Xuelian Liu

Nicolas Desilles

Laurent Lebrun
email: laurent.lebrun@univ-rouen.fr

Polyesters from renewable 1,4:3,6-dianhydrohexitols for food packaging: synthesis, thermal, mechanical and barrier properties

Keywords: Bio-based polyesters, Semicrystalline films, Food packaging, Barrier properties, Structure-property relationship

In the present work, biodegradable polyesters were synthesized by polycondensation reactions of renewable 1,4:3,6-dianhydrohexitols (isosorbide, IS; isomannide, IM) with aliphatic dicarboxylic acid chlorides containing 4, 6 and 10 carbons (namely, C4, C6 and C10) using a non-solvent method. All polyesters showed a thermal stability up to 320 °C and the glass transition temperature varied from 0 to 68 °C. Films were prepared by hot pressing.

The semi-crystalline ISC10 and IMC10 films showed good mechanical properties with Young's modulus E =127±20 MPa, E =125±10 MPa and elongation at break ε = 19±5%, ε = 16±3%, respectively. The water sorption (liquid and vapor) of films was slow and low due to their crystalline micro-structures. The films were strongly barrier for gases with permeability coefficients PN2 = 0.09, PO2 = 0.26, PCO2 = 1.32 barrer for ISC10, and PN2 = 0.18, PO2 = 0.46, PCO2 = 2.04 barrer for IMC10. Furthermore, the high CO2 and O2 selectivity coefficients of ISC10 (αCO2/O2 = 5.1) and IMC10 (αCO2/O2 = 4.4) were favorable for the fruits and vegetables preservation. ISC10 film showed a better quality for food packaging in terms of mechanical and gas permeation properties compared to IMC10 film. A detailed discussion of the relationships between microstructures and sorption/permeation properties was conducted.

Introduction

Food packaging is an indispensable part of our daily lives, and its demand is still growing. Traditionally, food packaging plastics (PET, PP, PE) are prepared from petroleum.

The increase use of this limited resource aggravated global warming. At the Paris climate conference (COP21) in December 2015, 195 countries adopted the first-ever universal, legally binding global climate deal. The agreement set out a universal action plan to limit global warming with the reorientation of the world economy towards a low carbon model. Among the proposed solutions, one would be to use natural, biodegradable materials made from renewable resources. According to statistics, the total market of biodegradable materials exceeded $1.1 billion in 2018 and is predicted to reach $1.7 billion in 2023, where food packaging and disposable tableware are the main driving forces [1]. The biopolymers currently used for industrial applications are poly(hydroxyalkanoate) (PHA) and poly(lactic acid) (PLA) but they often show low mechanical properties and thermal stability [START_REF] Farah | Physical and mechanical properties of PLA, and their functions in widespread applications -A comprehensive review[END_REF][START_REF] Ten | Mechanical performance of polyhydroxyalkanoate (PHA)-based biocomposites[END_REF]. Thus, the preparation of new bio-based materials, semi-crystalline, with better mechanical and thermal properties than PLA and PHA is scientific, industrial and environmental challenges.

The progresses in polymer science suggest to synthesize new bio-based polymers [START_REF] Fang | Mild synthesis of environment-friendly thermoplastic triblock copolymer elastomers through combination of ring-opening and RAFT polymerization[END_REF][START_REF] Kobayashi | Green polymer chemistry: new methods of polymer synthesis using renewable starting materials[END_REF][START_REF] Lavilla | High Tg bio-based aliphatic polyesters from bicyclic-mannitol[END_REF][START_REF] Zia | Recent developments and future prospects on bio-based polyesters derived from renewable resources: A review[END_REF].

However, thermal, mechanical and biodegradable properties of these new polymers must be studied before considering the replacement of conventional polymers [START_REF] Hu | Sustainable and rapidly degradable poly(butylene carbonate-co cyclohexanedicarboxylate): Influence of composition on its crystallization, mechanical and barrier properties[END_REF][START_REF] Kim | Sustainable terpolyester of high Tg based on bio heterocyclic monomer of dimethyl furan-2,5-dicarboxylate and isosorbide[END_REF][START_REF] Ouyang | A facile method to synthesize bio-based and biodegradable copolymers from furandicarboxylic acid and isosorbide with high molecular weights and excellent thermal and mechanical properties[END_REF]. Naturally bifunctional and thermally stable monomers can be extracted from biomass. Among them, 1,4:3,6-dianhydrohexitols (isosorbide, IS; isomannide, IM; isoidide, II) derived from cereals can be used as monomers for synthesizing bio-based polyesters [START_REF] Fenouillot | Polymers from renewable 1,4:3,6-dianhydrohexitols (isosorbide, isomannide and isoidide): A review[END_REF][START_REF] Galbis | Synthetic polymers from sugar-based monomers[END_REF][START_REF] Isikgor | Lignocellulosic Biomass: A sustainable platform for production of biobased chemicals and polymers[END_REF][START_REF] Naves | Enzymatic syntheses of unsaturated polyesters based on isosorbide and isomannide[END_REF][START_REF] Rose | Isosorbide as a renewable platform chemical for versatile applications-quo vadis?[END_REF].

As natural, chiral, rigid and non-toxic chemicals with functionalities, 1,4:3,6dianhydrohexitols attracted extensive interests in medical [START_REF] Taylor | Combination of isosorbide dinitrate and hydralazine in blacks with heart failure[END_REF][START_REF] Cohn | A Comparison of enalapril with hydralazine-isosorbide dinitrate in the treatment of chronic congestive heart failure[END_REF][START_REF] Cotter | Randomised trial of high-dose isosorbide dinitrate plus low-dose furosemide versus high-dose furosemide plus low-dose isosorbide dinitrate in severe pulmonary oedema[END_REF][START_REF] George | Spectroscopic (FT-IR, FT Raman) and quantum mechanical study on isosorbide mononitrate by density functional theory[END_REF][START_REF] Redfield | Isosorbide mononitrate in heart failure with preserved ejection fraction[END_REF][START_REF] Blair | Tolerability, safety and intermediary pharmacological effects of cilostazol and isosorbide mononitrate, alone and combined, in patients with lacunar ischaemic stroke: The LACunar Intervention-1 (LACI-1) trial, a randomised clinical trial[END_REF], chemical [START_REF] Dussenne | Synthesis of isosorbide: An overview of challenging reactions[END_REF][START_REF] Cao | Isosorbide production from sorbitol over porous zirconium phosphate catalyst[END_REF][START_REF] Kobayashi | Dehydration of sorbitol to isosorbide over H-beta zeolites with high Si/Al ratios[END_REF][START_REF] Yamaguchi | One-pot conversion of cellulose to isosorbide using supported metal catalysts and ion-exchange resin[END_REF][26] and polymer sciences [START_REF] Fenouillot | Polymers from renewable 1,4:3,6-dianhydrohexitols (isosorbide, isomannide and isoidide): A review[END_REF][START_REF] Saxon | Next-generation polymers: Isosorbide as a renewable alternative[END_REF][START_REF] Feng | Overview of advances in sugar-based polymers[END_REF][START_REF] Juais | Isosorbide polyesters from enzymatic catalysis[END_REF]. Since 1984, 1,4:3,6-dianhydrohexitols have been used to synthesize polyesters but only IS and IM were widely studied due to their commercial availability [START_REF] Thiem | Darstellung und gezielte polykondensation von anhydroalditolbausteinen aus stärke[END_REF]. Numerous patents and large amount of research articles based on these platforms, especially isosorbide, have been published. Within only the recent ten years, four review articles have covered all aspects of isosorbide chemistry [START_REF] Fenouillot | Polymers from renewable 1,4:3,6-dianhydrohexitols (isosorbide, isomannide and isoidide): A review[END_REF][START_REF] Rose | Isosorbide as a renewable platform chemical for versatile applications-quo vadis?[END_REF][START_REF] Saxon | Next-generation polymers: Isosorbide as a renewable alternative[END_REF][START_REF] Aricò | Isosorbide and dimethyl carbonate: A green match[END_REF]. In polymer chemistry, the sustainable, high-temperature resistant and biodegradable polymers based on 1,4:3,6-dianhydrohexitols are the ongoing hot topics. The combination of isosorbide with other renewable chemicals, such as aliphatic diacids and 2,5-furandicarboxylic acid to prepare thermosets have been reported as well [START_REF] Sousa | New unsaturated copolyesters based on 2,5-furandicarboxylic acid and their crosslinked derivatives[END_REF][START_REF] Dai | 2,5-Furandicarboxylic acid-and itaconic acid-derived fully biobased unsaturated polyesters and their cross-linked networks[END_REF][START_REF] Xu | Isosorbide as core component for tailoring biobased unsaturated polyester thermosets for a wide structure-property window[END_REF][START_REF] Ma | Hard and flexible, degradable thermosets from renewable bioresources with the assistance of water and ethanol[END_REF]. The industry has also developed commercial polycarbonates and polyesters based on isosorbide, such as Polysorb® PSA, LP and PA, produced by Roquette [START_REF]Isosorbide polycarbonate: POLYSORB® bisphenol a alternative, isosorbide based polycarbonate[END_REF]. However, the low reactivity of the secondary alcohols of IS and IM made it hard to obtain high molar mass polyesters. Chemical modifications to increase the reactivity of IS were tried but with complicated and costly steps [START_REF] Tamion | Synthesis of new chiral auxiliaries derived from isosorbide[END_REF]. General ways to obtain aliphatic polyesters by reacting a wide range of dicarboxylic acids, from 2 to 36 carbons, with IS or IM were presented [START_REF] Braun | 1,4:3,6-Dianhydrohexite als Bausteine für Polymere[END_REF][START_REF] Brandenburg | Process to produce polyesters which incorporate isosorbide[END_REF]. These polyesters were proved to be biodegradable [START_REF] Okada | Biodegradable polymers based on renewable resources. IV. Enzymatic degradation of polyesters composed of 1,4:3.6dianhydro-D-glucitol and aliphatic dicarboxylic acid moieties[END_REF][START_REF] Okada | Biodegradable polymers based on renewable resources: Polyesters composed of 1,4 : 3,6-dianhydrohexitol and aliphatic dicarboxylic acid units[END_REF] and compatible for using in implant biomedical materials [START_REF] Park | Synthesis and biocompatibility properties of polyester containing various diacid based on isosorbide[END_REF]. The tetrahydropyran rings were found to increase the chains rigidity and then the Tg [START_REF] Lavilla | High Tg bio-based aliphatic polyesters from bicyclic-mannitol[END_REF]. Multiple isosorbide-derived monomers for the synthesis of high thermal performance polyesters which could be used for hot-fill containers were also overviewed [START_REF] Feng | Overview of advances in sugar-based polymers[END_REF]. More recently, the effects of stereoisomerism of IS or IM and chain length of dicarboxylic acid units on the crystallization of the polyesters were investigated [START_REF] Marubayashi | Crystallization of polyesters composed of isohexides and aliphatic dicarboxylic acids: Effects of isohexide stereoisomerism and dicarboxylic acid chain length[END_REF][START_REF] Marubayashi | Crystal Polymorphism of Biobased polyester composed of isomannide and succinic acid[END_REF]. However, there is no investigation on gas and water permeation which is crucial to maintain the food shelf-life in packaging [START_REF] Schmid | Permeation of water vapour, nitrogen, oxygen and carbon dioxide through whey protein isolate based films and coatings-Permselectivity and activation energy[END_REF][START_REF] Hussein | Perforation-mediated modified atmosphere packaging of fresh and minimally processed produce-A review[END_REF].

In this work, bio-based polyesters were prepared by reacting IS or IM with aliphatic diacid chlorides (succinyl, adipoyl and sebacoyl chlorides) according to the melt polycondensation previously published by Okada et al. [START_REF] Okada | Biodegradable polymers based on renewable resources: Polyesters composed of 1,4 : 3,6-dianhydrohexitol and aliphatic dicarboxylic acid units[END_REF]. These polyesters were proved to be biodegradable [START_REF] Okada | Biodegradable polymers based on renewable resources. IV. Enzymatic degradation of polyesters composed of 1,4:3.6dianhydro-D-glucitol and aliphatic dicarboxylic acid moieties[END_REF][START_REF] Okada | Biodegradable polymers based on renewable resources: Polyesters composed of 1,4 : 3,6-dianhydrohexitol and aliphatic dicarboxylic acid units[END_REF]. The originality of this work was to prepare films and investigate their gas and water sorption/permeation properties, which are crucial for food packaging applications. Relationships between microstructures and sorption/permeation properties were established.

Experimental

Chemicals

Isosorbide (98%), isomannide (95%), succinyl chloride (95%), adipoyl chloride (98%), sebacoyl chloride (≥95%), ethylene glycol (> 99%) and decane (> 99%) were purchased from Sigma-Aldrich. Dichloromethane (reagent grade) was provided by Fisher Scientific.

Methanol (reagent grade) was supplied by VWR. All reagents were used as received without further purification.

Synthesis of the polyesters

IS (or IM) (60 mmol) was weighed into a 100 mL three-necked round-bottomed flask equipped with a mechanical stirrer and dipped into an oil bath (80 and 95 °C for IS and IM, respectively). Liquid succinyl chloride (60 mmol) was added when isosorbide was fully molten. The temperature was increased to 160 °C when the hydrogen chloride release decreased, and the mixture was kept under normal pressure for 6 h under N2 atmosphere.

Then, reduced pressure was applied: 50 mbar for 2 h, then 0.1 mbar for 15 h. After cooling to room temperature, the media was dissolved in dichloromethane (50 mL) and the polyester was precipitated in methanol (500 mL). The polymers were purified by 3 times repeated dissolution precipitation. Finally, they were dried for 2 days at 40 °C under reduced pressure.

All the polyesters were synthesized by the same way.

Preparation of the films

The films were processed by compression molding (thickness around 150 µm) using a hot press (Scamex, France) according to the following conditions: 5 min under 50 bars at different temperatures (given in Figure 5). The films were then stored at room temperature in a desiccator with P2O5 under room temperature for a constant time (60 days after hot pressing).

Characterization of the polymers 1.4.1 Polymer structure

The chemical structures were confirmed by FT-IR (VERTEX 70 from Bruker) and by 1 H NMR (300 MHz Bruker spectrometer) at room temperature in CDCl3. Chemical shifts (δ) were expressed in parts per million (ppm) and referenced to the internal deuterated solvents with tetramethylsilane as the internal standard.

Molar masses

The number average molar masses ( ) of the polyester samples were measured by steric exclusion chromatography (PL-GPC50 from Varian) at 27.5 °C. The solvent was dichloromethane and the analysis was carried out with PMMA calibration.

Thermogravimetric analysis (TGA)

The thermal decomposition of the polymers was investigated by TGA (TGA Q500 from TA Instruments) under nitrogen atmosphere. The sample (5-10 mg) was packed in a platinum pan and heated from 30 to 600 °C with a heating rate of 10 °C⋅min -1 .

Differential scanning calorimetry (DSC)

Differential Scanning Calorimetry measurements were performed with a DSC Q2000 from TA Instruments. The 5-10 mg samples were heated from -30 to 250 °C with 10 °C⋅min -1 heating rate. Tg were measured at the mid-point, Tm at the peak maximum and ∆Hm thanks to the melting peak area.

Characterization of films 1.5.1 Wide angle X-ray diffraction (WAXD)

The crystal structures and crystallinity (χc) were evaluated using an X-Ray diffractometer D5000 from Bruker AXS. The radiation was λCo = 1.789 Å powered by a current of 40 mA and a voltage of 40 kV. The data were acquired on a scanning range 2θ = 5°-55° with a scanning speed of 0.004° s -1 . Intensity correction, crystallinity determination and peak separation were performed using PeakFit ® software.

Polarized optical microscopy (POM)

Crystals in polyester films were observed using a DMLM Leica optical polarizing microscope. Image acquisition and analysis were performed with Archimed software.

Contact angle measurements

Contact angle (θ) and surface energy (γ) of films were determined at room temperature (23 °C), with three different liquids: water (MilliQ Millipore Water system), ethylene glycol and decane. The contact angle θ values were averaged from at least five measurements. The surface energy was calculated by Windrop++ Carrousel software according to Owens & Wendt method [START_REF] Owens | Estimation of the surface free energy of polymers[END_REF].

Mechanical properties

The tensile tests of polyester films were performed at room temperature (23 °C) using a ZwickRoell Z010 testing machine (TestXpert Ⅱ-V3.5 software). The load cell was 500 N, the crosshead speed was 10 mm⋅min -1 , and the distance between the grips was 40 mm. The dumbbell shaped specimens were 1BB type according to the ISO 527-2:2014-04 standard.

The tensile modulus (E) was determined in the linear part of the stress-strain curve. More than 10 tests were performed on each sample and the obtained values were averaged.

Liquid water sorption

The liquid water sorption measurements were performed at 25 °C by measuring the variation of the mass gain vs time of samples (initially dry) immersed in pure water (MilliQ Millipore Water system). Three measurements were carried out on each sample (A = 1 cm 2 ) using a balance (precision 0.1 mg). The mass gain at t time (Mt) was calculated according to Eq. 1.

M t = m -m d m d ×100 (Eq. 1)
with mt the film mass at t time and md the dry film mass. The accuracy on Mt values was estimated lower than 10%.

Water vapor sorption

Water vapor sorption measurements were conducted with the automatic gravimetric dynamic vapour sorption DVS1 Advantage (Surface Measurement Systems Ltd) as previously described [START_REF] Alix | Biocomposite materials from flax plants: Preparation and properties[END_REF]. Different water vapor activity steps (aw) ranging from 0 to 0.95 were applied at 25 °C on the film (m ≅ 15 mg). At each step, the mass gain was measured as a function of time until the equilibrium was reached. The mass gain at equilibrium Meq was calculated for each aw according to Eq. 2, which allowed the plotting of water vapor sorption isotherms.

M eq = m -m d m d ×100 (Eq. 2)
with meq the film mass at equilibrium.

The accuracy on the values was lower than 4%. Kinetic data were extracted from the normalized mass gain Mt/Meq vs reduced time (t 1/2 /L), where Mt is the mass gain at time t (Eq.1). Reduced time was used to allow data comparisons independently of the film thickness L.

Water vapor permeation

The water vapor permeation measurements were performed using the home-made permeation cell previously described [START_REF] Métayer | Diffusion of water through various polymer films: A new high performance method of characterization[END_REF]. The experimental conditions were: aw ≅ 0.20, 0.50, 0.55 and 0.60, T = 25 °C, surface of the film A = 3.6 cm 2 . The water vapor flux J through the film as a function of time t was measured. At steady state (J = Jst), the permeability coefficient P (expressed in g⋅m -1 ⋅d -1 ) was calculated from Eq. 3 [START_REF] Buquet | Permeation properties of poly(m-xylene adipamide) membranes[END_REF]:

P = J st × L ∆a w (Eq. 3)
where Δaw is the difference in water activity across the film.

Gas permeation

Carbon dioxide (CO2), oxygen (O2) and nitrogen (N2) permeation measurements were performed on films using the equipment previously described (25 °C, applied gas pressure 3 bar) [START_REF] Joly | Residual solvent effect on the permeation properties of fluorinated polyimide films[END_REF]. The quantity of transferred gas Q through the film was detected versus time t (s).

The permeability coefficient P (expressed in Barrer) was calculated according to Eq. 4:

P = L A ∆P dQ dt (Eq. 4)
with L and A, thickness and surface (11.34 cm 2 ) of the film; p ∆ , pressure difference between the two sides of the film; dQ/dt, slope of the experimental curve at stationary state.

The diffusion coefficient D (cm 2 ⋅s -1 ) was calculated from the time-lag tL value obtained from the extrapolation of the steady-state asymptote to the time axis [START_REF] Morontsev | A new approach to improvement of gas permeation properties of olefin metathesis derived poly(norbornenes): gemdifluorocyclopropanation of backbone double bonds[END_REF]:

D = L 2 6 t L (Eq. 5)
2

Results and discussion

Six aliphatic polyesters were synthesized by using a non-solvent polymerization according to Figure 1. The polyesters were identified as follows: the first two letters corresponded to the dianhydrohexitol used (IS for isosorbide and IM for isomannide), followed by CX, where X represented the number of carbon atoms coming from the aliphatic chain of the diacid chloride. The chemical structures of the polyesters were also characterized by 1 H NMR. The spectra of ISC10 and IMC10 were shown in Figure 2a and Figure 2b respectively, and the others were given in supplementary material (Figure S2). In Figure 2a

Molar masses

The polymer molar masses were determined by Steric Exclusion Chromatography (SEC) (Table 1). Except for IMC4, high yields were obtained and the molar masses ( ) of polyesters exceeded 10 000 g•mol -1 . The case of IMC4 can be ascribed to its high melting temperature around 173 °C (see section 2.1.4), higher than the reaction temperature (160 °C), which limited the polycondensation reaction [START_REF] Marubayashi | Crystallization of polyesters composed of isohexides and aliphatic dicarboxylic acids: Effects of isohexide stereoisomerism and dicarboxylic acid chain length[END_REF]. There is a general tendency that the molar mass increase with C atom number, probably due to the higher flexibility brought by the methylene units which would favor the motion of the growing chains in a very viscous bulk medium.

Thermal degradation

The thermal degradation of polyesters was investigated by thermal gravimetric analysis. The degradation curves were presented in Figure 3 and detailed data were recorded in Table S1. All polyesters showed a thermal stability up to 320 °C with a rapid decomposition around 400-450 °C. The degradation seemed to begin at higher temperatures when the length of the aliphatic diacid chain increased, with 5% weight loss temperature (T5%) values T5%(ISC4) < T5%(ISC6) < T5%(ISC10) and T5%(IMC4) < T5%(IMC6) < T5%(IMC10), which was probably due to the increased molar masses. Even if IMC10 showed the highest thermal stability, we cannot generally conclude that IM polyesters were more stable than IS homologues. It means that the diacid chain length has more influence on the thermal degradation than stereoisomerism. The results are consistent with literature values [START_REF] Park | Synthesis and biocompatibility properties of polyester containing various diacid based on isosorbide[END_REF][START_REF] Marubayashi | Crystallization of polyesters composed of isohexides and aliphatic dicarboxylic acids: Effects of isohexide stereoisomerism and dicarboxylic acid chain length[END_REF]. and the crystallization behavior of the polyesters. The first heating thermograms were presented in Figure 4 and detailed data were recorded in Table S2. It showed that only ISC10, IMC4 and IMC10 were semi-crystalline due to the presence of melting peaks. These melting peaks were complex, and could reveal not only melting / crystallization phenomena but more certainly crystalline polymorphism (as evidenced later from XRD patterns in Figure 7), as already observed in such polymers before [START_REF] Marubayashi | Crystallization of polyesters composed of isohexides and aliphatic dicarboxylic acids: Effects of isohexide stereoisomerism and dicarboxylic acid chain length[END_REF]. However, the crystals cannot be obtained back upon cooling (Figure S3) probably due to the very slow crystallization rate. These results are in good agreement with those obtained by Marubayashi et al. [START_REF] Marubayashi | Crystallization of polyesters composed of isohexides and aliphatic dicarboxylic acids: Effects of isohexide stereoisomerism and dicarboxylic acid chain length[END_REF] , except for IMC4, probably due to the low molar mass in our case. The Tg values decreased with the increase of the aliphatic carbon number, which is quite logical since more CH2 in the chain induced more flexibility and free volume. The highest Tg (65 °C) was observed for IMC4 and the lowest (0 °C) for IMC10. For a given aliphatic chain, considering almost the same molar masses of ISC10 and IMC10, it seemed that IM and IS polyesters showed only slight difference on Tg value, but larger differences on crystallinity. It means that the endo-exo and endo-endo stereoscopic difference affected the crystallization rather than the chain relaxation [START_REF] Marubayashi | Crystallization of polyesters composed of isohexides and aliphatic dicarboxylic acids: Effects of isohexide stereoisomerism and dicarboxylic acid chain length[END_REF].

Preparation of polymer films

The polymer films shown in Figure 5 were prepared by hot pressing under different temperatures depending on their thermal properties established by DSC. Since the obtained polymers could not recrystallize upon cooling, and the crystalline phase is well known to bring barrier properties, the films were sometimes prepared with partial melting.

Unfortunately, ISC4, ISC6, IMC4 and IMC6 films appeared too brittle, which was probably due to their short methylene units and lower molar masses. Only ISC10 and IMC10 films with

DSC characterization of polyester films

The melting behaviors of polyester films were investigated by DSC and compared to the corresponding polyesters in Figure 6. Both ISC10 and IMC10 films showed broad but less complex melting peaks compared to the corresponding polyesters. Furthermore, it seemed the initial melting temperature increased, which can correspond to an annealing process occurring during hot pressing and storage. The melting enthalpy (ΔHm) of ISC10 film was 26.1 J⋅g -1 , which is higher than that of ISC10 (ΔHm = 18.7 J⋅g -1 ), but the melting enthalpy of IMC10 (ΔHm = 15.9 J⋅g -1 ) and IMC10 film (ΔHm = 16.0 J⋅g -1 ) were almost the same. Despite the partial melting during film making, and the fact that no crystallization was observed during cooling in DSC for both polyesters, the recrystallization could effectively happen due to the annealing process during the hot pressing and the 60 days under room temperature storage.

ISC10 film was more crystalline with a higher melting temperature than IMC10 film.

Figure 6 DSC first heating of ISC10 and IMC10 polyesters and films 2. Two obvious diffraction peaks located at 2θ~6.5° and 2θ~22.5° corresponded to d1 = 1.60 nm and d2 = 0.46 nm respectively for both polyester films. This result was consistent with Marubayashi et al. [START_REF] Marubayashi | Crystallization of polyesters composed of isohexides and aliphatic dicarboxylic acids: Effects of isohexide stereoisomerism and dicarboxylic acid chain length[END_REF]. The dspacing d1 corresponded to the spacing parallel to the chain direction and d2 was that perpendicular to the chain direction (chain-chain packing). The crystallinity of ISC10 film (χc = 36.3%) is higher than IMC10 film (χc = 25.9%): ISC10 is more crystallized by chain-chain packing than IMC10. Besides, the diffractometric patterns show scattering peaks at very low 2θ values (5-10°) which can be related to liquid crystal regions as reported by Ding et al. [START_REF] Ding | Smectic liquid crystal Schlieren texture in rapidly cooled poly(butylene naphthalate)[END_REF].

XRD characterization of polyester films

Mechanical properties

The mechanical properties of ISC10 and IMC10 were gathered in Table 3. ISC10 had a better resistance to stretch than IMC10. The Young's modulus values of ISC10 and IMC10

were similar (127 ± 20 MPa and 125 ± 10 MPa respectively). Comparing the ultimate tensile strength and elongation at break, ISC10 (σb= 4.2 ± 0.8 MPa, εb = 19 ± 5 %) seemed tougher than IMC10 (σb= 1.0 ± 0.2 MPa, εb = 16 ± 3 %). Since their molar masses and Tg were very similar, it seemed that the higher crystallinity and larger crystal size of ISC10 brought more resistance at break. Both ISC10 and IMC10 had more toughness than PLA (elongation at break less than 10%) [START_REF] Farah | Physical and mechanical properties of PLA, and their functions in widespread applications -A comprehensive review[END_REF]. However, our ISC10 films were more rigid and more brittle than

Contact angle measurements

The surface energy (with dispersity and polar parts) and water contact angle values of polyester films were presented in Table 4. The surface energy values for ISC10 (31.0 mN⋅m -1 ) and IMC10 (29.6 mN⋅m -1 ) were similar. The water contact angle values, 85 ± 1.0° for ISC10 and 83 ± 0.5° for IMC10, highlighted the hydrophobicity of the films. The results showed almost no difference between ISC10 and IMC10, indicating that the surfaces properties of the films were not influenced by the stereoisomerism of IS and IM. The water contact angles of ISC10 and IMC10 were lower than the common used packaging materials PE (120 ± 0.1°) [START_REF] Kim | Plasma-modified polyethylene membrane as a separator for lithium-ion polymer battery[END_REF], PP (97 ± 0.4°) [START_REF] Thanakkasaranee | Preparation and characterization of polypropylene/sodium propionate (PP/SP) composite films for bread packaging application[END_REF], but comparable to semi-crystalline polyesters like PET (83.8 ± 1.0°) [START_REF] Gotoh | Wettability characteristics of poly(ethylene terephthalate) films treated by atmospheric pressure plasma and ultraviolet excimer light[END_REF] and PLA (80.0 ± 0.1° and 73.0 ± 2.0°) [START_REF] Farah | Physical and mechanical properties of PLA, and their functions in widespread applications -A comprehensive review[END_REF][START_REF] Fortunati | Investigation of thermo-mechanical, chemical and degradative properties of PLAlimonene films reinforced with cellulose nanocrystals extracted from Phormium tenax leaves[END_REF].

Liquid water sorption and water vapor sorption

Figure 9 showed that the sorption of liquid water is low whatever the sample. This behavior can be explained by the semi-crystalline micro-structures of the polyester films. The equilibrium was earlier reached (before 2800 min) for IMC10 than for ISC10 (after 2800 min), which meant IMC10 proceeded a quicker dissolution-diffusion process than ISC10. The sorption was less pronounced for ISC10 (Mt at equilibrium = Meq = 0.55 ± 0.05 %) than for IMC10 (Meq = 1.03 ± 0.04 %). This is due to the bigger size crystals and the higher crystallinity of ISC10, which bring more compact structure favorable for water barrier properties.

Figure 9 Liquid water sorption of ISC10 and IMC10 films

As observed before for liquid water sorption, water vapor sorption was less for ISC10 than for IMC10 (e.g. Meq = 0.58 % and Meq = 1.08 % at water activity aw = 0.95, respectively). Both ISC10 and IMC10 showed a constant sorption of water vapor at the lower aw then a larger increase (Figure 10a). The linear isotherm describes a Henry's type process corresponding to a random absorption (dissolution) and then a diffusion of the water molecules inside the film. This isotherm corresponded to the sorption of the water molecules in the free volume of the polymer. The slope of the straight line corresponds to the solubility coefficient S of water vapor in the film. ISC10 showed a constant sorption until aw = 0.8 (S = 5.3 x 10 -3 gH2O/gpol) and IMC10 until aw = 0.5 (S = 8.0 x 10 -3 gH2O/gpol). The convex part of the curve highlighted an adsorption of water molecules with formation of aggregates, which induced a larger swelling of the film. This accumulation of water molecules took place in the free volume of the amorphous phase, which means there are more free volume in IMC10 than in ISC10 films.

The kinetics of water sorption were presented by the normalized gain of mass (Mt/Meq) vs reduced time (t 1/2 L -1 ) for different vapor activities (Figure 10b). It is obvious that IMC10 showed a quicker dissolution-diffusion process of water vapor than ISC10, which agreed with the liquid water sorption. A slowing-down process was observed at high aw.

Water vapor permeation

Water vapor permeability coefficients (P) of ISC10 and IMC10 are given Table 5 for different water activities. ISC10 film showed the lowest P values whatever aw (e.g. P = 1.55 g⋅m -1 ⋅d -1 and P = 2.81 g⋅m -1 ⋅d -1 at aw ≅ 0. 5 for ISC10 and IMC10, respectively). These results are in accordance with those previously obtained by sorption measurements. The lowest P values obtained for ISC10 could be related to the lowest solubility coefficients S and diffusion coefficients D (P = D x S). Considering the same elements composition of ISC10 and IMC10, it could be concluded that the micro-structure of the ISC10 film brought more barrier for water molecules. ISC10 showed a comparable water vapor barrier referred to PLA (25 o C, aw=0.5, P= 1.39-1.63 g⋅m -1 ⋅d -1 ) [START_REF] Auras | Mechanical, physical, and barrier properties of poly(lactide) films[END_REF]. However, ISC10 was less barrier than PET (21 o C, aw=0.4, P= 0.47 g⋅m -1 ⋅d -1 ) [START_REF] Cava | Comparative performance and barrier properties of biodegradable thermoplastics and nanobiocomposites versus PET for food packaging applications[END_REF], which is probably due to its low glass transition temperature and the presence of the "Oxygen" in the IS molecular ring in the amorphous phase.

Gas permeation measurements

Table 6 presented the values of permeability (P) and diffusion (D) coefficients of ISC10 and IMC10 films. It could be easily observed that both ISC10 and IMC10 showed low gas permeability, which is probably due to the presence of crystals and liquid crystal regions, which are strongly barrier to gases [START_REF] Weinkauf | Gas transport properties of thermotropic liquid-crystalline copolyesters. II. The effects of copolymer composition[END_REF][START_REF] Kajiyama | Molecular motion-permeability relationships in polycarbonate/liquid crystal (EBBA) composites membrane[END_REF][START_REF] Hoshikawa | Liquid crystal display with barrier layer to reduce permeability[END_REF]. ISC10 showed higher barrier properties than IMC10 due to its higher crystallinity, as previously explained for water. The permeability coefficients P of ISC10 for N2, O2 and CO2 were 0.09, 0.26 and 1.32 barrer, respectively, which is similar to PLA (PO2= 0.16 barrer, PCO2 = 1.27 barrer) [START_REF] Ortenzi | Evaluation of crystallinity and gas barrier properties of films obtained from PLA nanocomposites synthesized via "in situ" polymerization of L-lactide with silanemodified nanosilica and montmorillonite[END_REF]. ISC10 and IMC10 appeared both more barrier to gases than the commonly used food packaging material PE (PO2=3.2 barrer, PCO2=11.7 barrer) [START_REF] Gholizadeh | Gas permeability measurement in polyethylene and its copolymer films[END_REF] but less barrier than PET (PN2 = 0.008 barrer, PO2= 0.03 barrer, PCO2 = 0.15 barrer) [START_REF] Gontard | Les emballages actifs[END_REF] and semi-crystalline PEF (PO2= 0.004~0.006 barrer, PCO2 = 0.01~0.02 barrer) [START_REF] Burgess | Penetrant transport in semicrystalline poly(ethylene furanoate)[END_REF]. To optimize a packaging, the selectivity [START_REF] Budd | Highly permeable polymers for gas separation membranes[END_REF] between CO2 and O2 (αCO2/O2) should be controlled because the respiration rate of fruits and vegetables should be slowed down. Thus, it is necessary to limit the oxygen content while increasing the concentration of carbon dioxide to keep the vegetables fresh as long as possible [START_REF] Bideau | Polypyrrole/nanocellulose composite for food preservation: Barrier and antioxidant characterization, Food Packag[END_REF].

However, the packaging must be a little permeable to oxygen because a too low concentration of oxygen triggers an anaerobic breathing process which induces bad taste and bad odors to vegetables, and then decrease their shelf-life. The selectivity coefficient αCO2/O2 for ISC10 (αCO2/O2= 5.1) was comparable to petroleum-based common food packaging polyester PET (αCO2/O2= 5.0) [START_REF] Gontard | Les emballages actifs[END_REF], but larger than for IMC10 (αCO2/O2= 4.4), bio-based potential food packaging polyester PLA (αCO2/O2= 1.0~2.0) [START_REF] Genovese | Design of biobased PLLA triblock copolymers for sustainable food packaging: Thermo-mechanical properties, gas barrier ability and compostability[END_REF] and PEF (αCO2/O2 ≅ 3) [START_REF] Burgess | Penetrant transport in semicrystalline poly(ethylene furanoate)[END_REF], which indicated that ISC10 is more favorable for the fruits and vegetables preservation.

The diffusion coefficient D was lower for ISC10 than for IMC10 whatever the gas (Table 6). These results were consistent with those obtained with water: the higher crystallinity and bigger crystals in ISC10 increased the tortuosity of the diffusion pathway, and also meant less free volume and less exposure of oxygen atoms in the ISC10 film, which could explain the lowest solubility to O2 and CO2 gases (S=P/D). P expressed in barrer (1 barrer = 10 -10 cm 3 STP⋅cm⋅cm -2 ⋅s -1 ⋅cmHg -1 ), D expressed in cm 2 ⋅s -1 .

Relationship between microstructure and permeation properties

As discussed before, the barrier properties of ISC10 and IMC10 may result not only from their semi-crystalline nature but also from the presence of liquid crystal regions, since these polymers contain rigid moieties (IS and IM) alternating with flexible PE-like sequences, which have a deep impact on barrier properties. The difference in mechanical and barrier properties for ISC10 and IMC10 films mainly depends on their crystal structures and crystallinity. The difference of crystallization ability between ISC10 and IMC10 can be attributed to the chain orientation degree as shown in Figure 11, where A refers to the exolinks and B refers to the endo-links. It is obvious that ISC10 repetition unit contains three kinds of links: exo-endo (A-C10-B), exo-exo (A-C10-A) and endo-endo (B-C10-B), while IMC10 repetition unit contains only endo-endo (B-C10-B) links. As predicted by Chemdraw ® Software, IMC10 has more regular bond connections in a short range that can crystallize, but the highly curved chains make it hard to crystallize by chain-chain packing. On the contrary, the exo-links in ISC10 induces a more oriented chain which can readily crystallize and give a more dense microstructure [START_REF] Morillon | Factors affecting the moisture permeability of lipid-based edible films: A review[END_REF][START_REF] Liu | Orientation-induced crystallization of isotactic polypropylene[END_REF]: the more the exo-exo links are repeated in the chain the easier the chains crystallize by chain-chain packing. This difference in chain orientation explained why ISC10 showed better barrier properties. Six aliphatic bio-based polyesters including isosorbide and isomannide were successfully synthesized by bulk polycondensation. The chain length of the diacid units influenced the degradation and glass transition temperature as well as the crystallization. Only polyesters that contained more CH2 repeat units (ISC10 and IMC10) showed good film forming properties. Crystallinity, with the presence of liquid crystal phase, was evidenced in both ISC10 and IMC10. The endo/exo stereoisomerism of isohexide units (IS or IM) slightly influenced the thermal degradation and glass transition temperature, but influenced the crystallization properties and then the mechanical and barrier properties. The exo-exo stereoscopic configurations in polyesters were more inclined to crystallize than the endo-endo configurations. Thus, polyesters prepared from isosorbide showed better tensile strength and better barrier properties to water and gases than isomannide polyesters. Furthermore, the ISC10 polyester showed more toughness and similar barrier properties for both water and gases compared to PLA, and was even more barrier to gases than the common food packaging material PE. Even if the barrier properties of ISC10 were lower than PET and PEF, the selectivity to CO2 and O2 was more favorable for fruits and vegetables preservation.

Isosorbide polyesters showed more advantages over isomannide polyesters in terms of price and permeation properties. Ongoing studies are focused on increasing glass transition and melting temperatures as well as crystallinity to enhance the thermomechanical and barrier properties of these promising food packaging materials.

Figure 1

 1 Figure 1 General route for the synthesis of aliphatic polyesters prepared from IS and IM with aliphatic dicarboxylic acid chlorides

  , the proton signals in the range from 5.22 to 3.68 ppm (8H) corresponded to isosorbide and the signals at 2.33 (4H), 1.61 (4H) and 1.29 (8H) ppm were attributed to methylene protons of sebacoyl chloride.Similarly in Figure2b, the signals in the range from 3.67 to 5.19 ppm (8H) were ascribed to isomannide and the signals at 2.37 (4H), 1.63 (4H) and 1.30 (8H) ppm belonged to sebacoyl chloride. The exo hydroxyl group connected with the second carbon (C-2) of isosorbide ring induced the irregular structure of ISC10 that resulted in stereo differences between ISC10 and IMC10, which could explain the peaks splited more in ISC10 spectrum.

Figure 2 1 H

 1 Figure 2 1 H NMR spectra of (a) ISC10 and (b) IMC10

Figure 3

 3 Figure 3 TGA curves of IS and IM polyesters

Figure 4 13 2. 2

 4132 Figure 4 DSC first heating of IS and IM polyesters

Figure 5

 5 Figure 5 IS and IM polyester films prepared by hot pressing

Figure 7

 7 Figure 7 XRD spectra of ISC10 and IMC10 polyester films

Figure 8

 8 Figure 8 Microscopy images of ISC10 film (a) and IMC10 film (b)

  those prepared by Park et al. [42] (E = 53 MPa and σb = 10.5 MPa). The differences may be due to the solvent casting technique used by Park et al. to prepare their films. With this technique, solvent molecules can be retained and plasticize the film.

  The detailed sorption kinetics data were interpreted in terms of rate of diffusion of water molecules by considering tr time to reach the equilibrium of sorption. For a short time (when Mt/Meq < 0.2), the slope (k1) of the linear regression of the curves (Mt/Meq) 2 = f(ttr) allowed the calculation of water diffusion coefficient D1: D1 = k1 π L 2 / 16. For a longer time (Mt / Meq > 0.7), the slope k2 of the linear regression of the curve (ln (1 -Mt/Meq) = f(t)) allowed the determination of the water diffusion coefficient D2: D2 = k2 L 2 / α1 (α1 = 2.40483)[START_REF] Alix | Biocomposite materials from flax plants: Preparation and properties[END_REF]. The changes of D1 and D2 for IMC10 and ISC10 as a function of aw were given Figure10c. The variation of the D1 and D2 values are in agreement with the isotherm. The D1 values appeared higher than D2 probably due to the aggregation of the water molecules during the sorption, which makes them less mobile. The D values were constant on large water activity range. This constant diffusion of the water molecules inside the film was in accordance with the Henry's behavior previously observed on the isotherms. The decrease of D1 and D2 coefficients at high aw (> 0.8) could be explained by the largest aggregation of water molecules. D1 and D2 values were higher for IMC10 than for ISC10, which is consistent with the kinetics. The larger size of the crystal structure and the higher crystallinity previously observed in ISC10 can explain this result. The low permeable crystal structures that stand in the way of the water molecules increased the tortuosity of the diffusion path, and consequently decreased D.

Figure 10

 10 Figure 10 Water vapor sorption of ISC10 and IMC10 films: (a) sorption isotherm (b) sorption kinetics (c) diffusion kinetics

Figure 11

 11 Figure 11 Polymer chains (gray) and the corresponding models (black) predicted by Chemdraw ® Software

Table 1

 1 The number average molar masses and dispersities of polyesters and yields of synthesis

	Polyester	1) (g⋅mol -1 )	Đ 2)	Yield (%)	Polyester	1) (g⋅mol -1 )	Đ 2)	Yield (%)
	ISC4	10 800	2.1	80	IMC4	4 300	1.8	62
	ISC6	10 600	1.7	88	IMC6	15 500	2.5	85
	ISC10	18 500	2.1	90	IMC10	19 000	2.0	89
	1) SEC conducted in CH2Cl2 with PMMA standards. 2) Dispersity

Table 2

 2 Crystallinity (χc) and d-spacing information for ISC10 film and IMC10 film

	Polyester film	χc (%)	d1 (nm)	d2 (nm)
	ISC10 film	36.3	1.60	0.46
	IMC10 film	25.9	1.60	0.46
	Figure 7 exhibited the XRD diffractograms of ISC10 film and IMC10 film. The

crystallinity (χc) and d-spacing were recorded in Table

Table 3

 3 Mechanical properties of ISC10 and IMC10 polyester films

	Polyester film	E (MPa)	σb (MPa)	εb (%)
	ISC10	127 ± 20	4.2 ± 0.8	19 ± 5
	IMC10	125 ±10	1.0 ± 0.2	16 ± 3

E, Young's modulus; σb, Stress at break; εb, Strain at break.

Table 4

 4 Surface energies and contact angles of ISC10 and IMC10 polyester films

	Polyester film	γ t (mN⋅m -1 )	γ d (mN⋅m -1 )	γ p (mN⋅m -1 )	θ ( o )
	ISC10	31.0	26.1	4.9	85.0 ± 1.0
	IMC10	29.6	24.1	5.4	83.0 ± 0.5

γ t , Total surface energy with the dispersive (γ d ) and polar (γ p ) parts; θ, water contact angle.

Table 5

 5 Water permeation coefficient (P) of ISC10 and IMC10 films at different water activities (aw).

	Polyester film	aw	P (g⋅m -1 ⋅d -1 )
		0.25	0.04
	ISC10	0.48 0.55	1.55 3.78
		0.65	5.10
		0.19	0.33
	IMC10	0.49 0.55	2.81 5.61
		0.60	5.95

Table 6

 6 Gas permeation (P) and diffusion (D) coefficients of ISC10 and IMC10 films

	Polyester film		N2		O2		CO2
		P	D x 10 6	P	D x 10 7	P	D x 10 8
	ISC10	0.09	1.26	0.26	2.78	1.32	1.98
	IMC10	0.18	2.50	0.46	3.86	2.04	2.54

Acknowledgments

The authors thank Pr. S. Marais for POM observations, Dr. K. Fatyeyeva for water vapor permeation measurements, B. Jiang, C. Legrand and J. Bouillon for helpful discussions. This manuscript is a tribute to the 50 year anniversary of the French Polymer Group (Groupe Français des Polymères -GFP).

Funding: This work was supported by the Chinese Scholarship Council [grant number 201701810120].

Data availability statement

The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations.


Evidence for the exchange effect in the beta decay of 241 Pu X. Mougeot, * M.-M. Bé, C. Bisch, and M. Loidl CEA, LIST, LNHB, Gif-sur-Yvette, F-91191, France.

(Dated: September 25, 2012)

The exchange effect has been previously given as a possible explanation for a significant deviation from an allowed shape observed at low energy in the 241 Pu beta spectrum. Calculations set out here confirm that this atomic effect explains a large part of this deviation. The equations needed to calculate the exchange effect are detailed, as well as the evaluation of the confluent hypergeometric function for complex arguments of large magnitudes. After a review of the possible other effects that could explain the remaining discrepancy at low energy, the screening correction using effective nuclear charges seems to be the best explanation. For radionuclides with high Z, this work has demonstrated the necessity to take into account the spatial variation of the nuclear charge experienced by the ejected electron to accurately correct for the screening effect.

I. INTRODUCTION

Metallic magnetic calorimeters have been shown to be a powerful experimental technique to study the energy spectra of beta particles having a maximum energy of about 1 MeV. Beyond this energy range, the bremsstrahlung effect has a significant contribution and studies have to be conducted to evaluate the relevance of this technique. The beta spectrum of 241 Pu was recently measured using a metallic magnetic calorimeter [1]. The source was enclosed in a gold absorber assuring a 4π solid angle and 100 % detection efficiency. The energy resolution, 29 eV at 5.9 keV, and the energy threshold, 300 eV, allow to test the theoretical beta spectra calculations with an accuracy never before achieved. The spectrum from [1] has already been compared to classical beta calculations [START_REF] Mougeot | LSC2010, International Conference on Advances in Liquid Scintillation Spectrometry[END_REF], and in this framework, a significant deviation of the experimental spectrum below 5 keV could not be explained. The exchange effect has been given as a possible cause, which is explained here in detail.

The radionuclide 241 Pu mainly decays by beta minus emission to the ground state of 241 Am, and the transition probability was evaluated to be P (β -) = 99.99756 (2) %, the remaining being α decays [START_REF] Bé | Table of Radionuclides -Monographie BIPM-5[END_REF]. The nature of the transition is first forbidden non-unique, with a maximum available energy of Q β -= 20.8 (2) keV. Its half-life T 1/2 = 14.33 (4) a was evaluated from published measurements which are independent of the beta spectrum.

A beta spectrum is the product of a weak interaction coupling constant, a statistical phase space factor which reflects the momentum distribution between the electron and the neutrino, the so-called Fermi function which corrects for the Coulomb effects, and a shape factor which contains all the remaining energy dependencies, as leptonic and nuclear matrix elements or atomic effects corrections. A calculation program of the shapes of beta spectra was already developed and described elsewhere for allowed and forbidden unique transitions [START_REF] Mougeot | LSC2010, International Conference on Advances in Liquid Scintillation Spectrometry[END_REF]. In this kind of beta transition, the energy dependence of the nuclear matrix elements can be factored out. This is not possible for forbidden non-unique transitions, hence the calculations become more complex. However, first forbidden non-unique transitions which fulfil the assumption 2ξ = αZ/2R E 0 can be treated as allowed ones [START_REF] Schopper | Weak interactions and nuclear beta decay[END_REF], with α the fine structure constant, R the radius of the daughter nucleus, Z its nuclear charge, and E 0 the maximum energy of the transition. This approximation, also called the ξ approximation, simply means that the Coulomb energy of the beta electron at the nuclear radius must be large compared to the total decay energy. The 241 Pu beta decay is a case study because of its low maximum energy (20.8 keV) compared to 2ξ

19.8 MeV. Therefore, this transition can be calculated as an allowed one.

Atomic effects play an important role for the shape of beta spectra, mainly at low energy. Atomic electrons partially screen the nuclear charge seen by the beta electron: this is the well-known screening effect. The sudden change of the nuclear charge can induce atomic excitations or internal ionizations, which are respectively called shake-up and shake-off effects and are not expected to have a contribution higher than 0.1 %. The exchange effect is also an atomic effect. It arises from the creation of a beta electron in a bound orbital of the daughter atom corresponding to one which was occupied in the parent atom. An atomic electron from the bound orbital makes simultaneously a transition to a continuum orbital of the daughter atom. This process cannot be distinguished from the direct decay to a final state containing one continuum electron.

The first study of the exchange effect was only focused on the 1s orbital. The conclusion was a destructive interference with the direct process, leading to a lower emission probability at low energy [START_REF] Bahcall | [END_REF]. Twenty years later, a specific study of the exchange effect in the tritium beta decay showed an enhancement at low energy, consistent with recent measurements [6]. The theoretical formalism of the exchange effect in the case of the allowed transitions was then extended to higher orbitals, including also the screening effect [7]. It was shown that the exchange effect should systematically enhance the decay probabil-ity at low energy.

Qualitative evidence of the exchange effect was first observed by the measurement of the beta spectrum of 107 Pd with a cryogenic micro-calorimeter [8]. A different type of cryogenic detector provided quantitative evidence in the beta spectrum of 63 Ni [9], but the contribution of the exchange effect is low above the energy threshold (8 keV). The beta decay of 63 Ni is well-suited for testing the atomic exchange effect because it is an allowed transition, which simplifies the calculations, and because of its low endpoint energy, 66.980 (15) keV [START_REF] Bé | Table of Radionuclides -Monographie BIPM-5[END_REF]. Recently, this spectrum was measured using the same technique as in [1], with an energy threshold of ∼ 200 eV [START_REF] Le-Bret | Evidence for exchange effect in the β spectrum of 63 Ni measured with cryogenic detectors[END_REF]. It was shown that the exchange correction factor determined following the formalism from [7] leads to a very good agreement between theory and experiment down to these very low energies.

II. FORMALISM A. General considerations

The formalism used in this publication is from Harston and Pyper [7]. The exchange effect is expressed as a correction factor which modifies the beta spectrum intensity I:

dI dE = dI N E dE 1 + η T ex (E) , (1)
where the superscript NE stands for 'no exchange', and η T ex (E) is the total exchange factor depending on the beta electron energy E. The exchange factor must involve the overlap between the electron radial wave functions of the bound and continuum orbitals.

For allowed transitions, only beta electrons created into an s bound orbital of the daughter atom take part in η T ex . Indeed, a beta transition is classified by the properties of the initial and final nuclear states: the total angular momentum change ∆J and the parity change ∆π. For an allowed transition, ∆J = 0, ±1 and ∆π = 0, which means that the electron-neutrino pair does not carry away any orbital angular momentum. Thus, only the ns orbitals are reachable by the beta electron.

In the first study of the exchange effect, Bahcall [START_REF] Bahcall | [END_REF] pointed out that in the hydrogenic approximation the probability density of the beta electron at the center of the nucleus is:

|Ψ ns (0)| 2 ∝ π -1 αZ n 3 , (2)
with n the principal quantum number and Z the charge of the daughter nucleus. The conclusion was that the largest contribution comes from the 1s orbital and the higher orbitals can be ignored. In their extended formalism, Harston and Pyper [7] underlined that the other ns orbitals could have a contribution of the same magnitude at low energy. They also highlighted an inconsistency in the wave functions used in Bahcall's calculations, giving a destructive interference of the exchange effect with the direct decay. Finally, they predicted an enhancement of about 10 % below 5 keV, from Z = 10 to Z = 90.

B. Expression

The total exchange factor can be written in terms of the subshell contributions η ns ex :

η T ex (E) = n η ns ex (E) + m,n (m =n) µ m µ n , (3)
and the exchange factor of the n-th subshell is given by:

η ns ex (E) = f s µ 2 n -2µ n . (4)
The parameters µ n and f s are expressed in terms of the bound and continuum electron radial wave functions according to:

µ n = Es |ns g b n,κ (R) g c κ (R)
and

f s = g c κ (R) 2 g c κ (R) 2 + f c κ (R) 2 .
(5) All the parameters are explained in the next section.

The second sum in equation ( 3) is not taken into account in [7] because of its expected smallness. Even if this is true over almost the entire energy range of the beta spectra, the contribution of this term can be about 1-3 % below 1 keV. It was therefore included in the present calculations.

C. Electron radial wave functions

The electron wave function is defined in [START_REF] Rose | Relativistic electron theory[END_REF] by its small and large radial components f and g:

Ψ κ ( r) = g κ (r) µ χ µ κ if κ (r) µ χ µ -κ , (6)
where the spin-angular functions χ µ κ are expanded into the orthonormal basis of the spherical harmonics Y µ-m l :

χ µ m = m C l 1 2 j; µ -m, m Y µ-m l χ m , (7)
with the Clebsch-Gordan coefficients C and the twocomponent spin eigenfunctions χ m . κ is the eigenvalue of the operator K = β( σ • L + 1) which appears by applying the theory of angular momentum to an electron in a Coulomb central field: β is the Dirac matrix, σ designates the Pauli matrices σ x,y,z and L is the orbital angular momentum operator.

The electron radial wave functions for the bound (f b n,κ , g b n,κ ) and continuum (f c κ , g c κ ) orbitals are the analytical solutions of the Dirac equation in the hydrogenic approximation given by Rose [START_REF] Rose | Relativistic electron theory[END_REF]. With κ = -1 for s orbitals, the following parameters are defined:

n = n -|κ|, ζ = αZ, γ = κ 2 -ζ 2 , λ = 1 + n + γ ζ 2 -1/2 , W n,κ = 1 + ζ n + γ 2 -1/2 , K = (2λr) γ-1 e -λr Γ(2γ + 1) 2λ 5 Γ(2γ + n + 1) n !ζ(ζ -λκ) 1/2
,

F 1 = n 1 F 1 (-n + 1, 2γ + 1, 2λr), F 2 = (κ -ζ/λ) 1 F 1 (-n , 2γ + 1, 2λr).
The Gamma function Γ is the extension of the factorial function to complex numbers. 1 F 1 is the confluent hypergeometric function, whose evaluation is described in section II E. The bound wave functions are then expressed as:

f b n,κ = -K 1 -W n,κ (F 1 -F 2 ), ( 8a
)
g b n,κ = -K 1 + W n,κ (F 1 + F 2 ). ( 8b
)
In the same way, the following parameters are defined:

W = 1 + E/m e c
2 the total energy of the electron in rest mass units,

p = √ W 2 -1 the corresponding momentum, y = ζW/p, Q = (2pr) γ e πy/2 |Γ(γ + iy)| 2r √ πp Γ(2γ + 1) ,
η the phase of the wave functions via

e iη = - κ -iy/W γ + iy 1/2 , C 1 = e -ipr+iη (γ + iy) 1 F 1 (γ + 1 + iy, 2γ + 1, 2ipr), C 2 = C * 1 .
The continuum wave functions are then expressed as:

f c κ = iQ √ W -1(C 1 -C 2 ), (9a)
g c κ = Q √ W + 1(C 1 + C 2 ). ( 9b
)
The overlap which appears in the left term of equation ( 5) is defined in the case of allowed transitions simply by:

Es |ns = Ω Ψ c * -1 ( r)Ψ b -1 ( r)dΩ. ( 10
)
It remains from the spin-angular functions a factor 1/ √ 4π. Thus, integrating over the angular parts and emphasizing the initial and final nuclear charge dependences, one has the explicit form:

Es |ns = Rmax Rmin r 2 g c -1 (Z f , r)g b n,-1 (Z i , r) dr + Rmax Rmin r 2 f c -1 (Z f , r)f b n,-1 (Z i , r) dr. (11)

D. Effective nuclear charges

When calculating a beta spectrum, the screening effect is generally corrected for using a Thomas-Fermi potential which is substracted from the total energy of the particle [START_REF] Good | [END_REF]. This method creates a non-physical discontinuity at the minimum energy defined by the potential [START_REF] Mougeot | LSC2010, International Conference on Advances in Liquid Scintillation Spectrometry[END_REF]. This mimimum energy is 20 keV, hence the discontinuity does not generally affect the practical use of the spectrum. Physically the influence of the atomic electrons is expected to be significant when the beta wavelength is comparable to the atom size, thus at low energy.

Bahcall [START_REF] Bahcall | [END_REF] and Haxton [6] did not consider the influence of the screening effect in their exchange calculations. Harston and Pyper [7] pointed out that this approximation was not valid for many-electron atoms since the screening distorts the beta wave functions and can contribute significantly to the overlap involved. Instead of a Thomas-Fermi potential, they preferred to use effective nuclear charges, also best suited for bound wave functions. Obviously, this does not concern wave functions evaluated at the nuclear radius, for which the full nuclear charge has to be used.

The effective nuclear charge Z eff is equal to the number of protons Z in the nucleus decreased by the electron density integral in a sphere of radius r:

Z eff (r) = Z -4π × 2 nlm r 0 |Ψ nlm (r )| 2 r 2 dr . ( 12
)
It is difficult to account for the spatial dependency in a simple model. Using the hydrogenic approximation framework, Harston and Pyper calculated effective nuclear charges for each atomic orbital from the mean radius of the orbital. In the non-relativistic case, the relation is well-known:

r = 3n 2 2Z eff - l(l + 1) 2Z eff . ( 13
)
As only s orbitals have to be considered, l = 0. For medium and large Z, a relativistic relation is needed [14]:

r = (n + γ) 3N 2 -κ 2 -κN 2N Z eff , (14)
with

N = (n + γ) 2 + ζ 2 .
According to equation ( 13) or ( 14), Harston and Pyper calculated the mean radii in the Hartree-Fock or the Dirac-Fock approximation using the GRASP code [15]. Atomic data tables including mean radii were preferred in this study. Within the Hartree-Fock approximation, values are available up to Z = 86 in [START_REF] Fischer | The Hartree-Fock method for atoms[END_REF], and up to Z = 102 in [START_REF] Mann | [END_REF]. They are available up to Z = 120 in [18] within the Dirac-Fock approximation. From these tabulated values, the effective nuclear charges were determined for the 106 Ru decay (Table I) and for the 241 Pu decay (Table II). These compare well with the ones calcutated by Harston and Pyper [7] for the first orbitals.

The discrepancies for the higher orbitals come from the fact that neutral atoms were assumed in the atomic data tables used here, whereas Harston and Pyper assumed ions Ru 1+ , Rh 2+ , Pu 3+ , Am 4+ .

E. Calculation of the confluent hypergeometric function

The evaluation of the confluent hypergeometric function is the main technical difficulty in the calculation of the exchange effect. The evaluation of 1 F 1 (a, b, z) is required for both cases of bound and continuum wave functions. The parameters required for the calculation of the bound wave functions are: a = -n + 1 or a = -n , with a ∈ N; b = 2γ + 1, with b ∈ R; and z = 2λr, with z ∈ R. Those required for the continuum wave functions are: a = γ + 1 + iy; b = 2γ + 1 (as above); and z = 2ipr. Since only s waves contribute to the exchange effect for the allowed transitions, then γ ∈ ]0, 1[, b ∈ ]1, 3[, (a) ∈ ]1, 2[. (a) has an energy dependence, e.g. for Z = 100, (a) 36.9 at 100 eV and tends to the fine structure constant at high energy. The parameter z is in an open interval. If the endpoint energy of a beta spectrum is less than 10 MeV, r might need to be as high as 10 4 (in atomic units) in order to achieve a good accuracy on the overlap calculation. Many methods were tested for the evaluation, the most precise of which were retained and are presented below.

Definition

The confluent hypergeometric function is a solution of the Kummer's differential equation:

z d 2 u(z) dz 2 + (b -z) du(z) dz -au(z) = 0, (15a)
and is defined by the following series:

1 F 1 (a, b, z) = ∞ n=0 (a) n (b) n z n n! , ( 15b
)
where a, b, z are complex numbers, and (a) n is the Pochhammer factorial:

(a) n = a(a + 1)(a + 2) . . . (a + n -1) = Γ(a + n) Γ(a) . (15c)
The numerical calculation of 1 F 1 , especially for complex arguments of large magnitudes, is poorly documented. There is no efficient calculational method available for all possible values of the parameters. A direct calculation is not possible because very large numbers are needed, whose products and ratios give a small quantity. The precision limit of the calculations is quickly achieved, even in double-precision.

Generalized Laguerre polynomials

For the bound wave functions, an exact solution exists in terms of generalized Laguerre polynomials:

L α n (x) = (α + 1) n n! 1 F 1 (-n, α + 1, x), (16a)
which satisfy the recurrence relation:

L α n (x) = 2 + α -1 -x n L α n-1 (x) -1 + α -1 n L α n-2 (x), (16b)
with the first two polynomials:

L α 0 (x) = 1 and L α 1 (x) = -x + α + 1. ( 16c
)
This method allows an exact and quick calculation. Even for Z = 120, no orbital beyond 8s needs to be considered.

Taylor and asymptotic series

The Taylor series method, based on equation (15b), defines A j = (a) j (b) j j! and S j as the sum of the first j terms of the series, which leads to the following recurrence relations:

A j+1 = A j • a + j b + j • z j + 1
and S j+1 = S j + A j+1 , [START_REF] Mann | [END_REF] with S 0 = A 0 = 1. This method is convenient for complex arguments with small magnitudes, but it quickly diverges.

For larger magnitudes, the asymptotic series method is more appropriate and is defined by:

1 F 1 (a, b, z) ∼ Γ(b) Γ(a) e z z a-b ∞ j=0 (b -a) j (1 -a) j z -j j! + e iπa z -a Γ(b -a) ∞ j=0 (a) j (1 + a -b) j (-z) -j j! . ( 18
)
Each of the two series in the right term of equation ( 18) is calculated using the Taylor series method described above.

Once again the use of double-precision is not sufficient because sometimes, and especially for the higher orbitals, the Taylor series diverges before the asymptotic series converges.

CONHYP and iRRAM: beyond double-precision

CONHYP is a Fortran numerical evaluator of 1 F 1 for complex arguments of large magnitudes [19]. Extended precision subroutines using large arrays allow calculations with a high accuracy. CONHYP was successfully tested up to magnitudes of about 1 000 but the calculation of the exchange effect often needs larger values, for which some limitations are binding. For a better compatibility with our code, it was translated into C++ and the limitations for small and large numbers (10 ±75 ) and maximal length of arrays (777) were removed. Unfortunately, this program cannot return any value for very large magnitudes, e.g. for the calculation of η 6s ex in 241 Pu. iRRAM is a C++ package for error-free real arithmetic [START_REF] Müller | RNC3: Third Conference on Real Numbers and Computers[END_REF]. The accuracy of the calculations is only limited by the size of computer memory. Common mathematical functions are defined for three specific data types (IN-TEGER, REAL, COMPLEX), which allow the package to be easily used. The accuracy of the calculation is controlled by redefining the standard outputs. The main drawback is the necessity to deal with memory instabilities.

The quick divergence of the generalized Laguerre polynomials leads us to calculate them exactly using the iR-RAM package. In other cases, the following strategy was adopted for the evaluation of 1 F 1 : with iRRAM, use of (i ) the Taylor series method for |z| < 100 and of (ii ) the asymptotic series method for |z| > 100; (iii) use of CON-HYP if no value is returned due to a memory problem.

A faster, simpler but less accurate method

Another method should be mentioned, which uses directly the asymptotic solutions of the electron wave func-tions given in [START_REF] Rose | Relativistic electron theory[END_REF]:

f = - W -1 πp 1/2 sin(pr + δ) r , ( 19a
)
g = W + 1 πp 1/2 cos(pr + δ) r . ( 19b
)
where the phase δ is defined by:

δ = y log 2pr -arg Γ(γ + iy) + η -πγ/2. (19c)
Thus, the evaluation of the confluent hypergeometric function is avoided for large pr values. For small pr values, 1 F 1 was calculated using the Taylor series method in double-precision. It turned out that a switch criterion from the latter method to the former is very difficult to set, since it may be correct for certain orbitals and certain radionuclides, but not consistently. Moreover, one has to check with a more exact calculation, which makes this method inappropriate.

III. RESULTS

Calculations were made initially for the two beta decays described in detail by Harston and Pyper, 106 Ru and 241 Pu. If the same orbitals and the same effective nuclear charges are used, the results from [7] are well reproduced. Figure 1 shows the total exchange factor for the 241 Pu decay, with the contribution of each orbital. The 7s orbital was included, as described later. The second term in equation ( 3) is also included in our calculation. Its contribution is ∼ 1.2 % at 100 eV and decreases quickly with increasing energy. Theoretical allowed beta spectra, calculated with and without exchange effect, were compared to the measured and Pyper [7] (see Table II).

spectrum of 241 Pu from [1]. The theoretical spectra were normalized to the data in the energy range 7-18 keV. The calculation of the total exchange factor was carried out using the effective nuclear charges given in Table II. Our hypothesis of a neutral atom led us to consider the 7s orbital, which has a significant contribution below 1 keV. The integration limits and steps used for the radial part in the calculation of the overlap [START_REF] Le-Bret | Evidence for exchange effect in the β spectrum of 63 Ni measured with cryogenic detectors[END_REF] are given in Table III. The main contribution to the overlap comes from the product of the large radial components of the wave functions. Thus, R max was chosen such as the amplitude of this product was 10 -12 to ensure an overlap precision of ∼ 10 -4 with r ∼ 10 4 , and r step was maximized such as a lower step yields an overlap change 10 -4 . Figure 2 shows the effect of the complete calculation.

Figure 3 presents the residuals relative to the allowed spectrum without exchange effect for the measured spectrum from [1]. Our calculation -neutral Pu atom with 7 orbitals (red solid line) -and the one from [7] -Pu 3+ ion with 6 orbitals (green long-dashed line) -are included and give very similar results. We have obtained a better agreement with the experimental spectrum adding the 7s orbital to Harston and Pyper's calculation. The result of this calculation is also included in figure 3 (blue chain-dashed line). The effective nuclear charge of the 7s orbital was determined with a second order polynomial fit from the lower orbitals, and is given in Table II. Obviously, the 7s orbital is not filled in the case of a Pu 3+ ion and should not have to be taken into account. It is just [1] and an allowed spectrum calculated with and without exchange effect. The effective nuclear charges used are the relativistic ones calculated for a neutral Pu atom (see Table II). a way of simulating another effect, which is discussed in the next section.

IV. DISCUSSION

The significant deviation from an allowed shape observed at low energy in the 241 Pu beta spectrum is reproduced in a large part by taking into account the exchange effect. But many effects, as described below, are not taken into account in the calculation, which could explain the remaining discrepancy.

a) Quality of the measurement

Commonly used in ionizing radiation metrology for activity measurements, the Liquid Scintillation Counting technique (LSC) requires the knowledge of the spectrum shape in the case of beta emitters [START_REF] Broda | [END_REF]. The influence of this shape was studied in detail in the case of 241 Pu decay [START_REF] Van Wyngaardt | Absolute standardization of 241 Pu by the TDCR technique and effect of the beta spectral shape[END_REF]. The authors demonstrated that, when the shape measured by [1] is used, LSC activity results are in better agreement with results obtained by techniques which are independent of the spectrum shape. The study underlines the quality of the 241 Pu beta spectrum from [1]. However, the remaining discrepancy at low energy could be due to the measurement itself. For example, the influence of the quality of the source was not yet studied in [1] and could yield distortion effects. (Color online) Residuals relative to the allowed spectrum for the measured spectrum of 241 Pu from Loidl et al. [1], for our calculation including the exchange effect (red solid line), for the one from Harston and Pyper [7] (green longdashed line), and for the latter one including the 7s orbital (blue chain-dashed line). The details of the calculations are explain in the text. This point must be clarified.

b) Contribution from the 241 Am decay

One might think the conversion and Auger electrons from the 241 Am α decay could have a great influence at low energy, because of their high emission probabilities [START_REF] Bé | Table of Radionuclides -Monographie BIPM-5[END_REF]. However, these electrons are emitted immediately after the α decay, and the α particle deposits so much energy that the detection system becomes saturated. Thus, this kind of event was easily rejected during the data analysis and the corresponding secondary electrons were not seen. Additionally, the 241 Pu source was purified during the preparation and the activity was about 8 Bq for a total acquisition time of 61 h, and as the T 1/2 ( 241 Am) = 432.6 (6) a, the 241 Am decays were extremely rare.

c) Compton background

Another influence at low energy could be the Compton background from the Kα and Kβ X-rays of the 55 Fe calibration source. The total counts in the two peaks is N (Kα + Kβ) 143 500. For gold, the ratio of the photoelectric cross section to the Compton cross section at 5 keV is ∼ 10 6 . Thus, the Compton effect contributes no more than 0.002 % which is completely insufficient to explain the remaining discrepancy.

d) Theoretical shape factor

The 241 Pu decay is considered as an allowed transition whereas it is a first forbidden non-unique transition. The accuracy of the ξ approximation is ∼ 1/ξ, better than 1 % in the present case. Moreover, the theoretical shape factor of this transition was already studied through complete calculation of the nuclear matrix elements [START_REF] Rizek | [END_REF]. Its influence increases linearly with the energy, not higher than 0.3 % at the endpoint and 0.1 % below 8 keV. Thus, the remaining discrepancy cannot be explained by the allowed transition hypothesis.

e) Chemical environment

The decaying atom is embedded in a chemical environment and the other atoms have an influence on the most external orbitals. A recent measurement of the 63 Ni allowed beta decay with a metallic magnetic calorimeter can be mentioned as an example [START_REF] Le-Bret | Evidence for exchange effect in the β spectrum of 63 Ni measured with cryogenic detectors[END_REF]. This radionuclide was electroplated on a gold absorber, ensuring a metallic form. Thus, it can be considered as a neutral atom. The spectrum calculation including the exchange effect exhibits an excellent agreement.

In the present case, the beta spectrum from [1] might be sensitive to chemical effects due to the accuracy of the measurement and the low energy threshold. However, the chemical composition of the 241 Pu source was not clear, being probably a mixture of PuO 2 with a small quantity of Pu(NO 3 ) 4 . Harston and Pyper estimated that chemical effects cannot be higher than 0.3 % by studying different ionic charges [25]. Indeed, Pu has the same oxidation number in PuO 2 as in Pu 4+ .

As it has been shown in the previous section with the analysis of figure 3, the calculated total exchange factor is very similar whether a neutral atom (red solid line) or an ion (green long-dashed line) is considered. Thus, ionic charges might not be the correct approach to simulate chemical effects, because the neutral atom hypothesis yields the same result. A more comprehensive study should consider molecular orbitals, following, for example, the formalism developed in [26].

f ) Screening

High Z nuclides, such as 241 Pu, require complete calculation of the effective nuclear charges, taking into account their spatial variations. This was already done for some nuclides fifteen years ago using the Dirac-Fock formalism [27]. The authors calculated the mean energy in the beta decay with and without exchange effect. Among the studied nuclides, they gave results for 63 Ni and 241 Pu decays.

For 63 Ni, we compare our theoretical calculations with the experimental spectrum measured in [START_REF] Le-Bret | Evidence for exchange effect in the β spectrum of 63 Ni measured with cryogenic detectors[END_REF]. Taking the exchange effect into account led to a mean energy lower by 2.3 % than the one without exchange effect. This is in good agreement with the measured result and also with the calculations carried out by [27]. Indeed, the screening correction is weak for such a low Z nuclide.

For 241 Pu, we calculate a mean energy lower by 2.5 % taking the exchange effect into account, significantly different from the 4.6 % obtained by [27]. The mean energy of the beta spectrum from [1] is lower by 3.9 % than the calculated one without exchange effect. This fact demonstrates our underestimation of the screening effect for high Z.

Assuming the measured 241 Pu spectrum from [1] as definitive, the remaining discrepancy at low energy seems to come from the screening correction more than any other effect. Then, the influence of the screening on the exchange effect must be studied preferably using the Dirac-Fock formalism, which is beyond the scope of this study.

V. CONCLUSION

All the equations that are necessary to calculate the total exchange factor for allowed beta transitions have been presented. The technical difficulty of the evaluation of the confluent hypergeometric function has been set out in detail. From the results shown above, the influence of the exchange effect on the spectrum shape is demonstrated in the case of low energy beta transitions, especially for 241 Pu decay.

However, the exchange effect does not seem sufficient to reproduce well the beta spectrum from [1] at low energy. The influence of possible other effects has been reviewed. The evaluation of the chemical effects by Harston and Pyper [25] has been shown to be questionable, but these are not expected to be sufficient enough even considering molecular orbitals. Furthermore, this work has demonstrated that the exchange correction is very sensitive to the screening, which seems to be underestimated by our use of effective nuclear charges. For radionuclides with high Z, it will be necessary to take into account the spatial variation of the nuclear charge experienced by the ejected electron, using electron radial wave functions generated in the Dirac-Fock formalism. This study also highlights the necessity of a new measurement of the 241 Pu beta spectrum. As mentioned in [1], an implantation of 241 Pu ions directly into the detector absorber is planned in order to limit the influence of the quality of the source.

FIG. 1 .

 1 FIG. 1. (Color online)Total exchange factor for the 241 Pu beta decay and contributions of each orbital. The effective nuclear charges used are the relativistic ones from Harston and Pyper[7] (see TableII).

  FIG. 2. (Color online)Comparison between the measured beta spectrum of 241 Pu from Loidl et al.[1] and an allowed spectrum calculated with and without exchange effect. The effective nuclear charges used are the relativistic ones calculated for a neutral Pu atom (see TableII).

  FIG. 3.(Color online) Residuals relative to the allowed spectrum for the measured spectrum of 241 Pu from Loidl et al.[1], for our calculation including the exchange effect (red solid line), for the one from Harston and Pyper[7] (green longdashed line), and for the latter one including the 7s orbital (blue chain-dashed line). The details of the calculations are explain in the text.

TABLE I .

 I Calculated effective nuclear charges compared to[7] for the 106 Ru → 106 Rh beta decay. ∆ is the difference.

	n Zi([7])	Zi	∆ (%) Z f ([7])	Z f	∆ (%)
	1 43.322 43.3226 0.001	44.320 44.3210 0.002
	2 39.841 39.8412 0.0005 40.819 40.8188 0.0005
	3 32.765 32.7656 0.002	33.727 33.7229 0.012
	4 22.841 22.7408	0.44	23.866 23.5974	1.1
	5 9.5087 a 9.67195	1.7	10.620 a 9.88093	7.5

a values extrapolated from the lower orbitals.

TABLE II .

 II Calculated effective nuclear charges compared to[7] for the 241 Pu → 241 Am beta decay. ∆ is the difference.

	n Zi([7])	Zi	∆ (%) Z f ([7])	Z f	∆ (%)
	1	93.222	93.2188 0.003	94.213	94.2188 0.006
	2	90.485	90.4844 0.0007	91.505	91.5039 0.001
	3	84.050	84.0488 0.001	85.108	85.1074 0.0007
	4	72.601	72.5986 0.003	73.695	73.6914 0.005
	5	56.388	56.3633 0.044	57.527	57.4670	0.11
	6	37.452	36.9865	1.3	38.860	37.7924	2.8
	7 13.0883 a 16.4458	20.4	14.6624 a 16.6616	12

a values extrapolated from the lower orbitals.

TABLE III .

 III Integration limits and steps for the radial part used in the calculation of the total exchange factor of the 241 Pu beta decay.

	n	1	2	3	4	5	6	7
	Rmax	45	90	150	250	500	900	2 700
	r step	0.05	0.05	0.1	0.1	0.2	0.2	1

ACKNOWLEDGMENTS

We wish to warmly thank Robert Vatin, retired from LNHB, and Mark A. Kellett, newly arrived, for fruitful discussions.


Impact of the Alex storm on the exchanges between the Roya River and its alluvial aquifer

Sandra Lanini 1,3 , Bernard Ladouche 1,3 , Benoit Dewandel 1,3 , Melissande Ibba 2 , Vincent Bailly-Comte 1,3 , and Marie Genevier 2 The alluvial aquifer of the transnational Roya watershed is exploited for drinking water to supply both French and Italian coastal cities. The Interreg project CONCERT-EAUX (2017-2020) aimed to improve the understanding of the functioning of this hydrosystem. Several wells were equipped in order to measure water level, temperature and conductivity temporal and spatial distribution within the aquifer. Two surface and groundwater sampling campaigns were also carried out to perform chemical and isotopic water analysis.

Near the Porra wellfield, vertical distribution of parameters proved the homogeneity of the alluvial formation in the first layer, and very different properties in the underlying less permeable alluvions. In the upper layer, temperature shows seasonal variations, whereas it is very stable in the underlying aquitard. The existence of advective transfers between the Roya River and the alluvial aquifer was proved both by the thermal transfer analysis and modelling, and by the isotope study. The CFCs and SF6 concentrations in groundwater confirmed that the aquifer is recharged with recent water. Thanks to meteorological and water level daily data, the aquifer recharge was estimated applying two different approaches. The results proved that for an annual average rainfall, the infiltration of the effective precipitation represents only 15 to 20% of the total aquifer recharge. The main part of the aquifer recharge comes directly from the Roya River.

A hydrogeological numerical model including the exchanges between the river and the aquifer has been developed using the finite difference model MARTHE©. Run in steady-state conditions, it allowed completing the water budget. It also highlighted the sections of the river which most contribute to the aquifer recharge.

All these elements helped us to understand the behaviour of the hydrosystem following the Alex storm in October 2020. An important volume of sediment has been transported by the river from the upper valley to the downstream reach. This lead to the clogging of the riverbed, which strongly reduced the aquifer recharge. Today, although trenches have been excavated immediately upstream of the wellfield to promote the artificial recharge of the aquifer by the Roya River, groundwater level still decreases due to pumping.

  , UMR 183, INRAE, CIRAD, IRD, AgroParisTech, Supagro, BRGM, Montpellier, France


Laurent Maurin

Pierre Ferdinand

Laurent Robert

J.-J Orteu

Jean-José Orteu

Development of an Integrated Fibre Bragg Grating Contact Pressure and Temperature Sensor for Composite Smart Manufacturing

.

. Experimental results of the sensor prototype, tested in a dedicated Resin Transfer Moulding (RTM) tool equipped with commercial reference sensors, demonstrate that the contact pressure variation information inside the mould can be measured with 0.1 bar resolution over the 10 bar range with such small sensors. However, for these first contact-pressure sensorprototypes, both hysteresis (10% full scale) and temperature cross-sensitivity shouldbe improved.

INTRODUCTION

Today, composite structural parts gain more and more importance in the primary structures of aircrafts (e.g.: sandwiches panel, monolithic stiffeners...) and their weight percentage rises from 5% on the Airbus A300, up to 10% on the A330/340, and greater than 15% on the A380. We anticipate that multi-technological and multimaterials parts, including sub-assemblies composites such as doors, fuselage..., will spread in the future. Similarly, an increasing number of primary structures is already planned to be made of composite materials. Among these, we should mention the A340-500 beam, and more recently the A380 centre wing box, which are two world first innovations in terms of composite-based structural parts highly loaded. As an indication, the number of composite parts manufactured per year for Airbus is huge: 4500 fins, flaps, and radomes, 400 central beams, 180 self-stiffened panels, etc., and these figures increased with both the A350 and the A400M military aircrafts. Taking into account the cost structure of a primary part roughly shared into: 30% (material), 65% (process), and 5% (NDT), it should be possible, from the perspective of a product validation based on a continuous recording of manufacturing parameters, to reduce the need for NDT (5% gain) as well as samples characterization (6% -10% gain), leading to an overall gain close to 12% of the element cost.

An instrumentation based on optical fibre sensing able to record temperature and contact pressure is a privileged way to access local and crucial information [START_REF] Maurin | High speed realtime contact measurements between a smart train pantograph with embedded fibre Bragg grating sensors and its overhead contact line[END_REF][START_REF] Ferdinand | High performance Fiber Bragg Grating systems for Smart Structures: Application to Overhead Contact Line monitoring in the railway industry[END_REF], since process parameters (temperature, [contact] pressure, degree-of-cure) are related to quality of the final product. Consequently, the availability of sensors which could be placed as close as possible to the composite parts is a matter of concern for a continuous improvement process based, for instance, on Statistical Process Control (SPC) methods. We claim that additional costs due to the integration of such instrumentation would be damped by reductions in NDT time and corresponding costs, NDT which therefore may be partially replaced by process parameters control.

The aim of the ELICo-CP project is to develop and qualify a monitoring system based on Fibre Bragg Grating (FBG)-based sensors, embedded on subsurface of manufacturing tools: such system will precisely enable in situ measurements (temperature, contact pressure) during the curing cycle. Technical goals on short term consist in providing a real-time system to production units for composite manufacturing process monitoring, enabling to i) ensure the production homogeneity, ii) in situ control of any deviation from nominal process (e.g.: raw material batch changes evidence...), iii) check that the polymerisation was locally done in the allowable tolerance, iv) anticipate potential areas of non-compliance, and v) provide a tool for decision making in case of incident during the manufacturing cycle.

SENSOR DEVELOPMENT AND LAB MEASUREMENTS

The need for a non-intrusive sensor able to measure the contact pressure within cavities or areas with small curvature radii requires the use of a sensitive surface, smaller in diameter than the curvature radius itself, typically lower than 3 mm.

It is reminded that a contact pressure variation equal to P = 0.1 bar applied on a steel made massive test body (Young's modulus E 210 GPa) leads to very small strain variations 0.048 µm/m, usually in the noise measurement, whatever the measurement technique used (figure 1). There are two main degrees-of-freedom to solve this sensitivity issue: i. to use a lever arm to amplify the strains of the test body, ii. to use a lower stiffness material for the test body, keeping in mind the elevated temperature constraint of the composite manufacturing process: up to 200°C during several hours.

The solution adopted consists in a tubular test-body (Ø 0.8 mm OD) equipped with two FBG strain gages, and a greater sensitive surface S in contact with the pressure P, leading to a strain amplification ratio greater than 14, and a sensitivity / P close to 7 (µm/m)/bar despite a contact surface diameter smaller than 3 mm, i.e. the maximum size within end-users' specifications (figure 2). Fibre Bragg Gratings (FBG) are, in such a case, very well suited in comparison with traditional strain gages: they can be easily inserted inside the tubular test body, and attached to it for strain measurements; moreover, additional FBGs can be inserted in close contact to the first FBGs for optimal temperature compensation 1 : temperature compensation is a key feature for accurate measurements (figure 3). 1 FBG are sensitive to both strain and temperature. The mechanical assembly leads to a small sensor which can be easily inserted inside a mould for contact pressure measurements (figure 4); this sensor will provide two kinds of information:

i. the temperature variations, thanks to the temperature compensation FBG transducer, ii. the contact pressure on the sensitive surface of the strain transducer, which could be: o the hydrostatic pressure if the material is liquid (this hydrostatic pressure can be measured with usual membrane sensors), o the contact pressure if there is a mix of solid and liquid, or only a solid: this contact pressure is different from the hydrostatic pressure, and cannot be measured with usual membrane sensors. This prototype has been first qualified in laboratory, using a very simple procedure consisting in putting calibrated weights on the sensitive surface, according to the equivalence relationship between the contact pressure P and the mass m: The experimental data, at ambient temperature, exhibit a high sensitivity in pressure, compatible with targeted 0.1 bar contact pressure resolution, provided that a high resolution FBG measurement system is used (figure 5). The hysteresis has been evaluated close to 10% full scale and appears to be reproducible from cycle to cycle; one explanation amongst others may be friction between the sensitive part of the sensor, and the sensor's body itself (figure 6).

EVALUATION OF THE SENSOR INSERTED IN AN RTM MOULD

A special test-bed has been designed in order to evaluate performances of the sensor. This experimental set-up is based on a Resin Transfer Moulding (RTM) process. A dedicated RTM mould has been designed and instrumented with the developed sensor and some additional sensors: a conventional strain-gage-based pressure sensor (XPCM10 from FGPSensors ® ) and PT100 platinum probes, for comparison purpose. Figure 7 shows the experimental demonstrator. The mould, lying on a heating table, is connected to a resin injection unit, which allows to set temperature and pressure steps. This mould is intended to produce injection resin-based composite parts, but for the first study reported in this paper, we have only injected air-pressure into the mould. Nevertheless, figure 8 (c) shows that both FBG sensors provide a temperaturecompensated pressure measurement. A linear fit leads to a sensitivity ratio equal to 2.66 for the temperature increase and 2.75 for the decrease. If needed, a quadratic interpolation leading to a better fit could be used.

Pressure calibration

Pressure calibration was based on increases and decreases of pressure, step-bystep, as depicted in figure 9. Pressure range is around 5 bar and pressure steps are around 0.5 bar. From figure 9 it can be seen that pressure variations are badly detected at the beginning of pressure increase and pressure decrease (see ellipses). Without considering these data, quasi linear responses are obtained for both pressure increase and decrease, with sensitivities values estimated to -18 pm/bar and -16 pm/bar, respectively. This leads to an hysteresis previously discussed, also evaluated close to 10% full scale and almost reproducible in all our tests.

CONCLUSION

We have developed a contact pressure and temperature sensor prototype that can be embedded into composite process moulds. Even if the sensor should be improved, the first results are very encouraging, and a fully operational sensor should be available later. With such innovative instrumentation, gains are anticipated in the short term, as follows: i) reduction of both time and costs of post production controls, ii) enhancement of traceability throughout the production life of composite parts, and iii) a decision support in case of incidents during the manufacturing process. Moreover, on the longer term, gains will also concern both the optimization of performances and the enhancement of products quality. On the long term also, such sensors will allow a smart monitoring of autoclaves, to optimize properties of structures, area by area, taking into account variations in shape and geometry of composite parts, as well as the variable thermal inertia behaviour of manufacturing tools. In particular, such new sensors should also participate to the local thermal control loop. For sure, such instrumentation will also be useful for the emergence and maturation of new non autoclave processes (e.g.: resin infusion).

Figure 1 .

 1 Figure 1. Massive test body equipped with conventional strain gage to measure contact pressure.

Figure 2 .

 2 Figure 2. Strain amplification and temperature compensation principle used in ELICo-CP sensor.

Figure 3 .

 3 Figure 3. Cylindrical test body developed in ELICo-CP project, with both strain and temperature compensation FBGs located inside.

Figure 4 .

 4 Figure 4. FBG contact pressure and temperature sensor.

  sensitive surface diameter (g = 9.81 m/s²)

Figure 5 .

 5 Figure 5. In lab tests with weights applied on the sensor's sensitive surface simulating an equivalent contact pressure.

Figure 6 .

 6 Figure 6. In lab tests: contact pressure hysteresis evaluation.

Figure 7 .

 7 Figure 7. Experimental RTM test-bed designed to evaluate performances of the sensor. The left picture shows the outside instrumented mould, the right picture shows both pressure sensors flushing with the inside surface of the mould.

Figure 8 .

 8 Figure 8. In situ temperature calibration.

Figure 9 .

 9 Figure 9. In situ pressure calibration.

ACKNOWLEDGMENT

The authors would like to thank the French National Research Agency (Agence Nationale de la Recherche, ANR) for its financial support within the framework of P2IC projects, and A. Hamasaiid, C. Vergnes, and J.-M. Mouys for their technical support.