Rubens Rodriguez

Ana Paula Batista

Cristina Silva Pereira

Didier Marion

Mathilde Marc

Regis Risani

Eric Desnoes

Xavier Falourd

Bruno Pontoire

Rúben Rodrigues

Rita Escórcio

Ana Paula Batista

Romain Valentin

Nathalie Gontard

Christelle Lopez

Eric Leroy

Denis Lourdin

Bénédicte Bakan
email: benedicte.bakan@inrae.fr

Co-Polyesters of Hydroxy-Fatty Acids Extracted from Tomato Peel Agro-Wastes and Glycerol with Tunable 16 Mechanical, Thermal and Barrier Properties

Keywords: Tomato cuticle, biomimetic, cutin polyester, hydroxy fatty acid, glycerol, semi-crystalline elastomer

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

Introduction

To face terrestrial life colonization, plants have developed a hydrophobic barrier, the cuticle, that covers the surface of their aerial organ. This ubiquitous interface between the plant and its environment fulfills multiple crucial functions such as mechanical supports against wounding and cracks, control of nonstomatal water loss, and permeation of gases [START_REF] Fich | The plant polyester cutin: biosynthesis, structure, and biological roles[END_REF].

The main biopolymer in this complex hydrophobic assembly is the cutin polyester, which is essential for the functional properties of the plant cuticle [START_REF] Nawrath | Unraveling the complex network of cuticular structure and function[END_REF][START_REF] Yeats | The formation and function of plant cuticles[END_REF]. Plant cutin is insoluble due to its polyester network architecture. It can be depolymerized mainly into a variety of longchain hydroxy fatty acids building blocks. Each of these carboxylic acid monomers contains 16 and-or 18 carbon atoms and one to three hydroxyl groups. Interestingly, the tomato fruit peels concentrate a high amount of cutin, whose monomer composition is strikingly homogenous, with about 80% by weight consisting of 9(10)-16 dihydroxyhexadecanoic acid (diOHC16). The remaining minor cutin components, especially phenolics, mono-and di-carboxylic acids probably play an important role in the insoluble polyester crosslinked network. Indeed, pure diOHC16 is an AB2 type molecule (with A being a carboxylic acid function and B a hydroxyl group). The polyesterification of such a single monomer would typically lead to hyperbranched macromolecules that are easily soluble in common organic solvents [START_REF] Feast | Synthesis, structure and properties of some hyperbranched polyesters[END_REF][START_REF] Testud | Hyperbranched polyesters by polycondensation of fatty acid-based ABn-type monomers[END_REF].

However, this w-hydroxylated molecule carrying midchain functionalities, presents high interfacial properties [START_REF] Fameau | Interfacial properties of functionalized assemblies of hydroxy-fatty acid salts isolated from fruit tomato peels[END_REF] and constitutes a unique opportunity for the development of innovative biosourced polymer materials from depolymerized tomato cutin feedstock [START_REF] Heredia-Guerrero | Cutin from agro-waste as a raw material for the production of bioplastics[END_REF].

Reaching this innovation would address the important landmarks set by the United Nations sustainable development goals because tomato cutin is a readily available residue of the tomato processing industries.

Tomato wastes, i.e., pomaces, are mainly composed of peels and seeds. 44 MT of tomato fruit are processed worldwide, to make derived tomato products (e.g., purees, juice, sauces) generating nearly 1.5 MT of animal feed and production of biogas whereas their transformation for development of new materials, from bioplastic to high-functional value products remains largely overlooked [START_REF] Fritsch | Processing, valorization and application of bio-waste derived compounds from potato, tomato, olive and cereals[END_REF]. Peels and seeds can be easily separated and dried, directly in the factory, and the hydroxy fatty acids can be easily extracted from the cutin of tomato peels [START_REF] Benítez | Valorization of tomato processing byproducts: fatty acid extraction and production of bio-based materials[END_REF]. Furthermore, glycerol, the main by-product of the biodiesel and oil industries [START_REF] Quispe | Glycerol: production, consumption, prices, characterization and new trends in combustion[END_REF], has been identified as a potentially influential minor component in tomato cutin. Esterified glycerol has been evidenced in the cutin polyester network structure [START_REF] Graça | Glycerol and glyceryl esters of omegahydroxyacids in cutins[END_REF] and typically represents only 0.6 to 0.7 % wt. of the cutin monomers obtained by depolymerization [START_REF] Philippe | Ester cross-link profiling of the cutin polymer of Wild-Type and cutin synthase tomato mutants highlights different mechanisms of polymerization[END_REF]. Besides, we have recently highlighted that the ratio of glycerol to hydroxy fatty acid can be modulated in tomato fruit. Indeed, in tomato fruit mutants affected in a cutin synthase (CUS1), an enzyme involved in the cutin biosynthesis, a 60% reduction in cutin deposition was induced without any significant negative impact either on the fruit development or the generation of cracks [START_REF] Girard | Tomato GDSL1 is required for cutin deposition in the fruit cuticle[END_REF]. Looking at the cutin polyester structure, we observed a significant increase of the glycerol to hydroxy fatty acid ratio. These results obtained with the mutants suggested that the inclusion of co-polymerized glycerol in the tomato cutin network could be involved in the mechanical adaptation of the fruit cuticle, in particular in the modulation of its extensibility. This is especially essential for resistance to turgor pressure during fruit growth and environmental constraints to prevent fruit from cracking, a phenomenon responsible for major crop losses [START_REF] Barker | Russeting (cuticle cracking) in glasshouse tomatoes in relation to fruit growth[END_REF]. Taking inspiration from nature, our leading hypothesis is that we can produce tunable bio-based co-polyesters that mimic potential adaptive events of cutin simply by playing on the glycerol/diOHC16 ratio.

Hereinafter, we describe a solvent-and catalyst-free polymerization process to generate copolyesters from two diOHC16 and glycerol, two readily available and renewable biological resources.

Tuning of the ratio of esterified glycerol to diOHC16 in a range similar to what it is observed in plant cutin effectively results in the modulation of the structure (amorphous or semi-crystalline), the properties (mechanical, thermal, and barrier properties) of the ensuing co-polyesters.

Experimental Section

Materials

Glycerol (99 % purity) and all other chemical ACS reagents were from Sigma-Aldrich (USA). The analytical grade solvents were obtained from Carlo Erba (Val de Reuil, France). The 16-10(9)-hydroxy-hexadecanoic acid-enriched fraction (referred to as "diOHC16" in the manuscript) was extracted from industrial tomato samples provided by the "Conserveries de Bergerac" (UNIPROLEDI, Bergerac, France). The peels and seeds of the pomaces were separated by decantation in a water tank. The floating peels were recovered, their water excess was squeezed out and then dried at room temperature. Dried peels (1kg) were grounded and dewaxed under the reflux of acetone: ethanol 1:1 (v:v) in a Soxhlet extractor and then dried in a fume hood. Cutin depolymerization was conducted at room temperature in 5% (wt.) KOH in ethanol 95%. After filtration on a Buchner funnel, about 90% of the filtrate was evaporated under vacuum and replaced by water. Hydroxy fatty acids were precipitated by adjusting pH at 3.5 with concentrated HCl. The hydroxy fatty acid precipitate was extensively washed with water, and finally freeze-dried.

The purity of the fatty acids was determined after phase partitioning in CHCl3/CH3OH/H2O 8:4:3 (v:v). The lower chloroform-rich phase containing only lipophilic compounds, mainly fatty acids and some pigments, was recovered, evaporated under vacuum, and weighed. The colored cutin monomer batches contain 98 ± 0.5% of lipophilic compounds. The fatty acid composition of the lipophilic fraction was determined by gas chromatography coupled to a mass spectrometer as previously described [START_REF] Philippe | Assembly of tomato fruit cuticles: a cross-talk between the cutin polyester and cell wall polysaccharides[END_REF]. A 60% yield of hydroxy fatty acids was obtained and the same batch of 500g of diOHC16 extract was used in the study. This batch was dark-red colored due to the remaining pigments and highly concentrated in diOHC16 (90% of the hydroxy fatty acids) (Supplemental Figure 1).

Preparation of the cutin-like co-polyesters films

Co-polyesters films were produced by a solvent-free and catalyst-free polycondensation. We developed a polymerization process to minimize the non-esterified glycerol within the copolyester films. Preparation of different weight ratio of diOHC16/ Glycerol (1.8g diOHC16 for PG0 ; 1.71g diOHC16/90 mg glycerol for PG2.3; 1.62gdiOHC16/180 mg glycerol for PG3.9; 1.53g diOHC16/ 270 mg glycerol for PG4.8 ; 1.44g diOHC16/ 360 mg glycerol for PG5.2 and 1.35 g of diOHC16/ 450 mg glycerol for PG6.2 respectively) were put in Teflon molds (5cm x 5cm) with a thickness of about 0.7-1 mm. The nomenclature of the co-polyesters refers to the glycerol weight ratio comprised within the co-polyester at the end of the process (Table 1).

Bulk poly-condensation was then conducted for 24h at 150°C in an oven (VaccuTherm, ThermoFisher, USA). To avoid bubbles and to minimize the presence of residual free glycerol, i.e., non-esterified, in the final material a 400mbar vacuum was applied during the first 2h of the polycondensation process. After 24h, the polymers were rapidly cooled on ice and stored at room temperature. Samples of the ensuing films were immersed under agitation in various solvents of diOHC16 and/or of glycerol, i.e., water, ethanol, and chloroform 16h at room temperature to assess their solubility.

Structural characterization of the cutin-like co-polyester films

Attenuated total reflectance Fourier-transform infrared spectroscopy (ATR-FTIR) spectra (200 scans) were recorded at a resolution of 2 cm -1 on a Nicolet Magna IR 550 spectrometer equipped with a liquid nitrogencooled mercury-cadmium-telluride detector. The instrument was continuously purged with dry air. Spectra of co-polyester films were obtained by attenuated total reflection (ATR) using a single reflection accessory fitted with a thermostated diamond crystal with a 45° angle of light incidence. All spectra (3 per co-polyester) were acquired in the 4000 to 700 cm -1 range at 4cm -1 resolution and accumulating 30 scans.

13 C solid-state NMR (CP-MAS NMR) was carried out on a Bruker AvanceIII-400 MHz spectrometer operating at 100.61 MHz for 13 C, equipped with a double-resonance H/X CP-MAS 4-mm probe for CP-MAS (cross-polarization magic-angle-spinning) solid-state experiments. Sixty mg of the co-polyester films were put packed into 4 mm zirconia rotors. The samples were spun at 12,000 Hz at room temperature. CP-MAS spectra were acquired with a contact time of 1 ms, a recycling delay of 10 s, and over-accumulation of 2048 scans. The carbonyl carbon was set to 176.03 ppm through external glycine calibration. NMR spectra deconvolution was performed using the PeakFit® software (Systat Software, Inc., USA). X-Ray Diffraction analysis was performed on a Bruker-AXS D8 Discover diffractometer. The X-ray beam was produced in a sealed copper tube at 40 kV and 40 mA. The 500-μm beam with a CuKα1 wavelength (1.5405 Å) was collimated and parallelized using two crossed-coupled Göbel mirrors. The Xray diffraction data were collected using a Vantec500 two-dimensional detector, previously calibrated with silver behenate, in the 2.θ-range: 3-70°. The samples are placed perpendicular to the X-ray beam. Recorded diffractograms I = f (2θ) were normalized to remove the influence of thickness variation among the samples. Temperature kinetics were performed using Linkam's HFS91 stage. The samples were placed in a mica window cell. The detector was positioned at a focusing distance of 8.6 cm from the sample surface.

It was in a direct beam position. The heating kinetic applied to the sample was 3°C.min -1 from 20°C to 100°C.

Chemical characterization of the cutin-like co-polyester films

Free glycerol was extracted from polymers (5mg in 1 mL of CH3OH), overnight at room temperature.

Esterified glycerol was released from polymers by mild methanolysis using a modified procedure [START_REF] Graça | Glycerol and glyceryl esters of omegahydroxyacids in cutins[END_REF]. Isolated pieces of the co-polyester films were stirred at room temperature in a mixture of 50mM sodium methoxide in dry methanol with the internal standard 1,2,3-butanetriol [START_REF] Shen | An improved GC-MS method in determining glycerol in different types of biological samples[END_REF]. Free and esterified glycerol extracts were dried with a nitrogen flow and analyzed by GC-MS and GC FID. For the quantification of the cutin hydroxy fatty acids, the co-polyesters were depolymerized through methanolysis during 6h until complete depolymerization. The depolymerized hydroxy fatty acids were silylated with 1% BSTFA/TMCS and analyzed by GC-MS and GC-FID as previously described [START_REF] Philippe | Ester cross-link profiling of the cutin polymer of Wild-Type and cutin synthase tomato mutants highlights different mechanisms of polymerization[END_REF].

Free OH groups were derivatized by benzyl-etherification resistant to alkaline depolymerization, according to an established method [START_REF] Philippe | Ester cross-link profiling of the cutin polymer of Wild-Type and cutin synthase tomato mutants highlights different mechanisms of polymerization[END_REF]. Briefly, co-polyesters (3mg) were mixed at 90°C in a screw-capped glass tube for 24h with 15mg of 2-benzyloxy-1-methylpyridinium triflate and 1.68 mg of magnesium oxide in 1mL of trifluorotoluene. The samples were then extensively washed with CH2Cl2 and dried before depolymerization and cutin monomer analysis. GC-FID surface response of the llabeleddiOHC16 (either in midchain position or in w-position) were compared between the different copolyesters and were expressed as equivalent of heptadecanoic acid, as an external standard.

Thermal and mechanical properties of the cutin-like co-polyester films

Differential Scanning Calorimetry (DSC) was performed on a DSC Q100 (TA Instruments, New Castle, DE, USA). The DSC instrument was calibrated with an indium standard. Analyses were made with 2-5mg of samples, using hermetically-sealed aluminum pans. An empty pan was used as a reference. Samples were cooled from 20°C to -50°C and heated at 3°C.min -1 until 80°C. The Glass transition temperature (Tg) of the co-polyesters was defined at the midpoint of heat capacity change on the thermograms recorded on the first scan. The melting temperature was determined at the maximum of the peak.

Dynamic Mechanical Thermal Analysis (DMTA) was performed on a DMTA MKIV (Rheometric Scientific, US). Rectangular specimens (20 mm × 4 mm) were cut from the films and thickness of about 0.7-1 mm was precisely measured with a micrometer. The samples were analyzed in the tensile mode at the frequency of 1 Hz with a strain amplitude of 0.1%. To keep the samples taut a static force superior by 10% to the dynamic force was applied to the sample. A scanning rate of 3°C min -1 from -50°C to 80 °C was used.

Each sample was analyzed in duplicate.

The tensile test of the films was carried out using MTS Synergie 100 (MTS Systems Corporation, USA). The film samples were cut into a dog-bone shape. The test was performed using a cross-head speed of 10 mm/min. The reported results, including Young's modulus, ultimate strength, and elongation at break, were the average values of five specimens. The standard deviations are indicated in Table 1.

Barrier properties of the cutin-like co-polyester films

Water permeances of the co-polyesters were determined at 25°C and under 100% relative humidity gradient using transpiration chambers as previously described [START_REF] Philippe | Ester cross-link profiling of the cutin polymer of Wild-Type and cutin synthase tomato mutants highlights different mechanisms of polymerization[END_REF][START_REF] Schreiber | Water permeability, Water and solute permeability of plant cuticles[END_REF]. Typically, co-polyesters (5 replicates) were mounted on the top of the transpiration chamber with a 0.6 cm diameter hole, subsequently filled with 300 μL of deionized water, inverted, and placed in a desiccator filled with a desiccant. Water transfer across the co-polyesters samples was determined by gravimetric analysis of the transpiration chamber every 8h during 4 days. Water permeance (P) was calculated as the ratio P=F/A.Dc, where F (Flow rate of water) is determined as the slope of water transfer across the samples (in g.s -1 ), A (in m 2 ) is the area across which transport has occurred, and Dc (in g.m -3 ) is the water gradient driving force.

The oxygen permeability of co-polyesters was determined in triplicate at 23°C and 0% relative humidity through an isostatic and dynamic method using gas-phase chromatography as previously described [START_REF] Motedayen | Gas barrier enhancement of uncharged apolar polymeric films by self-assembling stratified nano-composite films[END_REF]. The film samples (4 cm diameter) were placed inside sealed stainlesssteel permeability cells. The lower chamber was continuously spread by a 5 mL.min -1 flux of permeant gas (O2), and the upper chambers by the same flux of vector gas (helium), thus applying a permanent O2 partial pressure difference. The permeability cells were coupled to a gas chromatograph with a thermal conductivity detector (GC-TCD, Agilent 7890A, USA), equipped with an automatic valve to online analyze the evolution with time of O2 gas concentration in the upper chamber. The GC-TCD was equipped with two capillary columns -PoraPlot U (25 m × 0.530 mm id, Agilent) and HP Molesieve (30 m × 0.535 mm id, Agilent) -able to separate O2, CO2 and N2. Data were collected and processed using the ChemStation OpenLab Software (Agilent) and SRA Prochem interface (SRA Instruments). The O2 permeability, PO2 (mol.m -1 .s -1 .Pa -1 ), of the film samples was calculated as follows:

P " # = % × ( ) × ∆+ (equation1)
Where: J (mol.s -1 ) is the flux of gas that passes through the film, e is the film thickness (m), A is the permeation area (m²) and ΔP is the O2 pressure differential between the upper and lower chambers of the cell (Pa).

Microbiological analyses

S. aureus NCTC8325 and E. coli TOP 10 cells (10 5 cells•mL -1 ) in Mueller-Hinton broth (MHB) medium were exposed for 24h to 0.5 cm 2 of polyester films (ca. 10 to 20 mg•mL -1 ) at 37 °C, under orbital agitation (100 rpm). Upon incubation, the cellular morphology and the viability of the bacterial cells were visualized by light and fluorescent microscopy, respectively, using a Leica DM6000 B microscope equipped with an iXon EM+ 885 EMCCD camera (Andor Technology®). The fluorescent dye propidium iodide (PI) was used for labeling the dead cells (red fluorescence). Negative controls, i.e., bacteria growing in media devoid of cutinlike co-polyester films, were also carried out. Films from each tested condition were also collected to scan for the presence of bacterial cells on the film's surface. All assays were done in triplicate. The films were first washed with phosphate-buffered saline (PBS) to remove non-adherent bacteria, subsequently fixed with glutaraldehyde 2.5% (v/v) during 10 min. After 10 min, samples were rinsed with distilled water and dehydrated with aqueous solutions with increasing concentrations of ethanol (ranging from 70 to 100%).

Scanning electron microscopy (SEM) (microscope JEOL JSM-7001F, with an accelerating voltage set to 15 kV) was used to scan the co-polyester film's surfaces. All samples were coated with gold before analysis.

Results and Discussion

Sustainable production of dark black colored insoluble co-polyesters with different ratio of esterified-glycerol

The cutin-like polyesters were manufactured to meet green chemistry and sustainability criteria. Indeed, they were prepared through a catalyst-and solvent-free process using hydroxy fatty acids and glycerol from renewable resources, i.e. by-products of the tomato and oilseed processing industries. The biorefinery process designed for the recovery ion of the hydroxy fatty acids from tomato pomaces leads to higher yield and purity than the existing processes described in the literature [START_REF] Benítez | Valorization of tomato processing byproducts: fatty acid extraction and production of bio-based materials[END_REF][START_REF] Cogognigni | Extraction method of a polyester polymer or cutin from the wasted tomato peels and polyester polimer so extracted[END_REF]. First, fractionation of tomato pomaces into seeds and peels was conducted. This opens ways for the valorization of both seeds and peels and prevents seed lipids co-extraction. Indeed, seeds contain more than 30% oil (Giannelos el., 2005). In contrast with previous studies starting from raw pomaces, the extraction of fatty acids from peels provided a fatty acid fraction with a higher diOHC16 content (90% instead of 45% [START_REF] Tedeschi | Sustainable fabrication of plant cuticle-like packaging films from tomato pomace agro-waste, beeswax, and alginate[END_REF]. The hydrolytic process of the tomato cutin polyester was conducted at room temperature and used green solvents, i.e., water and ethanol, which can be recycled to improve the sustainability of the process. Interestingly, the ethanolic alkaline hydrolysis at room temperature gave rise to a high fatty acid yield (around 60% wt. of dry peels), instead of around 15% wt. for hot aqueous alkaline hydrolysis [START_REF] Cogognigni | Extraction method of a polyester polymer or cutin from the wasted tomato peels and polyester polimer so extracted[END_REF], thereby strengthening the sustainability of the process. Considering that tomato cuticle contains about 70% of cutin and 30% of polysaccharides [START_REF] Philippe | Assembly of tomato fruit cuticles: a cross-talk between the cutin polyester and cell wall polysaccharides[END_REF], this means that our biorefinery process recovered more than 85% of the available cutin monomers in the tomato pomaces. The cutin monomers extract consisted of ~90 % diOHC16 and were dark-red colored as for existing processes [START_REF] Benítez | Valorization of tomato processing byproducts: fatty acid extraction and production of bio-based materials[END_REF] due to the remaining pigments. According to the FTIR and UV spectra, these pigments contain phenolic compounds(Aleixandre-Tudo and du Toit, 2018; [START_REF] Benítez | Valorization of tomato processing byproducts: fatty acid extraction and production of bio-based materials[END_REF][START_REF] Heredia-Guerrero | Infrared and Raman spectroscopic features of plant cuticles: a review[END_REF][START_REF] Herrera | Accelerated photo-oxidation of concentrated pcoumaric acid in homogeneous solution. Mechanistic studies, intermediates and precursors formed in the dark[END_REF] (supplemental Figure 1). The cutin-like co-polyesters formed by polycondensation of the diOHC16-rich hydroxy fatty acid extract and glycerol (Figure 1) displayed a dark brown color. They were insoluble in water, ethanol, and chloroform (Table 1) as previously reported for polyesters synthesized from lower purity tomato hydroxy fatty acids extracts [START_REF] Benítez | Valorization of tomato processing byproducts: fatty acid extraction and production of bio-based materials[END_REF][START_REF] Benitez | Polyhydroxyester films obtained by non-catalyzed melt-polycondensation of natural occurring fatty polyhydroxyacids[END_REF]. The cutin monomers extracts were highly rich in lipophilic compounds (98%). Accordingly, the insolubility of our films suggests that, as in native plant cutin, the minor lipidic compounds, i.e., dicarboxylic fatty acids, and co-extracted phenolic molecules (supplemental Figure 1) could participate in the polymerization process. The participation of these compounds in the polyester crosslinked network should compensate for the AB2 structure of the diOHC16 which should normally lead to branched polymers soluble in organic solvents [START_REF] Feast | Synthesis, structure and properties of some hyperbranched polyesters[END_REF][START_REF] Testud | Hyperbranched polyesters by polycondensation of fatty acid-based ABn-type monomers[END_REF].

In previous works on fatty acid extracts containing 9,10,16-trihydroxypalmitic or 9,10dihydroxystearic acids [START_REF] Benitez | Polyhydroxyester films obtained by non-catalyzed melt-polycondensation of natural occurring fatty polyhydroxyacids[END_REF], the authors pointed out the capacity of such molecules to form two additional carboxylic acid functions by an oxidative diol cleavage reaction of the 9,10 neighbor hydroxyl groups. This would increase the acid/hydroxyl ratio and allow the formation of crosslinked insoluble polyester networks. In our case, despite oxidation reactions evidenced by the darkening that occurs during the polymerization process (Figure 1), no such fatty acids with vicinal hydroxyl groups are present in the diOHC16-rich extract (supplemental Figure 1).

Finally, the objective of the present study was to focus on the modulating effect of glycerol on the properties of the co-polyesters produced from the fatty acid extract from peels provided by tomato pomaces, in line with a vision of sustainable biobased and cutin-inspired chemistry. Therefore, the same cutin monomer batch was used and materials containing different amounts of esterified glycerol were produced. After the poly-condensation reaction, the added glycerol can be (i) free, i.e., non-esterified, acting as an external plasticizer, or (ii) esterified, constituting a polyester monomer. Determination of free and esterified glycerol contents revealed that more than 95% of the glycerol was esterified (

A gradual increase of the glycerol content induces a decrease in the branching of the copolyester films

The co-polyesters containing the lowest (PG0.4) and the highest (PG6.2) levels of esterified glycerol were characterized by ATR-FTIR and 13 C NMR (Figure 2). ATR-FTIR indicated the formation of esters bonds in these co-polyesters. Indeed, we observed the characteristic shift of the carbonyl band from 1705 cm -1 , assigned to the carboxylic acid, to the 1733 cm -1 band assigned to the esterified carbonyl [START_REF] Heredia-Guerrero | Infrared and Raman spectroscopic features of plant cuticles: a review[END_REF]. In addition, the 1715 cm -1 band assigned to the ester bond involved in hydrogen bonding was also evidenced, as previously observed in tomato cutin [START_REF] Girard | Tomato GDSL1 is required for cutin deposition in the fruit cuticle[END_REF]. In our present co-polyesters, the hydrogen bonds could involve the free hydroxyl groups either from the hydroxy fatty acids and/or from the esterified glycerol. Besides, the 1171 cm -1 band assigned to the stretching vibration of the C-O-C ester bond was observed in all co-polyesters. Likewise, the 13 C C-MAS (Figure 2B) data confirmed the formation of esters, evidenced by a single broad peak at 173 ppm and by the peak at 64 ppm, which is assigned to primary esters [START_REF] Tedeschi | Sustainable fabrication of plant cuticle-like packaging films from tomato pomace agro-waste, beeswax, and alginate[END_REF]. Furthermore, no significant free fatty acid signals were evidenced at 178 ppm (-COOH) and 35 ppm (-CH2-COOH). Finally, the comparison of the surface ratio of the 33.7 ppm and 37 ppm peaks, assigned to aliphatic carbons (-CH2-CHOR-CH2-) and (CH2-CHOH-CH2) respectively [START_REF] Benitez | Polyhydroxyester films obtained by non-catalyzed melt-polycondensation of natural occurring fatty polyhydroxyacids[END_REF], indicated a decrease (40%) in the esterification of the secondary hydroxyls of diOHC16 associated with the increase in the glycerol content. 1715 cm -1 1703 cm -1 1733 cm -1 Wavelength (cm -1 ) The reticulation of the co-polyesters was further monitored by the chemical labeling of the free OH groups by benzyl ether within the diOHC16-derived polyesters [START_REF] Philippe | Ester cross-link profiling of the cutin polymer of Wild-Type and cutin synthase tomato mutants highlights different mechanisms of polymerization[END_REF] (Figure 3).

According to the chemical structure of the diOHC16, both, linear and branched ester bonds, can be formed in the co-polyesters. After complete depolymerization, the release of diOHC16 containing labeled OH groups either in w-position or in midchain-position was compared in the different co-polyesters. In PG0.4, the ratio of labeled mid-chain OH/w-OH labeled in diOHC16 was around 9, which indicates that the primary OH groups in the w-position were preferentially esterified compared to the OH groups in the midchain position. This result agrees with the NMR data (Figure 2B) and the previous studies of polycondensation of polyhydroxylated fatty acids [START_REF] Benitez | Polyhydroxyester films obtained by non-catalyzed melt-polycondensation of natural occurring fatty polyhydroxyacids[END_REF]. A similar ratio was previously reported for tomato fruit cutin [START_REF] Philippe | Ester cross-link profiling of the cutin polymer of Wild-Type and cutin synthase tomato mutants highlights different mechanisms of polymerization[END_REF]. Furthermore, taking PG0.4 as a reference, the gradual introduction of glycerol in the co-polyesters resulted in a gradual increase (up to 3.5-fold) in the labeled OH groups, in particular in the midchain position. These results indicate that the introduction of glycerol in the co-polyesters induced a gradual decrease in the esterification of midchain hydroxyl groups involved in ester bonds and an increase in the linear polymerization scheme within the co-polyesters.

Glycerol content modulates the microstructure and thermo-mechanical properties of the co-polyesters

The thermal properties of all the produced co-polyesters were investigated by DSC on heating from -50°C to 80°Cat 3°C.min -1 . Characteristic thermograms are superimposed in Figure 4A. We checked that the cooling step at -50°C did not modify the crystallinity of the samples. The thermograms of the co-polyesters exhibited various events as a function of the composition, i.e., the amount of esterified glycerol in the samples. From -30°C to 0°C, depending on the glycerol content, the baseline fall was attributed to the glass transition characterizing the amorphous phase of the co-polyesters. The glass transition temperatures (Tg), determined at the midpoint of the baseline fall, were determined for each co-polyester (Table 1). The Tg value determined for the PG0.4 sample containing only traces of glycerol is -8.7°C, similar to that measured on polyester catalyst-produced (-7°C) from hydroxy fatty acids extracted from green tomato cutin [START_REF] Benitez | Polyhydroxyester films obtained by non-catalyzed melt-polycondensation of natural occurring fatty polyhydroxyacids[END_REF]. Interestingly, the Tg significantly decreased from -8.7 to -20.4°C with the gradual introduction of glycerol (up to 6.2% wt.) within the co-polyesters, suggesting that the mobility of the co-polyester chains increased giving rise to a more relaxed structure. This result is consistent with the observed reticulation pattern of the co-polyesters (Figure 3).

For the co-polyesters containing glycerol amounts lower than 4% wt. (PG0.4, PG2.3, PG3.9), the DSC thermograms recorded on heating did not exhibit thermal events above 0°C (Figure 4A). However, for the co-polyesters containing higher amounts of glycerol (PG4.8, PG5.2, and PG6.2), endothermic events were recorded and related to the gradual incorporation of glycerol in the materials. Indeed, for these copolyesters, two successive endothermic peaks were recorded around 17°C and 40°C. These endotherms may correspond to structural reorganizations, and/or changes in the physical state of molecules in the copolyesters. The second endothermic peak (40°C) exhibits a higher enthalpy than the first, and increases with the level of glycerol content (Table 1).

The X-ray diffraction experiments performed at 25°C showed at wide angles a bump centered at 2θ=19° (d=4.5 Å), corresponding to the amorphous state of the co-polyesters (Figure 4B). In the copolyesters containing at least 5 % wt. of glycerol (PG5.2, PG6.2), a peak at 2θ=21.2° was recorded (Figure 4B). This peak corresponds to a repeat distance of 4.2 Å between the acyl chains which is characteristic of a hexagonal packing of acyl chains. This single peak constitutes the XRD signature of the formation of a crystalline organization of the linear esterified acyl chains within the co-polyesters containing at least 5% wt. glycerol. This XRD peak was superimposed to the bump centered at 4.5 Å (Figure 4B), which corresponds to the coexistence of crystalline and amorphous states within the co-polyesters. The presence of glycerol in the samples, and the esterification occurring between the hydroxyl groups of glycerol (mainly in sn-1 and sn-3 positions) and the carboxyl groups of the fatty acids induced a linear organization of the chains or the formation of shorter chains. This spatial proximity between the linear and shorter copolyester chains may favor short-distance hydrophobic interactions between the acyl chains and result in a crystalline organization with a hexagonal packing below the melting temperature of the fatty acids. In absence of glycerol, the reticulation involves the hydroxyl groups both in midchain hydroxyl and w-positions. This prevents short-distance hydrophobic interactions between the linear chains formed by primary esterification, hence avoiding the formation of a crystalline organization in the co-polyesters.

The thermotropic phase behavior of the co-polyesters formed with very low (PG0.4) or high (PG6.2) amount of glycerol, were investigated by XRD at various temperatures ranging from -10°C to 80°C that have been selected according to the DSC results (Figure 4A). In presence of very low glycerol content (Figure 4C), the amorphous state of the polyester was characterized by a bump of RX diffusion, centered in a mean value that was affected by the temperature. The increase of temperature induced a shift toward lower angle values corresponding to higher distances between the acyl chains associated with an increase in the molecular mobility. In the presence of 6 % wt. of glycerol (Figure 4D), the XRD experiments were conducted at 25°C, then the samples were cooled to -10°C, heated to 30°C (i.e., above the first endotherm) and then finally to 80°C (i.e., above the second endotherm). The single peak at 2θ=21.2° (4.2 Å) was recorded at 25°C, confirming the results previously obtained. Decreasing the temperature to -10°C increased the intensity of the XRD peak at 2θ=21.2° related to a hexagonal phase and was interpreted as an increase in the amount of the crystalline phase formed in the co-polyesters. On heating at 30°C, the intensity of the XRD decreased. Increasing the temperature at 80°C, i.e., above the second endotherm recorded by DSC and then above the melting point of the crystalline structures, induced the disappearance of the XRD peak at 2θ=21.2°and to the recording of an XRD bump centered at 4.6 Å characteristic of acyl chains in a melted state. These XRD experiments evidenced the temperature-dependent behavior of the co-polyester, with the presence of various amounts of hexagonal phase as a function of temperature.

The two endotherms revealed successively by DSC correspond to independent packing events of the molecules both organized in a hexagonal form as revealed by XRD. The different melting temperatures may correspond to packing of molecules with various compositions, e.g. chain length.

The thermo-mechanical behavior of all co-polyesters was analyzed by DMTA. Figure 5A shows the plot of the storage modulus (E') and the loss factor tan d of the co-polyesters. For all samples, the large peak of tan d corresponds to the main mechanical relaxation a which is associated with the calorimetric glass transition from a glassy to a rubbery state. The relaxation temperature from the glassy to the rubbery state (Ta), determined at the maximum of the tan d curve, decreases from 8.6°C to 0.2°C when glycerol content increases (Table 1). In agreement with the Tg values obtained by DSC analyses (Table 1) such a depressor effect of glycerol on Ta further strengthens the impact of glycerol on the properties of the cutin-like co-polyesters.

Temperature (°C)

A second event appears on storage modulus which slightly falls in the temperature range 30-40°C

for the composition containing more than 4% (wt.) of glycerol (PG4.8, PG5.2, PG6.2) (Figure 5A). This transition is attributed to the melting of the crystalline phase according to DSC and XRD results (Figure 4).

Finally, the plateau of storage modulus is reached at about 50°C with a value which decreases from 5.867

MPa to 1.589 MPa when glycerol content increases from 0.4 to 6% wt. (Table 1).

These values are typical of rubber elasticity suggesting an elastomeric behavior of the co-polyesters. This was confirmed by the tensile tests performed at room temperature. The tensile strength, strain at break and, Young Modulus are reported in Table1, and the characteristic curves are plotted in Figure 6.

All films, either with low or high glycerol contents, show a quasi-linear and reversible stress-strain behavior with high elongation at break (> 100%) comparable to synthetic elastomers. It is striking that after breakage, the dogbone tensile test specimens show a complete elastomeric recovery of their initial shape (Supplemental Figure 2). The mechanical properties of the PG0.4 are comparable to those previously described for other polyhydroxylated fatty acid-based polyesters [START_REF] Benitez | Polyhydroxyester films obtained by non-catalyzed melt-polycondensation of natural occurring fatty polyhydroxyacids[END_REF][START_REF] Benítez | Long-chain polyhydroxyesters from natural occurring aleuritic acid as potential material for food packaging[END_REF].

Moreover, our present data show that the mechanical properties were strongly modified by the gradual introduction of glycerol in the co-polyesters which resulted in a decrease in the Young modulus (from 2.29 to 1. 19 MPa), as well as in the stress at break (from 2.1 to 1.5 MPa). Surprisingly, higher glycerol of 6% induces a slight increase of tensile strength simultaneous to the increase of strain at break. It is certainly due to the crystallinity evidenced by calorimetry and XRD measurements (Figure 4).

The most striking observation is the increase in strain at break from 124% to 217 % (for PG0.4 and PG6.2 respectively) induced by the esterified glycerol in the co-polyesters (Table1, Figure 6). This elongation value is comparable to some synthetic rubber such as ethylene-propylene-diene or polysulfidebutadiene rubbers [START_REF] Shanks | General purpose elastomers: structure, chemistry, physics and performance[END_REF]. This strain at break induced by a higher glycerol/hydroxy fatty acid ratio fully fits the CUS1 tomato phenotype which is affected in cutin deposition [START_REF] Philippe | Ester cross-link profiling of the cutin polymer of Wild-Type and cutin synthase tomato mutants highlights different mechanisms of polymerization[END_REF].

Importantly, the CUS1 tomato cutin polyester exhibited a fivefold increase in the glycerol ratio (vs hydroxy fatty acids), which probably account for the increase in its extensibility and the absence of cracks observed on the CUS1 fruits, regardless that their cuticle density shows a threefold reduction [START_REF] Girard | Tomato GDSL1 is required for cutin deposition in the fruit cuticle[END_REF].

Since our co-polyesters behave like elastomers displaying rubber elasticity, the Flory theory [START_REF] Flory | Elasticity of rubber, Principles of Polymer Chemistry[END_REF] may bring information about the macromolecular structure of the networks [START_REF] Tran | Natural rubber-based ionogels[END_REF].

Accordingly, the crosslink density n (mole.m -3 ) is given by: 𝜈 = -./0

(equation 2)

Where: R is the gas constant 8.32 J.mol -1 .K -1 , T is the temperature in Kelvin, and E the elastic modulus at small deformation. This model assumes that the material is amorphous and consequently, we will use the plateau storage modulus values measured in DMTA at 50°C (T = 323 K, that is to say above the melting point of the semi-crystalline samples.).

Therefore, the average molecular mass Mc (kg.mol -1 ) of elastically active chains between crosslinks is:

(equation 3)

Where: r is the density of the material in Kg.m -3 . The density of the co-polyesters at 50°C could not be measured. As a first approximation, we considered a constant density r= 1000 kg.m 3 for the calculations reported in Table 2 andFigure In a first approximation, the number of polymerized diOHC16 repeating units between crosslinks can be estimated by dividing these values of Mc by a repeating unit mass Mo= 270 g.mol -1 (this value assumes that elastically active chains consist essentially of polymerized diOHC16 monomer units in which only 1 acid and 1 hydroxyl group have reacted.)

The values of the ratio Mc/Mo calculated in Table 2 suggest that for samples PG0.4, PG2.3, and PG3.9 the number of polymerized diOHC16 repeating units should range from 5 to 7, while 19 to 21 repeating units are predicted for samples PG4.8, PG5.2, and PG6.2. For the latter, a few glycerol units are probably also present in the elastically active chains. Anyhow, such a loosening of the network induced by the gradual introduction of the glycerol is in full accordance with the increase in the labeling of free OH groups within the co-polyesters (Figure 3).

Besides, the plot of n and Mc as a function of this molar content of glycerol chemically bound to the network is shown in Figure 5B. A sharp transition from dense amorphous networks (PG0.4, PG2.3, and PG3.9) to loose semi-crystalline networks (PG4.8, PG5.2, and PG6.2) seems to take place around a threshold value of 12 mol% of glycerol.

The development of crystallinity for higher glycerol contents may be ascribed to the flexibility of the long-chain segments between two crosslinks which become able to fold into crystallites, while at lower glycerol content the shorter chains remain amorphous.

Glycerol can modify the barrier properties of the co-polyesters and their interaction with bacterial cells

Water permeance and oxygen barrier properties of the co-polyesters were analyzed. All the produced copolyesters showed low water vapor permeance properties (Figure 7A) around 10 -9 m.s -1 , in the same range as the plant cuticle permeance and some synthetic membrane, such as 3µm parafilm or 3µm polyethylene [START_REF] Zeisler-Diehl | Quantitative characterization of cuticular barrier properties: methods, requirements, and problems[END_REF]. The introduction of esterified glycerol and the associated modification of the branching of the co-polyesters induced a slight, but statistically not significant, increase in the water vapor permeability of the co-polyesters. This result is consistent with our previous data showing that a modification of the polymerization index did not significantly impact the water permeance of tomato cutins [START_REF] Philippe | Ester cross-link profiling of the cutin polymer of Wild-Type and cutin synthase tomato mutants highlights different mechanisms of polymerization[END_REF].

The oxygen permeability of the cutin-derived co-polyesters ranged between 0.7 and 1.4 x 10 -14 mol m -1 s -1 Pa -1 (Figure 7B). These values are in the same order of magnitude as synthetic polymers commonly used for food packaging such as low-density polyethylene (0.1-0.2 x 10 -14 mol.m -1 .s -1 .Pa -1 ) [START_REF] Durmuş | Intercalated linear low density polyethylene (LLDPE)/clay nanocomposites prepared with oxidized polyethylene as a new type compatibilizer: Structural, mechanical and barrier properties[END_REF][START_REF] Matar | Predicting shelf life gain of fresh strawberries 'Charlotte cv' in modified atmosphere packaging[END_REF][START_REF] Motedayen | Gas barrier enhancement of uncharged apolar polymeric films by self-assembling stratified nano-composite films[END_REF]. These values are also comparable to polylactic acid (PLA), a bio-based polyester, for which oxygen permeability values ranging between 1x10 -16 and 1x10 -14 mol. m -1 .s -1 .Pa -1 have been reported [START_REF] Lehermeier | Gas permeation properties of poly(lactic acid)[END_REF][START_REF] Mahmoodi | High-strength, low-permeable, and light-protective nanocomposite films based on a hybrid nanopigment and biodegradable PLA for food packaging applications[END_REF][START_REF] Palai | In situ reactive compatibilization of polylactic acid (PLA) and thermoplastic starch (TPS) blends; synthesis and evaluation of extrusion blown films thereof[END_REF]. Other polyesters such as fossil-based polyethylene terephthalate, or microbial synthesized polyhydroxyalkanoates (PHAs) have considerably higher barrier properties with oxygen permeability in the order of 1x10 -17 mol m -1 s -1 Pa -1 (Ambrosio- [START_REF] Ambrosio-Martin | Assessment of ball milling as a compounding technique to develop nanocomposites of poly(3hydroxybutyrate-co-3-hydroxyvalerate) and bacterial cellulose nanowhiskers[END_REF][START_REF] Crétois | Microstructure and barrier properties of PHBV/organoclays bionanocomposites[END_REF].

Oxygen Permeability

Interestingly, increasing the ratio of esterified glycerol in the co-polyesters improved the oxygen barrier properties of the films by reducing oxygen permeability by half (Figure 7B). This is probably associated with the increase of crystallinity, as observed for other elastomers [START_REF] Wang | Oxygen permeability in thermoplastic polyurethanes[END_REF] reducing the motion needed for oxygen diffusion.

Finally, the antibacterial activity of the co-polyesters containing the lowest and the highest levels of esterified glycerol (PG0.4 and PG6.2) was assessed on E. coli and S. aureus. After exposure to the copolyester films, no labeling of dead cells was revealed (Figure 8A), which indicates that the co-polyesters were devoid of bactericidal properties regardless of the glycerol content. This result contrasts with the biocide activity that was demonstrated with acyl glycerol esters of hydroxy fatty acids [START_REF] Cogognigni | Extraction method of a polyester polymer or cutin from the wasted tomato peels and polyester polimer so extracted[END_REF] or with films formed by self-assembly of these esters [START_REF] Ferreira | Unveiling the dual role of the cholinium hexanoate ionic liquid as solvent and catalyst in suberin depolymerisation[END_REF][START_REF] Garcia | Dissolution of cork biopolymers in biocompatible ionic liquids[END_REF] extracted by an ionic liquid catalyst from suberin, another hydrophobic plant polyesters. Since the co-polyesters are devoid of bactericidal capacity, we further analyzed the films' surface following their incubation with bacteria (Figure 8 B). For either co-polyester, the numbers of adherent E. coli cells were low and comparable. Remarkably over the surface of the PG0.4 co-polyester, S. aureus formed a compacted biofilm whereas PG6.2 hinder significantly the adhesion of this bacteria species. This result demonstrates that increased glycerol levels in the cutin-inspired co-polyesters lead to a significant reduction of bacterial fouling. (Figure 8 B). The observed differences between the two bacteria species might be due to the electronegativity of the cells surface that is higher in E. coli than S. aureus, regardless that their different morphologies and surface roughness (rods and cocci, respectively) may greatly influence their adhesion in the conditions used here. This observation, supported by the structural, thermal, and mechanical analyses, suggests that glycerol esterification altered the smoothness of the films' surface as well as their surface charge. One important finding is that by increasing further the glycerol levels in the cutin co-polyesters their anti-biofouling properties can be potentiated.

Conclusions

We report in this work, the production of glycerol/hydroxy fatty acid co-polyesters by a catalystand solvent-free polycondensation process. Glycerol and hydroxy fatty acids are biosourced molecules from by-products and wastes produced by the oilseed and tomato processing industries, respectively. These co-polyesters elastomers obtained are 100% biobased and bioinspired by the cutin polymer of plant cuticles, which is also composed of hydroxy-fatty acids and glycerol. Drawing on our previous studies on tomato fruit cuticle, we gradually increased the glycerol content while remaining in a concentration range observed in most plant cuticles. The co-polyesters obtained are insoluble and displayed tensile mechanical behavior comparable to that of some synthetic elastomers. By adjusting the level of esterified glycerol, we induced modifications of the reticulation pattern which was associated with variation in chain mobility and extensibility. Glycerol also induced the loosening of the elastically active chain network and the formation of crystalline domains. These modifications were associated with variations in the tensile mechanical and barrier properties of the co-polyesters as well as the adhesion of microorganisms.

The exact macromolecular structure of the co-polyesters probably involves not only glycerol and diOHC16 but also minor components present in the fatty acid extract. The elucidation of the role in the reaction scheme will be the subject of future work. Indeed, these hydroxy fatty acid/glycerol co-polyesters provide excellent templates to further delineate the relationships between the complex molecular and macromolecular architecture and the functional properties of plant cuticular barriers. This bioinspired approach also allows developing green polymer composites adapted to specific applications for food, packaging industry, or as an antifouling natural product. Furthermore, increasing the contents of glycerol led to available free hydroxyl groups for grafting and covalent substitutions for tailored functionalization, opening new routes for high-value applications (e.g., in the medical sector). A . ATR-FTIR spectra of hydroxy fatty acid (diOHC16) extracted from tomato cutin and the corresponding polymers with high (PG6.2) or low (PG0.4) glycerol. In the inset, the magnification of the C=O stretching bands at 1715 cm -1 and 1733 cm -1 are assigned to the ester group, involved or not in hydrogen bonds, respectively, whereas the vibration at 1703 cm -1 is assigned to the carboxylic group.

Figure Caption

GlycerolFigure 1 -

 1 Figure 1 -Solvent-and catalyze-free polycondensation of tomato cutin extract and glycerol

Figure 2 -

 2 Figure 2-ATR-FTIR and NMR spectra of the cutin-like polyesters films A . ATR-FTIR spectra of hydroxy-fatty acid (diOHC16) extracted from tomato cutin and the corresponding polymers with high ( PG6.2) or low ( PG0.4) glycerol. In inset, the magnification of the CO streching bands at 1733 cm -1 and 1715 cm -1 are assigned respectively to ester group, and ester bond interacting by hydrogen bond whereas the band at 1703 cm -1 is assigned to carboxylic group. B. Solid State 13 C CP-MAS NMR spectra of the polymers

Figure 3 .

 3 Figure 3. Relative proportion of free OH groups at the w-and midchain position of the the diOHC16 in the co-polyester films. Values are means of at least three replicates bars are standard deviations

  349

Figure 4 .

 4 Figure 4. DSC and XRD analyses of the different cutin-like diOHC16-glycerol co-polyester films

Figure 5 -

 5 Figure 5 -DMA analysis of the cutin-like diOHC16-glycerol co-polyestersA -Plot of the storage modulus (E') and the loss factor tan d of the copolyesters with different level of esterified glycerol content. B-Evolution of the crosslink density and the average mass between crosslinks as a function of the molar content of glycerol in the network. Dashed lines are just guides for the eyes.

Figure

  Figure 6-Typical Stress-strain curves of the cutin-like diOHC16-glycerol co-polyester films from 0.4 % wt. for PG0.4 to 6.2% wt. for PG6.2

  5B. PG0.4, PG2.3, and PG3.9 show the largest crosslink density values in the range of 500-to 730 mol.m -3 associated with small values of Mc in the 1300-2000 g.mol -1 range. Concurrently, much lower values of n (175 to 200 mol.m -3 ) are obtained for samples PG4.8, PG5.2, and PG6.2, with higher values of Mc in the 5000-5700 g.mol -1 range.

Figure 8 .

 8 Figure 8. Microscopy imaging of the cutin-like co-polyesters PG0.4 and PG6.2 after incubation with E. coli and S. aureus.A. Co-polyesters films activity against E. coli and S. aureus. Following exposure of bacteria to films pieces their morphology and viability were visualized by phase-contrast and fluorescence microscopy using Texas Red filter (dead cells are revealed by red fluorescence due to propidium iodide labelling), respectively. Controls was conducted without films. The scale bar in all images is 5μm.B.Cells adherent onto the surface of the cutin co-polyester films were observed by SEM microscopy. No differences were noticed in the numbers of adherent E. coli cells in either co-polyester films. Conversely, the number of S. aureus adherent cells (ex under arrows) are significantly higher in the PG0.4 co-polyester compared to the PG6.2. The scale bar in all images is shown at the bottom.

Figure 1 -

 1 Figure 1 -Solvent-and catalyze-free polycondensation of tomato cutin extract and glycerol

B.Figure 3 -

 3 Figure 3-Relative proportion of free OH groups at the w-and mid-chain position of the diOHC16 in the co-

Figure 5 -

 5 Figure 5-DMA analysis of the cutin-like diOHC16-glycerol co-polyesters

Figure 6 -Figure 7 -

 67 Figure 6-Typical Stress-strain curves of the cutin-like diOHC16-glycerol co-polyester films from 0.4 % wt. for PG0.4

Table 1 ). 296

 1

				Mechanical test	Breaking Strain at break Young's	stress (MPa) (%) modulus	(Mpa)	2.1 ± 0.5 124.0 ± 24.5 2.29 ± 0.05	2.0 ± 0.2 155 ± 12.7 1.74 ± 0.06	1.7 ± 0.2 180.7 ± 10.3 1.26 ± 0.2	1.4 ± 0.1 178.4 ± 8.6 1.11 ± 0.03	1.3 ± 0.1 197.3 ± 21.1 1.10 ± 0.01	1.5 ± 0.1 217.6 ± 14.2 1.19 ± 0.08
	W water, CF chloroform, Et Ethanol	Table 1 Composition, chemical, thermal and mechanical characterization of the cutin-inspired co-	polyesters	Sample Chemical characterisation DSC DMTA	glycerol total esterified free glycerol Insoluble fraction Insoluble fraction Insoluble fraction Glass transition Melting Melting Melting Melting Mechanical relaxation Storage	(%wt) glycerol (% wt) (%) (%) (%) temperature temperature Enthalpy temperature Enthalpy temperature Modulus at	(% wt) W CF Et (°C) 1 (°C) (J/g) 2 (°C) (J/g) (°C) 50°C (MPa)	PG0.4 0.43 ±0.24 0.38 ± 0.21 0.05 ± 0.04 100 100 98 -8.7 ----8.6 5.867	PG2.3 2.27 ±0.31 2.17 ± 0.32 0.10 ± 0,03 99 100 98 -13.1 ----5.4 5.035	PG3.9 3.93 ± 0.34 3.75 ± 0.31 0.18 ± 0,05 99 99 94 -15.7 --39.7 0.07 3.2 4.058	PG4.8 4.80 ± 0.82 4.57 ± 0.80 0.23 ± 0.15 99 97 95 -17.0 17.9 0.32 41.6 2.60 1.9 1.408	PG5.2 5.20 ± 0.31 4.87 ± 0.5 0.32 ± 0.2 99 94 91 -20.4 16.8 2.04 41.2 7.96 0.9 1.521	PG6.2 6.18 ± 0.39 5.89 ± 0.45 0.29 ± 0.1 100 88 80 -18.5 18.3 0.59 41,6 9.05 0.2 1.589

Figure 7-Barrier properties of the cutin-like co-polyesters

	A			Water permeance and permeability
	water permeance (m.s-1)	0,0E+00 2,0E-10 4,0E-10 6,0E-10 8,0E-10 1,0E-09 1,2E-09 1,4E-09
				PG0.4	PG2.3	PG3.9	PG4.8	PG5.2	PG6.2
	B
				2E-14
		Oxygene permeability (molO2 m-	1s-1Pa-1)	2E-15 4E-15 6E-15 8E-15 1E-14 1E-14 1E-14
				0E+00
				pA PG0.4	pD PG4.8	pF PG6.2
	Water permance (A) and Oxygen (B) barrier properties of the cutin-like diOHC16-glycerol
	co-polyesters containing increasing level of glycerol (0.4 %wt for PG0.4 to 6.2% wt for
	PG6.2 )

Acknowledgment

This work was supported by an INRAE Transfer grant (ELASTOMAT project) and the INRAE TRANSFORM department. RR is grateful to FCT funding for his Ph.D. scholarship (SFRH-BD-110467-2015).

The authors thank André Lelion for his technical assistance.

Water permeance (A) and oxygen (B) barrier properties of the cutin-like diOHC16-glycerol co-polyesters containing increasing levels of glycerol (0.4 %wt for PG0.4 to 6.2% wt for PG6.2)


Alin Bostan

Fernando Chamizo

Mikael Persson Sundqvist

On an Integral Identity

We give three elementary proofs of a nice equality of definite integrals, recently proven by Ekhad, Zeilberger and Zudilin. The equality arises in the theory of bivariate hypergeometric functions, and has connections with irrationality proofs in number theory. We furthermore provide a generalization together with an equally elementary proof and discuss some consequences.

INTRODUCTION.

The following infinite family of equalities between definite integrals was proven by S. B. Ekhad, D. Zeilberger, and W. Zudilin in [START_REF] Ekhad | Two definite integrals that are definitely (and surprisingly!) equal[END_REF], using the Almkvist-Zeilberger creative telescoping algorithm [START_REF] Almkvist | The method of differentiating under the integral sign[END_REF] for symbolic integration:

1 0 x n (1 -x) n ((x + a) (x + b)) n+1 dx = 1 0 x n (1 -x) n ((a -b) x + (a + 1) b) n+1 dx, (1)
for any reals a > b > 0 and any nonnegative integer n.

As pointed out in [START_REF] Ekhad | Two definite integrals that are definitely (and surprisingly!) equal[END_REF], these integrals "are not taken from a pool of no-one-cares analytic creatures": they are related to rational approximations to some logarithmic values [START_REF] Alladi | Legendre polynomials and irrationality[END_REF] and a trained eye could recognize in identity (1) a particular case of a known relation for Appell's bivariate hypergeometric function (see Section 5 below).

The proof provided in [START_REF] Ekhad | Two definite integrals that are definitely (and surprisingly!) equal[END_REF] is, without any doubt, elementary. It requires some clever (and at first sight, magic) auxiliary rational functions coming from the silicon guts of the first author of [START_REF] Ekhad | Two definite integrals that are definitely (and surprisingly!) equal[END_REF]. Of course, this is not objectable at all, and we are ourselves convinced that computer-assisted proofs are an increasingly important trend in mathematics. At the same time we think that, pour l'honneur de l'esprit humain, it is of some interest to offer a proof that a first-year undergraduate could not only follow but also create by hand.

In Sections 2-4 of this article we provide three elementary proofs of the identity (1). We also generalize the identity. In Section 5 we discuss some direct consequences, emphasizing the relation of the identities with known identities for hypergeometric functions. We also give a couple of combinatorical identities, and finally conclude that the Legendre polynomials are eigenfunctions of a certain differential operator.

FIRST PROOF: USING A RATIONAL CHANGE OF VARIABLES.

Note that the natural range to assure the convergence of the integrals is a, b > 0 and n ∈ R >-1 . Let us first demonstrate that an utterly simple change of variables proves the equality in this extended range and in fact an even more general equality:

Theorem 1. If a, b > 0, k, n ∈ R, and s, ∈ R >-1 , then 1 0 x (1 -x) s (x + a) k+1 (x + b) n+1 dx = (b + 1) s-n b n- 1 0 x s (1 -x) (x + b) n+k--s ((a -b) x + (a + 1)b) k+1 dx. (2)
Proof. Consider the rational change of variables

x = b(1 -u) b + u , and consequently dx = - b(1 + b) (b + u) 2 du.
Note that it is a promising change because it takes 0, 1, b(a + 1)/(b -a), and ∞, which are the singularities of the second integrand, into 1, 0, -a, and -b, respectively,

January 2014] ON AN INTEGRAL IDENTITY
which are singularities of the first integrand. The interval [0, 1] is preserved. Thus, a direct calculation shows that the integral in the left-hand side is

I = 1 0 b(1-u) b+u 1 -b(1-u) b+u s b(1-u) b+u + a k+1 b(1-u) b+u + b n+1 b(1 + b) (b + u) 2 du = 1 0 (1 -u) u s b +1 (1 + b) s+1 (u + b) k+n--s du b(1 -u) + a(b + u) k+1 b(1 -u) + b(b + u) n+1 = (b + 1) s-n b n- 1 0 u s (1 -u) (b + u) k+n--s (a -b)u + (a + 1)b k+1 du,
which proves the result.

We will come back to (2) and draw some conclusions from this more general identity in Section 5.

SECOND PROOF: USING INDEFINITE INTEGRATION.

We still consider the extended convergence range a, b > 0, but now n is a nonnegative integer. The following proof is based on the generating functions [START_REF] Wilf | ALIN BOSTAN is a computer scientist and mathematician working at Inria in France. His main research field is symbolic computation, notably the design of efficient algebraic algorithms. He blends an experimental mathematics approach with fast computer algebra algorithms[END_REF] of the sequence of integrals when n varies.

Proof of (1) for a, b > 0 and n ∈ Z ≥0 . Let I 1 and I 2 be the generating functions of each side of (1), i.e., multiplying by t n and summing from n = 0 to ∞. They clearly converge uniformly for small values of t and we have I j (t) = 1 0 dx/P j (x, t) where

P 1 (x, t) = (x + a)(x + b) -tx(1 -x) and P 2 (x, t) = (a -b)x + (a + 1)b -tx(1 -x). (3)
If P (x) = Ax 2 + Bx + C has no zero in [0, 1], and ∆ = B 2 -4AC > 0, then, using standard integration techniques, we obtain

1 0 dx P (x) = 1 r log B + 2C + r B + 2C -r with r = √ ∆.
Note that for t small enough, P 1 and P 2 fulfill these conditions. A calculation shows that both polynomials have the same discriminant ∆ and the same values of B + 2C.

Hence I 1 (t) = I 2 (t) in some interval containing the origin and then the integrals in [START_REF] Alladi | Legendre polynomials and irrationality[END_REF], which are their Taylor coefficients, are equal.

4. THIRD PROOF: CREATIVE TELESCOPING. We cannot resist the temptation to offer a third proof, in the spirit of the one in [START_REF] Ekhad | Two definite integrals that are definitely (and surprisingly!) equal[END_REF], but based on a different kind of "creative telescoping." The starting point is the same as in Section 3, namely that the family of identities ( 1) is equivalent to the fact that the two integrals I 1 (t) and I 2 (t) between x = 0 and x = 1 of the rational functions F 1 = 1/P 1 and F 2 = 1/P 2 , with P j as in (3), are equal.

Creative telescoping [START_REF] Almkvist | The method of differentiating under the integral sign[END_REF], this time in the classical "differential-differential" setting, shows that F 1 and F 2 satisfy the equalities1

(t -2ab -a -b)F j + t 2 -2t(2ab + a + b) + (a -b) 2 ∂F j ∂t + ∂ ∂x (F j R j ) = 0,
where R 1 (t, x) and R 2 (t, x) are the rational functions

R 1 (t, x) = (a + b + t + 2)x + 2ab + a + b -t x and R 2 (t, x) = (a -b) 2 -(2ab + a + b)t x + b(a + 1)(a -b + t) t .
Hence, by integration between x = 0 and x = 1, one obtains that both I 1 and I 2 are solutions of the differential equation

(t -2ab -a -b)I(t) + t 2 -2t(2ab + a + b) + (a -b) 2 I (t) + 2 = 0.
Therefore I 1 = I 2 by Cauchy's theorem, since I 1 -I 2 is the solution of a differential equation of first order with leading term nonvanishing at t = 0, and its evaluation at t = 0 is zero, as

I 1 (0) = I 2 (0) = 1 a-b ln a(b+1) (a+1)b .
5. SOME CONSEQUENCES.

Appell's identity. We will show that identity (2) from our Theorem 1 contains, as a particular case, a classical hypergeometric identity due to Appell. The relevant definitions needed to state this identity are the classical Gauss hypergeometric function 2 F 1 (α, β; γ; t) and the Appell bivariate hypergeometric function F 1 (α; β, β ; γ; x, y), given respectively by

∞ n=0 (α) n (β) n (γ) n t n n! and m,n≥0 (α) m+n (β) m (β ) n (γ) m+n x m y n m!n! ,
where (a) n denotes the rising factorial a(a + 1) • • • (a + n -1) for n ∈ N and it is assumed |t|, |x|, |y| < 1 to assure the convergence.

It is very classical that these hypergeometric functions admit the following integral representations, which hold as soon as β, β > 0 and γ := β + β > α > 0:

2 F 1 (α, β; γ; z) = Γ(γ) Γ(β)Γ(β ) 1 0 t β-1 (1 -t) β -1 (1 -tz) α dt (4)
and

F 1 (α; β, β ; γ; x, y) = Γ(γ) Γ(α)Γ(γ -α) 1 0 t α-1 (1 -t) γ-α-1 (1 -tx) β (1 -ty) β dt. (5)
Equation ( 4) is due to Euler [3, Theorem 2.2.1], and (5) to Picard [START_REF] Picard | Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques[END_REF]; see also [4, equation (9)]. With these notations, we are able to state the following hypergeometric function identity, which appears on page 8 of Appell's classical memoir [START_REF] Appell | Sur les fonctions hypergéométriques de plusieurs variables, les polynômes d'Hermite et autres fonctions sphériques dans l'hyperespace[END_REF]:

Corollary 2 (Appell's identity). If α, β, β > 0, β + β > α, |x| < 1, and |y| < 1 then 2 F 1 α, β; β + β ; y -x y -1 = (1 -y) α F 1 (α; β, β ; β + β ; x, y). (6)
January 2014]

ON AN INTEGRAL IDENTITY

Equivalently, in terms of integrals:

1 0 t β-1 (1 -t) β -1 ((y -x) t + 1 -y) α dt = Γ(β)Γ(β ) Γ(α)Γ(β + β -α) 1 0 t α-1 (1 -t) β+β -α-1 (1 -tx) β (1 -ty) β dt. (7)
Remark 3. Identity (1) is a particular case of [START_REF] Picard | Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques[END_REF], with α = β = β = n + 1.

Proof of Corollary 2. If we set n = + s -k in (2), then the identity becomes

1 0 x (1 -x) s (x + a) k+1 (x + b) +s-k+1 dx = (b + 1) k- b s-k 1 0 x s (1 -x) ((a -b) x + (a + 1)b) k+1 dx. ( 8
)
Next we replace the integration variable x with 1 -t in the first integral and with t in the second integral, then a with -1/x and b with -1/y, to deduce the following equivalent form for x, y < 0:

1 0 t (1 -t) s 1 -y-x y-1 t k+1 dt = (1 -y) +1 1 0 t (1 -t) s (1 -tx) k+1 (1 -ty) +s-k+1 dt.
This last integral can be analytically continued to the values of x and y as in the statement. Setting = α -1, k = β -1, and s = β + β -α -1, using the integral representations ( 4) and ( 5) and the obvious symmetry 2 F 1 (α, β; γ; t) = 2 F 1 (β, α; γ; t), we deduce [START_REF] Petkovšek | A = B[END_REF].

The close relationship between ( 8) and the univariate and bivariate hypergeometric functions is shown in the previous proof of Corollary 2. Taking into account that ( 7) is a formulation of this result not involving any hypergeometric function, it appears as a natural problem to provide a more direct proof of [START_REF] Picard | Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques[END_REF]. We present an independent proof involving basic real and complex analysis.

Alternative proof of [START_REF] Picard | Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques[END_REF]. We write γ = β + β as before. Changing t → 1 -t in the first integral of [START_REF] Picard | Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques[END_REF] and multiplying the identity by Γ(α)Γ(γ -α), we want to prove that the functions

G 1 (x, y) = Γ(α)Γ(γ -α) 1 0 t γ-β-1 (1 -t) β-1 (1 -ty -(1 -t)x) α dt and G 2 (x, y) = Γ(β)Γ(γ -β) 1 0 t α-1 (1 -t) γ-α-1 (1 -tx) β (1 -ty) γ-β dt coincide.
Both are analytic functions of x and y on Re(x), Re(y) < 1/2 (for instance by Morera's theorem applied separately in x and y). Then it is enough to prove

∂ n+m G 1 ∂x n ∂y m (0, 0) = ∂ n+m G 2 ∂x n ∂y m (0, 0) for every m, n ∈ Z ≥0 ,
because in this case their Taylor coefficients coincide. Noting that the kth derivative of

(1 -x) -δ is Γ(δ+k) Γ(δ) (1 -x) -δ-k
, we see that this is the same as

Γ(γ -α)Γ(α + m + n)B(γ + m -β, β + n) = Γ(β + n)Γ(γ + m -β)B(α + m + n, γ -α) with B(p, q) = 1 0 t p-1 (1 -t) q-1 dt.
Using the well-known [9, Ex. 6.3.7] elementary evaluation B(p, q) = Γ(p)Γ(q)/Γ(p + q), we complete the proof.

Combinatorial identities and Legendre polynomials. The equality [START_REF] Alladi | Legendre polynomials and irrationality[END_REF] and its generalizations can be interpreted in combinatorial terms. If we set a = b > 0, = s = n, and replace k by k -1, then we get

1 0 x n (1 -x) n (x + b) n+k+1 dx = 1 b k (b + 1) k 1 0 x n (1 -x) n (x + b) n-k+1 dx. (9)
Corollary 4. The functions f n (x) = x n (1 + x) n with n a nonnegative integer satisfy

f k f (n+k) n = (n + k)! (n -k)! f (n-k) n for 0 ≤ k ≤ n.
Proof. By Taylor expansion at x = -b, we have By essentially comparing coefficients, one gets a triple binomial identity:

x n (1 -x) n = (-1) n f n (-x) =
Corollary 5. For each k, , n nonnegative integers with , k ≤ n, k m=0 k m n m + 2m + 2 k + n = n 2 n -k .
Remark 6. This hypergeometric identity can be rewritten as

2 n + k 3 F 2 -k, -n, + 1
2 ; - k + n 2 + 1, 1 -n -k 2 + ; 1 = 2 n -k ,
so it is a particular case of the Pfaff-Saalschütz identity [8, §2.3.1] for the evaluation at 1 of well-poised 3 F 2 's. Alternatively, it can be automatically obtained using Zeilberger's creative telescoping algorithm [START_REF] Petkovšek | A = B[END_REF].

Proof. In Corollary 4 change the variable x to (x -1)/2 to obtain, after clearing denominators,

(n -k)!(x 2 -1) k d n+k dx n+k (x 2 -1) n = (n + k)! d n-k dx n-k (x 2 -1) n . (10)
We rewrite the left-hand side as

(n -k)! 1 (-1) k-1 k k -1 x 2 1 • (n + k)! 2 (-1) n-2 n 2 2 2 n + k x 2 2 -n-k ,
and the right-hand side as

(n + k)!(n -k)! (-1) n-n 2 n -k x 2 -n+k .
January 2014] Comparing coefficients and renaming 1 = k -m, 2 = m + yields the result.

Remark 7. It turns out that ( 10) is a known identity expressing a symmetry of the associated Legendre polynomials. This can be obtained from Rodrigues' formula and from the general Legendre equation; see [START_REF] Westra | Identities and properties for associated Legendre functions[END_REF] for a simple elementary proof. The one obtained here (Corollary 4) is comparatively simple.

The definition of the Legendre polynomials ensures that they are eigenfunctions of the differential operator d dx (x 2 -1) d dx . Although d dx and (x 2 -1) do not commute and apparently there is not a simple formula for the commutator of their powers, Legendre polynomials are also eigenfunctions of a simple operator composed by powers of these operators. Proof. Recalling that P n is proportional to d n dx n (x 2 -1) n (Rodrigues' formula), the result follows from taking the kth derivative of [START_REF] Westra | Identities and properties for associated Legendre functions[END_REF].

Actually, one could prove Corollary 4 from Corollary 8 by repeated integration, and noting that both sides in Corollary 4 are divisible by x k .

c

  m (x + b) m with c m = (-1) n+m f (m) n (b) m!.If we substitute this in (9), a term with log(b + 1) -log b appears in the left-hand side for m = n + k (the other terms are rational functions of b) and in the integral of the right-hand side for m = n -k. Hence c n+k = c n-k / b(b + 1) k .

Corollary 8 .

 8 Let P n be the nth Legendre polynomial. Then, for 0 ≤ k ≤ n,L[P n ] = (n + k)! (n -k)! P n where L = d k dx k (x 2 -1) k d k dx k .

We have obtained these equalities with the help of the commands Zeilberger and creative telescoping available in the Maple packages DEtools and Mgfun.

c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121

ACKNOWLEDGMENTS. We are grateful to the reviewers for their kind and constructive remarks. A. Bostan was supported in part by the ANR DeRerumNatura project, grant ANR-19-CE40-0018.