Marion Chenal

Cyril Véchambre

J.-M Chenal

Laurent Chazeau

Vincent Humblot

Laurent Bouteiller

Costantino Creton
email: costantino.creton@espci.fr

Jutta Rieger
email: jutta.rieger@upmc.fr

Jean-Marc Chenal

Mechanical properties of nanostructured films with an ultralow volume fraction of hard phase

Keywords: RAFT Polymerization, Film, Nanostructure, Core-shell latexes, Mechanical properties

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

Obtaining stiffer rubbery materials without sacrificing extensibility and reversibility of the deformation is a classical objective of polymer science. A very efficient way to do so is to design a percolating hard domain into a predominantly soft material. [1][2][3][4][5] At small deformations the stress is carried by the stiff network while as the deformation increases, the percolating structure breaks and the load is sustained by the soft domains which can be very extensible.

This general strategy has been used for example in double interpenetrated networks where two polymers are synthesized sequentially 6,7 but also in nanocomposites. 8,9 However for processing reasons it would be desirable to make such stiff but extensible materials in a single synthesis and drying step. One attractive strategy is to prepare materials through the selfassembly of colloidal systems, 10 synthesized by aqueous emulsion polymerization 11,12 to create a percolating structure made of thin but hard walls between deformable particles. This approach combines the attractiveness of replacing organic solvent by water, with an easy, efficient and straightforward method to create core-shell particles. 13,14 Although qualitatively the principle is demonstrated, it remains challenging to combine a high degree of stiffening (increase by one order of magnitude) while maintaining several hundred percent extensibility, using only a very low volume fraction of a hydrophilic hard phase and little or no surfactants to maintain water resistance.

We have recently shown that highly asymmetric poly(acrylic acid)-b-poly(n-butyl acrylate) block copolymers (PAA-b-PBA) of high molar mass can be synthesized in water thanks to a RAFT emulsion polymerization process. 15 During the course of the polymerization, such asymmetric and amphiphilic copolymers form and self-assemble in situ into soft PBA corehard PAA shell particles, according to the PISA (Polymerization-Induced Self-Assembly) principle. [START_REF] Lansalot | Polymerization-Induced Self-Assembly: The Contribution of Controlled Radical Polymerization to The Formation of Self-Stabilized Polymer Particles of Various Morphologies[END_REF][START_REF] Rieger | [END_REF][18] In a previous paper 2 , we reported that these core-shell latexes could be used to prepare, at room temperature, soft materials exhibiting an order of magnitude increase in stiffness, compared to non-structured films made of random P(BA-co-AA) copolymers latexes, while retaining an extensibility of several hundred percent. Small-Angle X-ray Scattering experiments (SAXS) and microscopy showed that the core-shell structure of the particles was retained in the film, giving rise to a percolating network of connected glassy shells. We proposed 2 that the high modulus of the films was controlled by the characteristics of the shell network, such as the shell thickness and strength, while the strain at break was mainly controlled by the deformability of the particle cores.

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In this paper we focus on the mechanical properties of the films in the small and large strain regime including not only simple tensile tests but also step-cycle tests to study the reversibility of the deformation, and fracture tests with notched samples to evaluate the fracture toughness. Furthermore a variety of modifications were performed both in the RAFT emulsion polymerization step such as the incorporation of a third, hard polystyrene block to the copolymer (to yield PAA-b-PBA-b-PS), and in the film forming step by strengthening the interactions between the shells through the addition of cationic additives (Fe 3+ or polymeric amines) in the water phase.

Experimental Part

Materials. Anhydrous acrylic acid (AA, >99 %, Fluka), N,N-dimethylaminoethyle acrylate (DMAEA, 98 %, Aldrich), 1,3,5-trioxane (>99 %, Fluka), ammonium hydroxide (NH 4 OH, 30 wt% in water, Carlo Erba), AIBN (>9 %, Fluka) and 1,4-dioxane (from VWR Rectapur) were used as received. 4,4′-Azobis-4-cyanopentanoic acid (ACPA, >98 %, Fluka) was purified by re-crystallization from 2-propanol. Deionized water was used for all emulsion polymerizations. n-Butyl acrylate (BA, 99 %, Acros Organics) and styrene (S, 99 %, Aldrich) were distilled under reduced pressure before use. The RAFT agent S-1-dodecyl-S′-(α,α′dimethyl-α″-acetic acid) trithiocarbonate (TTCA) was synthesized as described elsewhere. 19 Synthesis of PAA-b-PBA latexes. The PAA-b-PBA core-shell particles were synthesized as described elsewhere. 15 Briefly, a poly(acrylic acid) macroRAFT agent (PAA-TTC) is first synthesized in 1,4-dioxane in presence of an asymmetric RAFT agent (TTCA 19 ) and ACPA as an initiator. In a second step, the polymerization of butyl acrylate (23 wt%) is implemented in batch conditions at pH 5.4 (NH 4 OH) using the macroRAFT agent as both a control agent and stabilizer. Thus, poly(acrylic acid)-b-poly(n-butyl acrylate) copolymers (PAA-b-PBA) are obtained (theoretical M n of 100 kg.mol -1 ), which self-assemble in situ during polymerization into core-shell particles. Latexes composed of PAA-b-PBA core-shell particles with PAA= 3 kg.mol -1 and PBA = 100 kg.mol -1 are referred to as standard latexes. Preparation of PDMAEA polymer. PDMAEA was synthesized in 1,4-dioxane in presence of an asymmetric RAFT agent (TTCA 19 ) and AIBN as an initiator (See Supporting Information). M n = 3.7 kg.mol -1 , Đ = 1.39 (determined by SEC in DMF using a PMMA calibration).

Preparation

Film formation. The latexes (4.8 mL of a 23 wt% latex) were dried in flat rectangular polydimethylsiloxane (silicone) molds (68 mm 27.5 mm 30 mm) for 7 days at ambient temperature and for another 12 h at 100°C under reduced pressure to eliminate remaining water, ammonia and possibly residual monomer. After drying, the acid groups of poly(acrylic acid) were fully protonated (as checked by FTIR measurements) because of the complete evaporation of NH 3 . For the films containing PDMAEA polymer, the PDMAEA was firstly dissolved into deionized water before being added into a PAA-b-PBA latex at pH 8 (adjusted with NH 4 OH 30 wt% in water). The resulting latex was then dried as described above. For the films containing Fe 3+ ions, an Iron(III)(acetylacetonate) solution was prepared at 6 mM in deionized water before being added to a PAA-b-PBA latex. Then the resulting latex was dried as described above. [kg.mol -1 ] The force (F) and the local elongation in the tensile direction (λ = l/l 0 ) measured with the video extensometer were both recorded during the experiments. Nominal stress σ ே was defined as the tensile force per unit of initial area S 0 (Eq. 1):

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M n, PAA-b-PBA b) [kg.mol -1 ] / Đ % AA c) [wt%] σ B d) [kPa] ε B e) [%] E f) [MPa] Standard PAA-b-PBA
σ ே = ி ௌ బ Equation 1
Cyclic extension. Incremental loading and unloading cycles were performed with the same experimental setup than tensile tests. Seven cycles with no rest period were applied between nearly λ = 1.2 to λ = 2.1 with incremented maximum λ (λ i+1 >λ i ) and then two more cycles were performed at λ = 2.1. Here again, the force F and the extension ratio λ measured by the video extensometer were recorded all along the experiment.

SAXS.

The small angle X-ray scattering (SAXS) experiments were conducted at SOLEIL synchrotron on the SWING beamline in its USAXS configuration. Measurements were performed using a AVIEX CCD camera at an energy of 10keV with a sample-detector distance set at 6650mm, leading to a q-range of 0.0015 -0.1 Å -1 . The tensile test apparatus was kindly lent by the UMET laboratory (initial sample length = 6 mm, strain rate = 0.1 s -1 ).

Elongational rheology. Tensile rectangular specimens (20 mm x 5 mm x 0.5 mm) were cut from the dried films using a razor blade. Tensile stress-strain measurements were carried out at 25°C on a MCR Rheometer Series from Anton Paar, equipped with a SER extensional rheology device. This device consists of two counter-rotating drums with intermeshing gears and low-friction bearings. The sample, fixed to the drums with clamps, undergoes uniform stretching when the rotational movement of the rheometer is transferred to the drums. The extension rate was 0.06 s -1 . Three replicate experiments were performed for each sample.

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Fracture experiments. Tensile rectangular specimens (20 mm 5mm 0.5 mm) were cut from the dried films using a razor blade. A 1 mm long notch was then cut on one of the long sides of the sample with a razor blade. Uniaxial tensile stress-strain measurements of these notched samples were carried out at 25°C until the precrack propagated and broke the sample.

The nominal stress (σ ே ) and the extension ratio in the tensile direction (λ = l/l 0 ) measured with the video extensometer were both recorded during the experiments and the extension ratio at maximum stress was noted λ F . If the crack starts to propagate at a well-defined value of λ = λ F the fracture energy at initiation can then be obtained from the integral W (λ F ) of the stress strain curve of an unnotched sample stretched to λ F by using the approximate analysis of Greensmith 20,21 :

Γ = ௐሺఒ ಷ ሻ ඥఒ ಷ Equation 2
where c is the length of the notch, λ F is the extension ratio at fracture of the notched sample, and W(λ F ) is the strain energy per unit volume stored in an unnotched sample at λ = λ F .

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9

Results and Discussion

Rheology and structure of undeformed materials As reported in our previous study 2 , films made from PAA-b-PBA core-shell latexes (with PAA = 3 kg.mol -1 and PBA = 100 kg.mol -1 ) at room temperature (25°C) and at 0.1% strain have clearly the response of a viscoelastic solid (tan δ ~ 0.15, see Figure 1), although no chemical crosslinker is present in the film and only 3 wt% of the material is glassy. At 25°C, PBA of a molar mass of 100 kg.mol -1 is a viscoelastic fluid with a very strong frequency dependence of tan δ. This suggests that the glassy PAA shells form a percolating network.

Even if the nano-structured film obtained after drying the latex at room temperature is stable at ambient temperature, it is not at thermodynamic equilibrium. Figure 2 shows G' and G''as a function of temperature at a fixed frequency (1 rad.s -1 ). During the heating ramp, the elastic modulus is almost independent of temperature until about 110°C and then drops abruptly at around 120°C (1 decade loss). This modulus drop is irreversible as shown by the cooling ramp which is very different from the heating one. It is interesting to note however that even if the modulus drops one decade, the material remains a viscoelastic solid at small strains. Since the polymers are not reactive, this loss in modulus must result from a change in the film structure, which is confirmed by small angle X-ray scattering experiments. Before annealing, a well-defined interparticle distance is clearly observed in SAXS by a peak at q = 0.11 nm -1 corresponding to a characteristic repeat distance of 50 nm. 2 After thermal annealing, the characteristic distance decreases down to 35 nm and the degree of long range order, shown by the presence of higher order peaks, increases. Above 120°C the initial organization of the glassy shells into a continuous (honey-comb like) network is most likely lost, and the equilibrium organization should become that of a very asymmetric A-B block copolymer in the bulk, i.e. small PAA spherical domains dispersed into a PBA matrix 22 .

This new structure is no longer unusually stiff but retains some elasticity, at least at ω = 1 rad.s -1 .

M A N U S C R I P T A C C E P T E D

It should be noted that the change in structure and consequently in viscoelastic properties occurs over a narrow range of temperatures, with nearly no evolution below 110°C in contrast to less controlled core-shell structures resulting from non-covalently linked core-shell materials described by Dos Santos et al. 14 In the latter study, the elastic modulus was shown to decrease continuously during the heating ramp between -60°C and 110°C with a loss of 2 decades in modulus, before a sharp drop was observed around 110°C due to the T g of the glassy component. 14 This interesting property of our system must be correlated to the absence of a broad interphase of mixed composition replaced by a strong and very localized covalent link between the core and the shell chains, that was obtained thanks to the use of a controlled radical polymerization process.

In conclusion, below the T g of the PAA, the core-shell structure is stable, probably due to the strong hydrogen bonding between the PAA blocks of the shells. However above the T g of PAA, the PAA block softens and becomes mobile, leading to a film re-organization into a thermodynamically more stable structure, made of interdiffused soft PBA shells with a tiny hard PAA core.

Mechanical properties at large strain of a PAA-b-PBA copolymer based film

All particles discussed in the current article share the same structural features of a deformable soft core and a glassy shell, able to form a percolating network in the resulting films after drying at room temperature. The films nanostructure (shell thickness and connectivity between the shells) may be tuned via the macromolecular characteristics of the copolymers constituting the particles and the composition of the water phase (serum).

The large strain properties of the nanostructured films were characterized by elongational rheology. This technique, which is adapted to soft and slightly sticky materials, applies a uniaxial constant strain rate to the sample until fracture occurs. Figure 3 shows the nominal stress-strain curves obtained for the core-shell films, i.e. the as-cast and the annealed film. For strains below 10%, the cast Film 2 presents a linear behavior, with a tensile modulus of 1.0 MPa. 2 For larger strains, a non-linear regime can be observed, during which the film undergoes a progressive softening. Finally, above 300%, the sample breaks. This result shows that the particles must deform significantly before failure, suggesting the presence of entanglements between the block copolymers belonging to different particles. This deformation must cause a breakup of the shells into smaller fragments and consequently a drop of the modulus. In contrast, for the annealed film (Film 2-TA, Figure 3) the initial modulus is at the level of what would be expected for an entangled PBA film (0.2-0.3 MPa).

M A N U S C R I P T A C C E P T E D

The film extends to more than 10 times its initial length before failing, which is typically observed when acrylic acid groups are randomly incorporated in a PBA. 23 Both effects suggest again a significant change in structure due to the annealing step (above the glass transition temperature of the PAA block). To characterize the film structural changes under strain, SAXS measurements coupled to a tensile test were performed in order to monitor changes in the internal structure of the material as a function of strain (strain rate = 0.1 s -1 ). On Figure 4a and b, the diffraction patterns obtained for the undeformed film (λ = 1) and during stretching (λ = 1.55) are presented. Each pattern presents a distinct correlation peak. In the undeformed sample (Figure 4a), the peak appears at the same value of scattering vector in all directions, but as the sample is deformed, the extension of the sample induces a compression of the SAXS pattern in the direction of stretching and an expansion in the orthogonal direction (Figure 4b). In Figure 4c the characteristic distances in both directions (obtained from the 2π/q max ) are plotted as a function of the macroscopic stretch. It is shown that the characteristic distance increases parallel to the stretching direction while it decreases orthogonal to the stretching direction, as qualitatively expected from the SAXS pattern. In other words, as this characteristic distance represents the average interparticle distance, one can conclude that the particles shape evolves from an isotropic spherical shape to a compressed ellipsoid shape.

M A N U S C R I P T A C C E P T E D

When the evolution of the microscopic deformation is plotted as a function of the macroscopic extension ratio (Figure 4d), a deviation from the affine prediction (straight line) is clearly observed. This suggests that an additional mechanism (in addition to the expected particle deformation) must occur during stretching. The observed non-affine deformation

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14 through particle sliding suggests that entanglements between individual particles exist, that can partially relax and reform at that particular strain rate, causing some degree of particle slippage. This might be explained by the particular core-shell structure of the particles obtained by PISA and by the relatively low molar mass of the PBA block obtained by PISA which will only result in 3-4 entanglements per chain at the most (the average molar mass between entanglements M e for PBA is around 27 kg.mol -1 ). In this particular synthetic approach, the particles are segregated in a pure PBA core surrounded by a thin and covalently attached pure PAA shell. As the film is deformed, the shell breaks and entanglements between PBA blocks form (and prevent brittle fracture) but they remain rather dynamic allowing some relative slippage at the particle-particle interface. Note that annealed films are deformed more affinely (entanglements are preformed and the structure is lost) but remain sub-affine.

In contrast, for classic cellular films obtained from quasi statistical copolymers incorporating acrylic acid, an affine deformation has been reported 24 , since much longer lived entanglements between particles may occur due to the very high molar mass obtained by conventional emulsion polymerization.

To investigate the reversibility of the structural changes upon deformation, load-unload stepcycle tests were carried out, similarly to what is typically done for crosslinked rubbers 25 or semi-crystalline polymers 26,27 . The applied deformation history is shown in the inset of Figure 5b, the stress-stretch curves are reported in Figure 5a and the value of the initial modulus as a function of the applied λ max is shown in Figure 5b. The curves show a significant hysteresis between the loading and unloading curves for each incremental level of stretching and the modulus E decreases as the maximum applied stretch increases. However, observing the last three cycles carried out at the same maximum stretch level, the first cycle looks very different from the next two. This suggests two energy dissipating mechanisms: a permanent damage mechanism that depends on the maximum applied strain and a viscoelastic dissipation mechanism (visible in the second and third cycle at λ = 2.1) which will depend on strain rate and is characteristic of the material with a structure modified by the damage mechanism. This type of behavior is typical of the so-called Mullins effect in crosslinked rubbers 25 where the permanent damage is correlated to changes occurring in the filler network. Similarly, for our materials, we can explain the decrease in Young's modulus by assuming that the shells initially form a connected continuous (or percolating) filler network which progressively breaks as the material is stretched (around λ = 1.3-1.4, also visible in Figure 3). When all the hard PAA shells are broken, the Young's modulus stabilizes. The combined SAXS and mechanical data allow us to propose a deformation mechanism of the nanostructured film (Figure 6). When a uniaxial deformation is applied, the particles extend in the same direction as the stretching and contract orthogonal to it. When the deformation becomes too important (10-30%), the thin PAA shells progressively break, the continuous hard network fails, leading to a considerable drop in the Young's modulus.

M A N U S C R I P T A C C E P T E D

However this breakup of the thin shells does not lead to an overall failure. One can speculate

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that either entanglements between PBA chains in different particles exist in the undeformed film due to imperfect shells, or that PBA chains have enough time to re-entangle after the shells break. However the fact that the film can be extended to several hundred percent before breaking and this at a relatively high level of stress, suggests an additional mechanism of physical crosslinking along with entanglements, which could be created by nanoclusters of PAA chains as also observed for random P(AA-co-BA) copolymers when the mole fraction of PAA reaches 30%. In conclusion, the key parameters which control the properties of such nanostructured coreshell materials with an ultra-low volume fraction of hard shells are presumably the spatial organization (nanostructuration) of the shells in the material and their interactions. They have undoubtedly a crucial impact on the initial modulus (through the organization) and the breakup of the shells (interactions between shells). We now explore how both of these parameters can be fine-tuned.

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Interfacial physical crosslinking between particles, using Fe 3+ ions or PDMAEA polymer One strategy to delay the observed breakup of the percolating hard PAA network in our standard PAA-b-PBA diblock latex films is to strengthen the interfacial forces between the shells of different particles. Generally, the cohesion of the PAA network built by the PAA shells is guaranteed by the H-bonds of the carboxyl groups originating from different particles. 2 Other strong known interactions are ionic bonding, through oppositely charged ions. 28,29 To reach this objective, we have chosen to introduce water-soluble Fe III (acetylacetonate) in the water phase of the latex prior to filmification. During the film formation process, acetylacetone (T b = 140°C, at ambient pressure) and ammonia evaporate, generating Fe 3+ cations and carboxylate anions (Figure S3). Elongational rheology measurements carried out on films containing Fe 3+ (Figure 7) showed that the ionic interfacial crosslinking clearly increased the stress at which softening is observed, without however a clear impact on the modulus, suggesting that indeed the presence of Fe 3+ increases the strength of the interfaces but not the degree of organization. Yet once the PAA shells are broken into pieces, the stress remains higher for the highest Fe 3+ concentration, which suggests that the presence of Fe 3+ has an effect on the way the shell structure breaks up. Since the T g of the PDMAEA is only about 20°C, the mechanical stiffening at small strain must be due to the ionic interactions between the PAA and the PDMAEA rather than to Tg effects and the presence of the PDMAEA must also cause a stiffening at very large strain which causes the stable necking. Because of its outstanding stress and strain at break in simple extension experiments we investigated the mechanical properties of this last polymer in more detail. Load/unload cycles were carried out analogously to the PAA-b-PBA diblock copolymer latex films. Figure 11 shows the resulting stress-strain curves. Qualitatively the behavior is very similar to that of the PAA-b-PBA shown in Figure 5, but the stress values are much higher while retaining a reasonable recoverable deformation. This material possesses a glassy volume fraction in the Figure 12 shows the stress vs. strain curve for a notched and unnotched sample. The resulting fracture energy, Γ = 18 kJ/m 2 , is an extremely high value for a PBA based material. This very high value is due to the very large plastic deformation needed to stretch the material as seen in Figure 11 and as reported for high toughness gels by Sun et al. 30

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Conclusion

We have shown in this paper the potential of the PISA method to prepare out of equilibrium nanostructured films made of block copolymer particles. These films present unique properties through the combination of high stiffness at low strain and high toughness and extensibility at large strain. The key to such excellent properties is the combination of a percolating network made of the thin glassy PAA shells (only a few percent of the total mass), which gives the material a high stiffness, and the entanglements of the soft blocks in the particles, which gives the extensibility. In uniaxial extension we always observe a low strain (λ < 1.3) region where the material is stiff (E = 1-5 MPa) and elasticity is reversible, followed by a high strain region where the material is much softer but remains extensible and behaves similarly to a material made of statistical P(BA-co-AA) copolymers where interactions between PAA clusters prevent flow. Several strategies of stabilizing the nanostructure

1 ,

 1 of PAA-b-PBA-b-PS latexes. The PAA-b-PBA-b-PS particles were synthesized according to reference 15 . Styrene (0.60 g, 5.76 × 10 -3 mol) and AIBN (1.53 × 10 -3 g, 9.33 × 10 -6 mol) are added into a degassed PAA-b-PBA latex (5.47 g, M n, PAA = 2.3 kg.mol - M n, PAA-b-PBA = 115 kg.mol -1 , solid content = 15 wt%). After degassing for 12 h at 0°C with M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 5 argon bubbling, the polymerization is carried out at 70°C for 7 h in a septum-sealed flask introduced in a thermostated oil bath under gentle stirring to obtain the PAA-b-PBA-b-PS particles dispersion in water. The emulsion polymerization was quenched when reaching 90 % conversion (followed by gravimetric analysis) by immersion of the flask in an ice bath. The morphology of the aqueous self-assembled copolymers was analyzed by dynamic light scattering: the particles diameter increased from 70 nm to 96 nm. This was confirmed by TEM analyses. The number of particles N p remained constant (1 × 10 -8 L -1 -see SI), indicating that there was no secondary nucleation during styrene polymerization. After drying and methylation with trimethylsilyldiazomethane, the polymer was analyzed by size exclusion chromatography in THF. The number-average molar mass, M n , and the molar mass dispersity, Đ = M w /M n , were determined using a polystyrene (PS) calibration curve (M n, PAA-b-PBA-b-PS = 198 kg.mol -1 , Đ = 1.8).

a)

  Determined after methylation by SEC in THF with a PS calibration. The given value corresponds to the nonmethylated polymer (subtracting the -CH 3 contribution: × 0.84); b) Determined after methylation by SEC in THF with a PS calibration. The given value corresponds to the methylated polymer; c) Acrylic acid content = (m 0 (AA) / m 0 (BA)) 100; d) Stress at break measured by elongational rheology; e) Strain at break measured by elongational rheology; f) Young's modulus measured by elongational rheology; g) Control sample prepared from standard film after thermal annealing (160°C for 3 days); h) Films 4-FeIII were prepared adding 0.03 or 0.12 mol% of Fe(III)acetylacetonate relative to AA prior to film formation; i) Films 3-PDMAEA were prepared adding 0.16 or 0.47 mol% DMAEA units relative to AA prior to film formation; j) Film 5 was prepared from a triblock PAA-b-PBA-b-PS copolymer; k) The value corresponds to the PAA-b-PBA-b-PS triblock copolymer. Characterization Shear Rheology. Discs (diameter d = 8 mm, thickness t ~ 0.5 mm) were cut out from the dried films using a punch. Shear rheology measurements were realized with a parallel plate geometry, on a Physica MCR501 rheometer from Anton Paar equipped with the Rheoplus strain sweep was first carried out to identify the linear regime and then a temperature sweep was carried out at fixed frequency. Tensile tests. Tensile rectangular specimens (20 mm 5mm 0.5 mm) were cut from the dried films using a razor blade. Uniaxial tensile stress-strain measurements were carried out at 25°C using a standard tensile Instron machine (model 5565), equipped with a 10 N load cell and a video extensometer (model SVE) able to precisely follow the local displacement of two markers. Two black spots were drawn on the sample at approximately 10 mm initial distance to be tracked with the extensometer. Three replicate experiments were performed for each sample.

Figure 1 .

 1 Figure 1. Linear viscoelastic properties of a nanostructured standard film (Film 1). (a) Moduli G' ( ) and G'' ( ) as a function of frequency at 25°C and (b) Tan δ as a function of frequency at 25°C ( ) (ε = 0.1 %, F n = 0.1N).

10 Figure 2 .

 102 Figure 2.a) Elastic and loss moduli versus temperature for the nanostructured standard film (Film 1), b) value of tan δ at ω = 1 rad.s -1 (ε = 0.1 %, F n = 0.1N, 1°C.min -1 ). The heating step is marked in red, the cooling step in blue.

Figure 3 .

 3 Figure 3. Nominal stress-strain curve for the structured Film 2 and the annealed Film 2-TA (temperature = 25°C, strain rate = 0.06 s -1 ).(published in part in reference 2 )

Figure 4 .

 4 Figure 4. (a) SAXS pattern of the undeformed, nanostructured standard film (Film 3) (λ = 1), (b) SAXS pattern obtained at λ = 1.55, (c) characteristic distances parallel and orthogonal to the stretching direction plotted versus stretch, d) microscopic stretch as a function of macroscopic stretch for the pristine Film 3 and the annealed Film 3-TA.

Figure 5 .

 5 Figure 5. (a) Stress-strain curves for load-unload tensile cycles obtained for nanostructured standard film (Film 1) and (b) Young's modulus versus maximal elongation.

23

 23

Figure 6 .

 6 Figure 6. Proposed mechanism for the deformation of nanostructured PAA-b-PBA films.

Figure 7 .

 7 Figure7. Nominal stress-strain curve for the nanostructured standard film (Film 4), and films formed in the presence of 0.04 (Film 4-FeIII 0.04 ) and 0.12 (Film 4-FeIII 0.12 ) mol% (or equivalents, eq.) Fe 3+ (relative to AA) (temperature = 25°C, strain rate = 0.06 s -1 ).

Figure 8 . 20 Figure 9 .

 8209 Figure8. Nominal stress-strain curve for the nanostructured standard film (Film 3), and films formed in the presence of 0.16 (Film 3-PDMAEA 0.16 ) and 0.47 (Film 3-PDMAEA 0.47 ) mol% (or equivalents, eq.) DMAEA (relative to AA) (temperature = 25°C, strain rate = 0.06 s -1 ).

Figure 10 .

 10 Figure 10. Stress-strain curve for the standard PAA-b-PBA film (Film 1) and the triblock PAA-b-PBA-b-PS film (Film 5).

  , but the mechanical properties do not show the very distinctive strain hardening that is characteristic of thermoplastic elastomers.

Figure 11 . 22 Figure 12 .

 112212 Figure 11. Stress-strain curves for load-unload tensile cycles obtained for Film 5.

  have been explored. Remarkably the most interesting and effective method was the introduction of PS segments in the particle cores via the addition of a PS block at the end of the PAA-b-PBA chains leading to samples with an extremely high fracture resistance while retaining a solid character with significant elasticity.

Table 1 .

 1 Block copolymer characteristics and mechanical properties by elongational rheology of the resulting films.

	M n, PAA a)
	Film

  films

	Film 1	2.9	147 /1.7	3.0	460	550	4.5
	Film 2	3.1	88 / 1.8	3.2	360	360	1.0
	Film 2-TA g)	3.1	88 / 1.8	3.2	240	880	0.1
	Film 3	2.7	120 / 1.8	2.8	495	530	1.2
	Film 3-TA g)	2.7	120 / 1.8	2.8	-	-	-
	Crosslinking by Fe III
	Film 4	2.8	87 / 1.9	2.7	210	320	1.0
	Film 4-FeIII 0.03h)	2.8	87 / 1.9	2.7	270	320	1.0
	Film 4-FeIII 0.12h)	2.8	87 / 1.9	2.7	410	200	1.0
	Crosslinking by PDMAEA
	Film 3	2.7	120 / 1.8	2.8	495	530	1.2
	Film 3-PDMAEA 0.16i)	2.7	120 / 1.8	2.8	550	380	14
	Film 3b-PDMAEA 0.47i) 2.8	87 / 1.9	2.9	660	10	8.0
	PAA-b-PBA-b-PS films
	Film 5 j)	2.3	198 k) / 1.8	2.4	2500	1100	4.5

Acknowledgements

Financial support from the Ile-de-France region (M. Chenal) and the French ANR program ANR-10-BLAN-0801 Supradhesion (C. Véchambre) is acknowledged. F. Tanguy and B. Bresson are gratefully acknowledged for their help. We are indebted to the CLYM (Centre Lyonnais de Microscopiehttp://www.clym.fr) for the access to the Di3100 atomic force microscope, particularly to A. Malchère, Y. Liu, A. Descamps-Mandine and D. Albertini for their very helpful technical support. Thanks to SOLEIL synchrotron and particularly to G. Stoclet for the access to the SWING beamline.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

Highlights

• Nanostructured films containing a percolating network of PAA shells.

• High stiffness, extensibility and high fracture resistance.

• Tunable composition of block copolymer latex via the PISA approach.

• Tunable mechanical properties of the latex-derived films.

• Addition of cationic species in the serum modulates mechanical film properties.


Neumann problem with critical nonlinearity


Fabrice Bethuel

published or not. The documents may come     Multiple solutions to the Plateau problem for nonconstant mean curvature


come

An elliptic Neumann problem with critical nonlinearity in three dimensional domains


Mathieu Fabre
email: mathieu.fabre@epfl.ch

Sylvain Faure

Mathieu Laurière
email: mathieu.lauriere@nyu.edu

Bertrand Maury

Charlotte Perrin
email: perrin@instmath.rwth-aachen.de

NON CLASSICAL SOLUTION OF A CONSERVATION LAW ARISING IN VEHICULAR TRAFFIC MODELLING *

whether they are published or not. The documents may come

Introduction

Modelling traffic has become in the recent years a major challenge for the applied mathematics community. Among all the issues raised by vehicular traffic behaviours we are particularly interested in the modelling of congestion phenomenon, how traffic jams appear, persist and disappear. There are mainly three approaches to model traffic motion, each corresponding to a specific scale: microscopic scale (each entity is individually followed), macroscopic scale (the collection of entities is described by a local density), and mesoscopic scale (kinetic description). In this paper we will focus on the first two descriptions, the interested reader is referred to [START_REF] Bellomo | On the modelling of Traffic and Crowds: A Survey of Models, Speculations, and Perspectives[END_REF] for a review of the kinetic models.

The most direct way to describe the motion of a collection of vehicles is to represent each of them individually and to follow the trajectory of each agent, taking into account all the interactions with the surrounding agents. In one-dimensional situations (a one-lane road or a corridor), a simple description of the dynamics is provided by Follow-The-Leader models used for instance in [START_REF] Chandler | Traffic Dynamics: Studies in Car Following[END_REF]- [START_REF] Gazis | Nonlinear follow-the-leader models of traffic flow[END_REF]. In this context congestion is handled by introducing short-range correction terms which mimic some repulsion when an agent gets closer to the agent next to him. This is the approach adopted in Follow-The-Leader models, where the velocity is supposed to decrease with the inter-agent distance, and is taken to be zero at a minimal distance corresponding to the size of the agents, which is not negligible in the discrete description. Note that such models are not based on classical mechanics principles: interaction terms are not symmetric, since each agent is influenced by the one that lies in front of him, with no reciprocity.

Another mathematical description widely used consists in considering the assembly of vehicles as a continuous medium, and working with averaged variables: the average velocity of agents in an elementary volume (or length in the one-dimensional setting), and the density, which corresponds to the number of agents per unit volume. We then make an analogy with fluids by writing a balance equation expressing at the macroscopic level the conservation of the number of agents. If one wants to simulate the behaviour of a large number of agents, the macroscopic description may be attractive compared to the microscopic description for the computational cost and for the relative ease with which one can calibrate the model due to the low number of parameters. As we shall see, although such models are based on principles that are similar to the Follow-The-Leader approach, the "directionality of influence" that we previously mentioned will disappear in the expression of interaction terms. This very distinction between the two approaches will play a crucial role in the present paper.

Intuitively, discrete models like the Follow-The-Leader models should converge towards continuous conservation laws. This issue has been the object of few and relatively recent mathematical works. In [START_REF] Aw | Derivation of continuum traffic flow models from microscopic follow-the-leader models[END_REF], the authors prove a mathematical link between the discrete solutions of a second order (in time) Follow-The-Leader model for car traffic, and the semi-discretization in time of a macroscopic solution to a system of conservation laws called Aw-Rascle model. The formal idea consists in letting the number of vehicles go to infinity while imposing a particular scaling in time and space which ensures that the density and the velocity remain fixed.

Concerning first order Follow-The-Leader models, which neglect the effects of inertia in the motion, this question has been treated by Colombo and Rossi in [START_REF] Colombo | On the Micro-Macro Limit in Traffic Flow[END_REF] and by Di Francesco and Rosini in [START_REF] Francesco | Rigorous derivation of nonlinear scalar conservation laws from followthe-leader type models via many particle limit[END_REF]. In the latter, the authors prove that the discrete solutions satisfy a discrete inequality of Oleinik's type which yields strong controls of the semi-discrete density, and the strong convergence towards the unique entropy solution of the limit conservation law.

This limit passage between the microscopic and the macroscopic scales is also related to homogenization techniques. Such techniques have been interestingly used for instance by Forcadel, Imbert, Monneau [START_REF] Forcadel | Homogenization of some particle systems with two-body interactions and of the dislocation dynamics[END_REF] to prove the convergence of the primitive of the semi-discrete density towards a viscosity solution of a macroscopic Hamilton-Jacobi equation (which can be seen as a primitive of the LWR model). In a recent paper, Forcadel and Salazar [START_REF] Forcadel | A junction condition by specified homogenization of a discrete model with a local perturbation and application to traffic flow[END_REF] study this limit passage for a FTL model with a local perturbation depending on the position which leads at the macroscopic limit to a local flux limiting condition.

In this paper, we are interested in particular phenomena that can be observed and experimentally reproduced for cars, the typical situation we have in mind is the following: an accident takes place on a freeway, and instantaneously stops the traffic on both lanes. Traffic is frozen upstream the accident, and a jam is also created on the opposite lane due to drivers' curiosity. Although there is no physical obstacle on this lane, it can be commonly observed that this jam persists until every trace of the accident has been removed. As it will be detailed, macroscopic and microscopic standard models fail to reproduce this persistence of a jam. In [START_REF] Polson | Bayesian analysis of traffic flow on interstate I-55: The LWR model[END_REF], Polsen and Sokolov develop a particle filtering and learning algorithm to estimate and update the parameters of their macroscopic model. Applying it to the previously described situation, they are then able to identify a drop in the road capacity (i.e. the critical/maximal flow) and reproduce the formation of a jam on the opposite lane. Contrary to them, we want to keep the parameters of the road fixed. We propose here to model this type of phenomenon at the microscopic level by locally perturbing the classical Follow-The-Leader model. We shall consider that the behavior of the agents is modified in a small buffer zone, that corresponds to the zone in which the accident in the opposite lane can be seen. We can interpret the modification as follows: when entering in direct visual contact with the accident, people cannot help looking at it, and they no longer look carefully at the vehicle upfront, that may already be accelerating. Yet, because of the presence of police forces or because of moral considerations, they do not completely stop. We shall consider that they keep the velocity they had when entering in the buffer zone. This perturbation is different from the one introduced by Forcadel and Salazar in [START_REF] Forcadel | A junction condition by specified homogenization of a discrete model with a local perturbation and application to traffic flow[END_REF]. In their paper, the velocity is modulated by an additional function of the position, while here, the dynamics itself is modified in the buffer zone since the acceleration is imposed to be zero.

By letting the number of agents increase, the numerical solution is expected to converge to a solution of a macroscopic conservation law. The situation we will obtain, as expected, corresponds to what is observed in practice: a congested zone upstream the "accident", at low velocity, and a dilute zone downstream, at high velocity. Since characteristics go leftward on the left hand side of the discontinuity, and rightward on the right hand side, it corresponds to a non-classical solution, i.e. a non entropy solution, whereas the entropy solutions are usually considered as the "physical solutions" (see Ansorge [START_REF] Ansorge | What does the entropy condition mean in traffic flow theory?[END_REF]) and the most common numerical schemes are built to select this type of solutions. Non-classical solutions have been already considered for traffic (or pedestrian) flow modelling, but in other contexts: for instance when a additional flux constraint is imposed (see [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF], [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF]), or to describe panic phenomena in a crowd (see [START_REF] Colombo | Pedestrian flows and non-classical shocks[END_REF]).

Directly considering the macroscopic model, we will show that a local modification of a standard numerical scheme (originally consistent with the entropy conditions), is sufficient to capture stable traffic jams.

Microscopic models for traffic jams

We consider the one-dimensional domain T = [0, 2π] supplemented by periodic boundary conditions. By doing so we aim at modelling the dynamics of vehicles on a circular road. We will focus in this section on a first order Follow-The-Leader model (denoted FTL) where the vehicles instantaneously reach the desired velocity, which is a given function of the distance with the upfront vehicle. Starting with an initial congested distribution of vehicles on the circular road, we will show that the FTL model predicts a relaxation towards the uniform distribution with constant velocity for all the vehicles and constant distances between them. We will then try to perturb this dynamics by introducing a buffer zone which corresponds to the zone where the drivers look at the opposite lane and forget to follow the previous car. Finally, by letting numerically the number of cars increase, we want to approximate the continuous density of vehicles which is a solution to an appropriate conservation law.

The discrete Follow-the-Leader model

The Follow-The-Leader (FTL) model has been first introduced to model car traffic. It expresses the fact that each driver follows the car in front of him by adjusting his speed according to the distance between them. Let us consider N cars, whose positions and speeds are denoted respectively by x i and v i , i = 1, . . . , N , where the (i + 1)st car follows the ith car. Then the FTL model consists of N differential equations: for i = 1, . . . , N :

dx i+1 dt (t) = v i+1 (t) = ϕ(δ i (t)), (1)
where δ i = x i -x i+1 is the distance between the i-th car and the following one. By periodicity we define

δ N = x N -x 1 .
The function ϕ is assumed to be concave, increasing and with nonnegative values. Here, we will consider the particular function

ϕ(δ) = v max 1 - δ * δ 1 {δ>δ * } , (2)
where δ * > 0 represents the size of a car (all the cars are assumed to have the same size), and v max , the maximal velocity, is the velocity the cars would have if there were nobody in front of them. We have represented this function on Figure 1. The interested reader is referred to [START_REF] Gazis | Nonlinear follow-the-leader models of traffic flow[END_REF] concerning general FTL models.

The equations (1) can be rewritten as: for i = 1, . . . , N :

dδ i dt (t) = ϕ(δ i-1 (t)) -ϕ(δ i (t)).
Equilibrium of the basic system. Associated to this model, there exists a unique periodic solution which corresponds to the uniform distribution of the distances. All the distances and the velocities are then respectively equal to

δ eq = 2π N , v eq = v max 1 - δ * δ eq .
In the previous equations, we implicitly assume that the number of vehicles on the road is not too large, namely

N < 2π δ * . Introducing the discrete entropy S(∆) = i δ i log δ i , for ∆ = (δ i ) i=1,...,N ,
one can recover that the uniform distribution is a global attractor of the system. Indeed, one can show that the discrete entropy decreases in time and is minimal for the uniform distribution. Let us recall the main steps allowing to prove this fact. By replacing the distance δ i by δ i 2π we can assume that δ i = 1. Differentiating the entropy S with respect to time and using the periodicity, we have

d dt S(∆) = i (log δ i + 1) d dt δ i = i (log δ i + 1)(ϕ(δ i-1 ) -ϕ(δ i )) = i log δ i (ϕ(δ i-1 ) -ϕ(δ i )).
If the distribution is uniform, then we get 0. Otherwise, we can split the sum into two parts depending on the sign of δ i-1 -δ i :

I 1 = {(i -1, i) , δ i-1 < δ i }, I 2 = {(i -1, i) , δ i-1 ≥ δ i }.
The strict concavity of the function ϕ leads to

d dt S(∆) < ϕ (1) I1 log(δ i ) δ i-1 -δ i + ϕ (δ * ) I2 log(δ i ) δ i-1 -δ i .
Rearranging the sums we obtain

d dt S(∆) < ϕ (1) i log(δ i ) δ i-1 -δ i + (ϕ (δ * ) -ϕ (1)) I2 log(δ i ) δ i-1 -δ i < ϕ (1) i log(δ i ) δ i-1 -δ i < ϕ (1) i log δ i+1 δ i δ i .
Using the fact that i δ i = 1 and the concavity of the logarithm we finally obtain

d dt S(∆) < ϕ (1) log i δ i+1 δ i δ i = ϕ (1) log i δ i+1 = 0.
The entropy is therefore a global and strict Lyapunov functional for the considered equilibrium point, which asserts global asymptotic stability. Behaviour in presence of high density. Starting from an initial distribution where cars are concentrated in a small region, the system evolves towards the stable solution where the distances are uniform. In other words, a relaxation leads to the dissolution of the jam, as illustrated by Figure 2.

We have represented the trajectories of some cars, as well as the distances between them and the cars in front of them. We observe that after some time, the trajectories coincide with lines with a common slope which is the inverse of the uniform velocity v eq . After some oscillations, the distances seem to converge towards the uniform distance δ eq .

Inclusion of a buffer zone

In order to model the persistence of some traffic jams, we propose to include in the domain a buffer zone where the velocities are "frozen". In other words, drivers go through this zone with a constant velocity, equal to the velocity they had when they entered this buffer zone. This encodes the following behavior: the driver looks at the accident on the opposite lane, while maintaining his place. Let us denote Z F T L the normal zone where the drivers behave according to the FTL model, and Z f = T \ Z F T L the buffer zone.

The model writes: for every

i = 1, . . . , N          dx i+1 dt (t) = ϕ(δ i (t)) if x i+1 ∈ Z F T L , d 2 x i+1 dt 2 (t) = 0 if x i+1 ∈ Z f . (3)
Size of the buffer zone. In order to see an effect of this zone on the dynamics, we need to ensure that there is always at least one car whose velocity is frozen. The buffer zone then needs to be large enough compared with the size δ * of the cars, the expected velocity in traffic jam v i , and the time step ∆t.

In our simulations we have fixed the length of the frozen zone L f as

L f = 3δ * ,
and we have taken the time step ∆t such that it is much smaller than the critical time step ∆t c :

∆t ∆t c = L f v i .
We can thereby ensure that the cars spend several time steps in the buffer zone, which can modify the FTL dynamics.

Behaviour in presence of high density. Starting from an initial situation where the buffer zone is congested, we expect the system to remain congested. Intuitively, this is for the following reason: when a driver enters the buffer zone, he is stuck in the jam so his speed is very low. Then he slowly passes through the buffer zone, and even when the vehicle in front of him exits from the buffer zone and accelerates due to the dynamics imposed by the FTL model, he will keep the same speed. This maintains the jam behind him. Thus, locally freezing the car response can be expected to maintain the presence of a jam.

We observe on the simulations that we do not have any more the relaxation process as in the normal FTL model (see Figure 2), and it seems that the traffic jam is persistent. On Figure 3 are represented the trajectories of the cars. After some time, we observe periodicity of the movement which corresponds to a stable traffic jam. We have also represented on Figure 3 the velocities of four following cars from the time they enter into the traffic jam until they leave the buffer zone. We observe first an alignment of the velocities with the velocity imposed initially on the traffic jam. In this regime, the evolution of the velocities is continuous due to the FTL dynamics. Then, the driver enters into the buffer zone. When he finally leaves the buffer zone, he is subject to a speed jump which corresponds to the transition between the buffer zone and the FTL dynamics.

Macroscopic limit passage

Basing our numerical investigations on the recent paper of Colombo and Rossi [START_REF] Colombo | On the Micro-Macro Limit in Traffic Flow[END_REF], we introduce the micro-macro operator for a fixed integer N :

C N : X =    x 1 . . . x N    → ρ N = i δ * N x i -x i+1 1 [xi+1,xi] ,
where δ * N is the length of a car (and so it corresponds to the minimal distance between two cars). This operator C N associates to the discrete distribution X of the vehicles a piecewise constant density ρ N . We expect ρ N to converge almost everywhere when N → +∞ to the solution of a macroscopic conservation law. For our simulations we set

δ * N = 1 N , so that N δ * N is constant.
The size L f of the buffer zone is, as previously, related to the size of the vehicles by L f = 3δ * N . Note that this quantity converges to 0 as the number of cars N goes to infinity. Formally, we expect then that the microscopic model coupling FTL dynamics and the buffer zone converges towards the same macroscopic model as the single FTL model. This conservation law is

∂ t ρ + ∂ x (ρv(ρ)) = 0. (4)
In order to link this macroscopic velocity v with the velocity ϕ of the discrete FTL model we express the compatibility between the two descriptions on the discrete density ρ N (see for instance [START_REF] Colombo | On the Micro-Macro Limit in Traffic Flow[END_REF] eq.

(2.4)). Let x ∈ [0, 2π]; there exists i ∈ {1, . . . , N } such that ρ N (x) = δ * /δ i , replacing then δ i in the expression of the velocity of the FTL model we get

ϕ(δ i ) = ϕ δ * ρ N (x) = v max 1 - δ * δ * /ρ N (x) = v max (1 -ρ N (x)) .
The velocity involved in the limit conservation law is therefore v(ρ) = v max (1 -ρ).

We have represented in Figure 4 the discrete density ρ N computed from the model coupling the FTL dynamics and the buffer zone. The profile of this discrete density seems to be stable. It consists of a high density zone before and in the buffer zone corresponding to the traffic jam and of a low density zone outside the traffic jam.

We want to recover in the next section this type of solution directly on the limit conservation law.

Conservation laws for congestion

In the previous section we have let numerically the number of vehicles increase and thus constructed an approximate solution of a macroscopic model for a traffic jam. Historically, macroscopic systems based on the principles of the continuum mechanics have been first introduced for traffic flow modelling. The most famous model has been derived in 1955 by Lighthill and Whitham [START_REF] Lighthill | On kinematic waves II: A theory of traffic flow on long, crowded roads[END_REF] and completed one year later by Richards [START_REF] Richards | Shock waves on the highway[END_REF] who added the notion of shock wave in the model. This system, still widely used, is commonly called LWR model. It expresses the physical principle of mass conservation for the assembly of vehicles

∂ t ρ + ∂ x (ρu) = 0,
on the closed and bounded interval Ω, where ρ denotes the density of cars, u the mean velocity of the agents in an elementary volume. Both are considered as functions of time and space. We prescribe the velocity u as a function of the density: it will be the product of a desired velocity U and a term, denoted β(ρ), representing the effects of saturation

∂ t ρ(t, x) + ∂ x ρ(t, x)β(ρ(t, x))U = 0, (t, x) ∈ [0, T ] × Ω. (5)
The function β is then assumed to be strictly decreasing on [0, 1]. In the literature several expressions have been proposed for the function β. Among them one can cite those of

• Greenshields: β(ρ) = 1 -ρ, • Pipes-Munjal: β(ρ) = 1 -ρ α , α > 0, • Underwood: β(ρ) = e -ρ ,
• Greenberg (modified): β(ρ) = log 1 α -1 log 1 ρ+α , α > 0. The functions β are chosen to fit with the so-called fundamental diagram which represents the experimental flux of cars as a function of their density as explained in [START_REF] Jelic | Properties of pedestrians walking in line -Fundamental diagrams[END_REF] or [START_REF] Bellomo | On the modelling of Traffic and Crowds: A Survey of Models, Speculations, and Perspectives[END_REF]. Our previous study on the microscopic system (see § 1.3) leads us to consider a function of Greenshields type,

β(ρ) = 1 -ρ.
In the rest of this section we will first describe a new numerical scheme for dealing with congestion effects in the one-dimensional periodic setting. We prove that the numerical scheme is appropriate for entropy solutions and therefore does not capture the stable congested numerical solutions which has been exhibited in the previous section. We then propose a local modification of the scheme which reproduces at the macroscopic level the ideas of the microscopic buffer zone.

Upwind/downwind numerical scheme

We consider the domain Ω = T, which denotes the one-dimensional interval [0, 2π] with periodic boundary conditions as it has been done in the microscopic setting (see Section 1). It means that there is one entrance at x = 0 and one exit at x = 2π, with an inflow at x = 0 equal to the outflow at x = 2π. In this situation the desired velocity can be considered as a positive constant, which we take to be U = 1 for simplicity.

Our numerical scheme is a finite volume scheme which aims at reproducing the physical transport of the information. The information of the transport of the mass is propagating upwind (since the mass moves forward), whereas the information of congestion can be seen as an information propagating downwind. Our scheme is based on a separate treatment of these two features.

We discretize the interval [0, 2π] with a uniform grid of step size ∆x, and discretize the time scale [0, T ] uniformly with a step size ∆t, such that 1/∆x and T /∆t are two integers, denoted respectively by N x and N t . The points in space and time are indexed in {1, . . . , N x } and {0, . . . , N t } respectively. Let ρ n i ∈ R be the approximation of ρ on [(i -1)∆x, i∆x] at time n∆t. Our explicit scheme then writes: for each i ∈ {2, . . . , N x -1},

ρ n+1 i -ρ n i ∆t + 1 ∆x ρ n i β(ρ n i+1 ) -ρ n i-1 β(ρ n i ) = 0, (6)
with the periodic boundary condition

ρ n 1 β(ρ n 1 ) = ρ n Nx β(ρ n Nx ). (7)
Properties of the numerical scheme. Our numerical scheme is under conservative form as we can see by writing

ρ n+1 i = ρ n i - ∆t ∆x g n i+1/2 -g n i-1/2 , with g n i+1/2 = g(ρ n i , ρ n i+1 ) = ρ n i β(ρ n i+1
). We directly check that the scheme is consistent, i.e. g(ρ, ρ) = ρβ(ρ). Moreover, we argue that it is monotone under the CFL condition

∆t ∆x 1 + max w∈[0,1] |β (w)| ≤ 1 (8)
(for β(ρ) = 1 -ρ, the condition is simply: ∆t ≤ ∆x/2). Indeed, we recall that (see for instance the reference book written by Godlewski and Raviart [START_REF] Godlewski | Hyperbolic systems of conservation laws[END_REF]) a numerical scheme

ρ n+1 i = H(ρ n i-1 , ρ n i , ρ n i+1 ),
is said to be monotone if the function H : R 3 → R is an increasing function in each argument. In our case,

H(v 1 , v 2 , v 3 ) = v 2 - ∆t ∆x g(v 2 , v 3 ) -g(v 1 , v 2 ) .
The function g defining the numerical flux is increasing with respect to the first variable and decreasing with respect to its second variable since the function β is assumed to be decreasing. These two points prove that H is increasing with the first and the last variable. It remains to show that it is also increasing with the second variable. For that purpose, we differentiate H with respect to the second variable

∂ v2 H(ρ i-1 , v 2 , ρ i+1 ) = 1 - ∆t ∆x (β(ρ i+1 ) -ρ i-1 β (v 2 )) .
We observe that the CFL condition ensures the positivity of ∂ v2 H, and thus the monotonicity of our numerical scheme. As a by-product of the consistency and monotonicity properties, we get the L ∞ -stability and the consistency with the entropy condition. We refer for these two points again to [START_REF] Godlewski | Hyperbolic systems of conservation laws[END_REF], Theorem 3.2 and Theorem 4.2. Stationary solutions. We are interested now in the determination of the stationary solutions of our model with the periodic boundary conditions. The stationary solutions are characterized by a constant flux, denoted φ, such that

ρβ(ρ) = φ a.e.. (9)
By definition of β, the flux function ρ ∈ [0, 1] → ρβ(ρ) is strictly increasing on [0, 1/2] from 0 to 1/4 and strictly decreasing on [1/2, 1] from 1/4 to 0. Then, if φ < 1/4, there exist exactly two solutions to [START_REF] Forcadel | Homogenization of some particle systems with two-body interactions and of the dislocation dynamics[END_REF], say ρ -and ρ + , with ρ -< 1 2 < ρ + .

A unique "physical" (weak) solution to the conservation law is usually selected thanks to an additional entropy criterion (for the classical theory concerning classical entropy solutions we refer to Serre [START_REF] Serre | Systems of conservation laws. 1 & 2[END_REF], and for applications to vehicular traffic, to Ansorge [START_REF] Ansorge | What does the entropy condition mean in traffic flow theory?[END_REF], Herbin, Leclercq [START_REF] Herbin | A note on the Entropy Solutions of the Hydrodynamic Model of Traffic Flow" revisited[END_REF]). If we assume moreover that our flux is strictly concave on [0, 1], which is the case for β(ρ) = 1 -ρ, then the Oleinik criterion selects the admissible stationary shocks

ρ(t, x) = ρ -if x < x * ρ + if x > x * , x * ∈ (0, 1).
Therefore, with ρ -< ρ + , there are only three entropy stationary solutions of the conservation law which are

ρ = ρ -, ρ = ρ + , ρ = ρ -if x < x * ρ + if x > x * . This last case is possible only if ρ -(1 -ρ -) = ρ + (1 -ρ + ) = φ ∈ (0, 1/4).
As we have seen previously, our upwind/downwind numerical scheme satisfies a monotonicity property which ensures the consistency with the discrete entropy criterion. Therefore our numerical scheme should converge, as ∆x and ∆t go to 0, towards the unique entropy solution of the system, and this is what is observed on the numerical simulations, as shown in Figure 5. In order to investigate the properties of our scheme regarding stationnary solutions, we consider on the whole discrete space the semi-discrete equation: for all i ∈ Z,

∂ t ρ i + 1 ∆x ρ i (1 -ρ i+1 ) -ρ i-1 (1 -ρ i ) = 0. ( 10
)
The stationary solutions of the previous equation are characterized by a constant flux φ ∈ [0, 1/4] such that

ρ i (1 -ρ i+1 ) = φ, ∀i ∈ Z.
Proposition 2.1. Assume that φ ∈ (0, 1/4) and consider the densities

ρ ± = 1 ± √ 1 -4φ 2 . If ρ 0 = 1/2
, there exists a unique stationary solution of the scheme that corresponds to the flux φ. It connects the state ρ -at -∞ to the state ρ + at +∞. There does not exist any stationary solution if we exchange ρ -and ρ + .

Proof:

If ρ 0 = 1/2, then ρ -i = φ 1 -ρ -i+1
, ∀i ∈ N * , and the sequence converges as i → +∞ towards a fixed point of the map g : r → φ 1 -r , i.e. ρ + or ρ -. Since

g (ρ + ) = φ (1 -ρ + ) 2 = ρ + 1 -ρ + > 1, and |g (ρ -)| = ρ - 1 -ρ - < 1,
we deduce that the sequence (ρ -i ) i≥0 decreases towards ρ -. In a symmetric way (ρ i ) i≥0 converges towards ρ + as i → +∞.

One can see that the sequence (ρ i ) is nondecreasing, which excludes the non-entropic shocks.

We have exhibited in the previous section a microscopic stationary solution whose profile does not correspond to a stationary entropy solution of the conservation law [START_REF] Chandler | Traffic Dynamics: Studies in Car Following[END_REF]. It suggests that the notion of entropy solutions may not be relevant in some particular situations observed in traffic flows. Some realistic situations, as our stable traffic jam, are indeed automatically rejected by the entropy criterion. Indeed, we need to take into account that a driver does not behave like a gas molecule and is not driven by a Brownian motion at the discrete level. In the rest of this section we propose a local modification of the upwind/downwind scheme [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] for the macroscopic conservation law in order to capture the potential stable traffic jams.

Local modification of the numerical scheme

We aim here at designing a local (one-cell) modification of the numerical scheme to capture the traffic jam observed at the microscopic level. To that purpose, we come back to the idea of the buffer zone introduced in the microscopic FTL model. It was a zone in which the cars have a constant velocity, and do not take into account the possible effects of congestion. Here we only modify how the information of congestion is taken into account on a given cell i 0 : instead of imposing a downwind discretization of the saturation function β, we discretize in an upwind manner the whole flux. By doing so we preserve the consistency of the scheme with the conservation law [START_REF] Chandler | Traffic Dynamics: Studies in Car Following[END_REF]. The scheme writes as

                       ρ n+1 i -ρ n i ∆t = 1 ∆x ρ n i-1 (1 -ρ n i ) -ρ n i (1 -ρ n i+1 ) , ∀ i = i 0 , i 0 + 1, ρ n+1 i0 -ρ n i0 ∆t = 1 ∆x ρ n i0-1 (1 -ρ n i0 ) -ρ n i0 (1 -ρ n i0 ) , ρ n+1 i0+1 -ρ n i0+1 ∆t = 1 ∆x ρ n i0 (1 -ρ n i0 ) -ρ n i0+1 (1 -ρ n i0+2 ) . (11)
Forgetting for a while the periodic boundary conditions, given a flux φ ∈ (0, 1/4), one can check for instance that the discontinuity

ρ i = ρ + for i ≤ i 0 , ρ -for i > i 0 .
with ρ -and ρ + such that ρ

-(1 -ρ -) = ρ + (1 -ρ + ) = φ
is a stationary solution of the modified scheme.

Let us look at the stability of this equilibrium. We denote the above stationary solution ρ e and we consider the perturbation ρ = ρ e + ρ. The perturbation satisfies the equation

d dt ρ = ∇F (ρ e ).ρ,
where ∇F (ρ e ) is a tridiagonal matrix

∇F (ρ e ) = 1 ∆x          b 1 c 1 0 . . . 0 a 2 b 2 c 2 . . . . . . 0 . . . . . . . . . 0 . . . . . . a N -1 b N -1 c N -1 0 . . . 0 a N b N          . with a i = 1 -ρ i , b i = -1, c i = ρ i , ∀ i = i 0 , i 0 + 1, a i0 = 1 -ρ + , b i0 = -1 + ρ + , , c i0 = 0, a i0+1 = 1 -2ρ + , b i0+1 = -1 + ρ -, c i0+1 = ρ -.
Thanks to the Gershgorin circle theorem, we deduce that the eigenvalues are located in one of the following disks

D(-1, 1), D(-1 + ρ + , 1 -ρ + ), D(-1 + ρ -, 1 -2ρ + + ρ-).
Since, ρ -< ρ + , none of these disks intersects the half plane of positive real part, except in 0, which gives the stability in L 2 of ρ e (the eigenvalue 0 corresponds to a global translation). We argue that this numerical modification is the macroscopic counterpart of the buffer zone, or, in other words, that it is the residual microscopic effect of the buffer zone at the continuous limit. We can numerically compare the microscopic simulation with the buffer zone and the macroscopic simulation with local modification on one cell whose position corresponds to the discrete position of the buffer zone exit. To do so we start with a macroscopic initial density which is discretized theoretically via the operator (c.f. [START_REF] Colombo | On the Micro-Macro Limit in Traffic Flow[END_REF])

E N : ρ → x N = 2π x i = max x < x i+1 , xi+1 x ρ(y) dy = δ * N , ∀ i ∈ {1, . . . , N -1} . ( 12
)
where δ * N N = M , if the initial density ρ 0 satisfies ρ 0 (x) dx = M . Thanks to this local modification of the numerical flux, we are able to capture the stable traffic jam, equivalent to the one obtained with the microscopic coupled FTL-buffer zone scheme. The numerical results are shown on Figure 6. Stability under perturbations. In order to test the stability of a traffic jam we consider two types of perturbations: first a "physical perturbation" based on microscopic considerations, and then a random one. The physical perturbations that we impose on the downward jam consists in perturbing two neighbour cells by the same amplitude, but with opposite signs. Doing so, we ensure that the perturbed density has the same total mass as before. The density on the buffer cell remains unchanged. Physically, this perturbation expresses the following fact: if a driver reduces his distance to his predecessor, he will also increase the distance to his successor. The numerical results are represented on Figure 7. Both on microscopic and macroscopic simulations we recover after some time the traffic jam. The jam appears to be stable under such perturbations which keep unchanged the density at the buffer cells.

Remark:

The asymptotic behaviour of the solution is very sensitive to the initial density. As represented on Figure 8, if for instance we modify a little bit the value of the density on the buffer cell, the jam may disappear and the solution converges towards a constant state. Now, if we perturb the initial data randomly, stable traffic jams appear and are persistent. On Figure 9 we compare three random initial data with same mass but for different values in the neighbourhood of the buffer cell. We observe that different stable traffic jams are obtained at time T = 200. The jam is then characterized by the jump at the buffer cell. It seems then that even for small random perturbations of a stable traffic jam, we can obtain a different stable solution if the jump just after the buffer cell is modified. Thus, for a fixed total mass, the numerical simulations indicate that there exists a continuum of possible macroscopic traffic jams, characterized by the jump at the buffer cell. They seem stable under perturbations which do not affect the density in the neighbourhood of the buffer cell.

Figure 1 .

 1 Figure 1. Velocity of the FTL model. Parameters: v max = 1 and δ * = 0.1.

Figure 2 .

 2 Figure 2. Relaxation of the FTL model (N = 40 vehicles) which can be seen on the trajectories (on the left: for one out of every two vehicles), or on the evolution of the distances (on the right: for four cars: i = 1, 10, 20 and 40). Parameters: v max = 1 and δ * = 0.04.

Figure 3 .

 3 Figure 3. Trajectories (on the left: for one out of every two vehicles), and evolution of the velocities of four following vehicles (on the right). Parameters: N = 40, v max = 1 and δ * = 0.04.

Figure 4 .

 4 Figure 4. Discrete density ρ N for N = 200 cars, stationary state. Parameters: v max = 1 and δ * = 1/N = 0.005, end of buffer zone at x = 1.5.

Figure 5 .

 5 Figure 5. Initial traffic jam and relaxation towards a constant distribution, v max = 1.

Figure 6 .

 6 Figure 6. Stable jam from macroscopic (top) and microscopic (bottom) point of view. Parameters: N = 96, v max = 1 and δ * = 0.018.

Figure 7 .

 7 Figure 7. Perturbed initial density and convergence towards the stable jam from macroscopic and microscopic point of view. Parameters: N = 96, v max = 1 and δ * = 0.018.

Figure 8 .

 8 Figure 8. Comparison between two initial data differing only by the value on the buffer cell and leading to different stationary states. Top: ρ 0 [i 0 ] = ρ + = 0.7, bottom: ρ 0 [i 0 ] = 0.6.

Figure 9 .

 9 Figure 9. Comparison between random initial data of same mass 0.5

The authors would like to thank F. Lagoutière for his fruitful suggestions. This project is supported by ANR Project Isotace (ANR-12-MONU-0013).