Aurélien Mazzoni

Gérard Bernhart

Thierry Cutard

G Bernhart

Gerard Bernhart
email: gerard.bernhart@mines-albi.fr

Ceramic fibre reinforced refractory castables for very high temperature superplastic forming Keramikfaser verstärkte hochschmelzende Gießmassen für superplastisches Umformen bei sehr hohen Temperaturen

Keywords: Refractory castable, superplastic forming, ceramic dies, ceramic fibre reinforcement, microstructure, Schlüsselwörter: Hochschmelzende Gießmassen, superplastisches Umformen, Keramikgesenk, Keramikfaser Verstärkung, Mikrostruktur

    Ceramic fibre reinforced refractory castables for very high temperature superplastic forming

Introduction

Since several years, Fibre Reinforced Refractory Castables (FRRC) have been developed as an emerging material solution to manufacture tools in order to propose a low cost solution for Ti alloy sheet Superplastic Forming [1,2]. Non-reinforced refractory castables keep good mechanical properties until 1000 8C and are characterized by a good thermal shock resistance because of their low thermal expansion coefficient and low Young's modulus. Unfortunately, their mechanical behaviour remains quasi-brittle. It was demonstrated that the reinforcement with stainless steel fibres gives some ductility to the resulting FRRC and avoids a catastrophic failure after crack initiation.

Unfortunately when addressing higher temperatures, up to 1100 8C as those that may be required for TiAl alloy, Superpalloy or duplex steel sheet superplastic forming, classical metallic reinforcement solutions are no more possible. In order to overcome this limit, alumina based ceramic fibre reinforcement solutions were investigated. This change does not only introduce a composition difference but also a major morphological evolution; as a matter of fact ceramic fibres have a diameter close to 12 lm compared with the previous 500 lm for the metallic ones and aggregates sizes that are close to 3 mm.

After the description of fibre and castable matrix compositions, the processing route is shortly described and the high temperature mechanical properties under four-point bending conditions are reported. The non-reinforced castable is compared with two alumina fibre reinforcing solutions over the range from 700 8C to 1200 8C. Interrupted and post rupture microstructural investigations have been performed in order to get an understanding of the internal micro-mechanical damage and fibrematrix load transfer mechanisms.

Materials and experiments 2.1 Materials

A commercial grade of an andalusite based refractory concrete (And-NF: Andalusite No Fibres) is considered in this work [3][START_REF] Marzagui | Proceedings of the International Conference of Metallurgists COM2004[END_REF][START_REF] Marzagui | Proceedings of the International Conference of Metallurgists COM2004[END_REF]. It is made of aggregates with a 5 mm maximum size, of silica fume, of alpha-alumina and of a low content of a calcium aluminate cement. The main properties of the two alumina-rich fibres investigated as reinforcing solutions (Nextel 720m and Nextel 610m) are listed in Table 1. The as delivered long mineral fibre tows were first cut to produce 12 mm short fibres and then mixed in a planetary mixer with the castable raw material and with 5 wt% water addition to produce the test samples.

After casting in a wood mould, mould and samples are then immediately wrapped in plastic. They are kept and dried at room temperature during 24 h and then extracted from the moulds before a 110 8C -24 h drying step. Results on two grades of fibre reinforced andalusite concretes are reported in this paper and compared with the non reinforced one (And-NF):

-grade And-N7 : And-NF reinforced with 2 vol.% of Nextel 720m short fibres -grade And-N6 : And-NF reinforced with 2 vol.% Nextel 610m short fibres Before performing microstructural or mechanical tests, a firing cycle is applied with a heating and cooling ramp of 60 8C/min including an isothermal dwell of 5 hours at maximum firing temperature. Before high temperature mechanical testing samples are heated with the same heating rate of 60 8C/min and subjected to a half hour dwell.

Experiment procedures

The four-point bending tests were carried out on a MTS810 servo-hydraulic universal testing machine and more details can be found in previous papers [START_REF] Mazzoni | [END_REF][START_REF] Nazaret | Sixth RILEM Symposium on Fibre-Reinforced Concretes[END_REF][START_REF] Nazaret | [END_REF]. The samples, with a size of 150625625 mm 3 , were loaded at a constant crosshead speed of 0.2 mm/min. Applied load and beam deflection were recorded during the tests. Monotonic tests are performed at high temperature between 700 8C and 1200 8C using a resistive heating furnace and alumina testing fixtures. Fracture surfaces were observed at room temperature on a FEI-XL30-ESEM-FEG environmental scanning electron microscope.

High temperature mechanical behaviour

An increase of the non-linear domain of the behaviour curves is observed combined with an important increase of the strain capacity. Figure 1 shows that reinforcing the refractory concrete with alumina fibres modifies its behaviour over the whole temperature range. Moreover, this figure allows to notice that the mechanical characteristics of the refractory concrete is hardly modify with the testing temperature.

Figures 1a, 1b and1c show bending curves for the unreinforced And-NF refractory concrete and for Nextel 610m reinforced (And-N6) and Nextel 720m reinforced (And-N7) grades. Tests have been performed at 700 8C, 900 8C, 1000 8C, 1100 8C and 1200 8C. At 700 8C, the behaviour is quite similar for the three materials. After a linear elastic part, when the stress level increases, the behaviour moves to a non-linear one due to damage processes by microcracking. At higher stress levels, damage localizes and one or several macrocracks appear. After the stress peak, the behaviour curve exhibits an extended softening part. When comparing to stainless steel reinforced castables, fibre bridging is not so efficient and the post peak behaviour shows a rapid stress decrease, that is nevertheless not brittle [1,2]. At temperatures higher than 900 8C, general behaviour moves to a typical elasto-visco-plastic behaviour.

Reinforcing the And-NF concrete with Nextel fibres leads essentially to large modifications of the pre-peak behaviour. They mainly deal with an increase of the width of the pre-peak linear domain and with an increase of the stress peak and deflection peak levels. Nextel fibres can also be considered as delaying diffuse damage phenomena and to limit the visco-plastic behaviour of the concrete. Because of the high alumina content of Nextel fibres, the temperature range in which such phenomena can be obtained is largely increased. No benefit effect is nevertheless observed in the post-peak domain Table 2 and 3 gather respectively the peak stress and apparent elastic modulus relative variations of the reinforced And-NF castable and allow the quantification of the reinforcing effects. It can be seen that the addition of alumina-rich fibres increases the mechanical characteristics over the whole temperature range. The gain on the peak stress values varies between 9% and 96% for the And-N7 grade and between 41% and 117% for the And-N6 grade. For the apparent elastic modulus the gain varies between 12% and 110% for And-N7 grade and between 22% and 85% for And-N6 grade.

These first results of high temperature behaviour indicate that alumina-rich fibres have the capacity to delay the diffuse damage phenomena; fibres also modify the mechanisms that drive the visco-plastic concrete behaviour.

In order to study the visco-plasticity effect of the concrete, bending test were performed at 900 8C in modifying the crosshead speed from 0.2 mm/min to 0.02 mm/min, Figure 2. As it can be seen, dividing the displacement rate by a factor of 10 reduces the peak stress and the apparent modulus of the concrete without permitting a higher deformation capacity. For a given deflection value, a decrease of the strain rate causes a decrease of the stress values that has been related to the appearance of a liquid phase in the cementitious matrix [3]. The lower displacement rate allows a higher stress relaxation of the concrete. In Figure 2, the benefits of the alumina-rich fibre reinforcement can be observed, the higher the Al2O3 content the higher the load capacity: for the And-N6 grade tested at 0.2 mm/min, the stress gain is 80% for grade whereas for a rate of 0.02 mm/ min, the gain is as high as 150%.

Creep behaviour was also investigated under non-isothermal creep conditions: four-point bending samples were submitted to a constant 2 MPa stress level whereas temperature was raised from 20 8C to 1200 8C, Figure 3. Under such testing conditions, creep starts to occur between 750 8C and 800 8C whatever the grade. Results confirm that reinforcing the And-NF concrete with Nextel fibres increases its high temperature mechanical behaviour. Indeed, in all the temperature range, the measured deflection levels are really lower for the reinforced grade up to 1100 8C, the And-N7 grade showing the better performances. After 1100 8C, a rapid creep deflection was observed for the And-N6 grade whereas And-N7 grade retains his creep resistance.

High temperature microstructural evolution

In order to identify the phenomena involved in the high temperature range, SEM observations were performed on samples after testing. For the 900 8C tested samples, Figure 4 shows a fracture surface with fibres that have been pulled out of the concrete. This indicates that debonding and sliding mechanisms act at the fibre/concrete interfaces. Such mechanisms must be considered as having a major role on the high temperature behaviour of the reinforced grades.

Previous studies have shown the microstructural evolution of the matrix concrete at high temperature (starting at 900 8C) and in particular the apparition of liquid phases in the matrix [START_REF] Marzagui | Proceedings of the International Conference of Metallurgists COM2004[END_REF][START_REF] Marzagui | Proceedings of the International Conference of Metallurgists COM2004[END_REF]. In order to observe the interaction between liquid phase and fibres, high magnification observations were completed on samples ruptured in the range 700 8C -1200 8C. Figure 5 shows clearly the importance of the matrix microstructural evolution from a still cementitious matrix at 700 8C towards a matrix with high liquid phases at higher temperatures. These images show that Nextel fibres reinforcement process involves also liquid phases at fibre/concrete interface.

During high temperature testing, the fibres modify the natural network of microcracks of the concrete. This can be seen on Fig-

ure 6 showing microstructural observations on four-point bending test samples submitted to interrupted tests at 900 8C (test stop took place before maximum stress peak). For both grades (And-N7 (a) and And-N6 (b)) numerous microcrack deviations, crack branching and bridging and fibre/matrix debonding configurations are visible (white arrows). As a consequence the presence of fibres slows crack propagation and opening. Debonding phenomena take extensively place both at matrix/aggregate interfaces and at matrix/fibre interfaces. The combination of all these phenomena is beneficial and partly explains the overall non-linear behaviour that is a mix between elastic-damageable and viscous contributions. However, at this scale, the specific effect related to the presence of liquid phase is difficult to observe especially after return to room temperature for observations.

Figure 7 highlights the role of fibres in the visco-plastic behaviour of the concrete at 1200 8C and before a high level of damage. At this temperature, liquid phase amount of matrix is important and some vitreous phases could be seen. The observations partly help to explain the macroscopic viscoplastic deformation as a sliding mechanism between solid particles immersed in a large amount of liquid matrix phase. Such mechanisms are consistent with the increase of the strain capacity of the concrete and the stress accommodation phenomena in this high temperature testing domain. As a consequence the reinforced concrete becomes sensitive to strain rate. Moreover, many vitreous bridges are observed at fibres/matrix interfaces or between fibres. From this it can be considered that fibres can slow down or stop the relative displacement between the solid constituents of the matrix and modifying the rheology of the matrix at high temperature.

Conclusion

Ceramic fibre reinforced refractory castable have been studied in order to propose material solutions for die manufacturing for very high temperature Superplastic Forming. Two grades of Alumina based ceramic fibres with a diameter close to 12 lm have been investigated as reinforcement solutions for an andalusite castable refractory and for temperatures higher than 900 8C.

Fig. 1 .

 1 Fig. 1. Four-point bending behaviour curves of concrete fired and tested at high temperature between 700 8C et 1200 8C: a) unreinforced (And-NF), b) Nextel 610m (And-N6) and c) Nextel 720m (And-N7).

Fig. 2 .

 2 Fig. 2. Crosshead speed effect on four-point bending behaviour at 900 8C of the three grades: (a) 0.2 mm/min and (b) 0.02 mm/min.

Fig. 3 .

 3 Fig. 3. And-NF, And-N6 and And-N7 grades anisothermal creep behaviour under 2 MPa (four-point bending) until 1200 8C.

Fig. 4 .

 4 Fig. 4. Fracture surface of an And-N7 sample after a bending test at 900 8C (SEM micrograph).

Fig. 5 .

 5 Fig. 5. Fracture surfaces of Nextel 720m fibres and surrounding matrix after four-point bending tests in the temperature range 700 8C to 1200 8C.

Fig. 6 .

 6 Fig. 6. Observation of diffuse damage developed during a 900 8C fourpoint bending test (interrupted before the peak stress) in an andalusite based castable reinforced with 2 vol.% of: (a) Nextel720m fibres (b) Nextel610m fibres. White arrows indicate microcracks deviation phenomena to fibres.

Fig. 7 .

 7 Fig. 7. Observation of And-N7 grade at room temperature after an interrupted four-point bending test at 1200 8C.

Table 1 .

 1 Designation and properties of reinforcing fibres

	Parameter	Nextel 720m	Nextel 610m
		(N7)	(N6)
	Diameter [lm]	12	12
	Length [mm] Main constituents [wt%]	12 85% Al2O3	12 A 99% Al2O3
		15% SiO2
	Young's modulus [GPa]	260	380
	Tensile strength [MPa]	2100	3100
	Maximum use temperature [ 8C]

Table 2 .

 2 Peak stress relative variation at high temperature (four-point bending test)

	Testing	700 8C	900 8C	1000 8C 1100 8C 1200 8C
	temperature
	And-N7	86%	79%	96%	52%	31%
	And-N6	117%	65%	73%	51%	41%
	Unreinforced concrete is the reference

Table 3 .

 3 Apparent elastic modulus relative variation at high temperatures (four-point bending test)

	Testing	700 8C	900 8C	1000 8C 1100 8C 1200 8C
	temperature
	And-N7	44%	16%	110%	12%	49%
	And-N6	36%	22%	85%	49%	49%
	Unreinforced concrete is the reference

UniversitØ de Toulouse; INSA, UPS, Mines Albi, ISAE; ICA (Institut ClØment Ader), Albi cedex, France

High temperature mechanical tests were performed up to 1200 8C in four-point bending conditions followed by after testing microstructural and rupture surface observations. Results show that significant mechanical and creep improvement were obtained up to 1100 8C when compared to non reinforced refrac-tory castables. These material solutions are now ready for industrial trial. At high temperatures, these materials exhibit a macroscopical visco-plastic behaviour, that has been related to microcracking branching and fibre/matrix debonding phenomena as well as numerous liquid phase appearance.

6 References


Natassia Lona Batista
email: natassia_lb@hotmail.com

Philippe Olivier

Gérard Bernhart

Mirabel Cerqueira Rezende

Edson Cocchieri Botelho

Mirabel Cerqueira Rezende

Correlation between degree of crystallinity, morphology and mechanical properties of PPS/carbon fiber laminates

Keywords: CF/PPS, Thermoplastic composites, Crystallinity

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

PPS is a semi-crystalline thermoplastic with a symmetrical rigid backbone chain consisting of recurring para-substituted rings and sulphur atoms [START_REF] Anagreh | Low-pressure plasma pretreatment of polyphenylene sulfide (PPS) surfaces for adhesive bonding[END_REF][START_REF] Díez-Pascual | Mechanical and electrical properties of carbon nanotube/poly(phenylene sulphide) composites incorporating polyetherimide and inorganic fullerene-like nanoparticles[END_REF] . This high-performance polymer exhibits a glass transition temperature (T g ) of about 90°C and a melting temperature (T m ) around 285°C [START_REF] Naffakh | Synthesis and characterization of nitrated and aminated poly(phenylene sulfide) derivatives for advanced applications[END_REF] . It presents outstanding physical and chemical properties such as high-temperature stability, inherent flame resistance, good chemical resistance, anti-aging, high hardness and rigidity and excellent friction properties [START_REF] Díez-Pascual | Mechanical and electrical properties of carbon nanotube/poly(phenylene sulphide) composites incorporating polyetherimide and inorganic fullerene-like nanoparticles[END_REF][START_REF] Kim | Material selection windows for hybrid carbons/poly(phenylene sulfide) composite for bipolar plates of fuel cell[END_REF] . Due mainly to its low melt viscosity (~200 Pa.s), the PPS thermoplastic polymer can be molded with high filler or reinforcement contents [START_REF] Naffakh | Synthesis and characterization of nitrated and aminated poly(phenylene sulfide) derivatives for advanced applications[END_REF][START_REF] Yilmaz | Geometric parameters and chemical corrosion effects on bearing strength of polyphenylenesulphide (PPS) composites[END_REF] . Most PPS products are reinforced with carbon or glass fibers for high-performance applications, including electronic, automotive and aerospace ones [START_REF] Yang | Morphology control of nanofillers in poly (phenylene sulfide): A novel method to realize the exfoliation of nanoclay by SiO 2 via melt shear flow[END_REF] . Carbon fiber reinforced PPS (CF/PPS) is mainly used in aircraft structural applications, such as in J-Nose wing substructures of the Airbus A340-500/600 [START_REF] Naffakh | Tuning the properties of carbon fiber-reinforced poly(phenylene sulphide) laminates via incorporation of inorganic nanoparticles[END_REF] .

PPS is also well known for being able to attain a high level of crystallinity, with the commercial grades achieving crystallinities of 60% [START_REF] Vieille | Influence of temperature on the behavior of carbon fiber fabrics reinforced PPS laminates[END_REF] . Crystallinity degree in high performance thermoplastics is important, because it has a strong influence on chemical and mechanical properties: the crystalline phase tends to increase stiffness and tensile strength, while the amorphous phase is more effective in absorbing impact energy [START_REF] Vieille | About the influence of stamping on thermoplastic-based composites for aeronautical applications[END_REF] . When a semi-crystalline thermoplastic is cooled from the melt state, it normally forms spherical structures so called "spherulites". Spherulites have been described as approximately radially symmetric semi-crystalline structures.

Considered from a central point, each spherulite consists of a radial assembly of thin crystalline lamellae which are separated by amorphous layers. The crystalline domain is formed by regularly ordered molecular chains, while the amorphous phase consists of an assembly of disordered macromolecules, which are morphologically constrained by the neighboring crystalline lamellae [START_REF] Van Dommelen | Micromechanical modeling of intraspherulitic deformation of semicrystalline polymers[END_REF] . The dimension range of spherulites varies from micrometers to millimeters, depending on the structure of the polymer chain and the crystallization conditions, such as cooling rate, crystallization temperature, and the content of the nucleating agent [START_REF] Li | Direct observation of growth of lamellae and spherulites of a semicrystalline polymer by AFM[END_REF] . When these structures are viewed between cross polarizers, they exhibit a characteristic "Maltese cross" extinction pattern [START_REF] Ma | Effect of starch fractions on spherulite formation and microstructure[END_REF] .

For semi-crystalline polymer composites, the fillers can have the potential to induce or alter polymer crystallization, acting as nucleating agents. Thus, an increase in crystallization rate and crystallization temperature as well as a decrease in crystal size is generally observed [START_REF] Ning | Realizing the enhancement of interfacial interaction in semicrystalline polymer/filler composites via interfacial crystallization[END_REF] . The crystalline formation at the interphase region depends on a number of factors, such as matrix morphology, substrate surface condition, as well as the presence of reactive functionalities [START_REF] Gao | Cooling rate influences in carbon fibre/ PEEK composites. Part 1. Crystallinity and interface adhesion[END_REF] .

Some fibers may induce crystallization along the polymer/fiber interface due to the high density of active nuclei on its surface, which hinder the full extension of spherulites and force the crystal growth in the perpendicular direction [START_REF] Ning | Realizing the enhancement of interfacial interaction in semicrystalline polymer/filler composites via interfacial crystallization[END_REF] . This particular behavior can results in a columnar crystalline layer, known as transcrystallinity (TC) [START_REF] Quan | On transcrystallinity in semi-crystalline polymer composites[END_REF] . Since it was first reported, TC has attracted tremendous attention, because it may be an effective and economical method to improve the interfacial adhesion between polymer matrix and fiber. To date, TC has been reported to occur at the interface between several semi-crystalline polymers, such as PPS and PEEK [START_REF] Ning | Realizing the enhancement of interfacial interaction in semicrystalline polymer/filler composites via interfacial crystallization[END_REF] .

Several different routes have been developed for manufacturing continuous fiber-reinforced thermoplastic composites. These composite materials can be obtained by using pre-impregnation techniques, such as hot-melt impregnation, fluid/bed impregnation and solvent-based impregnation; or post-shaping impregnation techniques, such as powder suspension impregnation, film stacking and commingling of thermoplastic fibers with reinforcing fibers [START_REF] Fujihara | Influence of processing conditions on bending property of continuous carbon fiber reinforced PEEK composites[END_REF][START_REF] Dai | Processing of single polymer composites with undercooled polymer melt[END_REF][START_REF] Díez-Pascual | Influence of carbon nanotubes on the thermal, electrical and mechanical properties of poly(ether ether ketone)/glass fiber laminates[END_REF] . Hot compression molding process is a widely used technique to produce thermoplastic-matrix-composite laminates from these raw materials. The consolidation is achieved by placing the plies with proper sequence into the die cavity, heating it up and keeping the material under pressure at temperatures higher than the thermoplastic melting/softening point until the impregnation process between the plies is completed. Finally, the composite is cooled down to the room temperature [START_REF] Naffakh | Tuning the properties of carbon fiber-reinforced poly(phenylene sulphide) laminates via incorporation of inorganic nanoparticles[END_REF] . For non-flat dies, the post-shaping impregnation techniques are more suitable due to the drapeability of the fabrics obtained by this process.

The aim of this work is to bring a better understand of the crystallization process, under three different cooling rates, of PPS polymer on the carbon fiber (CF) surfaces and evaluate the mechanical properties of CF/PPS composites. The influence of crystallinity on the laminates was evaluated by thermal and mechanical characterization applying differential scanning calorimetry (DSC), dynamic mechanical analysis (DMA), interlaminar shear strength (ILSS) and acoustic vibration damping.

Material and methods

Material

Thermoplastic laminates were processed by PPS films (supplied by Curbell Plastics) and plain weave carbon fiber fabric (provided by Hexcel) film stacking technique. The final laminates were produced with 2.5mm of thickness, attending the reinforcement/matrix volume content of 60/40 (v/v). In order to obtain this result, 15 layers of reinforcement and polymer films were required.

Polarized light optical microscopy

The crystallization process of PPS around a microfilament of carbon fiber was observed on thin films using a polarized light optical microscope (Laica-DMLS) with automatic heating and cooling stages (Linkam THMS600). The samples, placed between two glass covers, were heated up from room temperature to 315 °C at 10 °C.min -1 , kept in this temperature during 10 min and cooled down in the following different rates: 0.5, 1, 5, 10 and 20 °C.min -1 .

Rheology

Rheological tests were performed on PPS film samples using a parallel plate rheometer with controlled deformation (Thermo Scientific HAAKE MARS CTC), in order to establish the maximum period of time that the material could be kept inside the die before cooling. The linear rheological response was obtained from one hour long analysis by heating the polymer (10 °C.min -1 ) until 315 °C with frequency of 1Hz and 1% of deformation.

Hot compression molding process

The composite laminate consolidation process was carried out in a steel die with dimensions of 300 mm x 300 mm. Firstly, the material was heated from room temperature (~25 °C) up to 315 °C at approximately 10 °C.min -1 , held in this temperature for 30 min under 1.2 MPa of pressure and then cooled down to room temperature. These parameters were stipulated according to the literature found on this subject [START_REF] Anagreh | Low-pressure plasma pretreatment of polyphenylene sulfide (PPS) surfaces for adhesive bonding[END_REF][START_REF] Naffakh | Synthesis and characterization of nitrated and aminated poly(phenylene sulfide) derivatives for advanced applications[END_REF][START_REF] Kim | Material selection windows for hybrid carbons/poly(phenylene sulfide) composite for bipolar plates of fuel cell[END_REF][START_REF] Yilmaz | Geometric parameters and chemical corrosion effects on bearing strength of polyphenylenesulphide (PPS) composites[END_REF][START_REF] Yang | Morphology control of nanofillers in poly (phenylene sulfide): A novel method to realize the exfoliation of nanoclay by SiO 2 via melt shear flow[END_REF][START_REF] Naffakh | Tuning the properties of carbon fiber-reinforced poly(phenylene sulphide) laminates via incorporation of inorganic nanoparticles[END_REF][START_REF] Hu | RF properties of BST-PPS composites[END_REF][START_REF] Stoeffler | Polyphenylene sulfide (PPS) composites reinforced with recycled carbon fiber[END_REF] and preliminary performed tests. In this work, three different cooling rates were applied in order to vary the final degree of crystallization, a fast rate of about 10 °C.min -1 , a slow rate of approximately 1°C.min -1 and an air cooling, obtained by leaving the press, die and laminate cool naturally in air. Figure 1 shows the processing cycle used to obtain the PPS/carbon fiber laminates.

Differential Scanning Calorimetry (DSC)

The crystallization of the processed composites was investigated by DSC using a Seiko Exstar 6000 -DSC 6220 differential scanning calorimeter operating under nitrogen flow. Samples (10mg) were cut from CF/PPS laminates and sealed in aluminum pans. A heating rate of 10 °C.min -1 was used to heated the samples from room temperature (25 °C) up to 315 °C.

The melting enthalpy of each sample was obtained from the area under the melting peaks. The levels of crystallinity were calculated by dividing the melting enthalpy of the samples by the melting enthalpy of PPS 100% crystalline is 80 J.g -1 3 . However, since the specimens contain polymer and carbon fiber, the reinforcement mass had to be subtracted from the total. Therefore, the levels of crystallinity of PPS composites were determined according to Equation 1:

( ) ( ) c c m H X % 100% H 1 x ∆ = × ∆ °- (1)
where: X C is the degree of crystallinity, ΔH C is obtained by the melting peak area, ΔH°M is the melting enthalpy of PPS 100% crystalline, and x is fiber weight fraction.

Dynamic Mechanical Analysis (DMA)

The dynamic mechanical performance of the composite samples was analyzed using a Seiko SII Exstar 6000 dynamic mechanical analyzer. Experiments were performed in dual cantilever bending mode at frequency of 1 Hz. A dynamic force of 2 N was used at fixed frequency and amplitude of 10 μm. The results were recorded in the temperature range of 30 to 200°C, at a heating rate of 3 °C.min -1 . The dimensions of the specimens analyzed were approximately of 50 mm x 14 mm x 2.5 mm.

Interlaminar Shear Strength (ILSS)

Interlaminar shear strength measurements were carried out using a Shimadzu autograph AG-X series precision universal machine at room temperature, using a constant cross-speed of 1 mm.min -1 and a load cell of 5 kN. The tests were performed according to ASTM D2344 standard, using at least five specimens with dimensions of 15 mm x 5.5 mm x 2.5 mm.

Determination of Young's modulus

The impulse excitation of vibration was used to determine Young's modulus from the natural frequency, using a rectangular specimen under flexure mode. At the technique applied, the specimen is lightly stroked, the acoustic response is captured and this is processed by software, which calculates the Young's modulus. The impulse excitation technique was applied using Sonelastic® equipment, developed by ATCP -Physical Engineering, at Brazil. This non-destructive test was performed according to ASTM 1876 standard, using specimens with dimension of 50 mm x 14 mm x 2.5mm.

Results and discussion

Crystallization morphology

It is well known that the crystallization process plays a crucial role on the properties of semi-crystalline polymers [START_REF] Ma | Effect of starch fractions on spherulite formation and microstructure[END_REF][START_REF] Ning | Realizing the enhancement of interfacial interaction in semicrystalline polymer/filler composites via interfacial crystallization[END_REF][START_REF] Gao | Cooling rate influences in carbon fibre/ PEEK composites. Part 1. Crystallinity and interface adhesion[END_REF] and that the presence of carbon fibers as reinforcement may also influence this process [START_REF] Quan | On transcrystallinity in semi-crystalline polymer composites[END_REF][START_REF] Fujihara | Influence of processing conditions on bending property of continuous carbon fiber reinforced PEEK composites[END_REF][START_REF] Dai | Processing of single polymer composites with undercooled polymer melt[END_REF] . However, the images obtained by polarized light optical microscopy in this work do not show any evidence of nucleation under the cooling rates performed, when analyzed in the range of 0.5 to 20 °C/min. The crystal growing seemed to take place in a homogenous manner, even for the slowest cooling, as it can be observed in Figure 2.

Although the PPS crystallites formation was highly homogeneous, a different behavior was observed for the cooling performed at 5 °C/min. Due to the formation of a bubble on the surface of a carbon fiber microfilament, the very low thickness of the created polymer film allowed the visualization of crystallites nucleation on the microfilament surface, as shown in Figure 3. As mentioned, it is believed that the observation of nucleation was only possible owing to the thickness of the film, since there were no crystallites overlapping. Moreover, the crystal growing continues to develop as before, in a very homogenous manner.

In addition to nucleation, a transcrystalline layer can also be seen on the carbon fiber surface (Figure 4(a)). Such phenomenon was not visualized on the bubble boundary far from the fiber, as showed in Figure 4(b), which excludes the possibility that the layer formed was a result of the bubble presence.

Rheological investigation

The rheological analysis performed on the PPS films (Figure 5) revels that before 30 min, at a constant temperature of 315°C, no significant increase on the viscosity is observed. This behavior not only facilitates the polymer percolation during the consolidation process, but also indicates that thermal degradation is not likely to occur in this period of time. Therefore, the crystallization process would not be affected by thermal degradation.

Melting behavior and crystalline content

The DSC melting curves of the different manufactured laminates, concerning the cooling rate, are displayed in Figure 6. The melting peak area of the DSC curves increases with the decrease of the cooling rate due to the higher crystalline content into the PPS matrix. This behavior can be better observed in Table 1, where the melting enthalpy (or melting peak area) is used to obtain the crystallinity percentage of PPS in the PPS/carbon fiber composite.

In addition to the alteration in the area, two other differences in the peak are observed. Firstly, it is observed that the melting temperature increases from fast to in air cooling, which indicates that the solid polymer matrix resulting from the latter cooling rate has better ordered crystallites resulted from the longer cooling time. Therefore, longer cooling time favors the alignment and folding of the polymer chains forming regions more orderly. Secondly, the curve shape for the fast cooled samples also differs from the others, not showing the shoulder observed in the other two curves. The two-steps melting observed in the DSC curves suggest the presence of two different ordering of the crystallites. Probably one refers to the melting of the crystallites in the polymer bulk and the other to the transcrystallinity phase, respectively. Therefore, it is possible to conclude that the surface of the carbon fiber favored the crystallites nucleation when slowly cooled.

Dynamic mechanical properties

DMA tests were performed in order to evaluate the viscoelastic behavior of the composites as a function of temperature. Dynamic mechanical tests are sensitive to several transitions and relaxation processes of the polymer in the composite, and provide information about the fiber/ matrix interface [START_REF] Díez-Pascual | Influence of carbon nanotubes on the thermal, electrical and mechanical properties of poly(ether ether ketone)/glass fiber laminates[END_REF] . Figure 7(a) shows the temperature dependence of the storage modulus (E'). The storage modulus is indicative of the elastic energy stored in the material and it is affected by changes in the morphology of the material, induced, for example, by the crystalline content. Regarding the different laminates, significant E' increments are found for the samples with higher crystalline content. This result shows that slower cooling rates favored the stiffness increasing. Furthermore, it can be concluded that different cooling rates affected the crystallinity content of PPS matrix and consequently the mechanical behavior of the PPS/carbon fiber composite. Figure 7(b) shows the variation of the loss modulus (E") with the temperature. The loss modulus is related to the energy dissipation mechanisms in the material. Therefore, composites with poor interfacial bonding tend to dissipate more energy. The experimental data indicates that the composite laminates with higher crystalline content (cooled slowly and in air, respectively) display lower loss modulus, suggesting that an improvement on the fiber/matrix interface occurred mainly as a result of PPS polymer crystallites nucleation on the carbon fibers.

The ratio between loss modulus and storage modulus is defined as tan δ and it is an indicative parameter of the mechanical damping ability of the material. The damping properties provide the balance between the elastic and viscous phases in a polymeric structure. The evolution of tan δ as a function of temperature is shown in Figure 7(c). In comparison with the fast cooled CF/PPS, the height of tan δ peak decreases for both the slow cooled and in air cooled laminates, respectively. Therefore, the results reveal a decrease of the damping ability with the increase of the crystalline content. As the damping is an indicative of the amount of energy used to deform the material that is directly dissipated into heat, these results are found to be coherent.

Young's modulus measurements

The Young's modulus (E), measured by impulse excitation vibration, was also investigated. The results obtained are shown in the summary of the mechanical test results (Table 2). The Young's modulus of the laminates cooled slowly and in air experience a raise of about 5% and 9%, respectively, when compared with the fast cooled. These increments are attributed to the higher stiffness promoted by the increase of the crystallinity content. This result is in agreement with the behavior observed for E', obtained by DMA. Thus, the increase of crystallinity corroborates to a higher modulus, storage and Young's, and, consequently, to an improvement on the mechanical resistance.

Interlaminar shear strength

To evaluate the fiber/matrix interface of the laminates, ILSS tests were performed and the quantitative results included in Table 2. The interlaminar shear strength of the CF/PPS laminates increased about 14% when the cooling mode changed from fast to in air. However, statistically speaking, no variation in strength was observed concerning the cooling change from fast to slow. Thus, the cooling in air probably favored the nucleation of crystallites on the surface of the carbon fibers and the crystallization formation along the polymer/fiber interface, also known as transcrystalline layer. Comparing the slow and in air cooling curves on Figure 1, it is possible to suppose that the transcrystallinity formation was not only due to the low cooling rate, since both profiles did not present a significant variation in the first 30 min of temperature fall. The absence of the cooling system utilization probably contributed to promote a stable environment, favoring the transcrystallinity formation. A similar behavior was observed by polarized light microscopy, as depicted in Figure 3, when the bubble on the surface of the fiber provided not just a thinner layer of polymer but also a more stable environment, favoring the crystallites nucleation and growth. As a result of the transcrystalline layer, the interface of the laminates was enhanced, leading to an improvement in the ILSS values. This result is in line with previous E" analysis, where an improvement in a mechanical property caused by the fiber/matrix interface enhancement was also inferred.

Overall, this study has shown the determinant effect of the cooing rate on the degree of crystallinity of PPS matrix in CF/PPS laminates and, as a consequence, its fundamental role on the improvement of the mechanical and thermo-mechanical performances of the processed composite laminates.

Conclusions

The morphological aspects and the thermal and mechanical properties of CF/PPS laminates with different crystalline degrees have been investigated. Previous works have discussed about transcrystallization in CF/ PPS composites and the degree of crystallinity influence on the mechanical properties [START_REF] Schulz | Effect of transcrystallization in carbon fiber reinforced poly(p-phenylene sulfide) composites on the interfacial shear strength investigation with the single fiber pull-out test[END_REF][START_REF] Boey | Effect of matrix crystallinity on the buckling failure of a PPS thermoplastic composite[END_REF][START_REF] Vieille | Influence of temperature on the behavior of carbon fiber fabrics reinforced PPS laminates[END_REF][START_REF] Boey | Annealing effects on the dynamic mechanical properties of aromatic polyphenylene sulphide fibre reinforced composite[END_REF] . In the present work, morphological observations revealed a highly homogeneous crystallization process for PPS, which posed some difficulties in the visualization of crystallites nucleation on the surface of the carbon fiber. Thus, the identification of nucleation and transcrystalline layer formation was only possible in a more stable environment with a considerably thinner polymer film. DSC results confirmed the increase of the crystallinity contents for the samples obtained with lower cooling rates and showed that the crystallization occurred in two steps for the specimens with higher crystalline degree (slower cooling rate). This result indicates the presence of different ordering of the crystallites, probably due to the presence of them in the polymer bulk and also the transcrystalline layer formation. DMA experiments showed an increase in the storage modulus and a decrease in both the loss modulus and tan δ with the increment of the crystalline degree. This behavior was a result of the stiffness increase and the improvement of the fiber/matrix interface. The Young's modulus and the ILSS also increased mainly due to the improvements in stiffness and fiber/matrix interface, respectively. Overall, by changing only one parameter of the PPS/carbon fiber processing, the cooling rate, it was possible to induce a significant improvement in the laminate mechanical resistance. The mechanical tests demonstrated enhancements in stiffness, strength and fiber/matrix interface for the laminates processed with lower cooling rates.

Figure 1 .

 1 Figure 1. Consolidation cycle used for laminate manufacturing.

Figure 2 .

 2 Figure 2. Polarized light optical micrographs of melted PPS cooling at 0.5 °C/min in the presence of a carbon fiber microfilament: (a) 277°C; (b) 276°C and (c) 275°C.

Figure 3 .

 3 Figure 3. Polarized light optical micrographs of melted PPS cooling at 5 °C/min in the presence of a carbon fiber microfilament: (a) 250 °C; (b) 235 °C; (c) 230 °C and (d) 225 °C.

Figure 4 .

 4 Figure 4. Polarized light optical micrographs of crystallized PPS cooled at 5 °C/min in the presence of a carbon fiber microfilament: (a) transcrystallinity formation; (b) superior boundary limit of the formed bubble.

Figure 5 .

 5 Figure 5. Viscosity behavior of melted PPS at 315°C.

Figure 6 .Table 1 . 9 Figure 7 .

 6197 Figure 6. Effect of the cooling rate on the DSC curves of CF/PPS composites. Table 1. Effect of the cooling rate on the crystallinity content of PPS matrix in the composite. Cooling mode X c % Fast 51.1±1.0 Slow 58.5±0.8 In Air 61.9±1.9 Figure 7. Effect of the cooling rate on the: (a) storage modulus; (b) loss modulus; and (c) tan δ values of PPS/carbon fiber composites.

Table 2 .

 2 Young's modulus and ILSS values of the CF/PPS composites obtained with different cooling rates.

	Cooling mode	E (GPa)	ILSS (MPa)
	Fast	63.8±0.6	27.5±2.1
	Slow	67.2±0.7	28.1±3.3
	In Air	69.7±0.9	31.4±1.9

Acknowledgements

The authors would like to thank CNPq (Process numbers 303287/2013-6 and 303559/2012-8), FAPESP (Process number 2012/12207-6) and CAPES (Process number 12357-13-8) for the financial support.


Gloria Faccanoni

Hélène Mathis

ADMISSIBLE EQUATIONS OF STATE FOR IMMISCIBLE AND MISCIBLE MIXTURES * , * *

Keywords: Two-phase flows, equation of state, thermodynamic of equilibrium, phase transition, homogeneous equilibrium and relaxation model, hyperbolicity. 2010 MCS. 76T10, 80A15, 35Q79, 35L

This paper addresses the construction of admissible Equations of State (EoS) for compressible two-phase flows. We investigate two approaches. In the first one, the mixture is treated as a single fluid with a complex thermodynamic. Most of the time the available EoS are determined experimentally and are often incomplete EoS, i.e. we know only the pressure as a function of the volume and the temperature. We present here a general framework to compute a complete EoS based on such an incomplete EoS. In the second approach, each phase is depicted by its own EoS. Following the Gibbs formalism, the mixture entropy is the sum of the phasic entropies which achieves its maximum at equilibrium. Depending on the miscibility of the mixture, one gets different geometrical properties on the resulting mixture entropy. Eventually we address the coupling of mixture EoS with the dynamic of the fluid. Homogeneous Equilibrium and Relaxation Models (HEM and HRM) are introduced for an immiscible and a miscible two-phase mixture. Hyperbolicity is ensured taking advantage of the concavity properties of the mixture entropies.

Introduction

Compressible multiphase flows are involved in many industrial applications, notably in nuclear framework, for instance in loss of flow or loss of coolant accidents in pressurized water reactor. For the purpose of CFD calculations, thermodynamic properties must be calculated over a wide range of states. Thus it is mandatory to provide a consistent Equation of State (EoS) to give a qualitatively and quantitatively correct material description. This EoS should represent phase equilibria over wide temperature and pressure ranges, in pure region (liquid or vapor) as well as in the VLE (Vapor-Liquid Equilibrium) region.

The construction of coherent EoS raises several difficult questions:

(1) how to obtain an accurate (and useful in CFD) description of a pure region?

(2) how to compute a given thermodynamic state by performing calculations with different couples of input variables? (3) how to describe accurately the transition from a pure region to the VLE region? (4) CFD requires extremely fast algorithms to compute thermodynamic properties because such algorithms are frequently used in the inner iteration cycles of the process calculations. How to balance accuracy and reasonable computing times? A classic strategy for the last question ( 4) is to use an analytical EoS. So the question is to construct such an analytical EoS while respecting constraints 1-3.

Focusing on a macroscopic description of the compressible two-phase flow, we consider that the dynamic of the fluid is depicted by an Euler-like system. The two-phase mixture is treated as a single fluid with a complex thermodynamic. 1 In order to define the complex EoS, we consider two main approaches.

• The first possibility is to use an analytical EoS, valid in a wide range of temperatures and pressures. Such equations are often deduced from measurements and are usually incomplete in the sense that they define solely the pressure p as a function of the volume τ and the temperature T . However, in the CFD calculations (based on the Euler equations), it is necessary to specify the pressure as a function of the volume and the specific internal energy e. To obtain these relation, an idea is to construct an entropy function (τ, e) → s, which is actually a complete EoS. It is then possible to derive all incomplete EoS, including the pressure and the temperature. We present this general construction in Section 2. The section begins by the definitions of the different thermodynamic potentials both in the extensive and intensive settings. Then in paragraph 2.2 we build a complete EoS starting from the incomplete law (τ, T ) → p. The principle is to determine a temperature (τ, e) → T and thus an entropy (τ, e) → s thermodynamically consistent, i.e. they satisfy the first principle of the Thermodynamics and a differential relation of the form T ds = de + pdτ .

• The second approach we present consists in assuming that each phase is depicted by its own complete EoS (for instance its entropy). Out of equilibrium the mixture entropy is the sum of the phasic entropies.

According to the second principle of Thermodynamics, the mixture entropy reaches its maximum at equilibrium, under the constraints of the mass and energy conservation and a volumic constraint. Depending on the miscibility of the mixture, the latter constraint changes leading to different geometrical properties of the mixture entropy. We present in Section 3 the construction principle in the case of binary mixture. Note that it can be extend to multiphase mixture, see for instance [START_REF] Hurisse | A homogeneous model for compressible three-phase flows involving heat and mass transfer[END_REF][START_REF] Mathis | A thermodynamically consistent model of a liquid-vapor fluid with a gas[END_REF] for three-phase framework. A particular attention is paid on the characterization of the thermodynamic equilibrium and on the concavity properties of the mixture entropy. In Section 4, we address the coupling of the admissible EoS computed in Section 3 with the dynamic of the fluid. We present two Homogeneous Equilibrium Models (HEM) depending on the miscibility of the twophase mixture. They consist in Euler type models with the closure laws computed in Section 3. The concavity properties of the mixture entropies ensure that the models are hyperbolic. However the closure laws is often difficult to compute and may not be analytical, even for simple phasic EoS. It is then convenient to approximate the model by a relaxation procedure, see Section 5. One obtains an Homogeneous Relaxation Model (HRM), with extra equations and relaxation source terms toward the thermodynamic equilibrium, which comply with the entropy growth criterion. Again we present two models according to the miscibility of the two-phase mixture.

1 Note that a way to avoid this definition of the complex EoS is to consider a two-fluid description, which treats each phase as a separated fluid depicted by balance laws of phasic mass conservation, phasic momentum and energy which are coupled through exchanging terms and nonconservative terms. Then the modelling difficulties are shifted on the definition of the exchanging terms, see [START_REF] Baer | A two phase mixture theory for the deflagration to detonation (ddt) transition in reactive granular materials[END_REF][START_REF] Hérard | A class of compressible multiphase flow models[END_REF].

Single fluid thermodynamics

Extensive and intensive notations

Let consider a fluid of mass M ≥ 0 and internal energy E ≥ 0 occupying a volume V ≥ 0. Let us denote W def = (M, V ) and W def = (V, M, E). As the fluid is homogeneous and at rest, its thermodynamic behaviour is entirely described by its entropy function:

S : (R + ) 3 → R W → S(W ).
This entropy function S is uniquely and smoothly determined by the state vector W ∈ (R + ) 3 . Standard Thermodynamic requires that S is a concave function of W . Then it is classic to extend it by -∞ outside the close convex cone (R + ) 3

S(W ) = S(W ), W ∈ (R + ) 3 , -∞, elsewhere.
We adopt the assumptions stated in [START_REF] Callen | Thermodynamics and an Introduction to Thermostatistics[END_REF] and [START_REF] Evans | A survey of entropy methods for partial differential equations[END_REF]:

Assumption 1. (Extensive assumptions) The entropy S : (R + ) 3 → R ∪ { -∞ } is such that (i) the set of admissible states C def = W ∈ (R + ) 3 S(W ) > -∞ is a non-empty close convex domain, (ii) S is a concave function of W ∈ C, (iii)
S is extensive, i.e. Positively Homogeneous of degree 1 (PH1), that is

∀λ ∈ R + * , ∀W ∈ C, S(λW ) = λS(W ),
(iv) S is upper semi-continuous (u.s.c.) that is

∀W 0 ∈ C, lim W →W0
sup S(W ) ≤ S(W 0 ), (v) S is of class C 2 on C and its partial derivative with respect to the internal energy E is strictly positive

∀W ∈ C, ∂S ∂E > 0.
Assumptions (ii) and (iii) are equivalent to assume (-S) sub-linear [START_REF] Rockafellar | Convex analysis[END_REF] or equivalently that the epigraph of (-S) is a closed set in R 3 × R. The set C is not necessarily convex, as it has been studied in [START_REF] Helluy | Interpolated pressure laws in two-fluid simulations and hyperbolicity[END_REF][START_REF] Jung | Schémas numériques adaptés aux accélérateurs mutlicoeurs pour les écoulements bifluides[END_REF]. Observe that the extensive entropy S cannot be strictly concave since it is PH1.

The fundamental extensive thermodynamic relation is

T dS = dE + pdV -µdM, (1)
where W → T is the temperature, W → p the pressure and W → µ the chemical potential given by

T = ∂S ∂E M,V -1 , p = T ∂S ∂V M,E , µ = -T ∂S ∂M V,E . (2)
As derivatives of a PH1 function, this functions are PH0 or intensive potentials. Since S is PH1, it satisfies the Euler's relation ∇ W S • W = S which leads to another characterization of the chemical potential

M µ = -T S + pV + E. (3)
Remark 2.1. Following [START_REF] Evans | Entropy and partial differential equations[END_REF], the formula S = S(W ) is called the fundamental equation (or complete equation of state) of our system, and by definition contains all thermodynamic informations. An identity involving other derived quantities (such T or p for instance) is an incomplete equation of state, which typically does not contain all thermodynamic information.

Intensive properties -specific description

By homogeneity of the extensive entropy function, one may introduce the specific entropy s as

s def = 1 M S(W ) PH1 = S W M = S V M , 1, E M . (4)
It corresponds to a complete EoS which is uniquely determined by the specific volume τ def = V /M and the specific internal energy e def = E/M :

s : (R + ) 2 → R (τ, e) → s(τ, e). (5)
The fundamental intensive thermodynamic relation is

T ds = de + pdτ, (6)
where the temperature T and the pressure p are now functions of w def = (τ, e):

w → T = ∂s ∂e τ -1 , w → p = T ∂s ∂τ e (7)
and the chemical potential satisfy µ = -T s + pτ + e.

Assumptions 1 admit their intensive counterpart.

Assumption 2. (Intensive specific assumptions) The specific entropy s is such that

• the set of admissible states c def = w ∈ (R + ) 2 s(w) > -∞ is a non-empty close convex domain, • s is of class C 2 on c such that T = ∂s ∂e τ -1 > 0, (8)
• its Hessian matrix is negative definite, that is

det(H s ) def = ∂ 2 s ∂τ 2 e ∂ 2 s ∂e 2 τ - ∂ 2 s ∂τ ∂e 2 > 0, ∂ 2 s ∂τ 2 e < 0, ∂ 2 s ∂e 2 τ < 0, (9)
which implies that s is strictly concave (but the converse is wrong).2

Intensive properties -volumic description

The homogeneity of the extensive entropy allows to introduce also the volumic entropy σ satisfying

σ def = 1 V S(W ) PH1 = S W V = S 1, M V , E V . ( 10
)
It is also a complete EoS uniquely determined by the density ρ def = M/V = 1/τ > 0 and the volumic internal energy

ε def = E/V = ρe such that σ : (R + ) 2 → R (ρ, ε) → σ(ρ, ε). (11)
As a function of w def = (ρ, ε), it complies with the following total differential

T dσ = dε -µdρ, (12)
where the temperature T and the chemical potential µ are now functions of w:

w → T = ∂σ ∂ε ρ -1 , w → µ = -T ∂σ ∂ρ ε , (13)
and the pressure satisfy

p = T σ + ρµ -ε. ( 14
)
Assumption 3. (Intensive volumic assumptions) The volumic entropy σ is such that

• the set of admissible states c = { w ∈ (R + ) 2 |σ( w) > -∞} is a non-empty close convex domain, • σ is of class C 2 on c such that T = ∂σ ∂ε ρ -1 > 0, (15)
• its Hessian matrix is negative definite, that is

det(H σ ) def = ∂ 2 σ ∂ρ 2 ε ∂ 2 σ ∂ε 2 ρ - ∂ 2 σ ∂ρ∂ε 2 > 0, ∂ 2 σ ∂ρ 2 ε < 0, ∂ 2 σ ∂ε 2 ρ < 0, (16)
which implies that σ is strictly concave (but the converse is wrong). 3

From an incomplete to a complete EoS

It is very common to specify the EoS of a material by providing the pressure as a function of the volume and the temperature: (τ, T ) → p. As noted in Remark 2.1, such an expression does not embody the full range of thermodynamic informations available from the fundamental equation S = S(W ) (or s = s(w)). However, in the CFD calculations (based on the Euler equations), one need to specify the pressure as a function of the volume and the internal energy: (τ, e) → p. To do so, an idea is to construct a complete EoS in the form of a specific entropy (τ, e) → s, based on the knowledge of the incomplete EoS (τ, T ) → p. Once the complete EoS is built, all other quantities can be computed thanks to the fundamental thermodynamic relation [START_REF] Dellacherie | Relaxation schemes for the multicomponent Euler system[END_REF] and the Maxwell relations (see appendix A).

General construction

In this section, we focus on the specific quantities and present a general method to deduce a complete EoS (τ, e) → s from an incomplete EoS (τ, T ) → p . This construction involves determining a temperature (τ, e) → T and thus a specific entropy (τ, e) → s, which are thermodynamically consistent in the sense that it complies with the fundamental relation [START_REF] Dellacherie | Relaxation schemes for the multicomponent Euler system[END_REF].

Let us consider a general incomplete EoS (τ, T ) → p of the form

(τ, T ) → p = rT τ -b -aQ(τ ), (17)
where a and b ≥ 0 are material-specific constants and r > 0 is the universal gas constant. The choice Q(τ ) = 1 corresponds to the (Noble-Abel)-Stiffened-Gas (NASG) law [START_REF] Chiapolino | Extended noble-abel stiffened-gas equation of state for sub-and-supercritical liquid-gas systems far from the critical point[END_REF][START_REF] Métayer | Elaborating equations of state of a liquid and its vapor for two-phase flow models[END_REF][START_REF] Métayer | The noble-abel stiffened-gas equation of state[END_REF], while the choice Q(τ ) = τ -2 corresponds to the van der Waals law [START_REF] Callen | Thermodynamics and an Introduction to Thermostatistics[END_REF][START_REF] Landau | A Course of theoretical physics[END_REF].

Since the specific entropy s is a total derivative which satisfies the fundamental relation [START_REF] Dellacherie | Relaxation schemes for the multicomponent Euler system[END_REF], its cross derivatives are equal:

∂ p T ∂e τ = ∂ 1 T ∂τ e . ( 18
)
Using the pressure definition [START_REF] Hérard | A class of compressible multiphase flow models[END_REF], one can exhibit an hyperbolic PDE satisfied by the temperature (τ, e) → T :

∂ τ T + aQ(τ )∂ e T = 0. ( 19
)
Let introduce the characteristic curves τ → χ(τ ) associated with the transport equation ( 19). In the (τ, e)-plane they are solutions of the Cauchy problem

χ (τ ) = aQ(τ ), χ(τ 0 ) = χ 0 , (20)
that is

χ(τ ) = χ 0 + a τ τ0 Q(x)dx. (21)
The temperature T def = T (τ, e = χ(τ )) is constant along the characteristic curves, hence it holds T (τ ) = T (τ 0 ) and yields

T (τ, e) = T τ 0 , e -a τ τ0 Q(x)dx . ( 22
)
For illustrative purposes, we choose the temperature as

T (τ, e) = e -a τ τ0 Q(τ )dτ c v , (23)
with c v the specific heat (which is assumed constant here for sake of simplicity). Finally a complete EoS can be obtained by integrating the fundamental relation ( 6)

ds = p T dτ + 1 T de = c v e -a Q(τ )dτ p(τ, e) dτ + c v e -a Q(τ )dτ de. (24)
Using ( 8) and treating τ as a constant, by integrating the relation [START_REF] Lagoutière | Modélisation mathématique et résolution numérique de problèmes de fluides compressibles à plusieurs constituants[END_REF] with respect to e gives

s(τ, e) = 1 T de + ω(τ ) = c v e -a Q(τ )dτ de + ω(τ ) = c v ln e -a Q(τ )dτ + ω(τ ). ( 25
)
In order to determine the function ω, one has to derive [START_REF] Landau | A Course of theoretical physics[END_REF] with respect to τ . It holds

∂s ∂τ e = p T = r τ -b - c v e -a Q(τ )dτ aQ(τ ).
Using the general form of the pressure [START_REF] Hérard | A class of compressible multiphase flow models[END_REF] and the temperature definition [START_REF] Jung | Schémas numériques adaptés aux accélérateurs mutlicoeurs pour les écoulements bifluides[END_REF], by identification, the function ω(τ ) satisfies ω (τ ) = r τ -b and consequently ω(τ ) = r ln(τ -b). Finally the specific entropy reads

s(τ, e) = c v ln e -a Q(τ )dτ + r ln(τ -b) + s 0 , (26)
where s 0 is a numerical constant of integration. We underline that, given τ → Q(τ ), the definition of the temperature T is not unique (and, as a consequence, the specific entropy is not unique as well).

Applications

Considering the aforementioned choices of Q(τ ), one obtains the following complete EoS:

• the Noble-Abel-Stiffened-Gas law corresponds to Q(τ ) = 1 which leads to s(τ, e) = c v ln (e + aτ + const.) + r ln(τ -b) + s 0 .

Getting const.=-q + ab where q is a constant, we recover the Noble-Abel-Stiffened-Gas EoS described in [START_REF] Chiapolino | Extended noble-abel stiffened-gas equation of state for sub-and-supercritical liquid-gas systems far from the critical point[END_REF][START_REF] Métayer | Elaborating equations of state of a liquid and its vapor for two-phase flow models[END_REF][START_REF] Métayer | The noble-abel stiffened-gas equation of state[END_REF]. This specific entropy s is a strictly concave function of (τ, e) and complies with Assumptions 2. • The van der Waals law is obtained setting Q(τ ) = τ -2 and gives

s(τ, e) = c v ln e + a τ + const. + r ln(τ -b) + s 0 .
Classically const.= 0, see [START_REF] Callen | Thermodynamics and an Introduction to Thermostatistics[END_REF][START_REF] Landau | A Course of theoretical physics[END_REF]. However (τ, e) → s is not concave everywhere. Thus the van der Waals specific entropy fits into the foregoing framework (Assumption 2) only if we restrict attention to regions where (τ, e) → s is concave. If we apply the well-know Maxwell area construction, which is equivalent to replace s by its concave hull (the smallest concave function greater that or equal to s in some region), the graph contains a line and this geometrical property is related to the physical interpretation of phase transition (see [START_REF] Callen | Thermodynamics and an Introduction to Thermostatistics[END_REF]). In [START_REF] Helluy | Pressure laws and fast Legendre transform[END_REF][START_REF] Mathis | Étude théorique et numérique des écoulements avec transition de phase[END_REF] the authors investigate the link between the Legendre Transform (LT) and the sup-convolution and the concave hull. The sup-convolution is transformed into an addition by the LT and applying twice a LT to any function is an elegant way to compute its concave hull. A fast LT algorithm is proposed in [START_REF] Helluy | Pressure laws and fast Legendre transform[END_REF] in order to compute tabulated Equation of State of mixture with application to the van der Waals law.

Computing admissible EoS for two-phase mixture

We consider an ideally isolated system of fixed mass M ≥ 0, volume V ≥ 0 and internal energy E ≥ 0, composed of two-phases A and B which are entirely described by their entropy functions S k , function of their masses 0

≤ M k ≤ M , volumes 0 ≤ V k ≤ V and internal energies 0 ≤ E k ≤ E, k ∈ K def = { A, B }. In the following we denote W k def = (V k , M k , E k ) and W k def = (M k , E k ).
Mass and energy conservations ensure that

M = M A + M B , (27)
E = E A + E B . (28)
Depending on the miscibility of the mixture, the volume constraint should differ:

• as the two phases are immiscible, they are separated by an interface, leading to

V = V A + V B , (29)
so that W = W A + W B ; • when the two phases are miscible, we assume they are completely intimate and no interface arises, meaning that they behave like ideal gases in the sense that

V = V A = V B , (30)
so that W = W A + W B .
Remark 3.1. Note that in both cases, we assume that no vacuum appears in the model. Otherwise one should consider that

V ≥ V A + V B .
Remark 3.2. As the two phases are immiscible, one should consider V A + V B ≤ V . As explained in [START_REF] Helluy | Relaxation models of phase transition flows[END_REF], if the phasic pressures p k , k ∈ K, are always strictly positive, then the constraint is necessarily saturated and boils down to

V = V A + V B .
In a perfectly miscible case, one should consider V A ≤ V and V B ≤ V . Again as the phasic pressures and the temperatures are non-negative, the constraint is saturated in the sense that

V = V A = V B .

Characterization of the thermodynamic equilibrium

Since the fluid system is ideally isolated, the second principle of Thermodynamic states that all internal processes lead to an increase on the total extensive entropy of the system. When the extensive entropy reaches its maximum value, the system is at thermodynamic equilibrium. In this section we address the definition of the extensive equilibrium entropy of the mixture depending on its miscibility.

Out of equilibrium, the mixture entropy S is the sum of the phasic entropies

S k . For (W A , W B ) ∈ C A × C B , it reads S (W A , W B ) = S A (W A ) + S B (W B ) = S A (V A , M A , E A ) + S B (V B , M B , E B ). ( 31
)
Definition 3.3. (Extensive equilibrium entropy of an immiscible mixture) Let W def = (V, M, E) ∈ (R + ) 3 be the state vector of the two-phase immiscible mixture. The equilibrium entropy is:

S immi (W ) def = max (W A ,W B )∈Ω immi S (W A , W B ), (32)
where 28) and ( 29) hold

Ω immi def = { (W A , W B ) ∈ C A × C B | (27), (
} = { (W A , W B ) ∈ C A × C B | W = W A + W B } . Definition 3.4. (Extensive equilibrium entropy of a miscible mixture) Let V = V 0 > 0 fixed. Let W def = (M, E) ∈ (R +
) 2 be the state vector of the two-phase miscible mixture. The equilibrium entropy is

S mi (V 0 , W ) def = max ( W A , W B )∈Ω mi S ((V 0 , W A ), (V 0 , W B )), (33)
where

Ω mi def = ( W A , W B ) ∈ C A × C B (27) and (28) hold = ( W A , W B ) ∈ C A × C B W = W A + W B .
Remark 3.5. In [START_REF] Helluy | Relaxation models of phase transition flows[END_REF] the authors highlight that the maximization processes ( 32) and ( 33) are called supconvolution operations. This operation is well-known in convex analysis, see for instance [START_REF] Rockafellar | Convex analysis[END_REF], where it is denoted . It reads

S immi (W ) = S A S B (W ) = sup W A ∈C A S A (W A ) + S B (W -W A ), S mi (V 0 , W ) = S A V0 S B (V 0 , W ) = sup W A ∈ C A S A (V 0 , W A ) + S B (V 0 , W -W A ).
Note that in the miscible case, the sup-convolution is partial and acts only on the mass and energy variables, the volume V 0 being fixed. The sup-convolution shares many properties. Among them, it ensures the existence and the concavity of the equilibrium mixture entropies.

Remark 3.6. If one assumes that no mass transfer arises between the two phases, then the phasic masses are fixed. In the case of an immiscible mixture without phase transition, the mixture entropy is also concave with respect to V, M, E and M A as soon as the phasic entropies S k are sub-linear, see [START_REF] Jung | Schémas numériques adaptés aux accélérateurs mutlicoeurs pour les écoulements bifluides[END_REF]. As miscible mixture without phase transition is concerned, the mixture entropy is also a concave function of (V, M, E) since it corresponds to a linear transformation of a concave function.

Following [START_REF] Jung | Schémas numériques adaptés aux accélérateurs mutlicoeurs pour les écoulements bifluides[END_REF], one can prove that the extensive entropies S immi , S mi :

C → R ∪ { -∞ } are PH1 concave functions of W ∈ C, where C = C A + C B is a non empty convex cone.
When the equilibrium entropies S λ , λ = immi or mi, are differentiable with respect to the volume V and the energy E, one can define the mixture temperature, pressure and chemical potential at equilibrium as follows:

T def = ∂S λ ∂E M,V (W ) -1 , p def = T ∂S λ ∂V M,E (W ), µ def = T ∂S λ ∂M V,E (W ), (34)
leading to the extensive fundamental relation

T dS λ = dE + pdV -µdM. ( 35
)
We now turn to the characterization of the equilibrium. In the definition of the mixture entropy, the maximizer may be not unique but the upper value is unique. Besides if the phasic entropies S k are upper semicontinuous, then the supremum is actually a maximum. Proposition 3.7. When the maximum is reached in the interior of the set of constraints, one has:

• the maximum is always characterized by the thermal equilibrium

T = T A = T B ; (36)
• the maximum defining S immi in the immiscible case is characterized by

p = p A = p B , (37)
while, in the miscible case with S mi , it corresponds to the Dalton's law [START_REF] Müller | Fundamentals of thermodynamics and applications[END_REF]:

p = p A + p B ; (38)
• as phase transition occurs in the miscible and miscible mixture, the equilibrium is also characterized by

µ = µ A = µ B . (39)
We refer to [START_REF] Allaire | A strictly hyperbolic equilibrium phase transition model[END_REF][START_REF] Helluy | Relaxation models of phase transition flows[END_REF][START_REF] Jaouen | Étude mathématique et numérique de stabilité pour des modèles hydrodynamiques avec transition de phase[END_REF] for a proof for the immiscible case and to [START_REF] Helluy | Pressure laws and fast Legendre transform[END_REF] for both miscible and immiscible cases. Here we focus on the equilibrium reached in the interior of the domain of S λ . It can also be reached on the boundary of the set of constraints. In that case, one should characterized the maximum by inequalities on the pressures and temperatures. Since the immiscible case is treated in [START_REF] Jung | Schémas numériques adaptés aux accélérateurs mutlicoeurs pour les écoulements bifluides[END_REF] (considering the mass M A is fixed), we just give some intents in the miscible case, that is V = V A = V B = V 0 > 0 fixed. If the maximum is reached at the point (M , E), the concavity of the entropy leads to

∇ (M A ,E A ) (S A (V 0 , M A , E A ) + S B (V 0 , M -M A , E -E A ))((M A , E A ) -(M , E)) ≤ 0. ( 40
)
Assume that the energy constraints is saturated, then one has

• either E = E and 1 T A - 1 T B ≥ 0;
• or E = 0 and 1

T A - 1 T B ≤ 0.
Conversely, if the mass constraint is saturated, one gets

• either M = M and - µ A T A + µ B T B ≥ 0;
• or M = 0 and -

µ A T A + µ B T B ≤ 0.

Intensive entropies and geometrical interpretation

We now turn to the definition of intensive quantities. The system is now entirely described by its intensive specific entropy s defined by ( 5) as a function of w def = (τ, e) ∈ (R + ) 2 or by its intensive volumic entropy σ defined by [START_REF] Faccanoni | Étude d'un modèle fin de changement de phase liquide-vapeur[END_REF] as a function of w def = (ρ, ε) ∈ (R + ) 2 . We assume that they comply with Assumptions 2 and 3 respectively.

Let us introduce the volume fraction α k , the mass fraction ϕ k and the energy fraction z k of the phase k ∈ K defined respectively by

α k def = V k V , ϕ k def = M k M , z k def = E k E , (41)
which belong to [0, 1].

• Each phase k ∈ K is depicted by its specific phasic entropy s k defined by

s k (τ k , e k ) def = S k (τ k , 1, e k ). (42)
It is a function of the specific volume τ k and a specific internal energy e k such that

τ k = V k M k = α k ϕ k τ, e k = E k M k = z k ϕ k e, (43)
satisfying the phasic fundamental relation

T k ds k = de k + p k dτ k . (44)
• Eventually the phase k ∈ K may be depicted by its volumic phasic entropy σ k defined by

σ k (ρ k , ε k ) def = S k (1, ρ k , ε k ). ( 45
)
It is a function of the density ρ k and the volumic internal energy ε k such that

ρ k = M k V k = ϕ k α k ρ = 1 τ k , ε k = E k V k = z k α k ε = z k α k ρe = e k τ k , (46)
satisfying the phasic fundamental relation

T k dσ k = dε k -µ k dρ k . ( 47
)
The extensive constraints ( 27)- [START_REF] Müller | Fundamentals of thermodynamics and applications[END_REF] admit their intensive counterparts. The mass and energy conservations ( 27) and [START_REF] Mathis | Étude théorique et numérique des écoulements avec transition de phase[END_REF] give the following constraints on the fractions:

1 = ϕ A + ϕ B , (48) 1
= z A + z B . (49)
• In the case of an immiscible mixture, the extensive volume constraint (29) translates into

1 = α A + α B , (50)
and, associated to (43), leads to

τ = ϕ A τ A + ϕ B τ B , (51)
e = ϕ A e A + ϕ B e B . (52)
Out of equilibrium, the extensive mixture entropy (31) can be written in intensive specific formulation using the homogeneity property of the extensive entropies S k , k ∈ K:

s(w A , w B ) def = S M = ϕ A s A (w A ) + ϕ B s B (w B ). ( 53
)
• In the miscible case, the extensive volume constraint (30) translates into

1 = α A = α B , (54)
and, associated to (46), we have

ρ = ρ A + ρ B , (55)
ε = ε A + ε B . (56)
Out of equilibrium, the extensive mixture entropy (31) can be written in intensive volumic formulation using the homogeneity property of the extensive entropies S k , k ∈ K:

σ( w A , w B ) def = S V = σ A ( w A ) + σ B ( w B ). ( 57
)
We now turn to the description of the intensive mixture entropy at equilibrium. The optimization processes ( 32) and (33) can be derived in an intensive formulation as follows, leading to an elegant geometrical interpretation.

• As the immiscible case is considered, at equilibrium, the optimization problem (32) reads under its intensive formulation s immi (τ, e) = max

1=ϕ A +ϕ B w=ϕ A w A +ϕ B w B s(w A , w B ) , (58)
under constraints (48), ( 51) and (52). One observes that the formulation (58) defines nothing else but the concave hull of the maximum of the two partial entropies s k , k ∈ K:

s immi (w) = concave hull of max s A (w), s B (w) . ( 59
)
Therefore the mixture entropy is concave but not strictly concave. In a region corresponding to the vaporliquid equilibrium, the graph of s immi contains a line of plane. This characterization is highlighted for instance in [START_REF] Allaire | A strictly hyperbolic equilibrium phase transition model[END_REF][START_REF] Barberon | Finite volume simulation of cavitating flows[END_REF][START_REF] Faccanoni | Étude d'un modèle fin de changement de phase liquide-vapeur[END_REF][START_REF] Faccanoni | Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium[END_REF][START_REF] Helluy | Pressure laws and fast Legendre transform[END_REF][START_REF] Helluy | Relaxation models of phase transition flows[END_REF]. • In the miscible case, the total volume V is fixed and equal to V 0 > 0. Then the optimization process allows to define the volumic mixture entropy as

σ mi ( w) = max w= w A + w B σ( w A , w B ) , (60)
under the mass and energy conservation constraints (55) and (56). Observe that the intensive mixture entropy remains a sup-convolution operation of the phasic entropies

σ mi (ρ, ε) = σ A σ B (ρ, ε), (61)
see Remark 3.5. The properties of the sup-convolution operation ensure that if the phasic entropies are strictly concave, then the mixture entropy is strictly concave as well [START_REF] Hiriart-Urruty | Fundamentals of convex analysis[END_REF][START_REF] Rockafellar | Convex analysis[END_REF]. Note that the thermodynamic equilibrium is still characterized by Proposition 3.7 that is the equilibrium of the temperature and the Dalton's law on the pressures. The equilibrium entropies properties are stated in the following statement: Proposition 3.8. Consider the thermodynamic equilibrium with the equilibrium temperature T and the equilibrium chemical potential satisfying (36) and (39), i.e. T = T A = T B and µ = µ A = µ B .

• If the mixture is immiscible, the equilibrium pressure p satisfies (37) and the intensive equilibrium entropy s immi is a concave function of (τ, e) and satisfies

T ds immi = de + pdτ. ( 62
)
• If the mixture is miscible, the equilibrium pressure p complies with the Dalton's law (38) p = p A + p B and the volumic equilibrium entropy of the mixture is a concave function of (ρ, ε) and satisfies

T dσ mi = dε -µdρ. (63)
Proof.

• We multiply the phasic Gibbs relations (44) T k ds k = de k + p k dτ k by the mass fractions ϕ k and we sum over k ∈ K:

0 = k∈K (ϕ k T k ds k -ϕ k de k -ϕ k p k dτ k ) (36)(37) = T k∈K (ϕ k ds k ) - k∈K (ϕ k de k ) -p k∈K (ϕ k dτ k ) = T k∈K d(ϕ k s k ) - k∈K d(ϕ k e k ) -p k∈K d(ϕ k τ k ) - k∈K (T s k -e -pτ k ) dϕ k = T ds immi -de -pdτ + k∈K µ k dϕ k (39) = T ds immi -de -pdτ + µd k∈K ϕ k = T ds immi -de -pdτ + µd(1) = T ds immi -de -pdτ.
• We sum the phasic relations (47) for k ∈ K: since the temperatures are equal, see (36), as well as the chemical potentials, see (39), then

0 = k∈K (T k dσ k -dε k -µ k dρ k ) (36)(39)
= T dσ mi -dε -µdρ,

and k∈K p k = k∈K (ρ k µ k + T k σ k -ε k ) (36)(39) = µ k∈K ρ k + T k∈K σ k - k∈K ε k (55)(56)(57) = µρ + T σ -ε = p

Homogeneous Equilibrium for two-phase flows

We now take into account the dynamic of the two-phase mixture assuming that the two-phases have the same velocity. The aim of this section is to provide an Homogeneous Equilibrium Euler's system, called HEM model. The HEM model has to be endowed with appropriate closure laws in agreement with the thermodynamic equilibria studied in Section 3. Emphasis is given to the hyperbolicity and the entropy structure of the models. For the sake of simplicity we consider mono-dimensional case.

Immiscible case

At thermodynamic equilibrium the immiscible two-phase flow is depicted by the multicomponent Euler system

                 ∂ t ρ + ∂ x (ρu) = 0, ∂ t (ρu) + ∂ x (ρu 2 + p immi ) = 0, ∂ t (ρE) + ∂ x ((ρE + p immi )u) = 0, E = 1 2 u 2 + e, ∀k ∈ K : p k = p k (τ k , e k ), τ k = ρ -1 k , (64)
where the flow has a density ρ (we also define the specific volume τ = 1/ρ), a velocity u, a pressure p immi , and an internal energy e (E being the total energy). The phase k ∈ K is depicted by its mass fraction ϕ k , its pressure p k , its specific volume τ k and its specific internal energy e k , see Section 3.2. All the phases evolve at the same velocity u. The multicomponent Euler system admits eight equations and has thirteen unknowns which are ρ, u, E, p immi , e, ϕ A , ϕ B , τ A , τ B , e A , e B , p A , p B Thus one has to provide five closure laws. The first two closure laws are given by the constraints (51) and (52). The three remaining closure laws are given by Proposition 3.7, that is

     T immi = T A = T B , p immi = p A = p B , µ = µ A = µ B , (65)
where T immi and p immi are the thermodynamic temperature and pressure of the two-phase flow and p immi = p immi (1/ρ, e).

To prove the hyperbolicity of the system, one would like to apply the Godunov-Mock theorem using the function η : (τ, u, E) → -s immi (τ, E -u 2 /2) as a Lax entropy of the system, see for instance [START_REF] Dellacherie | Relaxation schemes for the multicomponent Euler system[END_REF][START_REF] Lagoutière | Modélisation mathématique et résolution numérique de problèmes de fluides compressibles à plusieurs constituants[END_REF][START_REF] Mathis | A thermodynamically consistent model of a liquid-vapor fluid with a gas[END_REF]. Unfortunately, since the equilibrium entropy s immi is not a strictly concave function of its arguments, this argument does not apply in our case. One has to study the eigenvalues of the system and to prove that the speed of sound of the mixture is real. First notice that, for any state vector w def = (τ, e), the speed of sound of the system can be written as

c 2 (w) = -τ 2 T (w) (p(w), -1) H s immi (w) p(w) -1 , (66)
where H immi s (w) denotes the Hessian matrix of s immi . Because s immi is the concave hull of the phasic entropies s k , k ∈ K, it follows that

• for all w in pure phase regions

v H s immi (w)v < 0, ∀v = 0, (67)
• for all w in the VLE region it exists a unique v(w) = 0, such that (v(w))

H s immi (w)v(w) = 0. ( 68
)
The hyperbolicity results is stated in [START_REF] Allaire | A strictly hyperbolic equilibrium phase transition model[END_REF][START_REF] Faccanoni | Étude d'un modèle fin de changement de phase liquide-vapeur[END_REF][START_REF] Faccanoni | Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium[END_REF] by proving the following theorem:

Theorem 4.1. For any state vector w in the VLE region,

v(w) ≡ (p(w), -1) , (69)
so that the system (64) is strictly hyperbolic.

Miscible case

We now turn to the miscible case. At thermodynamic equilibrium the miscible two-phase flow is depicted by the multicomponent Euler system

                 ∂ t ρ + ∂ x (ρu) = 0, ∂ t (ρu) + ∂ x (ρu 2 + p mi ) = 0, ∂ t (ρE) + ∂ x ((ρE + p mi )u) = 0, E = 1 2 u 2 + ε ρ , ∀k ∈ K : p k = p k (ρ k , ε k ). (70)
The system (70) admits six equations and thirteen unknowns which are {ρ, u, E, p mi , ϕ A , ϕ B , τ A , τ B , e A , e B , p A , p B }.

Hence one has to provide seven closure laws. The first two closure laws are given by the constraints (55) and (56). The characterization of the equilibrium provides the five remaining closure laws:

     T mi = T A = T B , p mi = p A + p B , µ mi = µ A = µ B , (71)
where p mi is the pressure of the two-phase flow and p mi = p mi (ρ, ε).

The hyperbolicity of the system (70) relies on its entropy structure. Indeed in Section 3.2 we have shown that the volumic equilibrium entropy is strictly concave as soon as the phasic entropies are. In this case the Godunov-Mock theorem applies (see for instance [START_REF] Dellacherie | Relaxation schemes for the multicomponent Euler system[END_REF][START_REF] Lagoutière | Modélisation mathématique et résolution numérique de problèmes de fluides compressibles à plusieurs constituants[END_REF][START_REF] Mathis | A thermodynamically consistent model of a liquid-vapor fluid with a gas[END_REF]) and we have the following theorem: Theorem 4.2. The strictly convex function η(ρ, u, E) → -σ(ρ, ρE -ρu 2 /2) is a Lax entropy for the system (70) associated to the entropy flux f η (ρ, u, E) = -uσ. thus the system (70) is strictly hyperbolic.

Homogeneous Relaxation two-phase models

The equilibrium two-phase models presented in the previous section are difficult to use for practical computations. Indeed the determination of the equilibrium mixture pressure law may be tedious, and even worse, it is not analytical when considering standard phasic EoS such as stiffened gas law, see for instance [START_REF] Faccanoni | Étude d'un modèle fin de changement de phase liquide-vapeur[END_REF][START_REF] Faccanoni | Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium[END_REF][START_REF] Mathis | Étude théorique et numérique des écoulements avec transition de phase[END_REF]. In [START_REF] Helluy | Pressure laws and fast Legendre transform[END_REF] the authors propose to use a Fast Legendre Transform algorithm to compute admissible mixture pressure laws (see also Remark 3.5). Again the resulting mixture pressure law is not analytical but solely a tabulated one.

In order to avoid the manipulation of a complex pressure law, it is convenient to approximate the equilibrium Euler system by a Homogeneous Relaxation Model (HRM), following [3, 11-13, 16, 19, 20]. It consists in adding convection equations of additional variables and to modify the pressure to make it depend on these variables. Relaxation source terms complete the new convection equations. The relaxation structure should be consistent with the second principle of Thermodynamics. In particular, the source terms have to comply with the entropy growth criterion and bring the system to thermodynamic equilibrium.

Immiscible case

Following [START_REF] Barberon | Finite volume simulation of cavitating flows[END_REF][START_REF] Faccanoni | Étude d'un modèle fin de changement de phase liquide-vapeur[END_REF][START_REF] Hurisse | Application of an homogeneous model to simulate the heating of two-phase flows[END_REF], the additional variables we consider are the fractions of mass, volume and energy of the phase A: Y def = (ϕ A , α A , z A ). Note that one may have chosen another set of variables as well. When no transfer occurs between the liquid and the vapor, the fractions should be perfectly convected, that is

∂ t Y + u∂ x Y = 0. ( 72
)
Combining with the conservation of total mass and adding the source terms to the fraction equations gives

         ∂ t (ρY ) + ∂ x (ρuY ) = Q, ∂ t ρ + ∂ x (ρu) = 0, ∂ t (ρu) + ∂ x (ρu 2 + p) = 0, ∂ t (ρE) + ∂ x ((ρE + p)u) = 0, (73)
with the closure law p = p(1/ρ, e, ϕ A , α A , z A ).

(74)

The system (73) admits the concave function

s : (τ, e, ϕ A , α A , z A ) → ϕ A s A α A ϕ A τ, z A ϕ A e + ϕ B s B 1 -α A 1 -ϕ A τ, 1 -z A 1 -ϕ A e ,
already defined in (53), as an entropy function. Recall that its concavity is deduced from the concavity of the phasic entropies s k , k ∈ K. Moreover it complies with

ds = k∈K z k T k de + k∈K α k p k T k dτ + k∈K ∂ ϕ k sdϕ k + k∈K ∂ α k sdα k + k∈K ∂ z k sdz k . (75)
We can then identify the mixture pressure (74) and the mixture temperature by

p(1/τ, e, ϕ A , α A , z A ) T (1/τ, e, ϕ A , α A , z A ) = k∈K α k p k α k ϕ k τ, z k ϕ k e T k α k ϕ k τ, z k ϕ k e , 1 T (1/τ, e, ϕ A , α A , z A ) = k∈K z k T k α k ϕ k τ, z k ϕ k e ,
under the constraints (48), ( 49) and (50) on the fractions. Weak entropy solutions of the system satisfy

∂ t (ρs) + ∂ x (ρus) ≥ 0. ( 76
)
As relaxation towards the equilibrium is infinitely fast, one recovers the equilibrium fractions which satisfy

Y immi (τ, e) def = argmax (ϕ A ,α A ,z A ) s(τ, e, ϕ A , α A , z A ). ( 77
)
As a result the equilibrium pressure is

p immi (τ, e) = p(τ, e, Y immi (τ, e)). (78)
We now turn to the definition of the relaxation source terms involved in the fraction equations in the system (73). Actually there is no consensus on their definition [START_REF] Drew | Theory of multicomponent fluids[END_REF][START_REF] Drew | Mathematical modeling of two-phase flow[END_REF]. But they must relax towards the correct thermodynamic equilibrium and comply with the entropy growth criterion (76). Following for instance [START_REF] Barberon | Finite volume simulation of cavitating flows[END_REF][START_REF] Faccanoni | Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium[END_REF][START_REF] Helluy | Verification of a two-phase flow code based on an homogeneous model[END_REF][START_REF] Helluy | Relaxation models of phase transition flows[END_REF], one may choose Q(Y ) def = Y immi (τ, e) -Y . Thus one has

∂ t (ρs) + ∂ x (ρus) = ∇ Y s • (∂ t Y + u∂ x Y ) = ∇ Y s • (Y immi (τ, e) -Y ) ≥ s(τ, e, Y immi ) -s(τ, e, ϕ A , α A , z A ) ≥ 0,
by the concavity of the mixture entropy s. This choice is then in agreement with the entropy growth criterion.

Miscible case

In the miscible case, the volume fractions are equal to one, see equation (54). Then we consider the additional advection equations on the mass and energy fractions of the phase A, Y def = (ϕ A , , z A )

∂ t Y + u∂ x Y = 0. ( 79
)
Combining with the conservation of total mass and adding the source terms to the fraction equations gives

         ∂ t (ρ Y ) + ∂ x (ρu Y ) = Q, ∂ t ρ + ∂ x (ρu) = 0, ∂ t (ρu) + ∂ x (ρu 2 + p) = 0, ∂ t (ρE) + ∂ x ((ρE + p)u) = 0, (80)
with the closure law

p = p(1/ρ, e, ϕ A , z A ). ( 81
)
The system (80) admits the concave function

s : (τ, e, ϕ A , z A ) → ϕ A s A τ ϕ A , z A ϕ A e + ϕ B s B τ 1 -ϕ A , 1 -z A 1 -ϕ A e ,
as an entropy function, which complies with

d s = k∈K z k T k de + k∈K p k T k dτ + k=A,B ∂ ϕ k sdϕ k + k∈K ∂ z k sdz k . ( 82
)
We can then identify the mixture pressure (81) and the mixture temperature by p(1/τ, e, ϕ A , z A )

T (1/τ, e, ϕ A , z A ) = k∈K p k τ ϕ k , z k ϕ k e T k τ ϕ k , z k ϕ k e , 1
T (1/τ, e, ϕ A , z A ) = k∈K z k T k τ ϕ k , z k ϕ k e ,
under the constraints (48), ( 49) and (54) on the fractions. Weak entropy solutions of the system satisfy

∂ t (ρ s) + ∂ x (ρu s) ≥ 0. ( 83
)
As relaxation towards the equilibrium is infinitely fast, one recovers the equilibrium fractions which satisfy

Y mi (τ, e) def = argmax (ϕ A ,z A ) s(τ, e, ϕ A , z A ). ( 84
)
As a result the equilibrium pressure is

p mi (τ, e) = p(τ, e, Y mi (τ, e)). (85)
Like in the immiscible case, a possible choice for the relaxation source terms on the fractions is Q( Y ) = Y mi (τ, e) -Y which guarantees the entropy growth criterion thanks to the concavity of the mixture entropy s.

Conclusion & Perspectives

We address in this paper the construction of admissible equations of state for two-phase flows. Considering the mixture as a single fluid, it is possible to construct a complete EoS based on the knowledge of an incomplete EoS. This construction allows for instance to build the van der Waals entropy function starting from the pressure law (τ, T ) → p. However this law does not comply with convexity constraints required by standard Thermodynamics. Another construction consists in defining the mixture equilibrium entropy as an optimization problem on the phasic entropies. Depending on the mixture is immiscible or not, the mixture entropy corresponds to the concave hull of the phasic entropies or to their sup-convolution. Both operations guarantee concavity properties and, as a consequence, hyperbolicity of the multicomponent Euler system. However the mixture pressure laws are often difficult to compute and are, at best, tabulated EoS, which are difficult to manipulate and CPU consuming. One comes back to the remaining question 4 of the Introduction: "How to balance accuracy and reasonable computing times?" A solution is to consider rather HRM models. Hence the relaxation source terms have to relax towards the correct thermodynamic equilibrium while complying with the entropy growth criterion. which q = ∇f (r). In this case we have f * (q) = r(q) • q -f (r(q)) with r = r(q) solving q = ∇f (r). Furthermore ∇f * (q) = r + (q -∇f (r))∇ q r = r and so r = ∇f * (q).

In classic Thermodynamics it is convenient to define a complete EoS as the function (s, τ ) → e instead of (e, τ ) → s as in the present paper. Because the temperature positivity (8) holds, we can solve s = s(e, τ ) and use e as a C 1 function of (s, τ ), then we have de = T ds -pdτ and the Hessian matrix of e is positive definite, that is

det(H e ) def = ∂ 2 e ∂τ 2 s ∂ 2 e ∂s 2 τ - ∂ 2 e ∂τ ∂s 2 > 0, ∂ 2 e ∂τ 2 s > 0, ∂ 2 e ∂s 2 τ > 0. ( 86
)
It implies that the internal energy e is strictly convex in (s, τ ). There are three possible Legendre transforms of (s, τ ) → e, according to whether we transform: in the variable s only, in τ only, or in (s, τ ) together.

Because it is customary in Thermodynamics to take the opposite of the mathematical Legendre transform (i.e. where s = s(T, p) and τ = τ (T, p) solve T = ∂e ∂s τ and -p = ∂e ∂τ s ; the Gibbs free energy µ is strictly concave in (T, p). These functions are called "Thermodynamic potentials". They are complete EoS describing the behaviour of a system as a function of the so-called "natural variables". In Table 1 we resume these relations. The Maxwell relations are obtained by exploiting the exactness of differentials (i.e. the order of the mixed second partial derivatives does not matter). Using Legendre's transform of (e, τ ) → s, we can compute three other potentials (and associated Maxwell's relations, see Table 2) called Massieu-Planck Functions [START_REF] Nicholas W Tschoegl | Fundamentals of equilibrium and steady-state thermodynamics[END_REF]:

-sup { • • • } = inf { -• • • }),
• the function ( Following similar ideas for (ρ, σ) → ε and (ρ, ε) → σ we obtain tables 3 and 4. The function ♣ is sometimes called Kramer function [32, p.63].

The other functions are unnamed in our knowledge. Note that, since ε is strictly convex, its conjugate function is precisely -p, function of (T, µ) = ∇ε(σ, ρ). Note that, since σ is strictly concave, its conjugate function is precisely p T , function of -µ T , 1 T = ∇σ(ρ, ε).

∂σ ∂ρ ε dσ = 1 T dε -µ T dρ - ∂ µ T ∂ε ρ = ∂ 1 T ∂ρ ε = σ -1 T ε ( 1 T , ρ) ε = - ∂ ∂ 1 T ρ µ T = - ∂ ∂ρ 1 T d = -εd 1 T -µ T dρ ∂ µ T ∂ 1 T ρ = ∂ε ∂ρ 1 T ♠ = σ + µ T ρ (ε, µ T ) 1 T = ∂♠ ∂ε µ T ρ = ∂♠ ∂ µ T ε d♠ = 1 T dε + ρd µ T ∂ρ ∂ε µ T = ∂ 1 T ∂ µ T ε ♣ = σ -1 T ε + µ T ρ ( 1 T , µ T ) ε = - ∂♣ ∂ 1

  the relevant formula are actually the following: • the Helmholtz free energy (T, τ ) → f (sometimes noted a) is f (T, τ ) = inf s (e(s, τ ) -T s) = e(s, τ ) -T s, where s = s(T, τ ) solves T = ∂e ∂s τ ; the Helmholtz free energy f is strictly concave in T , strictly convex in τ ; • the enthalpy (s, p) → h is h(s, p) = inf τ (e(s, τ ) + pτ ) = e(s, τ ) + pτ, where τ = τ (s, p) solves -p = ∂e ∂τ s ; the enthalpy h is strictly concave in p, strictly convex in s; • the Gibbs potential (a.k.a. chemical potential or free enthalpy) (T, p) → µ is µ(T, p) = inf (s,τ ) (e(s, τ ) + pτ -T s) = e(s, τ ) + pτ -T s,

1 T

 1 , τ ) → j = inf e s(e, τ ) -e T = s(e, τ ) -e T where e = e( 1 T , τ ) solves 1 T = ∂s ∂e τ ; • the function (e, p T ) → = inf τ s(e, τ ) -p T τ = s(e, τ ) -p T τ where τ = τ (e, p T ) solves p T = ∂s ∂τ e ; • the function ( 1 T , p T ) → r = inf (e,τ ) s(e, τ ) -p T τ -e T = s(e, τ ) -p T τ -e T where τ = τ ( p T , e) and e = e( p T , 1 T ) solve p T = ∂s ∂τ e and 1 T = ∂s ∂e τ . The function j was first introduced by Massieu and it is called Massieu's potential. The function r was introduced by Planck and is called Planck's potential.

1 T 3 .Table 4 .

 134 Table Thermodynamic potentials based on complete EoS (σ, ρ) → ε. Note that ♦ = ρf and ♥ Thermodynamic potentials based on complete EoS (ε, ρ) → σ. Note that = ρj and ♥ = -T ♣, that is ♣ = -p/T

Table 1 .

 1 Thermodynamic potentials based on complete EoS (s, τ ) → e

	Potential	Natural variables	Conjugate variables	Differential		Maxwell relations
	e	(s, τ )	T =	∂e ∂s τ	p = -	∂e ∂τ s	de = T ds -pdτ	-	∂p ∂s τ =	∂T ∂τ s
	f = e -T s	(T, τ )	s = -	∂f ∂T τ	p = -	∂f ∂τ T	df = -sdT -pdτ	∂p ∂T τ =	∂s ∂τ T
	h = e + pτ	(s, p)	T =	∂h ∂s p	τ =	∂h ∂p s	dh = T ds + τ dp	∂τ ∂s p =	∂T ∂p s
	µ = e -T s + pτ	(T, p)	s = -	∂µ ∂T p	τ =	∂µ ∂p T	dµ = -sdT + τ dp	∂τ ∂T p = -	∂s ∂p T

Table 2 .

 2 Thermodynamic potentials based on complete EoS (e, τ ) → s. Note that j = -f /T and r = -µ/T

	Potential	Natural variables	Conjugate variables	Differential	Maxwell relations
	s	(e, τ )

Note that the concavity of s may be deduced from the concavity of the extensive entropy S and its homogeneity.

Note that the concavity of σ may be deduced from the concavity of the extensive entropy S by homogeneity, or otherwise, from the concavity of the specific entropy s, since σ = ρs.

* This work was partially funded by the CNRS project call NEEDS (nuclear, energy, environment, waste and society) for which the LMNC/LMNC2 projects were selected in 2014-2018. * * The second author was partially supported by the ANR project MOHYCON (ANR-17-CE40-0027-01) and would like to thank the Centre Henri Lebesgue ANR-11-LABX-0020-01 for creating an attractive mathematical environment.

A. Legendre Transform and Maxwell Relations -intensive description

The entropy s (or the energy e) are not directly physically measurable, whereas certain of the intensive variables (e.g. T or p) are. It is consequently convenient to convert the entropy into functions of various intensive variables. Since s is concave, we can employ ideas from convex analysis to rewrite various formula in terms of the intensive variables T and p. The main tool will be de Legendre transform.

Assume that f : R n → (-∞, +∞] is a convex, lower semicontinuous (l.s.c.) and proper function r → f (r). The Legendre transform of f is the function f * : R n → (-∞, +∞] defined by f * (q) = sup r∈R n (r • q -f (r)). The function f * is likewise convex, l.s.c. and proper. Furthermore (f * ) * = f . Moreover, if f is C 2 and strictly convex then, given q, there exists a unique r which maximizes (r • q -f (r)), namely the unique r = r(q) for