMC by SPS and, as a result, limits possible fiber damage.

To minimize fiber degradation the pressure was applied only when the sintering aids began to soften, but soon enough (1300°C) to limit CO (g) evolution and thus to inhibit any possible reaction of Si 3 N 4 or sintering aids with the pyrocarbon interphase or the fibers [16].

The sintered composites were subsequently cut and ground into 45x12x2.5 mm 3 specimens. The flexural strength () is defined as the strength when the specimen fails in the 3-point bending test. It is calculated according to the equation (1):

 = 3 F L / 2w t 2 (equation 1)
Where F is the breaking force of the specimen, L the support span (35 mm), w the width (12 mm) and t the thickness (2.5 mm). We used a universal testing machine (Model 5569, Instrom Corp) at cross-head speed of 0.5 mm/min at room temperature.

The polished cross sections of the composites before bending tests were characterized by scanning electron microscopy (Quanta 400 FEG, FEI) and energy-dispersive X-ray spectroscopy (EDAX). Some composites were oxidized under air at 850°C during 13 h and observed to study their self-healing behavior. Samples sintered under N 2 reach density closed to the theoretical value (3.54 g/cm 3 ) on the contrary of samples sintered under Ar (Figure 2). This is due to the fact that sample sintered 6 minutes under N 2 is constituted of β-Si 3 N 4 , α-Si 3 N 4 and TiB 2 whereas sample sintered during 6 minutes under Ar is constituted of β-Si 3 N 4 , Si, TiN, and BN phases (Figure 3 and Figure 4). Thus, under Ar atmosphere, undesired chemical reactions occur in the sample.

Results and discussion

Elaboration of the ceramic monolithic matrix

Possible explanation for the formation of TiN is that Si 3 N 4 decomposes forming Si and N 2 according to reaction 1 and then N 2 gas reacts with TiB 2 to form TiN and BN (reaction 2). (reactions 1, 2 and 3) do not adequately explain the formation of TiN [13]. As TiN was not formed adjacent to BN phase, they suggested that they were nucleated from the liquid at separate sites and that it was the activity and the kinetics of migration of species through the inter-granular liquid phase that control the rate and type of reaction that occurs.

In samples sintered by SPS at 1500°C as the formation of TiN is only detected under Ar atmosphere and not under N 2 atmosphere, it is unlikely that reaction 3 is the main reaction explaining the formation of TiN.

From our results, there is no evidence that other mechanisms, as suggested by Jones et al., cannot happen to explain the formation of TiN.

Sun et al. fabricated Si 3 N 4 + TiB 2 samples by HP at 1800°C for 1h in N 2 atmosphere [14]. They detected the presence of TiSi that they explained by the reaction of Si 3 N 4 , TiB 2 and N 2 according to reaction 4.

Si 3 N 4 + 3 TiB 2 + N 2 = 3 TiSi + 6 BN (4)

As TiSi was not detected in our samples sintered by SPS, reaction 4 is unlikely to happen in our materials.

As we want to maintain TiB 2 (one of the healing compound generator), the sintering conditions chosen to densify by SPS the Si 3 N 4 + TiB 2 matrix are a thermal treatment at 1500°C applied during 3 minutes at 30

MPa, under N 2 atmosphere.

Elaboration of CMC by SPS

After the under-vacuum slurry infusion step, the inter-tows porosity of the preform is filled homogenously by the mineral charges as can be seen on Figure 5.

The microstructures of the composites were studied in term of fiber degradation, densification of the inter and intra-tows matrix, microcracks inside the matrix, decohesions between the fibers and the matrix and will be successively discussed.

The inter-tows matrix of composites is fully dense (Figure 6) except in some located areas (Figure 6d) between crossing tows for C-fibers composites with a SiC coating around the fibers. This lies in the fact that the SiC coating withstands the transmission of the pressure to the matrix located at the crossing of the tows [START_REF] Magnant | Fiberreinforced ceramic matrix composites processed by a hybrid technique based on chemical vapor infiltration, slurry impregnation and spark plasma sintering[END_REF].

Inside the tows the fibers are bonded by the pyrocarbon and eventually by the SiC coating. Carbon fiber composites without SiC coating have some intra-tow areas densified by the Si 3 N 4 -based matrix. For carbon fiber composites with the SiC coating, the accessibility to the intra-tow porosity is blocked by the SiC layer prior to powder impregnation.

The composites made of C f /PyC i /(Si 3 N 4 +TiB 2 ) matrix exhibit microcracks after fabrication (Figure 6).

These microcracks are the result of the CTE mismatch between the carbon-fibers and the Si 3 N 4 +TiB 2 matrix. Sabouret et al. have computed around 450 MPa the internal stress inside the matrix in unidirectional carbon fiber-reinforced Si 3 N 4 matrix composite (with E C-fiber =250 GPa) [START_REF] Sabouret | Composites à matrice Nitrure de Silicium et fibres de carbone : Elaboration, comportement mécanique[END_REF]. In our multidirectional C-fiber arrangement composites studied, the repartition of the thermal stresses inside the matrix is more complicated than in unidirectional composites but theses stresses are higher than the intrinsic stress at failure of the matrix, which leads to the formation of cracks inside the matrix.

No important decohesion between the fibers and the interphase/coating is observed. The thickness of PyC interphase can accommodate some strain differences between the constituents of the composite and, furthermore carbon fibers show no radial strain. Some decohesions or cracks between the PyC interphase or the SiC coating and the matrix are observed. These decohesions may be the result of thermal stresses produced by the CTE mismatch between the C-fiber and the matrix or to some preferential shrinkage of the matrix not followed by the displacement of the fiber-tows.

The stress-displacement curves obtain during bending mechanical tests at ambient temperature are presented in Figure 7. C-fibers composites exhibit a damageable behavior with the maximum stress at failure around 300 MPa which is closed to the one of C f /SiC m composite processed by CVI process [START_REF] Eberling-Fux | Matériaux composites à matrice nanostructurée[END_REF].

The presence of the SiC coating around the fibers enhances the mechanical properties. The hypotheses formulated to explain this point are that (i) the strength of the interface bond between the SiC coating and the matrix phases (more precisely the intergranular SiO 2 -Y 2 O 3 -Al 2 O 3 glassy phase) is stronger than the bond between the PyC interphase and the amorphous matrix phase, (ii) as the strain at failure of the composites with and without SiC coating are more or less the same, the increase in ultimate stress for the composite with SiC coating is due to the increase of the elastic modulus of the composite by the SiC coating and (iii) the SiC coating prevented the fibers from chemical degradation during the fabrication.

However, as fiber degradations were not observed in the material without the SiC coating, the last hypothesis is improbable.

Oxidation behavior of the CMC

At 850°C under air, the (TiB 2 and Si 3 N 4 ) matrix oxidation is described by the following reactions:

2TiB 2 + 5O 2 = 2TiO 2 + 2B 2 O 3 (5) Si 3 N 4 + 3O 2 = 3SiO 2 + 2N 2 (6)
At this temperature, another reaction between B 2 O 3 and SiO 2 occurs and leads to an amorphous phase with a relatively low melting point: the healing fluid. On the Figure 8 it appears that the cracks seem to be healed inside the matrix since they cannot be observed after 13h under air at 850°C. Moreover, the oxidation of TiB2 leads to the formation of TiO 2 grains (reaction 5) which are covering the matrix surface (due to an increase of 1.64 in the molar volume [START_REF] Garrite | Etude de l'oxydation/corrosion des composites céramiques[END_REF]) and can act as an environmental barrier.

Further characterizations on the self-healing behavior of this material are under progress.

CONCLUSION

Ceramic matrix composites with a complex matrix were successfully processed by a short time hybrid process based on chemical vapor infiltration fiber coating, on the introduction of Si 3 N 4 +TiB 2 +sinteringaids inside the fiber preform by slurry impregnation and on the consolidation of the composite by spark plasma sintering. This process is achieved at 1500°C for 3 minutes under 75 MPa mechanical pressure and N 2 atmosphere rising to a dense matrix without fiber degradation. The ceramic matrix composite shows a high bending stress at failure (≈300 MPa) and a self-healing behavior in oxidizing conditions. showing the cracks healing.

List of figure and table captions

Table I. Some characteristics of the powders used.

Figure 1

 1 Figure 1 shows the shrinkage and shrinkage rate curves of the matrix (66.5 wt% Si 3 N 4 + 20 wt% TiB 2 + 9 wt% Y 2 O 3 + 4.5 wt% Al 2 O 3 ) during sintering up to 1750°C at 30 MPa under N 2 atmosphere. The displacement rate curve clearly shows two maxima around 1260°C and 1500°C. The first important increase in shrinkage observed around 1260°C can be attributed to particles rearrangement due to liquid phase formation and the second one around 1500°C to the dissolution-diffusion-reprecipitation densification mechanisms.

  formation of TiN, Jones et al. suggested the above reactions (1 and 2) as well as the reaction between Si 3 N 4 and TiB 2 according to reaction 3 Jones et al. did not detect Si or Si containing phases (e.g. titanium silicides) in their samples sintered by HIP at temperature bellow 1750°C, they suggested that the above mechanisms alone

Figure 1 .

 1 Figure 1. Temperature and shrinkage rate of the sample during the SPS cycle.

Figure 2 .

 2 Figure 2. Variation of the matrix density with the SPS processing parameters at 1500°C, 30 MPa for different dwelling periods and atmosphere.

Figure 3 .

 3 Figure 3. Variation of the XRD patterns of the matrix after SPS at 1500°C, 30MPa according to different dwelling periods under Ar or N 2 atmosphere.

Figure 4 .

 4 Figure 4. Back-scattered SEM micrographs of 66.5 wt% Si 3 N 4 + 20 wt% TiB 2 + 4.5 wt% Al 2 O 3 + 9 wt% Y 2 O 3 samples sintered at 1500°C during 6 min (a) under Ar, and (b) under N 2 atmosphere. In (a) silicon and titanium nitride correspond to the white phase contrast, silicon nitride to the grey contrast and boron nitride is in black. In (b) titanium diboride are the light grey inclusions and the silicon nitride corresponds to the dark grey contrast.

Figure 5 .

 5 Figure 5. Micrographs of fiber preforms (a) before impregnation by the mineral powder and (b) after impregnation. The powders correspond to the grey contrast between the tows.

Figure 6 .Figure 7 .Figure 8 .

 678 Figure 6. SEM micrographs of Spark Plasma Sintered composites (a) and (b) C f / PyC i / Si 3 N 4m +TiB 2m , (c) and (d) C f / PyC i + SiC c / Si 3 N 4m +TiB 2m Figure 7. Stress/displacement curves from bending test for composites sintered by SPS with a 75 MPa mechanical pressure applied at 1300°C Figure 8. SEM back-scattered micrographs of C f /Si 3 N 4 +TiB 2 after oxidation under air at 850°C-13h

Figure 1 Figure 2 Figure 3 N

 123 Figure 1

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Table I Si 3

 I3

		N 4	TiB 2	Y 2 O 3	Al 2 O 3
	Provider	HC	HC	HC	Alfa Aesar
		Starck	Starck	Starck
	grade	M11	F	C	-
	SSA	12-15	2.7	10-16	10
	(m².g -1 )

ACKNOWLEDGEMENTS

This work was supported by DGA (Direction Générale pour l'Armement from France) and Herakles (Safran Group) through a grant. The authors acknowledge E. Philippe and S. Bertrand (Herakles) for material supply; G. Chevallier, C. Estournes and G. Raimbeaux (PNF 2 , University of Toulouse, France) for Spark Plasma Sintering experiments.


Barbara Pascal

Nelly Pustelnik

Patrice Abry

Marion Serres

Valérie Vidal

Joint estimation of

Keywords: Total Variation, Primal-dual proximal algorithm, Texture segmentation, Strong convexity, Multiphase flow

published or not. The documents may come

INTRODUCTION

Texture characterization -A texture is a perceptual attribute whose characterization has been envisaged with various mathematical models involving geometric or statistical attributes such as e.g., local variance or local spectral histograms. Amongst state-of-the-art texture segmentation techniques, one can refer to Yuan et al. [START_REF] Yuan | Factorization-based texture segmentation[END_REF] relying on Gabor coefficients as features followed by a selection procedure based on a matrix factorization step or to the multiscale contour detection procedure whose descriptors are the concatenation of brightness, color and texture (using textons) information followed by the computation of an oriented watershed transform [START_REF] Arbelaez | Contour detection and hierarchical image segmentation[END_REF]. In the present work, particular attention is granted to another class of multiscale techniques relying on scale-free local features [START_REF] Wendt | Wavelet leaders and bootstrap for multifractal analysis of images[END_REF], well-suited for multiphase flow texture modeling. Challenges in multiphase flow characterization -Understanding and predicting the dynamics of multiphase flows is a major issue in geosciences (soil decontamination, CO2 sequestration) and in the industry (enhanced oil recovery, catalytic reactions) [START_REF] Reddy | Effects of soil heterogeneity on airflow patterns and hydrocarbon removal during in situ air sparging[END_REF][START_REF] Hessel | Gasliquid and gas-liquid-solid microscructured reactors: Contacting principles and applications[END_REF]. Such phenomena all involve a joint gas and liquid flow inside a porous medium. Quantifying the contact areas between the different phases, where chemical reactions take place, is of tremendous importance for analyzing and predicting the efficiency of such processes [START_REF] Kreutzer | Multiphase monolith reactors: Chemical reaction engineering of segmented flow in microchannels[END_REF]. However, the porous medium produces a global, multiscale texture Work supported by Defi Imag'in SIROCCO and by ANR-16-CE33-0020 MultiFracs, France. which makes it difficult to extract gas-liquid interfaces. Segmentation techniques based on morphological tools used so far to differentiate phases in multiphase flows [START_REF] Serres | On the stability of taylor bubbles inside a confined highly porous medium[END_REF] present severe limitations: arbitrary threshold setting, non-physical irregular bubble contour, non detection of small bubbles. In addition, recent development in highresolution imaging yield large-size images thus bringing forward issues in memory and computational costs. Model -In the present study, we consider a broad class of textured images x = (xn)n∈Ω ∈ R |Ω| of size |Ω| = N1 × N2, piecewise monofractal textures, characterized by piecewise constant statistical properties of their multiscale coefficients X (j) n n∈Ω,j∈N * , where 2 j denotes the scale. These multiscale coefficients consist of either the absolute value of wavelet coefficients or wavelet leaders of texture x [START_REF] Wendt | Wavelet leaders and bootstrap for multifractal analysis of images[END_REF], and behave locally as

X (j) n sn2 jhn when 2 j → 0 (1)
where s ∈ R |Ω| , the local variance, and h ∈ R |Ω| , the local regularity, are assumed to be piecewise constant fields.

To model multiphase flow textures, it is not assumed a priori that both s and h have edges at same locations. In this context, performing texture segmentation thus consists in estimating sharp changes in s and h from image x. State-of-the-art -Model (1) is of common use for modeling scalefree textures [START_REF] Pustelnik | Combining local regularity estimation and total variation optimization for scale-free texture segmentation[END_REF] and has proven its efficiency to caracterized homogeneous textures as encountered in art investigation [START_REF] Abry | When Van Gogh meets Mandelbrot: Multifractal classification of painting's texture[END_REF] or medical imaging [START_REF] Marin | Mammographic evidence of microenvironment changes in tumorous breasts[END_REF], i.e. sn ≡ s0 and hn ≡ h0. The common approach to estimate s and h naturally involves the use of (weighted) linear regression across scales as log 2 X (j) n log 2 sn + jhn as 2 j → 0.

(

When homogeneous texture are analysed, the multiscale coefficients are first averaged across space and then linear regressions on those averages lead to accurate estimates of h0 and s0, notably using wavelet leaders [START_REF] Wendt | Wavelet leaders and bootstrap for multifractal analysis of images[END_REF]. However, when the objective consists in obtaining local estimates to capture changes in h or s, other strategies need to be deployed. In [START_REF] Pustelnik | Combining local regularity estimation and total variation optimization for scale-free texture segmentation[END_REF], estimation relies on a two-step procedure (as often in texture segmentation) consisting in (i) estimating hReg by linear regression across scales from X (j)

n and (ii) obtaining from hReg a piecewise constant estimate of h denoted hTV by means of total-variation denoising. In [START_REF] Pustelnik | Combining local regularity estimation and total variation optimization for scale-free texture segmentation[END_REF], we also proposed to combine both steps by incorporating the regression weight estimation into the optimization process leading to: minimize h,w n∈Ω j

w (j) n log 2 X (j) n -hn 2 + λΩ(h, w)
where λ > 0 denotes a regularization parameter and Ω a convex regularization term inducing flexibility in the choice of the weights (w.r.t unbiased estimator) and where h is assumed to be piecewise constant. This procedure lead to satisfactory estimation performance at the price though of high computational and memory costs. Contribution -In this work, we propose a new estimation/segmentation procedure that perform jointly the estimation of h of s. The proposed method aims (i) to estimate simultaneously the local variance and the local regularity, (ii) to rely on a fast algorithmic procedure and (iii) to involve a limited amount of memory. In Section 2, we formulate joint estimation as an optimization problem. The algorithmic solution devised to solve the optimization problem is presented in Section 3. Performance of the proposed strategies are assessed and compared to disjoint estimation procedure on synthetic piecewise constant monofractal textures in Section 4. Results achieved on multiphase flow images are detailed in Section 5 and compared to those obtained from state-of-the-art segmentation methods.

SEGMENTATION AS AN OPTIMIZATION PROCEDURE

We adopt a variational approach to estimate jointly s and h from the multiresolution quantity X :

minimize s∈R |Ω| ,h∈R |Ω| F (s, h; X ) + G(s, h). (3)
where

F (•, •; X ) : R |Ω| × R |Ω| →] -∞, +∞] is a data-fidelity term reminiscent of (2) and G : R |Ω| × R |Ω| →] -∞, +∞
] is a penalization favoring piecewise constant estimates, based on the use of total variation. From ( 2), a natural choice for the data-fidelity term F reads:

F (s, h; X ) = 1 2 J 2 j=J 1 log 2 X (j) -log 2 s -jh 2 2 , (4)
where 1 ≤ J1 < J2. To manipulate convex data-term, we set v = log 2 s and deal with the minimization problem over (v, h) rather than over (s, h) leading to

( v, h) ∈ Argmin v,h j log 2 X (j) -v -jh 2 2 + ηTV(h) + ζTV(v)
(5) with s = 2 v , and where η > 0, ζ > 0 are regularization parameters. TV models the isotropic total variation. Let D : R |Ω| → R |Ω|×2 computing the horizontal and vertical variations of intensity at each location n = (n1, n2) ∈ Ω,

(Dy) n 1 ,n 2 = yn 1 ,n 2 +1 -yn 1 ,n 2 yn 1 +1,n 2 -yn 1 ,n 2 , (6)
the total variation is defined as, for every y ∈ R |Ω| , TV(y) = ||Dy||2,1 where for every z = (z1, z2) ∈ R |Ω|×2 : 2 + ϕ(ỹ) (and similarly for ψ), have closed form expressions, this problem can be solved using Chambolle-Pock primal-dual algorithm [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF], belonging to the class of proximal algorithms [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert Spaces[END_REF][START_REF] Combettes | Proximal splitting methods in signal processing[END_REF] and particularly efficient when dealing with TV penalties.

z 2,1 = N 1 -1 n 1 =1 N 2 -1 n 2 =1 (z1) 2 n 1 ,n 2 + (z2) 2 n 1 ,n 2 . ( 7
However, when working with large size data, as e.g. in multiphase flows, the basic form of Chambolle-Pock primal-dual algorithm suffers from too high and hence redhibitory computational cost. To deal with high dimensionality problems, it is necessary to take advantage of additional properties of the functions ϕ and ψ, such as strong convexity. A function ϕ is said to be µ-strongly convex, for a given µ > 0, when the function y → ϕ(y) -µ 2 y 2 2 is convex.

When ϕ is µ-strongly convex, it is proven in [START_REF] Chambolle | A first-order primal-dual algorithm for convex problems with applications to imaging[END_REF] that it is possible to design an accelerated algorithm, relying on adaptive stepsizes, to solve [START_REF] Pustelnik | Combining local regularity estimation and total variation optimization for scale-free texture segmentation[END_REF]. This algorithm is detailed in Algorithm 1 where the sequence (y [k] ) k∈N converges to the solution of (8).

Algorithm 1: Accelerated Chambolle-Pock algorithm.

Initialization : Set y [0] , z [0] = z[0] = Ly [0] . Set δ0 > 0 and ν0 > 0 such that δ0ν0 L 2 < 1. for k ∈ N * do y [k+1] = prox δ k ϕ y [k] -δ k L * z[k] z [k+1] = prox ν k ψ * z [k] + ν k Ly [k+1] ϑ k = (1 + 2µδ k ) -1/2 , δ k+1 = ϑ k δ k , ν k+1 = ν k /ϑ k z[k+1] = z [k+1] + ϑ k z [k+1] -z [k]

Strong convexity of the data-fidelity term

A differentiable function ϕ is µ-strongly convex if and only if

(∀ (y, z) ∈ H × H) ∇ϕ(y) -∇ϕ(z), y -z ≥ µ y -z 2
where . , . denotes the usual scalar product. For the data-fidelity term considered here that is,

H = R |Ω| ×R |Ω| and ∀y = (v, h) ∈ H, ϕ(y) = F (v, h; X ) = J 2 j=J 1 log 2 X (j) -v -jh 2 2 , (9)
one directly finds

∇ F (v, h; X ) = M (v, h) T , with M = S0I S1I S1I S2I is block matrix defined from Sm = J 2 j=J 1 j m , ( 10
)
and I is the identity matrix of R |Ω| . Since the gradient ∇ F is linear, the condition for F to be µ-strongly convex rewrites:

∀(v, h) ∈ H, ∇ F (v, h; X ), (v, h) = M (v, h) T , (v, h) T ≥ µ (v, h) 2
Moreover M is symmetric and positive definite, thus, denoting by β its lowest eigenvalue, β > 0 and

M (v, h) T , (v, h) T ≥ β (v, h) T , (v, h) T , (11)
we conclude that the function F (v, h; X ) is µ-strongly convex w.r.t the variables (v, h), with µ = β.

Proposed algorithm for texture segmentation

Given the strong convexity result of the previous section, it is thus possible to use the general accelerated primal-dual Algorithm 1 for solving Problem [START_REF] Hessel | Gasliquid and gas-liquid-solid microscructured reactors: Contacting principles and applications[END_REF]. The iterations are customized to the minimization of (5) in Algorithm 2 setting y = (v, h). The proximity operator of the data-fidelity term is provided in Proposition 1 while the proximal operator of the conjugate function of the mixed 2,1-norm is derived in [START_REF] Fadili | Total variation projection with first order schemes[END_REF]. The convergence of v [k] , h [k] k∈N to a minimizer of Problem ( 5) is insured by δ0ν0 D 2 < 1.

Algorithm 2: PLOVER (Piecewise constant LOcal Vari-ancE and local Regularity joint estimation)

Initialization v [0] , h [0] ∈ R |Ω| , u [0] = ū[0] = Dv [0] [0] = ¯ [0] = Dh [0] ∈ R |Ω|×2 ; δ0, ν0 > 0, s.t. δ0ν0 D 2 < 1 for k ∈ N * do v [k+1] h [k+1] = prox δ k F v [k] h [k] -δ k D * ū[k] D * ¯ [k]    u [k+1] = prox ν k ζ • * 2,1 u [k] + ν k Dv [k+1] [k+1] = prox ν k η • * 2,1 [k] + ν k Dh [k+1] ϑ k = (1 + 2µδ k ) -1/2 , δ k+1 = ϑ k δ k , ν k+1 = ν k /ϑ k ū[k+1] ¯ [k+1] = u [k+1] [k+1] + ϑ k u [k+1] [k+1] - u [k] [k]
Proposition 1 (Computation of prox F ). Let F be defined as in [START_REF] Abry | When Van Gogh meets Mandelbrot: Multifractal classification of painting's texture[END_REF].

Let S = j log 2 X (j) , T = j j log 2 X (j) and N = (1 + S0)(1+S2)-S 2 1 , {Sm} m∈{0,1,2} defined in [START_REF] Marin | Mammographic evidence of microenvironment changes in tumorous breasts[END_REF]. For every Because it is now well known that TV-based estimation suffers from large biases [START_REF] Deledalle | Clear: Covariant least-square refitting with applications to image restoration[END_REF] that precludes the recovery of exact values for s and h in the different regions. To overcome this difficulty, we first perform a K-means segmentation from the estimates ( s•, h•) and re-estimate from the multiscale coefficients X the values of local variance and regularity for the segmented regions by global averages within segmented regions. Achieved performance are displayed in the fourth and fifth columns, λ, η and ζ are chosen as those leading to the best SNR on these re-estimates. Local regularity and variance estimation is clearly improved by the proposed joint estimation PLOVER procedure, compared to disjoint estimation. This is significantly the case for the local regularity estimates, a significant outcome as local regularity is a notoriously difficult quantity to estimate. Moreover, edges of estimates are found to be more regular for the proposed PLOVER procedure, an outcome of practical importance for multiphase flow analysis, where bubble perimeter estimation if of critical importance. For piecewise monofractal textures, it has been shown that texture segmentation state-of-the-art methods such as those in [START_REF] Yuan | Factorization-based texture segmentation[END_REF][START_REF] Arbelaez | Contour detection and hierarchical image segmentation[END_REF] do not yield satisfactory results (cf. [START_REF] Pustelnik | Combining local regularity estimation and total variation optimization for scale-free texture segmentation[END_REF]Fig.3]), the corresponding comparisons not reported here for space reasons further confirm such findings.

(v, h) ∈ R |Ω| × R |Ω| , (p, q) = prox F (v, h) with p = ((1 + S2)(S + v) -S1(T + h)) /N , q = ((1 + S0)(T + h) -S1(S + v)) /N .

MULTIPHASE FLOW LARGE-SIZE IMAGE ANALYSIS

Data acquisition -Experiments of joint gas and liquid flow through a porous medium were performed in a quasi-2D vertical cell (Hele-Shaw cell of width 210 mm, height 410 mm and gap 2 mm). The porous medium is an open cell solid foam of NiCrFeAl alloy, with a typical pore diameter of 580 µm. Constant gas and liquid flow rates are injected at the bottom of the cell through nine injectors (air) and a homogeneous slit (water). Images of the multiphase flow are acquired by a high-resolution camera (Basler, 2048×2048 pixels) at 100 Hz [START_REF] Serres | Etude hydrodynamique d'un écoulement gazliquide dans un milieu poreux confiné[END_REF]. The size of the images to analyze is 1677 × 1160. An example is provided in Fig. 3(a), showing that the gas phase (blue or yellow structures) is textured because of the solid porous medium. The liquid phase (in green) is also textured though at smaller scales (cf. Fig. 3(b)). Discussion -Figs. 3(c)-(d) clearly show that the local variance s and local regularity h provide information different in nature: h captures well all gas bubbles and it can be conjectured that the value of h is linked to bubble thickness variations ; s brings forward bubbles located in the foreground only (yellow structures in Fig. 3(b)), thus providing rich information on the bubble distribution in the cell gap.

Comparisons with state-of-art -Fig. 3 (2nd row) compares segmentation outcomes of the proposed PLOVER procedure to those obtained from state-of-the-art texture segmentation methods: Fig. 3(e) illustrates the segmentation obtained with the oriented watershed transform of Arbelaez et al [START_REF] Arbelaez | Contour detection and hierarchical image segmentation[END_REF] and Fig. 3(f) shows the segmentation obtained following Yuan et al [START_REF] Yuan | Factorization-based texture segmentation[END_REF] method based on matrix factorization. The results of disjoint TV estimation procedure [START_REF] Pustelnik | Combining local regularity estimation and total variation optimization for scale-free texture segmentation[END_REF] are displayed on Fig. 3(g). For disjoint TV and PLOVER results, the segmentation is obtained by performing K-means over h• and s•. The proposed method PLOVER yields better determination of individual objects (bubbles), this is critically the case for the smaller size bubbles, of great importance in a forthcoming quantification of phase contact surfaces, critical features in multiphase flow characterization. In addition, PLOVER produces an enhanced separation of the foreground and background bubble populations, which yield an improved description of multiphase flows. Further, some state-of-the-art approaches are so demanding computational and memory-resource wise, that they could not be applied to the large-size images directly while the current PLOVER implementation permits a fast analysis of such large-size images. Results were hence compared for cropped versions of actual images.

CONCLUSION

These computational efficiency of PLOVER combined to the significant improvements in multiphase flow large-size image analysis pave the way towards a systematic use in this application. Analyses on much larger simulated and real-world image datasets are under current investigations together with the automated tuning of the hyper-parameters λ, η and ζ.

) 3 .

 3 STRONG CONVEXITY AND FAST ALGORITHM 3.1. Primal-dual algorithms and strong convexity Problem (5) can be rewritten in a general form y ∈ Argmin y∈H ϕ(y) + ψ(Ly)(8)where ϕ : H → R and ψ : G → R are proper lower semicontinuous convex functions defined on Hilbert spaces H and G, and L : H → G is a bounded linear operator. When the proximal operators of functions ϕ and ψ, defined as prox ϕ (y) = argmin

Fig. 1 :

 1 Fig. 1: Estimates for local variance (top) and regularity (bottom) : linear regressions (column 1), disjoint TV estimation (column 2), PLOVER (column 3). Average re-estimates for disjoint TV and PLOVER (columns 4 and 5 resp.).

1 2

 1 4. PERFORMANCE ASSESSMENTSimulation settings -To evaluate the performance of the proposed piecewise constant local variance and local regularity estimation PLOVER procedure (cf. Algorithm 2), we generate piecewise monofractal textures fully characterized by their (piecewise) local variance s and (piecewise) local regularity h. An example is displayed in Fig.2(a), while underlying masks for local variance and local regularity are represented on Figs. 2(b) and (c). Synthetic textures consist of three regions embedded in a background, characterized with sn ≡ 0.3 and hn ≡ 0.1: Region 1(left) corresponds to a change in local regularity h (w.r.t. the background) but no change in local variance s (sn ≡ 0.3, hn ≡ 0.6) ; Region 2 (middle) corresponds to changes in both local variance and regularity (sn ≡ 0.6, hn ≡ 0.6) ; Region 3 (right) corresponds to changes in local variance only (sn ≡ 0.6, hn ≡ 0.1). The masks consisting of three different size disks permit to test the impact of region sizes on performance. Wavelet leaders, local suprema of wavelet coefficients [3] are chosen as multiresolution quantities X . Scales involved in the minimization range from J1 = 1 to J2 = 5. Performance evaluation -Fig. 1 reports estimation performance compared i) when performing linear regressions equivalent to solve Problem (5) when η = ζ = 0 leading to ( vReg, hReg) and with sReg = 2 v Reg (column 1) ; ii) when applying a simple TV denoising on vReg and hReg separately, i.e., solving the minimization problem yTV = argmin y y -yReg 2 2 + λyTV(y) where y = {s, h} (column 2) ; iii) when applying the proposed PLOVER procedure (column 3).

  Synthetic texture x (b) s mask (c) h mask

Fig. 2 :

 2 Fig. 2: (a) Synthetic piecewise monofractal textures using piecewise local variance (b) and regularity (c).

Fig. 3 :

 3 Fig. 3: Multiphase flow image (a) and zooms (b). PLOVER estimates for local variance (c) and regularity (d). State-of-the-art segmentations applied on x (e) and (f), K-means based segmentation from disjoint estimates (g) and PLOVER (h).