Claire Hotton
email: claire.hotton@sorbonne-universite.fr

Juliette Sirieix-Plénet

Guylaine Ducouret

Thomas Bizien

Alexis Chennevière

Lionel Porcar

Laurent Michot

Natalie Malikova
email: natalie.malikova@sorbonne-universite.fr

Organisation of clay nanoplatelets in a polyelectrolyte-based hydrogel

Keywords: Hydrogel, clay, polyelectrolyte, ionene, SAXS, SANS

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Hydrogels have attracted a lot of attention due to their numerous applications especially in the biomedical field where they provide the basis for stimuli-responsive artificial tissues, for vehicles in drug delivery or for biosensors [START_REF] Zhang | Advances in engineering hydrogels[END_REF][START_REF] Ortony | Self-Assembly of an Optically Active Conjugated Oligoelectrolyte[END_REF][START_REF] Takahashi | Control superstructure of rigid polyelectrolytes in oppositely charged hydrogels via programmed internal stress[END_REF][START_REF] Leong | Patterned prevascularised tissue constructs by assembly of polyelectrolyte hydrogel fibres[END_REF]. The combination of hydrogels and colloidal dispersions opens an even wider window of potential applications [START_REF] Wang | High-water-content mouldable hydro-785 gels by mixing clay and a dendritic molecular binder[END_REF][START_REF] Rose | Nanoparticle solutions as adhesives for gels and biological tissues[END_REF]. More specifically, association of clay nanoparticles and hydrogels has a great potential in the design of materials that exhibit additional mechanical, optical, swelling or barrier properties [START_REF] Zhao | Recent advances in clay mineral-containing nanocomposite hydrogels[END_REF]. Smectites, especially montmorillonite and laponite, a synthetic hectorite, are the most widely used clay additives [START_REF] Shen | Interactions between smectites and polyelectrolytes[END_REF]. Smectites are layered aluminosilicates that exhibit a negative charge on the surface coming from partial isomorphous substitutions in the crystalline structure [START_REF] Bergaya | Chapter 1 General Introduction: Clays, 800 Clay Minerals, and Clay Science[END_REF]. The charge is neutralised by alkali and alkaline earth cations, which become mobile with an increasing water content of the system. This leads to high cation exchangeability, absorption, and swelling properties of clay minerals [START_REF] Zhao | Recent advances in clay mineral-containing nanocomposite hydrogels[END_REF]. Further, clay particles are highly anisotropic (plate-like shape), which leads to a much richer set of possible organisations inside the gel.

The use of a polyelectrolyte compared to a neutral polymer to form nanocomposite hydrogels, leads to a richer 25 set of possible interactions due to the introduction of electrostatics forces in addition to hydrogen bonding, hydrophobic interactions and van der Waals forces [START_REF] Shen | Interactions between smectites and polyelectrolytes[END_REF]. Moreover, the influence of local effects, such as ion-specificity or exact charge locations on the interactions between the clay 30 particles and the polyelectrolyte chains can be critical. These interactions thus determine the structure of claypolyelectrolyte based hydrogels. The polyelectrolytes can wrap around clay platelets, be intercalated inside clay layers, induce flocculation or exfoliation of the platelets [10, 35 11, 12]. A set of key parameters influences this structure including the molecular weight [START_REF] Takeno | Effects of molecular mass of polymer on mechanical properties of clay/poly (ethylene oxide) blend hydrogels, and comparison between them and clay/sodium poly-820 acrylate blend hydrogels[END_REF][START_REF] Takeno | Structural and mechanical properties of composite hydrogels composed of clay and a polyelectrolyte prepared by mixing[END_REF], the chain-charge density and the concentration of the polyelectrolyte [START_REF] Sakhawoth | Aggregation of Plate-like Colloids Induced by Charged Polymer Chains: Organization at the Nanometer Scale Tuned by Polymer Charge Density[END_REF]; the size [START_REF] Takeno | Mechanical Properties and Structures of Clay-Polyelectrolyte Blend Hydrogels[END_REF], the clay concentration [START_REF] Zhao | Recent advances in clay mineral-containing nanocomposite hydrogels[END_REF], and the type [START_REF] Munteanu | The Effect of Clay Type on the Physicochemical Properties of New Hydrogel Clay Nanocomposites[END_REF] of the clay platelets. Nonetheless the set of requirements 40 to obtain one or another structure remains unclear. The need to estimate quantitatively the degree of intercalation and exfoliation in such systems is still challenging in order to establish the link between structure and mechanical, swelling or barrier properties [START_REF] Chen | A critical appraisal of polymer-clay nanocomposites[END_REF]. Moreover, most of these studies have been done with anionic polyelectrolytes or neutral polymers and only a few relate to cationic polyelectrolytes [START_REF] Janek | Interaction of a cationic surfactant with bentonite: a colloid chemistry study[END_REF][START_REF] Alemdar | Interaction between a tertiary amine methacrylate based polyelectrolyte and a sodium montmorillonite dispersion and its rheological and colloidal properties: Sodium Montmorillonite Dispersion[END_REF].

In the current study, we are concerned with a physically cross-linked hydrogel formed with a cationic poly-50 electrolyte, a modified ionene, and different smectite clays (Arizona montmorillonite, Wyoming montmorillonite and beidellite) of specific size fractions and counterions as additives. Ionenes [START_REF] Noguchi | Reactions of n,n,n',n'-tetramethyl-diaminoalkanes with dihaloalkanes[END_REF] are cationic polyelectrolytes composed of charged ammonium quaternary centres on the 55 main chain and separated by methylene groups, Figure 1 (A). Ionenes are strong polyelectrolytes, which means that the charge is pH independent, and their charge density is not statistical, but regular and tuneable by synthesis. Therefore, they are considered as model cationic polyelectrolytes and they have been used in several studies related to antimicrobial agents [START_REF] Strassburg | Cross-Linking of a Hydrophilic, Antimicrobial Polycation toward a Fast-Swelling, Antimicrobial Superabsorber and Interpenetrating Hydrogel Networks with Long Lasting Antimicrobial Properties[END_REF], humidity detectors [START_REF] Erdmenger | Microwave-assisted synthesis of imidazolium ionenes and their application as humidity absorbers[END_REF] or ion-exchange resins [START_REF] Raskop | Polystyrene immobilized ionenes as novel stationary phase for ion chromatography[END_REF]. While conventional ionenes do not form gels, introducing linkers with benzene and amide moieties into the ionene backbone opens the possibility of hydrogelation, where the π-π interactions between benzene groups and the hydrogen bonding between amide groups provide the necessary chain crosslinking [25] [26]. The modified ionene structure is presented in Figure 1 (B). An aqueous solution of the modified ionene needs a thermal treatment to induce gelation. While gelation of modified ionenes has been reported previously [START_REF] Misawa | Hydrogels Based on Surfactant-Free Ionene Polymers with N,N'-(p-Phenylene)dibenzamide Linkages[END_REF], the structural changes in the polyelectrolyte network taking place during the gelation process are shown here for the first time.

Ionenes have already proved their ability to act as a flocculating agent for clay suspensions in a liquid state [START_REF] Sakhawoth | Flocculation of Clay Colloids Induced by Model Polyelectrolytes: Effects of Relative Charge Density and Size[END_REF][START_REF] Sakhawoth | Aggregation of Plate-like Colloids Induced by Charged Polymer Chains: Organization at the Nanometer Scale Tuned by Polymer Charge Density[END_REF]. Introduction of benzene and amide moieties into the ionene backbone can lead to several possible interactions with clay. In this paper, we will focus on a single chain charge density for the modified ionene. The 6-Cl modified ionene (6-Cl mod ionene) is chosen as it displays the best gelation ability [START_REF] Misawa | Hydrogels Based on Surfactant-Free Ionene Polymers with N,N'-(p-Phenylene)dibenzamide Linkages[END_REF]. We follow here the structural changes in the polyelectrolyte network on gel formation and give an experimental evidence of the π-π interactions. Then, the focus turns towards the study of clay nanoplatelet organisation throughout the process of gelation and in the final 6-Cl mod ionene based hydrogel. The 6-Cl mod ionene/clay based hydrogel is studied here at the nanoscale by means of small angle X-Ray scattering 90 (SAXS) and small angle neutron scattering (SANS). We concentrate on the influence of the clay volume fraction, the clay type and clay-compensating counterions on the hydrogel structure as well as its rheological properties. The 6-Cl mod ionene and its precursors were synthesized via a two-step reaction with a protocol adapted from the one described previously [START_REF] Misawa | Hydrogels Based on Surfactant-Free Ionene Polymers with N,N'-(p-Phenylene)dibenzamide Linkages[END_REF]. All chemical reagents were purchased from Sigma-Aldrich and used without fur-100 ther purification. All solvents were purchased from Sigma-Aldrich and VWR.

Experimental section

2.1.1. First step: synthesis of 1,4-bis-[4-(chloromethyl)benzamido]benzene All vials used were dried in a oven at 80 • C the day be-105 fore. p-phenylenediamine (1.08 g, 10 mmol, 1 eq) was dissolved in freshly distilled dichloromethane (100 mL) into a 250 mL round-bottom flask with a magnetic stirrer and placed under an inert argon atmosphere. The solution in the round-bottom flask was then cooled to 0 • C by using an 110 ice-water bath and triethylamine (3.5 mL, 25 mmol, 2.5 eq) was added. A solution of 4-(chloromethyl)benzoyl chloride (3.78 g, 20 mmol, 2 eq) dissolved in distilled dichloromethane (70 mL) was added dropwise with an isobar dropping funnel. After the addition, the ice-water bath was removed, 115 and the system was warmed up to room temperature. The system was stirred for 24h to ensure complete conversion. The obtained white precipitate was vacuum filtrated and then washed with distilled water (2 x 80 mL). The product was dried under vacuum at 90 • C. The product was 120 obtained in high yield (85%). Structural analysis was carried out by 1 H-NMR in DMSO-d 6 .

Second step: synthesis of 6-Cl mod ionene

1,4-bis-[4-(chloromethyl)benzamido]benzene (2.06 g, 5 mmol) was dissolved in dimethylformamide (200 mL) at 125 80 • C in a 500 mL round-bottom flask with a magnetic stirrer. Then, N,N,N',N'-tetramethyl-1,6-hexanediamine (1.07 mL, 5 mmol) was added. The system was stirred at 80 • C for 48h. The obtained yellow precipitate was vacuum filtrated and washed subsequently with dimethyl-130 formamide (100 mL), acetonitrile (100 mL) and dichloromethane (100 mL). The product was dried under vacuum at 50

Clay fraction, purification and cations exchange

Three different clay minerals (two Na-montmorillonites, Arizona (SAz-1) and Wyoming (SWy-2), and a Na-beidellite from Idaho (SBId)) were purchased from the Source Clays Minerals Repository of the Clay Mineral Society at Purdue University, Indiana. Clay suspensions were purified and separated in size fractions by successive centrifugation following the previously established procedure [START_REF] Paineau | Aqueous Suspensions of Natural Swelling Clay Minerals. 1. Structure and Electrostatic Interactions[END_REF][START_REF] Michot | Phase Diagrams of Wyoming Na-Montmorillonite Clay. Influence of Particle Anisotropy[END_REF]. Each fraction has a defined average platelet size with a reduced size polydispersity. Here we used one size fraction of Wyoming montmorillonite and Beidellite platelets (denoted T3) and three size fractions of Arizona montmorillonite (denoted T2, T3 and T4). The characteristic sizes and charge densities of the clay particles we are interested 150 in are summarized in Table 1. Counterion exchange of the SAz1-T3-Na suspension was carried out with aqueous solutions of La(NO 3 ) 3 and CaCl 2 at 0.1 M and 0.5 M respectively. The salt solutions were added to the clay suspension and the mixtures were 155 stirred for 24h. The mixtures were then centrifuged at 7000 rpm for 15 minutes. The supernatant was removed and the flocculent was re-dispersed in salt solutions. These steps were repeated three times. The final suspensions were washed ten times against ultrapure water. The 6-Cl mod ionene gel without clay were formed by heating the ionene solution at 70 • C for 10 minutes followed by cooling to room temperature. The critical gelation concentration (CGC) was determined to be around 165 10 g/L via vial inversion method and rheology measurements. Regarding the formation of the 6-Cl mod ionene gel with clay, the 6-Cl mod ionene was first added into a clay suspension and dispersed by using a sonicator (Fisher Scientific FB 15051) for 15 minutes. Then the mixture was heated at 70 • C for 10 minutes followed by cooling to room temperature to form the hydrogel (see pictures of the hydrogels on Supplementary Material S1). The sonication step was seen as essential for producing visually transparent gels. The CGC remains unchanged by the addition of clay.

Small Angle X-ray Scattering (SAXS)

SAXS experiments were conducted at the SWING beamline of the SOLEIL synchrotron (Orsay, France) using a fixed energy of 16 keV with two samples-to-detector dis-180 tances of 0.5 m and 6 m. Thus the q range available, q being the diffusion vector, is 0.002 Å-1 < q < 2 Å-1 . Cylindrical glass capillaries of 1 mm diameter were filled with the sample in solution state (at 70 • C) and flame sealed. The hydrogels are then formed inside the capillaries by 185 cooling down to room temperature. Two-dimensional scattering patterns were recorded on an Eiger 4M (Dectris) detector. The values of the scattering length density and the contrast of the main components of the system are presented in Table 2. The size of the symbols of the SAXS 190 curves reflects the errors bars. 6-Cl mod ionene SAz1-T3-Na X-ray SLD (cm -2 ) 6.9 10 10 2.12 10 11 ∆ρ 2 (cm -4 ) with H2O 6.33 10 20 1.82 10 22

Small Angle Neutron Scattering (SANS)

SANS experiments under the condition of full contrast (in D 2 O solvent) were conducted at the Laboratoire Léon Brillouin (LLB) in CEA Saclay (France) on the PAXY fa-195 cility. Three different sample-to-detector distances of 1, 3 and 5 m were used together with a neutron wavelength of 4, 5 and 8.5 Å respectively. The q range available was thus 0.006 Å-1 < q < 0.6 Å-1 . SANS measurements were carried out with samples inside quartz cells of path length 200 of 2 mm. The values of the scattering length density and the contrast of the main components of the system are presented in Table 3. SANS measurements performed under the contrast matching condition (in H 2 O/D 2 O solvent mixture) for clay platelets were conducted at the Insti-205 tute Laue-Langevin (ILL) in Grenoble (France) on the D22 spectrometer. Two different sample-to-detector distances of 3 and 17 m were used. The q range available was thus 0.002 Å-1 < q < 0.6 Å-1 . SANS measurements were carried out with samples inside quartz cells of path length of 210 1 mm.

Table 3: Values of the scattering length density (SLD) and of the contrast (∆ρ 2 ) of the main components of the system for SANS.

6-Cl mod ionene SAz1-T3-Na

Neutron SLD (cm -2 )

1.33 10 10 3.6 10 10 ∆ρ 2 (cm -4 ) with D2O 2.51 10 21 6.06 10 20

Rheology

Rheology measurements were carried out on a straincontrolled rheometer, ARES TA instruments (LS1) equipped with a cone-plate geometry of 20 mm in diameter, an angle sweeps were performed in the linear viscoelastic regime at 1% in strain in a pulsation range from 1 to 100 rad/s. The samples were introduced into the geometry one day after the gel formation. The duration of the measurement is less than the sample drying time. Each measurement was repeated twice to ensure reproducibility.

UV-visible absorption spectroscopy

UV-vis spectroscopy measurements were performed using a UVIKON XL Secoman UV spectrometer. The samples were subjected to photon radiation with a wavelength range between 180 nm and 900 nm. The samples are placed between two quartz plates separated by 0.1 mm.

Results and discussion

Formation of the modified ionene based gel

To have a complete overview of the system we investigate, we have first studied the formation and the structure of the 6-Cl mod ionene based gel without clay nanoplatelets. The gel formation consists in heating the ionene solution to 70 • C, a gel is formed after cooling back to room temperature. Heating to 70 • C is an important step leading to good solubilisation of the 6-Cl mod ionene chains. While classical ionenes are highly soluble in water and show the characteristics of hydrophilic chains across a wide range of charge densities [START_REF] Malikova | Aqueous solutions of ionenes: interactions and counterion specific effects as seen by neutron scattering[END_REF][START_REF] Malikova | On the crossroads of current polyelectrolyte theory and counterion-specific effects[END_REF], the introduction of the benzene moieties into the ionene backbone makes the modified ionene chains partially hydrophobic. For this experiment, we used SANS measurements under conditions of full contrast (in D 2 O solvent) because the 6-Cl mod ionene polyelectrolyte have a very good contrast with deuterated water (refer to Table 3). In Figure 2, we present the scattering intensity of the 6-Cl mod ionene at both high temperature (70 • C) and room temperature, i.e., in solution and in the gel states.

In the solution state, we observe the presence of a shoulder resembling the signal of a polyelectrolyte peak from classical ionenes and other polyelectrolytes [START_REF] Malikova | Aqueous solutions of ionenes: interactions and counterion specific effects as seen by neutron scattering[END_REF] in the high q region. The shoulder is not as sharp as the peak we usually observe for polyelectrolyte systems. On the other hand, the scattered intensity cannot be fully described by an Ornstein-Zernike (OZ) function which is the typical behaviour for uncharged polymer solutions [START_REF] Ornstein | Proc. Acad. Sci[END_REF]. The shoulder can be fitted with a Lorentz peak function within the broad peak model, referring to strongly charged polyelectrolytes [START_REF] Morishima | Small-angle scattering study of tetra-poly(acrylic acid) gels[END_REF]:

I(q) = I 0 1 + (q-q * ) B 2 (1)
where q * is the interchain correlation position and B the width of the peak. However, as the peak width is difficult to determined due to the upturn in the low-q region (usually interpreted as the presence of large-scale aggregates or heterogeneities in polyelectrolyte solutions [START_REF] Dubois | Conformation of Poly(styrenesulfonate) 910 Polyions in the Presence of Multivalent Ions: Small-Angle Neutron Scattering Experiments[END_REF]), q * value is not best determined using the Lorentz peak model. Rather, we consider the value of the limit between the q -2 dependence of the curve in the middle q-region and the low-q region. Thus, the shoulder can be evaluated 270 around 0.09 Å-1 . Previous studies on classical ionenes in aqueous solution have already shown a polyelectrolyte peak in this q-region for equivalent molar concentration in charge [START_REF] Malikova | On the crossroads of current polyelectrolyte theory and counterion-specific effects[END_REF]. The typical "mesh-size" ξ = 2π q * of the chain network in solution is determined as 7 nm.

275

It has been shown in previous studies [START_REF] Shibayama | 14 Small Angle Neutron Scattering on Gels[END_REF] that the scattering intensity I tot (q) of polymer gels is the sum of a scattering function I s (q) due to thermal concentration fluctuations and corresponding to the scattering intensity of the polymer solution, and an excess scattering function 280 I d (q) due to heterogeneities introduced by cross-linking:

I tot (q) = I s (q) + I d (q) (2)
In the gel state, we observe in Figure 2 an additional shoulder in the low wave-vector region, around 0.03 Å-1 , correponding to the I d (q) contribution. This contribution is 285 also best described with a Lorentz peak function. The formation of the gel is thus accompanied by the appearance of an additional larger characteristic mesh-size Ξ estimated as 21 nm. This additional mesh-size corresponds to the distance between physical cross-linking nodes of the 290 hydrophobic groups in the 6-Cl mod ionene chains. Moreover, a slight upturn in I(q) at very low q region appears in the gel state. This upturn in I(q) is confirmed by the SAXS signal of the 6-Cl mod ionene, where we can access to the scattered intensity for smaller q values (see the Supporting 295 Information, Figure S2). This type of upturn in the small q-range is characteristic of the heterogeneities introduced by cross-linking. The shoulder corresponding to the I s (q) contribution is weakly visible around 0.1 Å-1 . Hence, in the long wave vector limit the scattering intensity can be 300 reduced to the I s (q) contribution.

It has been evoked in the literature that the gel formation for ionene-based systems is possible thanks to hydrogen bonds and π-π stacking [START_REF] Misawa | Hydrogels Based on Surfactant-Free Ionene Polymers with N,N'-(p-Phenylene)dibenzamide Linkages[END_REF][START_REF] Bachl | Synergistic Computational-Experimental Approach to Improve Ionene Polymer-Based Functional Hydrogels[END_REF]. Hydrogen bonding between amide groups can be evidenced by FT-IR [START_REF] Misawa | Hydrogels Based on Surfactant-Free Ionene Polymers with N,N'-(p-Phenylene)dibenzamide Linkages[END_REF].

However, π-π stacking interactions deserve further clear experimental evidence in this system. UV-visible absorption measurements have been made to follow the evolution of π-π stacking interactions in the 6-Cl mod ionene gel and solution. UV-visible absorption measurements are performed at different concentrations of 6-Cl mod ionene before and after the critical gelation concentration (CGC). The absorbance represented in Figure 3 is not an absolute absorbance due to the difficulty to obtain a constant thickness between the quartz plates from one sample to another. From Figure 3, we note the presence of three absorption maximums corresponding to π-π * transitions. Two absorption maximums are observed at 199 nm and at 233 nm regardless of the concentration of 6-Cl mod ionene which can be attributed respectively to the two E-320 bands (ethylenic). The third absorption maximum is between 299 nm and 318 nm and can be assigned to B-band (benzoic). Before the CGC at 10g/L, the third absorption peak shifts towards higher wavelengths with increasing concentration of the 6-Cl mod ionene. We therefore observe a bathochromic effect. Indeed, when π-π stacking interactions occur in the system, the transition energy from the highest occupied π orbital to the lowest unoccupied π orbital decreases due to overlaps of the π-π molecular orbitals of the aromatic groups. A decrease in the transition energy results in a shift of the λ max of the absorption band towards higher wavelengths [START_REF] Chen | Small Angle Neutron Scattering Studies of the Counterion Effects on the Molecular Conformation and Structure of Charged G4 PAMAM Dendrimers in Aqueous Solutions[END_REF]. After the CGC, the wavelength at which we observed the third absorption peak remains constant. It has been suggested that the hydrogen bonding interaction between the amide 335 groups could be made indirectly through Cl -counterions and water molecules [START_REF] Misawa | Hydrogels Based on Surfactant-Free Ionene Polymers with N,N'-(p-Phenylene)dibenzamide Linkages[END_REF]. Thus, we could suggest that the presence of a shoulder instead of a sharp peak in the SANS signal may be due to the cross-linking attractive interactions as part of the charge could participate to this 340 cross-linking. Moreover anion-π (Cl --π) interactions and cation-π (N + -π) interactions could screen the electrostatic repulsion responsible of the polyelectrolyte peak. In SAXS experiments, the scattered intensity depends on the electron density of the materials analyzed. Thus, in our system, the scattered intensity is mostly due to clay nanoplatelets because of their higher X-ray scattering contrast in H 2 O compared to the one of 6-Cl mod ionene 350 (refer to Table 2). The SAXS experiment therefore makes it possible to study the organisation of clay platelets in the gel. Before describing our findings on the structure of the system, it is meaningful to recall some previous results regarding the components of our system. In very diluted clay 355 suspensions, the scattering patterns present a monotonous decay of the scattering intensity that scales as a q -2 law. This is characteristic of the scattering from non-interacting disks in the intermediate regime where 2π D < q < 2π e , D being the average lateral dimension of the disk and e the 360 thickness [START_REF] Borsali | Soft-Matter Characteriza-920 tion[END_REF]. Thus, the q -2 law represents the form factor of the system and the q 2 I(q) representation can be plotted to emphasise the structural features superimposing onto the q -2 decay. For clay suspensions, depending on the volume fraction and the ionic strength, two major 365 variations of the scattered intensity can be distinguished. The first one is the appearance of a periodic modulation in the middle-q region (from 0.005 Å-1 to 0.1 Å-1 ) due to long or short-range positional order of the platelets. The second one is the appearance of a peak in the high-q 370 region corresponding to distances comparable to the thickness of one or several water layers and reflecting the interlamellar distance within dense face-to-face clay aggregates also called tactoids. In our study, the q 2 I(q) representation shall give access to the local organisation of the clay 375 platelets within the hydrogel as variations in the scattering intensity inform us on characteristic distances within the system. The range of the clay volume fraction considered here is φ = 0.07% to φ = 0.38%. At this volume fractions, pure clay suspensions of montmorillonite or beidellite are 380 isotropic liquids [START_REF] Paineau | Liquid-Crystalline Nematic Phase in Aqueous Suspensions of a Disk-Shaped Natural Beidellite Clay[END_REF] [START_REF] Michot | Liquid-crystalline aqueous clay suspensions[END_REF]. Figure 4 features the SAXS patterns of the 6-Cl mod ionene/SAz1-T3-Na hydrogels at a constant 6-Cl mod ionene concentration and varying the clay volume fraction. In the gel state, i.e. when the 6-Cl mod ionene concentration is higher than the CGC, the 385 SAXS patterns are isotropic and exhibit a periodic modulation of the scattered intensity. This means that the addition of clay platelets into the 6-Cl mod ionene network involves the emergence of a middle-range positional order of the platelets. The average interparticle distance 390 d can be predicted from the positions of the maxima of these oscillations q max , 2 q max , 3 q max , etc, according to d = 2π qmax . Surprisingly, the average interparticle distance is independent of the clay volume fraction. Indeed, at 30g/L of 6-Cl mod ionene, the average interparticle distance d is 18 nm whatever the clay volume fraction. This interparticle distance is much higher than the interlamellar distance of 1.43-1.46 nm in clay/ionene tactoids [START_REF] Sakhawoth | Aggregation of Plate-like Colloids Induced by Charged Polymer Chains: Organization at the Nanometer Scale Tuned by Polymer Charge Density[END_REF] and lower than if the platelets only experienced electrostatic or steric repulsions as in the nematic clay suspensions at volume fractions of the same order of magnitude (around φ = 0.5%) [START_REF] Paineau | Liquid-Crystalline Nematic Phase in Aqueous Suspensions of a Disk-Shaped Natural Beidellite Clay[END_REF]. Another important feature that we need to emphasize is the intensity upturn in the low q region. It is clearly visible in the q 2 I(q) representation. Such evolution suggests the presence of heterogeneities in the distribution of the platelets in the system. Indeed, at the same clay volume fraction, pure clay suspensions present a q 0 dependence in the low q region. Overall, the data suggests a strong face-to-face correlation indicating that platelets inside the hydrogel form stacked discs. These stacks have to be distinguished from tactoids by the interparticle distance which is significantly higher (∼ 20 nm compared to ∼ 1.5 nm in tactoids) and by the fact that in this case, clay platelets are in the osmotic swelling region. It might be of interest to confirm this organisation at larger scales using optical microscopy techniques.

Figure 5 (A) presents the SAXS signals of the 6-Cl mod ionene/SAz1-T3-Na gels at a constant clay volume fraction of φ = 0.15% and at different 6-Cl mod ionene concentrations. Interestingly, at a constant clay volume fraction, the structure of the hydrogel is different depending on the 6-Cl mod ionene concentration. At the CGC, the results reveal that the clay platelets display less pronounced correlation peaks compared to higher 6-Cl mod ionene concentration. When the 6-Cl mod ionene concentration is increased, the correlation peaks are shifted towards high q regions meaning that the average interparticle distance d decreases from 27 nm to 16 nm for 10g/L to 40g/L of 6-Cl mod ionene respectively. Overall, we observe that the 6-Cl mod ionene concentration is the parameter that dictates the local platelet organisation in the system. The distance between clay platelets in a stack is thus mediated by the 6-Cl mod ionene network. Figure 5 (B) shows the SANS curves of the 6-Cl mod ionene pure hydrogel from 10 g/L to 40g/L in ionene. In the pure hydrogel, we observed characteristic mesh sizes between the ionene chains very similar to distances between the clay particles, from 33 nm to 18 nm for 10 g/L to 40 g/L of 6-Cl mod ionene respectively.

Figure 6 compares the evolution of (1) the mesh size between the 6-Cl mod ionene chains in pure hydrogels, (2) the mesh size between the 6-Cl mod ionene chains in hydrogel with clay additives and (3) the average interparticle distance of clay platelets as a function of the 6-Cl mod ionene concentration for a fixed clay volume fraction of φ = 0.15%. We consider the SAXS (Figure S3) and the SANS scattering intensity of the 6-Cl mod ionene in water at different concentrations. Even though the absolute intensities are different between SAXS and SANS due to the difference of contrasts, the overall scattering profiles 450 are the same. It appears that the typical distance Ξ describing the mesh-size of the hydrogel seems to follow a dependence on c -0.47 . Regarding the distance between clay platelets, the exponent of the power law changes slightly to -0.4. Thus the decrease of the typical interparticle dis-455 tance with the increase of the ionene concentration is less important compare to the one of the mesh size of the pure hydrogel. Moreover, the average interparticle distance between platelets in the system is smaller than the mesh size in the pure hydrogel. To provide information on the con-460 formation of the 6-Cl mod ionene network inside the 6-Cl mod ionene/SAz1-T3-Na gels, we have done small-angle neutron scattering (SANS) measurements under the contrast matching condition using a solvent with a 35%:65% H 2 O/D 2 O composition for montmorillonite and 38%:62% 465 H 2 O/D 2 O for beidellite (Figure S4). The characteristic distance between the 6-Cl mod ionene chains within the 6-Cl mod ionene/SAz1-T3-Na gels is lower than the one in the 6-Cl mod ionene gel for 10 g/L (0.017 mol/L) in ionene. From 20g/L (0.033 mol/L), the distance between the 6-Cl mod ionene chains within the 6-Cl mod ionene/SAz1-T3-Na gels and in the 6-Cl mod ionene gels is the same. Overall, only at low ionene concentration does the presence of clay nanoplatelets modify sligthly the final hydrogel mesh size. Thus we can suggest that the clay platelets are part of the hydrogel structure without changing its characteristic mesh size. Due to electrostatic interactions, the lateral surfaces of the clay particles are most likely covered by a layer of polyelectrolyte chains and as such inserted into the hydrogel structure.

made in full contrast meaning that the scattering intensity results from both 6-Cl mod ionene and the clay component. In the solution state at high temperature, before gel formation, the introduction of clay nanoplatelets leads to the appearance of periodic correlation peaks with a first peak around 0.05 Å-1 . This reflects that clay platelets already form stacks in solution at high temperature. In the gel state, this first correlation peak appears in the low q-region around 0.035 Å-1 . The average interparticle distance changes with the gel formation as the average char-495 acteristic distance between platelets increase from 12.5 nm to 18 nm from solution to gel. Thus, when the additional mesh-size of the chain network appears on gel formation (as seen previously in Figure 2 for the 6-Cl mod ionene pure hydrogel), the face to face spacing of platelets in-500 creases. Due to electrostatic interactions, the negatively charged basal surfaces of the clay particles are already covered by the ionene chains in solution. On the formation of gel, these covered particles are inserted into the hydrogel structure. 505

Influence of clay type

The different types of smectite clays investigated here are two montmorillonites (SAz1-T3-Na and SWy2-T3-Na) and one beidellite (SBId-T3-Na). The SBId-T3-Na is a tetrahedrally substituted clay whereas the SAz1-T3-Na 510 and the SWy2-T3-Na, are octahedrally substituted clays. It has already been shown that the charge location is a key parameter in platelet interactions [START_REF] Paineau | Aqueous Suspensions of Natural Swelling Clay Minerals. 1. Structure and Electrostatic Interactions[END_REF]. Indeed, the repulsive potential between individual clay platelets in suspension is sensitive to the exact location of the charge 515 within the individual clay platelets. The tetrahedrally substituted clays are then more repulsive than the octahedrally substituted ones. In the system investigated here, the positional order of SBId-T3-Na platelets differs significantly from the one of SAz1-T3-Na and SWy2-T3-Na. At 520 the CGC (10g/L), for a constant clay volume fraction of φ = 0.15%, it appears that SBId-T3-Na already displays clearly marked periodic correlation peaks whereas with the SAz1-T3-Na, the pattern displays less pronounced correlation peaks as shown in Figure 7 (A). The correlation peaks of the SWy2-T3-Na are much broader than those for SBId-T3-Na. This difference could not be attributed to the difference in platelet lateral dimension between montmorillonite and beidellite platelets as the correlation peaks do not get any sharper with the increase of the platelet lateral dimension for the scattering patterns of hydrogels with SAz1-T2-Na, SAz1-T3-Na and SAz1-T4-Na (Supporting information, Figure S6). When the 6-Cl mod ionene concentration is increased, the difference between the clay type is less pronounced as seen in Figure 535 7 (B). The structure factors of the system with beidellite and Arizona montmorillonite are then roughly the same. However, in the case of Wyoming montmorillonite, the correlation peaks are much broader. It is worth mentioning that the average interparticle distance remains the same in the system whatever the clay type for a given 6-Cl mod ionene concentration higher than the CGC. It appears also that the characteristic distance between 6-Cl mod ionene chains within the 6-Cl mod ionene/clay gels is quite the same for the three clays investigated (Figure S4). As a consequence, the evolution of the average interparticle distance d in the 6-Cl mod ionene/clay hydrogel as a function of the 6-Cl mod ionene concentration and for a fixed clay volume fraction of φ = 0.15% diplayed in the Figure 6 is valid for all clay types investigated.

550

To better describe the structure and the difference in the sharpness of these systems, we use the Stacked Discs model [START_REF] Guinier | Small-angle scattering of x-rays[END_REF][START_REF] Kratky | [END_REF][START_REF] Higgins | Polymers and neutron scattering[END_REF], a clay platelet being assimilated to a disc, and we compare it with the scattering signal from a fully exfoliated monodisperse disc model. The scatter-555 ing length density of the disc and of the water solvent, the size and the thickness of the disc are fixed parameters, with values presented in Table 1 and Table 2. The position of the correlation peak depends on the d-spacing value, so we fix it to 2π qmax = 25 nm. We have thus simulated the 560 scattering profile of the system using this model by varying the number of platelets per stack. We can see from Figure 8 that the number of platelets within the stacks strongly influences the sharpness of the correlation peaks. From 2 to 6 platelets per stack, the peak becomes increas-565 ingly sharper. To fit the experimental curve, we have to combine the fully exfoliated monodisperse disc model with the stacked disc model. To reproduce the scattered intensity of the first peak, a combination of stacks with a different number of platelets per stack is needed. For the 570 6-Cl mod ionene hydrogel at 10g/L with SBId-T3-Na (φ = 0.15%), the best simulated curve is composed of 20% of the stacked dics model with 2 platelets per stack, 10% with 3 platelets per stack, 5% with 4 platelets per stack and of 65% of the fully exfoliated monodisperse disc model 575 as represented by the black curve in Figure 8. This means that the sharpness of the correlation peak is also reflecting the polydispersity of the system. The correlation peaks of the second and the third order are less intense than the first order peak. The simulated curve from the model pre-580 sented cannot reproduce the intensity of the higher order peaks to a satisfactory level. This is partly due to the fact that clay platelets are oriented in the same direction but with a lateral disorder, as in a nematic phase. The hydrogel with beidellite thus leads to a stronger 585 ordering of the platelets and a higher number of platelets within a stack than the hydrogels with the two montmorillonites indicated by the sharp correlation peaks. This stronger ordering of the beidellite platelets have already been observed in clay suspensions [START_REF] Paineau | Aqueous Suspensions of Natural Swelling Clay Minerals. 1. Structure and Electrostatic Interactions[END_REF]. Paineau et al. [START_REF] Paineau | Aqueous Suspensions of Natural Swelling Clay Minerals. 1. Structure and Electrostatic Interactions[END_REF] 590 concluded that the repulsive interactions between the beidellite platelets are stronger than for the other types of clay, inducing a better organisation of the beidellite particles in suspension. In our system, the 6-Cl mod ionene possibly interacts electrostatically more strongly with bei-595 dellite nanoplatelets than with montmorillonite. However, it is more difficult to explain the difference in the structure factors for Arizona montmorillonite and Wyoming montmorillonite because they have similar chemical formulas and the same charge location (octahedral). The difference 600 in the broadening of the peak could not be explained just on the basis of electrostatic considerations. The possi- ble explanation of this phenomenon could be based on the shape of the clay platelets. Indeed, the SWy2-T3-Na could be more flexible than the SAz1-T3-Na and thus lead to a less organised system. Additional investigations would be necessary to confirm this hypothesis.

Rheological properties

The elastic modulus G' and the viscous modulus G" of hydrogels with clay addition is independent of the frequency and the value of G' is always a decade higher than G" as shown in Figure 9 (A). This corresponds to the behaviour of a gel. The variation of the elastic modulus G' of the hydrogels as a function of the clay volume fraction for the three different clays investigated is presented in Figure 9 (B) and Figure 9 (C). Rheological measurements show that depending on the clay type, the introduction of clay nanoparticles either preserves or improves the elastic properties of the gel. Indeed, at 30g/L of 6-Cl mod ionene, the elastic modulus increases up to 45% of the initial value. The increase of the elastic modulus G' is clearly visible for the Wyoming montmorillonite and for the beidellite whereas we do not observe a significative difference with the Arizona montmorillonite. At 10g/L of 6-Cl mod ionene, the elastic modulus increases much more with the addition of platelets than at 30g/L of ionene. Indeed, G' increases from 100% to 300% of the initial value depending on the clay type. The most important increase in G' for the hydrogel is with the addition of beidellite. The lowest increase in G' stays with the addition of Arizona montmorillonite. Thus, at a ionene concentration close to the gelation point, the strongest increase in G' coincides with the most organized clay structure. This suggests that the improvement of the positional order of the platelets leads to the reinforcement of the mechanical properties of the gel. This can be explained by the possibility that the clay particles stiffen the 6-Cl mod ionene network.

Hydrogel as a function of clay counterions

Clay counterion exchange was performed with the SAz1-T3-Na against CaCl 2 and La(NO 3 ) 3 . Whereas the SAz1-640 T3-Na suspension, the SAz1-T3-Ca and the SAz1-T3-La in water are not exfoliated systems but form tactoids and bigger aggregates at the volume fractions investigated. The hydrogel systems containing SAz1-T3-Ca and the SAz1-T3-La provide very different scattering patterns compare 645 Figure 10: (A) Variation of the scattering intensity as a function of the wave vector q of the system 6-Cl mod ionene/SAz1-T3 for Na + , Ca 2+ and La 3+ counterions (30g/L of 6-Cl mod ionene and φ = 0.15%). Dashed straight line in black is a guide corresponding to a q -2 dependence. Continuous line in black is a guide corresponding to a q -2.5 dependence. (B) Evolution of the elastic modulus (G') as a function of SAz1-T3 volume fraction with Ca 2+ and La 3+ counterions for a concentration of 6-Cl mod ionene at 30g/L.

to SAz1-T3-Na as summarized in Figure 10 (A). Indeed, at 30g/L of 6-Cl mod ionene, the slope in the low-q region loses the q -2 dependence and the exponent increases to -2.5 for SAz1-T3-Ca and SAz1-T3-La. The periodic modulations of the scattering intensity in the middle-q region are no longer observed but a correlation peak appears in the high-q region. This correlation indicates clearly the presence of clay tactoids with a small interlamellar distance (1-2 nm at most). The position of the peak maximum corresponds to this distance within the tactoid as d = 2π qmax as 655 seen previously. For both La 3+ and Ca 2+ counterions, the interlamellar distance is estimated around 2.1 nm. Studies on coagulation of montmorillonite platelets by inorganic salt of La(NO 3 ) 3 and Ca(NO 3 ) 2 [START_REF] Michot | Coagulation of Na-Montmorillonite by Inorganic Cations at Neutral pH. A Combined Transmission X-ray Microscopy, Small Angle and 940 Wide Angle X-ray Scattering Study[END_REF] also highlight the presence of a correlation peak corresponding to distances around 1.9 nm. This corresponds to the formation of faceto-face stacks of layers with a repetition distance equivalent to the intercalation of three water layers. However, the slope of the curve for aggregates of montmorillonite platelets induced by those inorganic salt or in SAz1-T3-

665

Ca and the SAz1-T3-La flocculates is much higher than in our system, around -2.8 to -3.

As the hydrogel structure is completely different depending on the counterions, it is interesting to link these features with the corresponding rheological properties. The These observations allow us to conclude that in order to improve the mechanical properties of the gel with the addition of clay particles, the osmotic swelling of clay platelets 685 must be achieved, i.e. the presence of tactoids must be avoided.

Conclusion

We have investigated the organisation of clay nanoplatelets within a modified ionene-based hydrogel. The 690 addition of clay into the three-dimensional network of the hydrogel involves the emergence of a face-to-face positional order of the platelets with a repeat distance of about 20 nm. This feature is evidenced by a clear appearance of a periodic modulation of the scattered intensity. We have first 695 shown that the structure of the clay network is dictated by the 6-Cl mod ionene concentration. The average interparticle distance varies only with the 6-Cl mod ionene concentration and is independent of the clay volume fraction. At first sight, it is surprising that this distance does 700 not depend on the clay volume fraction. This distance does not reflect the distance between individual platelets but the one between two platelets in a stack mediated by the 6-Cl mod ionene network. We have then investigated the influence of the clay type (and thus of the clay charge 705 location) on the structure and properties of the hydrogel. When the charge is located at the surface of the platelet as for beidellite, the 6-Cl mod ionene interacts electrostatically more strongly with clay platelets and it results in a more ordered system. Further, the short-range clay 710 stacking tendency (i.e. tactoid formation) within the gel structure can be tuned via the nature and the valency of clay-compensating cations. With Ca 2+ and La 3+ clay counterions, the clay platelets are inserted in the form of tactoids, with an inter-layer distance comparable to three 715 water layers [START_REF] Sakhawoth | Aggregation of Plate-like Colloids Induced by Charged Polymer Chains: Organization at the Nanometer Scale Tuned by Polymer Charge Density[END_REF]. This distance is the same as for clay tactoids formed in the presence of inorganic salts (Ca(NO 3 ) 2 and La(NO 3 ) 3 ) [START_REF] Michot | Coagulation of Na-Montmorillonite by Inorganic Cations at Neutral pH. A Combined Transmission X-ray Microscopy, Small Angle and 940 Wide Angle X-ray Scattering Study[END_REF].

For the case of the monovalent clay counterions and thus in the absence of tactoids, the introduction of the 720 platelike clay nanoparticles into the hydrogel network enhances their mechanical performance. Moreover, the more the system is ordered, the better the mechanical properties. With multivalent counterions, the addition of clay significantly affects the mechanical performance of the hydrogel. Indeed, the elastic properties of the gel are somewhat inferior on addition of clay platelets in form of tactoids. These results show that the enhancement of the mecanical properties of the hydrogel can be achieved only if the osmotic swelling of clay platelets occurs.

Beyond the nature of the clay counterions, the properties of the ionene based gel can be further tuned via the counterions of the polyelectrolyte chains themselves. Striking counterions specifics effects in aqueous solutions of classical ionene have already been observed [START_REF] Malikova | Aqueous solutions of ionenes: interactions and counterion specific effects as seen by neutron scattering[END_REF] [START_REF] Malikova | On the crossroads of current polyelectrolyte theory and counterion-specific effects[END_REF].

The interaction between individual ionene chains depends dramatically on the nature of the counterion and the trends observed follow the empirical Hofmeister series [START_REF] Salis | Models and mechanisms of Hofmeister effects in electrolyte solutions, and colloid and protein systems revisited[END_REF]. Thus, it may be relevant to investigate the platetelet organisation in the ionene based hydrogel with other ionene compensating counterions. A highly versatile hydrogel system is potentially at hand. experimental work. CH, JSP, LM and NM jointly contributed to the design of the work. JSP, GD, LM and NM supervised the work. All the authors contributed to the manuscript revision and have approved the manuscript.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

awarding the beamtime on beamline SWING. We would like also to thank the Laboratoire Léon Brillouin for awarding the beamtime on PAXY beamline and Mesut Demirelli for his help. We are grateful to the Institut Laue-Langevin for the beamtime allocation on fast access (DOI: 10.5291/ILL-

Figure 1 :

 1 Figure 1: (A) Structure of a classical ionene polymer, x and y represented the number of methylene groups separating the charged quaternary ammonium centers. Charges are compensated by Br - counterions. (B) Structure of an n-Cl modified ionene, n represents the number of methylene groups separating the charged quaternary ammonium centers. In this paper n is fixed at n = 6. Charges are compensated by Cl -counterions.

95 2 . 1 . 6 -

 216 Cl mod ionene and precursor synthesis

160 2 . 3 .

 23 Preparation of the 6-Cl mod ionene/clay hydrogels

Figure 2 :

 2 Figure 2: SANS profiles of 6-Cl mod ionene at 30g/L at 70 • C in solution state and at room temperature (RT) after gelation. The figure shows the coherent SANS intensity versus the wave vector q of the samples in D 2 O.

Figure 3 :

 3 Figure 3: UV-visible measurements of 6-Cl mod ionene in water at different concentrations. The inset features the evolution of the third absorption maximum as a function of the 6-Cl mod ionene concentration.

3. 2 .

 2 Effect of polymer concentration and clay volume fraction on 6-Cl mod ionene/clay hydrogels 345

Figure 4 :

 4 Figure4: (A) Scattered intensity I(q) as a function of q and (B) q 2 I(q) representation of 6-Cl mod ionene/SAz1-T3-Na hydrogels at a fixed ionene concentration of 30g/L and at different volume fractions of SAz1-T3-Na. Dashed straight line in black is a guide corresponding to a q -2 dependence. SAXS signals of the 6-Cl mod ionene gel at 30g/L without clay and of a SAz1-T3-Na suspension at φ = 0.15% are displayed in (A) as references.

Figure 5 :

 5 Figure 5: (A) q 2 I(q) representation of 6-Cl mod ionene/SAz1-T3-Na hydrogels at different 6-Cl mod ionene concentrations and at a fixed volume fraction of SAz1-T3-Na (φ = 0.15%). The curves are shifted along the y-axis for a better visualization. (B) SANS profiles of 6-Cl mod ionene pure hydrogels at different concentrations. The figure shows the coherent SANS intensity versus the wave vector q of the samples in D 2 O.

Figure 6 :

 6 Figure 6: Evolution of the mesh size Ξ of the 6-Cl mod ionene chains in the pure hydrogel and in the 6-Cl mod ionene/clay hydrogel as a function of the 6-Cl mod ionene concentration, for a fixed clay volume fraction of φ = 0.15%. Evolution of the average interparticle distance d in the 6-Cl mod ionene/clay hydrogel as a function of the 6-Cl mod ionene concentration, for a fixed clay volume fraction of φ = 0.15%. Lines indicate a c -0.47 and a c -0.4 dependence of Ξ and d respectively.

Figure 7 :

 7 Figure 7: q 2 I(q) representation of 6-Cl mod ionene hydrogels at 10g/L (A) and 30g/L (B) with addition of SAz1-T3-Na, SBId-T3-Na and SWy2-T3-Na (φ = 0.15%).

Figure 8 :

 8 Figure8: Scattering intensity I(q) as a function of the wave vector q of the 6-Cl mod ionene hydrogel at 10g/L with addition of SBId-T3-Na (φ = 0.15%). The simulated curves are from the stacked disc model with different number of platelets per stack (2, 3, 4 and 6) and the green curve is the simulated curve from the fully exfoliated monodisperse disc model (single disc model).

Figure 9 :

 9 Figure 9: (A) Evolution of the elastic (G') and viscous (G") moduli with oscillation frequency for a 6-Cl mod ionene hydrogel at 30g/L with SAz1-T3-Na at φ = 0.15%. The size of the symbols reflects the errors bars. (B) Evolution of the elastic modulus (G') as a function of clay volume fraction for a concentration of 6-Cl mod ionene at 30g/L. (C) Evolution of the elastic modulus (G') as a function of clay volume fraction for a concentration of 6-Cl mod ionene at 10g/L

670

  variation of the elastic modulus G' with clay volume fraction for SAz1-T3-Ca and SAz1-T3-La is presented in Figure 10 (B). Rheological measurements show that the elastic properties of the gel are slightly weakened on addition of clay platelets. Indeed, the elastic modulus decreases down 675 to 45% of the initial value in the range of clay volume fractions investigated. However, these systems still display the behaviour of a gel. The decrease of the G' implies that the structure of the 6-Cl mod ionene cross-linking network is perturbed by the clay particles when tactoids are present. 680 Thus, the ionene network should have a lower cross-linking density.

  • C. The 6-Cl mod ionene was obtained in good yield (75%). Structural analysis was carried out by 1 H-NMR in D 2 O. The average molecular weight of the 6-Cl mod

	135
	ionene was previously determined as 7.84 10 4 Da [26].

Table 1 :

 1 Main features of different clay particles used in this study: the lateral dimension D, the polydispersity in diameter, the average thickness e and the cation-exchange capacity (CEC).

			Saz-1		SWy-2 SBId
		T2	T3	T4	T3	T3
	D (nm)	150	95	60	100	209
	Polydispersity (%)	42	19	-	25	38
	e (nm)	0.7	0.7	-	0.7	1
	CEC (mmol/100g)	124.8 122.5 124.6	96.4	94.3

Table 2 :

 2 Values of the scattering length density (SLD) and contrast (∆ρ 2 ) of the main components of the system for SAXS.

of 0.04 rad and a gap of 0.043 mm at 25 • C. The frequency

Acknowledgments

The authors acknowledge the SOLEIL synchrotron for

We can also wonder if clay platelets possess such structure only in the gel or also before the gel formation, i.e. in solution at high temperature. SANS measurements of the system as a function of temperature are shown in the Supporting Information, Figure S5. The experiment is

Author contribution

CH drafted the manuscript and conducted the investigation. CH, JSP, GD, TB, AC, LP, LM and NM performed DATA.[86516]) for the contrast matching SANS measurements.

Appendix A. Supplementary material

The following are the Supplementary data to this article: Supplementary Material.pdf


A Tayeb

A Makrem

A Hamdi

A new phenomenological behavior laws for elastomers: logarithmic and classical measure of deformation

Keywords: free energy density, classical measure of deformation, logarithmic measure of deformation, Phenomenological approach

Elastomers (or rubber-like) materials are widely used in different engineering applications such as civil engineering, automotive and aerospace thanks to their high deformability and recoverability after deformation. Their incompressible hyperelastic material behavior is presented by a free energy density which depends only on principal strain invariants. In literature, several approaches have been investigated to determine this free energy density which remains a subject for several researches.

A phenomenological study of biaxial test data in the literature leads to a separation of the free energy density into two functions, each one depends only on one principal strain invariant. This study has been carried out with two different measures of deformation: classical and logarithmic measure of deformation. For the first one, this study was based on Rivlin and Saunders' biaxial experimental data. Then, the capability of the model proposed by Diani on predicting Treloar uniaxial tests is highlighted. For the second one, this study was based on Kawabata's biaxial tests. Thus, a new free energy density has been proposed combined with an identification procedure using several uniaxial tests.

Introduction

Isotropic hyperelastic materials are conveniently characterized by a free energy density per unit undeformed volume w .which is a function of principle stretches or deformation tensor invariants. Adopting the phenomenological point of view, there are two different approaches for the determination of the free energy density: the first approach is to determine the form of the energy using uniaxial tests while the second is to determine the derivatives of the energy with respect to the deformation tensor invariants using biaxial tests. Our purpose here is to analyse the approach proposed by Rivlin and Saunders [START_REF] Rivlin | Large elastic deformations of isotropic materials. Experiments on the deformation of rubber[END_REF] and criticize the model proposed by Diani and co-workers [START_REF] Diani | Christian Rey, New phenomenological behavior laws for rubbers and thermoplastic elastomers[END_REF] in the prediction of the experiments of Treloar [START_REF] Treloar | Stress-strain data for vulcanised rubber under various types of deformation[END_REF] for the classic measure of deformations. Then, we present the same work with the logarithmic measure of deformations using Kawabata [START_REF] Kawabata | Experimental survey of the strain energy density function of isoprene rubber vulcanizate[END_REF] biaxial experiments. Results of these two approaches are highlighted in this text.

Classic measure of deformation

The free energy density is a function of invariants  are the corresponding stresses respectively.

2 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 1 1 2 1 1 2 1 2 2 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 2 2 1 2 1 2 2( )( ) 2( )( ) 1 1 2( )( ) 2( )( ) W I W I                                                   (1)

Rivlin and Saunders work:

Rivlin and Saunders [START_REF] Rivlin | Large elastic deformations of isotropic materials. Experiments on the deformation of rubber[END_REF] did an experiment of biaxial extension for a rubber-like material. They exploited the equation( 1) in order to plot the derivatives of the free energy density against the two invariants. The results are plotted in Figure 1. Analysing this figure one leads to the following conclusions:

model of Diani [2]

Based on these experiments Diani [START_REF] Diani | Christian Rey, New phenomenological behavior laws for rubbers and thermoplastic elastomers[END_REF] proposed a model in which she separates the dependence of the derivatives of the free energy on the invariants i.e only). This model is presented in equation ( 2) in which 1  and 2  are two functions of the principal stretch depending of the type of experiment. This model has a good agreement with Treloar [3] test data for the lower value of the stretch but it can't predict the behaviour of the material for large strain domain i.e( 4

 

). This result is exposed in Figure 2.

1 1 2 2 1 1 2 2 00 ( ) ( ) ( ) ( ) ( ) exp ( 3) ( ) exp (ln( )) nm ii ii ii f I g I f I a I g I b I                         (2)
Figure 2: prediction of Treloar [START_REF] Treloar | Stress-strain data for vulcanised rubber under various types of deformation[END_REF] experimental data by the model of Diani [START_REF] Diani | Christian Rey, New phenomenological behavior laws for rubbers and thermoplastic elastomers[END_REF]: uniaxial extension

Logarithmic measure of deformation:

The measure of deformation in this section is the logarithm of the right Cauchy Green tensor i.e

( 1 ln( ) 2 qC 

). This strain tensor has two invariants in the incompressible case: 2 K and 3 K . The derivatives of the free energy density with respect to these two invariants are exposed in equation [START_REF] Treloar | Stress-strain data for vulcanised rubber under various types of deformation[END_REF].

                    1 1 2 3 2 2 1 2 2 2 3 1 3 2 3 1 3 2 1 1 2 3 2 2 1 3 3 2 3 1 3 2 3 1 3 . . Y Y Y Y W K Y Y Y Y K W K Y Y Y Y                                             (3)
Exploiting the biaxial experiments of Kawabata [START_REF] Kawabata | Experimental survey of the strain energy density function of isoprene rubber vulcanizate[END_REF] we plotted the derivatives of the energy against the two invariants; the curves are exposed in Figure 3. Thus, we postulate the formula of the free energy which is a function of the two invariants 2 K and 3 K (see equation ( 4))

 

2 3 2 3 2 , ( ). ( ) W K K f K K g K  (4)
Figure 3: derivatives of the energy against the logarithmic invariants.

Conclusion

The aim of this work is to compare two different approaches in the determination of the free energy density for isotropic hyperelastic materials. For the classical measure of deformation the model proposed by Diani [START_REF] Diani | Christian Rey, New phenomenological behavior laws for rubbers and thermoplastic elastomers[END_REF] was criticised, and for the logarithmic measure a new form of the free energy was motivated using Kawabata [START_REF] Kawabata | Experimental survey of the strain energy density function of isoprene rubber vulcanizate[END_REF] biaxial experiments

1 I and 2 I 1  and 2 

 1212 of the right Cauchy Green tensor C . Thus, for an incompressible hyperelastic material undergoing biaxial extension the derivatives of the energy density with respect to the invariants take the form in the equation[START_REF] Rivlin | Large elastic deformations of isotropic materials. Experiments on the deformation of rubber[END_REF]. In this equation denote the principal stretches whereas 1  and 2

  with respect to two invariants 1 I and 2 I .

Figure 1 :

 1 Figure 1: derivatives of the energy against the invariants.


Adel Tayeb

Makrem Arfaoui

Abdelmalek Zine

Adel Hamdi

Jalel Benabdallah

Mohamed Ichchou

Tayeb Adel

Arfaoui Makrem

Zine Abdelmalek

Hamdi Adel

Benabdallah Jalel

Ichchou Mohamed

Identification of nonlinear viscoelastic model

Keywords: modelling, identification, linear least square, reduced time

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

The behaviour of rubber-like materials is known to be hyperviscoelastic at different deformation levels. In this paper we carry out a systematic identification procedure of fitting an incompressible hyperviscoelastic constitutive law [START_REF] Adel | Mohamed: modeling and identification of a class of hyperviscoelastic material behaviour[END_REF] to experimental data. To this end we use an experimental characterization performed with Bromobutyl rubber (BIIR) [START_REF] Jridi | Predeformation and frequency-dependence Experiment and FE analysis[END_REF] and a linear and nonlinear least square optimization tool in Matlab and Abaqus. Thus, a numerical discretization of the stress-strain integral relation was performed and validated with homogeneous transformation tests. Shear relaxation tests were used to identify the relaxation shear module, tests of simple shear and simple extension at equilibrium were exploited to identify the hyperelastic part of the behaviour whereas monotonic tests were utilized to identify the strain shift function. Results of the identification are highlighted and capacity of the model to predict the behaviour of the material of interest is discussed.

Hyperviscoelastic model

In figure 1 is plotted the shear relaxation module for two different levels of deformation: 10% and 50% . The material of interest shows a significant dependence of the relaxation shear module upon deformation level. Motivated by this fact, we developed a nonlinear viscoelastic model taking into account the dependence upon the deformation via a deformation shift function and thermodynamic principles [START_REF] Adel | Mohamed: modeling and identification of a class of hyperviscoelastic material behaviour[END_REF]. In what follows we expose a systematic procedure to identify these three functions.

Identification procedure

The identification procedure contains three main steps:  Identification of the shear relaxation module G using shear relaxation tests  Identification of the hyperelastic potential 0  using equilibrium curves for pure shear and simple extension experiments  Identification of reduced time function  using monotonic tests of simple extension and the discretized stress-strain relation.

Identification of the shear relaxation module

In literature there are several decay functions to describe the shear relaxation module. In this work we adopted the decay exponential functions (equation ( 2)). For a nonlinear viscoelastic model it's more convenient to identify the relaxation function at low level of deformation [seddiq]. Thus, the shear relaxation at 10% and the identification module of Abaqus software are used to fit the relaxation module.

  0 1 1 exp i i i t G t G g                  (2)
In expression (2) 0 G is the initial shear module whereas i g and i  are the coefficients of the Prony series, these coefficients are reported in figure 1. Coefficients

Identification of the hyperelastic potential

The identification of hyperelastic potential relies on two main steps which are the choice of the free energy density to use in the first place and the check of the stability of the potential in the second place. Whether the hyperelastic potential is linear or nonlinear in terms of its parameters a linear or nonlinear least square optimization is used respectively. In this work we adopted the Mooney-Rivlin [START_REF] Rivlin | Large elastic deformations of isotropic materials. Experiments on the deformation of rubber[END_REF] polynomial free energy density for an isotropic incompressible material (equation ( 3)). In

ij ij c i j          (4)
In the above expression (equation ( 4)

)   ,, ij
 is a nonlinear function of  , for a simple extension experiment it takes the following form

  1 2 2 1 2 2 2 1 1 , , 2 3 2 3 2 1 1 2 3 2 3 ij ij i j i j                                                          (5)
In order to obtain more useful form of equation ( 4) with respect to the identification procedure, it's possible to express the material's parameters ij c with the vector [START_REF] Ciambella | Experimental Testing and Nonlinear Viscoelastic Modeling of Filled Rubber[END_REF] In light of expression (7), the identification process reduces to the minimization of expression (8),  is the vector of experimental nominal stresses. The condition of stability is expressed in terms of empirical inequalities (9). Hence, we use a constrained linear least square minimization via "lsqlin" tools in Matlab software. In figure 2 and3 are plotted the relative error of the Cauchy stress for simple extension and pure shear respectively.

Identification of the reduced time function

The identification of the reduced time function is obtained thanks to the discretization of the behaviour law equation ( 1), thus, at each experimental point k t the minimization of the Cauchy stress error between the model and the and the recoded stress gives the corresponding reduced time k  . The experiment used here is the monotonic simple extension test at different strain rates combined with a dichotomy algorithm implemented in Matlab software. As a result we got the reduced time as a function of real time, the nominal stress error for the three strain rates 10%,100% and 200% are reported in figure 4, 5 and 6 respectively.

Figure 1 :

 1 Figure 1: dependence of the relaxation module upon deformation level

  this expression ij c are the parameters to be identified, 1 I and 2 I are the first and the second invariant of the right Cauchy-Green strain tensor C. of such model is expressed as follow in term of the principle stretch  :

  ...., , ,........, , ......... , k is the number of experimental points, hence we have the following expression for the discrete nominal stress c

Figure 2 :

 2 Figure 2: Cauchy stress relative error for simple extension

Figure 3 :

 3 Figure 3: Cauchy stress relative error for pure shear

Figure 4 :

 4 Figure 4: Cauchy stress relative error for simple extension at 1 10% mn 

Figure 5 :

 5 Figure 5: Cauchy stress relative error for simple extension at 1 100% mn 

Figure 6 : 1 200%

 61 Figure 6: Cauchy stress relative error for simple extension at 1 200% mn 

Table 1 :

 1 Prony series coefficientsUsing the values in table 1 one could simply calculate the relative error between experimental and fitted shear relaxation module, this error is about 0.001% .


Matthieu Py
email: matthieu.py@univ-amu.fr

Mohamed Sami Cherif
email: mohamed-sami.cherif@univ-amu.fr

Djamal Habet
email: djamal.habet@univ-amu.fr

A Proof Builder for Max-SAT

Keywords: Max-SAT, Proof, Max-SAT Resolution

Complete Max-SAT solvers are able to return the optimal value of an input instance but they do not provide any certificate of its validity. In this paper, we introduce for the first time a Max-SAT proof builder, called MS-Builder, which generates Max-SAT proofs under the particular form of a sequence of Max-SAT equivalence-preserving transformations. To generate a Max-SAT proof, MS-Builder iteratively calls a SAT oracle to get a SAT refutation which is handled and adapted into a sound refutation for Max-SAT. We also propose an extendable tool, called MS-Checker, able to verify the validity of any proof using Max-SAT inference rules.

Introduction

Given a Boolean formula in Conjunctive Normal Form (CNF), the Maximum Satisfiability (Max-SAT) problem consists in determining the maximum number of clauses that it is possible to satisfy by an assignment of the variables. Max-SAT is an optimization extension of the Satisfiability (SAT) problem and a natural way to model many real world and crafted problems [START_REF] Xu | sub-SAT: a formulation for relaxed Boolean satisfiability with applications in routing[END_REF][START_REF] Guerra | Reasoning over Biological Networks Using Maximum Satisfiability[END_REF][START_REF] D'almeida | Model-based diagnosis with default information implemented through MAX-SAT technology[END_REF] making it a well studied problem in theory as well as in practice. Different complete solving paradigms for Max-SAT have seen the day in recent years including Branch and Bound algorithms [START_REF] Abramé | Ahmaxsat: Description and Evaluation of a Branch and Bound Max-SAT Solver[END_REF][START_REF] Li | New inference rules for Max-SAT[END_REF][START_REF] Küegel | Improved exact solver for the weighted max-sat problem[END_REF], SAT-based algorithms [START_REF] Ignatiev | RC2: an efficient maxsat solver[END_REF][START_REF] Martins | Open-WBO: A Modular MaxSAT Solver[END_REF][START_REF] Narodytska | Maximum Satisfiability Using Core-Guided MaxSAT Resolution[END_REF] and reduction to other problems (such as ILP [START_REF] Davies | Solving MAXSAT by Solving a Sequence of Simpler SAT Instances[END_REF], Max-ASP [START_REF] Andres | Unsatisfiability-based optimization in clasp[END_REF] and WCSP [START_REF] De Givry | Solving max-sat as weighted csp[END_REF]).

Recent years have also witnessed a particular interest in proof systems for Max-SAT [START_REF] Bonet | A complete calculus for MAX-SAT[END_REF][START_REF] Bonet | Resolution for Max-SAT[END_REF][START_REF] Larrosa | Resolution in Max-SAT and its relation to local consistency in weighted CSPs[END_REF][START_REF] Larrosa | Augmenting the Power of (Partial) MaxSat Resolution with Extension[END_REF][START_REF] Larrosa | Towards a better understanding of (partial weighted) maxsat proof systems[END_REF][START_REF] Bonet | Equivalence Between Systems Stronger Than Resolution[END_REF][START_REF] Py | Towards bridging the gap between sat and max-sat refutations[END_REF]. In particular, Max-SAT resolution [START_REF] Bonet | A complete calculus for MAX-SAT[END_REF][START_REF] Bonet | Resolution for Max-SAT[END_REF][START_REF] Larrosa | Resolution in Max-SAT and its relation to local consistency in weighted CSPs[END_REF] was one of the first known complete systems for Max-SAT and was later extensively used in the context of Max-SAT solving [START_REF] Li | New inference rules for Max-SAT[END_REF][START_REF] Abramé | Ahmaxsat: Description and Evaluation of a Branch and Bound Max-SAT Solver[END_REF][START_REF] Narodytska | Maximum Satisfiability Using Core-Guided MaxSAT Resolution[END_REF]. However, generating proofs establishing the optimum cost of Max-SAT formulas remains an unexplored topic in practice. Indeed, current Max-SAT solvers are not able to output certificates as it is the case for SAT solvers. This is in part due to the variety of paradigms and techniques for Max-SAT solving which make it difficult to devise a generalized approach to compute certificates.

In this paper, we devise an independent proof builder for Max-SAT, called MS-Builder, which builds proofs for Max-SAT by iteratively calling a SAT oracle to get a resolution refutation. The builder relies on recent work [START_REF] Py | Towards bridging the gap between sat and max-sat refutations[END_REF] to adapt the SAT refutation into a Max-SAT refutation which is then applied to the current formula. Moreover, we introduce an extendable Max-SAT proof checker, called MS-Checker, to verify the validity of any proof using Max-SAT inference rules. Both tools are experimentally evaluated on the unweighted partial benchmark of the 2020 Max-SAT Evaluation [START_REF] Bacchus | MaxSAT Evaluation[END_REF].

This paper is organized as follows. Section 2 includes some necessary definitions and notations. Section 3 recalls related work on the adaptation of SAT refutations to Max-SAT refutations. MS-Builder and MS-Checker are respectively presented in Sections 4 and 5 and their experimental evaluation is detailed in Section 6. Finally, we conclude and discuss future work in Section 7.

Preliminaries

Definitions and Notations

Let X be a set of propositional variables. A literal l is a variable x ∈ X or its negation

x. A clause c is a disjunction of literals (l 1 ∨ l 2 ∨ • • • ∨ l k ). A formula φ in Conjunctive Normal Form (CNF) is a conjunction of clauses φ = c 1 ∧c 2 ∧• • •∧c m .
An assignment I : X -→ {true, f alse} maps each variable to a boolean value and can be represented as a set of literals. A literal l is satisfied (resp. falsified) by I if l ∈ I (resp. l ∈ I). A clause c is satisfied by I if at least one of its literals is satisfied by I, otherwise it is falsified by I. The empty clause contains zero literals and is always falsified. A clause c opposes a clause c if c contains a literal whose negation is in c , i.e. ∃l ∈ c, l ∈ c . For a given CNF formula, solving the Satisfiability (SAT) problem consists in determining whether there exists an assignment I (called model) that satisfies it. The cost of an assignment I is the number of clauses falsified by I. For a given CNF formula φ, solving the (plain) Max-SAT problem consists in determining the maximum number of satisfied clauses in φ.

Resolution Refutations in SAT

To certify that a CNF formula is satisfiable, it is sufficient to exhibit a model of the formula. On the other hand, to prove that a CNF formula is unsatisfiable, we need to refute the existence of a model. A well-known SAT refutation system is based on an inference rule for SAT called resolution [START_REF] Robinson | A Machine-Oriented Logic Based on the Resolution Principle[END_REF]. The resolution rule, defined below, deduces a clause called resolvent which can be added to the formula from two opposed clauses.

Definition 1 (Resolution [START_REF] Robinson | A Machine-Oriented Logic Based on the Resolution Principle[END_REF]). Given two clauses c 1 = (x ∨ A) and c 2 = (x ∨ B), the resolution rule is defined as follows:

c 1 = (x ∨ A) c 2 = (x ∨ B) c 3 = (A ∨ B)
A resolution refutation is a sequence of resolutions leading to an empty clause. Many restricted classes of resolution refutations have been studied in the literature namely linear resolution [START_REF] Loveland | A linear format for resolution[END_REF], unit resolution [START_REF] Hertel | Algorithms and complexity results for input and unit resolution[END_REF], input resolution [START_REF] Hertel | Algorithms and complexity results for input and unit resolution[END_REF],

regular resolution [START_REF] Urquhart | A near-optimal separation of regular and general resolution[END_REF], read-once resolution [START_REF] Iwama | Intractability of read-once resolution[END_REF] and tree (or tree-like) resolution refutations [START_REF] Ben-Sasson | Near optimal separation of treelike and general resolution[END_REF] among others. In particular, a resolution refutation is tree-like if every intermediate clause, i.e. resolvent, is used at most once in the proof. Similarly, a resolution refutation is read-once if each clause is used at most once in the proof. Clearly, read-once resolution refutations are also tree-like since they form a restricted class of tree resolution refutations. It was shown in [START_REF] Iwama | Intractability of read-once resolution[END_REF] that there exists unsatisfiable CNF formulas which cannot be refuted using read-once resolution. Finally, a resolution is regular if each branch (path from a clause of the initial formula to the empty clause) contains at most one resolution per variable.

Example 1. We consider the CNF formula φ

= (x 1 ∨ x 3 ) ∧ (x 1 ) ∧ (x 1 ∨ x 2 ) ∧ (x 2 ∨
x 3 ). The resolution refutation of φ, represented in Figure 1, is tree-like (and) regular, but not read-once because of clause (x 1 ).

x 1 ∨ x 3 x 1 x 1 ∨ x 2 x 2 ∨ x 3 x 3 x 2 x 3
Fig. 1. Resolution refutation

Proofs for Max-SAT

In the last fifteen years, the study of inference rules for Max-SAT has led to major results in Max-SAT theory and solving. In particular, one of the first proof systems for Max-SAT is based on an inference rule called Max-SAT resolution, which is an extension of the resolution rule. Max-SAT resolution was shown sound and complete for Max-SAT, i.e. it is sufficient to prove the optimum cost of a given CNF formula.

Definition 2 (Max-SAT resolution [START_REF] Bonet | A complete calculus for MAX-SAT[END_REF][START_REF] Bonet | Resolution for Max-SAT[END_REF][START_REF] Larrosa | Resolution in Max-SAT and its relation to local consistency in weighted CSPs[END_REF]). Given two clauses

c 1 = x∨a 1 ∨• • •∨a s and c 2 = x∨b 1 ∨• • •∨b t with A = a 1 ∨• • •∨a s and B = b 1 ∨• • •∨b t ,
the Max-SAT resolution rule is defined as follows:

c 1 = x ∨ A c 2 = x ∨ B c 3 = A ∨ B cc 1 = x ∨ A ∨ b 1 . . . cc t = x ∨ A ∨ b 1 ∨ • • • ∨ b t-1 ∨ b t cc t+1 = x ∨ B ∨ a 1 . . . cc t+s = x ∨ B ∨ a 1 ∨ • • • ∨ a s-1 ∨ a s
where c 3 is the resolvent clause and cc 1 , . . . , cc t+s are compensation clauses.

In recent work, Max-SAT resolution was augmented with other rules such as the split rule [START_REF] Larrosa | Towards a better understanding of (partial weighted) maxsat proof systems[END_REF][START_REF] Py | Towards bridging the gap between sat and max-sat refutations[END_REF] defined below or the extension rule [START_REF] Larrosa | Augmenting the Power of (Partial) MaxSat Resolution with Extension[END_REF]. It was shown that the addition of such rules to Max-SAT resolution can improve its efficiency in generating shorter proofs [START_REF] Larrosa | Towards a better understanding of (partial weighted) maxsat proof systems[END_REF][START_REF] Larrosa | Augmenting the Power of (Partial) MaxSat Resolution with Extension[END_REF] or allow, given a resolution refutation for SAT, to generate a Max-SAT resolution refutation [START_REF] Py | Towards bridging the gap between sat and max-sat refutations[END_REF].

Definition 3 (Split rule). Given a clause c 1 = (A) where A is a disjunction of literals and x a variable, the split rule is defined as follows:

c 1 = (A) c 2 = (x ∨ A) c 3 = (x ∨ A)
Remark 1. Unlike the resolution rule, the Max-SAT resolution rule and the split rule replace the premise(s) by the conclusion(s).

To be more exhaustive, we must also mention that other Max-SAT proof systems exist like the Clause Tableau Calculus [START_REF] Li | A Clause Tableau Calculus for MaxSAT[END_REF]. If these proofs systems have been extensively studied in theory, generating proofs remains an unexplored topic in practice. Hence, this work aims to contribute to this topic by proposing tools to build and check Max-SAT proofs.

Related work

In this section, we briefly recall recent results established in [START_REF] Py | Towards bridging the gap between sat and max-sat refutations[END_REF] on the adaptation of resolution refutations to Max-SAT refutations. One of the main results deals with tree resolution refutations showing that a linear adaptation to a Max-SAT refutation is possible in this case. Indeed, if the resolution refutation is tree-like, it is possible to transform it into a smaller refutation which is tree-like regular by iteratively eliminating irregularities (sequences of successive resolutions whose first and last are on the same variable) [START_REF] Urquhart | The complexity of propositional proofs[END_REF]. To adapt tree regular refutations, Max-SAT resolution is augmented with the split rule to fix the non-read-once clauses, i.e. clauses which are used more than once in the proof. The split rule is applied on a non-read-once clause to augment it with the variable resolved on in the junction point of all the branches starting from it. Thus, the obtained clauses can replace the non-read-once clause as a premise without affecting the validity of the proof. The same treatment is applied until all non-read-once clauses are fixed. Finally, when the proof becomes read-once, it is sufficient to replace each resolution step by a Max-SAT resolution to get a valid Max-SAT refutation [START_REF] Iwama | Intractability of read-once resolution[END_REF]. An adaptation of tree regular resolution refutation is showcased in Example 2.

Example 2. We consider the tree regular resolution refutation in Example 1, represented in Figure 1. We observe that the clause (x 1 ) is used two times as a premise of a resolution step. The junction point of the left and right branches eliminates variable x 3 . Thus, we apply the split rule on clause (x 1 ) to get (x 1 ∨x 3 ) and (x 1 ∨x 3 ) and we replace (x 1 ) by (x 1 ∨x 3 ) and (x 1 ∨x 3 ) respectively on the left and right branches. Finally, we replace all resolutions by Max-SAT resolutions to obtain the complete Max-SAT refutation represented in Figure 2.

x 1 x 1 ∨ x 3 x 1 ∨ x 3 x 1 ∨ x 3 x 1 ∨ x 2 x 2 ∨ x 3 x 3 x 2 ∨ x 3 x 1 ∨ x 2 ∨ x 3 x 1 ∨ x 2 ∨ x 3 x 3

Fig. 2. Adaptation of a tree-like regular resolution refutation

In the generic case where the resolution refutation is not tree-like, it is also possible to adapt it into a Max-SAT refutation but with an exponential cost. To adapt the refutation, the entire proof leading to each non-read-once intermediate clause is duplicated, thus generating as many copies of the clauses as needed to render the proof tree-like. Then, the tree refutation is adapted into a Max-SAT refutation as explained above.

MS-Builder

In this section, we describe a Max-SAT proof builder, called MS-Builder based on the adaptation of resolution refutations for Max-SAT recalled in Section 3. The idea is to iteratively call a SAT oracle in order to get a resolution refutation for the current formula, adapt it into a Max-SAT refutation and apply it to the formula. The proof builder repeats this step until the SAT oracle returns a model for the final formula.

For practical reasons, we add an additional treatment to the Max-SAT proof builder. If the resolution refutation is not read-once, we first try to fix the effect of unit propagation in the non-read-once part of the proof. To do that, we discard the non-read-once unit clauses and we re-inject them at the end of the resolution refutation. Indeed, some resolution refutations are non-read-once simply because of the effects of unit propagation. It is the case of the resolution refutation proposed in Example 1, which is in fact read-once after fixing the unit propagation as showcased in the following example.

Example 3. We consider the tree regular resolution refutation in Example 1 (represented in Figure 1). We fix the unit propagation by discarding (x 1 ) and reinjecting it at the end of the resolution refutation to get the read-once resolution refutation represented in Figure 3.

x 1 x 1 ∨ x 3 x 1 ∨ x 2 x 2 ∨ x 3 x 1 ∨ x 3 x 1

Fig. 3. Fixing unit propagation to get a read-once resolution refutation

The adaptation of any resolution refutation into a Max-SAT refutation is integrated into MS-Builder which can be seen as a core-based Max-SAT proof builder whose correctness is guaranteed by the correctness of the adaptation proposed in [START_REF] Py | Towards bridging the gap between sat and max-sat refutations[END_REF].

Example 4. Let φ = (x 1 ∨ x 3 ) ∧ (x 1 ) ∧ (x 1 ∨ x 2 ) ∧ (x 2 ∨ x 3
). The SAT oracle returns the resolution refutation in Figure which is adapted into the read-once one in Figure 3 and to a Max-SAT one (by replacing resolutions by Max-SAT resolutions) and then applied to the formula. Now we have φ

= ∧ (x 1 ∨ x 2 ∨ x 3 ) ∧ (x 1 ∨ x 2 ∨ x 3
) and the SAT oracle returns that the current formula (without considering the empty clause) is satisfied by the assignment I(x 1 ) = I(x 2 ) = I(x 3 ) = 0. MS-Builder thus returns the proof described in Figure 5.

MS-Checker

In this section, we present our extendable Max-SAT checker, called MS-Checker, which requires two input files: a formula and a proof. The formula has to be given in the standard WCNF format, either in the old or new format [START_REF] Bacchus | MaxSAT Evaluation[END_REF]. The proof file must start with a sequence of Max-SAT transformation lines. A Max-SAT transformation line must start with 't' and must include the name of the inference rule (msres for Max-SAT resolution and split for the split rule) and its premise(s) (between '< >'). For the split rule, the variable to split on is specified as a parameter after its name. Then, the proof file must contain a line (starting with 'o') with the announced optimum cost of the formula. Finally, it must contain a line (starting with 'v') with a truth assignment satisfying the final formula (without the empty clauses). After reading the formula, MS-Checker verifies that the proposed inference rules are correct and that the premises are still in the formula then applies the transformation. Finally, it checks if the truth assignment satisfies the final formula without considering the empty clauses.

c Formula of Example 1 1 -1 3 0 1 1 0 1 -1 2 0 1 -2 -3 0 Fig. 4. Formula file format t msres < 1 -1 -2 | 1 -2 -3 > t msres < 1 -1 3 | 1 -1 -3 > t msres < 1 1 | 1 -1 > o 1 v 000

Experiments

We have implemented MS-Builder and MS-Checker in C++1 . Resolution Refutations are computed using Booleforce [START_REF] Biere | Booleforce[END_REF] and Tracecheck [START_REF] Biere | TraceCheck[END_REF][START_REF] Biere | PicoSAT Essentials[END_REF]. We consider the benchmark of the unweighted partiel track of the 2020 Max-SAT Evaluation [START_REF] Bacchus | MaxSAT Evaluation[END_REF]. The experiments are performed on Dell PowerEdge M620 servers with Intel XeonSilver E5-2609 processors (clocked at 2.5 2.6 GHz) under Ubuntu 18.04. Each solving process is allocated a slot of 1 hour and at most GB of memory per instance.

MS-Builder has succeeded to construct full proofs for 163 instance while MSchecker has succeeded to check 575 complete or partial proofs over 576 in total. The running time for building and checking instances are plotted respectively in Figures 6 and7. Proof checking is obviously much easier than proof building except on rare formulas with an important number of clauses (in the input file or after applying some transformations) which can make difficult the linear operation of extracting a premise to the formula used in MS-Checker. MS-Builder has also succeed to build at least half of the proofs (with respect to the number of empty clauses) of 302 instances over 463 instances for which the optimum cost is known. This is illustrated in Figure 8 which reports the percentage of empty clauses built per solved instance. The sizes of the computed proofs vary from few bytes to 1 Gigabyte as illustrated in Figure 9. Notice how empty (incomplete) proofs are computed for some very hard instances for which the timeout is not sufficient to even compute the first Max-SAT refutation. On the other hand, there are some instances, usually with an optimum cost of 1, which have very small proofs. Finally, we can observe in Table 1 that read-once resolution refutations and resolution refutations which are read-once after fixing unit propagation appear very often. However, there are many instances such that the last resolution refutation met are the largest and the hardest (i.e. unrestricted) and that is why the last resolution refutations are often the most difficult to adapt and the timeout often stops on these resolution refutations.

Conclusion

In this paper, we proposed two tools, MS-Builder and MS-Checker, to respectively generate and check Max-SAT proofs. MS-Builder builds proofs by iteratively calling a SAT oracle and adapting the obtained SAT refutations into Max-SAT refutations. MS-Builder has succeeded in building a substantial amount of proofs for unweighted partial instances of the 2020 Max-SAT Evaluation. However, unrestricted resolution refutations are usually hard to adapt due to the exponential overhead caused by duplicating parts of the proofs.

As future work, it would be interesting to include more advanced techniques such as core reduction or minimization [START_REF] Ignatiev | RC2: an efficient maxsat solver[END_REF][START_REF] Marques-Silva | Minimal unsatisfiability: Models, algorithms and applications[END_REF] in order to improve the efficiency of these tools. Furthermore, It would be relevant to study the possibility of extending the UP-fixing mechanism to non-unit clauses. Finally, we are also working on extending our tools to build and check proofs for weighted partial Max-SAT formulas.

Fig. 5 .

 5 Fig. 5. Proof file format

Fig. 6 .

 6 Fig. 6. Running time (in seconds) for building complete proofs

Fig. 7 .

 7 Fig. 7. Running time (in seconds) for checking proof

Fig. 8 .Fig. 9 .

 89 Fig. 8. Percentage of proved per instance

Table 1 .

 1 Encountered types of resolution refutations in the whole benchmark

	Type of resolution refutation Number Percentage
	read-once	169,239 83.7 %
	read-once after UP-fixing	24,556 12.1 %
	tree-like regular	2,879 1.4 %
	tree-like	1,795 0.9 %
	unrestricted	3,799 1.9 %

The source code is available on https://pageperso.lis-lab.fr/matthieu.py/en/software.html.