Clément Fosse
email: clement.fosse1@univ-rouen.fr

Aurélie Bourdet
email: aurelie.bourdet@univ-rouen.fr

Esteve Ernault
email: esteve.ernault@univ-rouen.fr

Antonella Esposito
email: antonella.esposito@univ-rouen.fr

Nicolas Delpouve
email: nicolas.delpouve1@univ-rouen.fr

Laurent Delbreilh
email: laurent.delbreilh@univ-rouen.fr

Shanmugam Thiyagarajan
email: shanmugam.thiyagarajan@wur.nl

Rutger J I Knoop
email: rutger.knoop@wur.nl

Eric Dargent
email: eric.dargent@univ-rouen.fr

Determination of the equilibrium enthalpy of melting of two-phase semi-crystalline polymers by fast scanning calorimetry

Keywords: fast scanning calorimetry, rigid amorphous fraction, enthalpy of melting, PEF, PBF -

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

INTRODUCTION

Since polymers can crystallize to different extents but never entirely, an extrapolated value of enthalpy, the so-called equilibrium enthalpy of melting ∆Hm 0 [J g -1 ], can be theoretically defined as the enthalpy that would be obtained from the melting peak of one gram of a 100% crystalline material. According to its own definition, this information is not directly accessible because polymers are made of macromolecules which are way too big to perfectly fold and entirely fit in a regularly repeated crystal lattice; it is however essential to give an estimate of ∆Hm 0 , for it allows estimating the degree of crystallinity of a semi-crystalline polymer by direct comparison with the enthalpy of melting measured by DSC for any given semicrystalline microstructure. The debate is still open on the method used to determine ∆Hm 0 , and probably because the method may be different from a research group to another, the values of ∆Hm 0 found in the literature are sometimes diverging, as shown in Table 1 for a selection of polymers. In addition, some polymers are subjected to polymorphism; to our knowledge, so far only Righetti and co-workers [START_REF] Righetti | Enthalpy of melting of α′and α-crystals of poly(l-lactic acid)[END_REF] took it into account for the calculation of ∆Hm 0 in the case of poly(lactic acid). Besides, new polymers are continuously synthesized, such as polyfuranoates [START_REF] Thiyagarajan | Biobased furandicarboxylic acids (FDCAs): effects of isomeric substitution on polyester synthesis and properties[END_REF] and many other polyesters, and a value of ∆Hm 0 will certainly have to be found for each of them to make preliminary characterizations complete [START_REF] Bourdet | Molecular Mobility in Amorphous Biobased Poly(ethylene 2,5-furandicarboxylate) and Poly(ethylene 2,4-furandicarboxylate)[END_REF][START_REF] Androsch | Cold-crystallization of poly(butylene 2,6-naphthalate) following Ostwald's rule of stages[END_REF]. It is therefore of great interest to find a method that is robust and efficient to determine the equilibrium enthalpy of melting for semi-crystalline polymers, which would eventually help closing the debate or at least provide further elements for discussion.

Table 1 Values of the equilibrium enthalpy of melting ∆Hm 0 [J g -1 ] found in the literature for a selection of crystallizable polymers.

Polymer

Abbreviation ∆Hm 0 [J g -1 ] Poly(ethylene) PE 289 [START_REF] Chiang | Equilibrium between Crystalline and Amorphous Phases in Polyethylene 1[END_REF], 282 [START_REF] Wunderlich | Heat of fusion of polyethylene[END_REF], 307 [START_REF] Atkinson | Thermodynamic properties of ideally crystalline polyethylene[END_REF], 281 [START_REF] Cebe | Heat of fusion of polymer crystals by fast scanning calorimetry[END_REF] Isotactic poly(styrene) iso-PS 86 [START_REF] Danusso | Fusion enthalpy and entropy of isotactic polypropylene[END_REF], 80 [START_REF] Dedeurwaerder | Enthalpie et entropie de fusion du polystyrolène isotactique[END_REF], 96 [START_REF] Abu-Isa | Specific Heat of Synthetic High Polymers. XII. Atactic and Isotactic Polystyrene 1[END_REF] Isotactic poly(propylene) iso-PP 63 [START_REF] Gee | Proceedings of the Chemical Society[END_REF], 260 [START_REF] Danusso | Fusion enthalpy and entropy of isotactic polypropylene[END_REF], 234 [START_REF] Schaefgen | Estimation of the heat and entropy of fusion of some polyhydrocarbons[END_REF], 183 [START_REF] Newman | On the characterization of stereoregular polymers. II. Polypropylene[END_REF], 65 [START_REF] Ke | Characterization of polyolefins by differential thermal analysis[END_REF], 147 [START_REF] Wilkinson | Specific heat of synthetic high polymers. X. Isotactic and atactic polypropylene[END_REF], 188 [START_REF] Passaglia | Specific Heat of Atactic and Isotactic Polypropylene and the Entropy of the Glass[END_REF], 138 [START_REF] Fatou | Melting temperature and enthalpy of isotactic polypropylene[END_REF] Polyamide 6

PA6 188 [START_REF] Marx | Specific Heat of Synthetic High Polymers. IV. Polycaprolactam[END_REF], 155 [START_REF] Gogolewski | Crystallization of polyamides under elevated pressure: 2. Pressure-induced crystallization of nylon-6 (polycapramide) from the melt[END_REF] Poly(ethylene terephthalate) PET 140 [START_REF] Pyda | ATHAS Data Bank[END_REF], 125 [22] Poly(butylene terephthalate) PBT 145 [START_REF] Conix | Crystallization behavior and melting properties of mphenylene group containing polyesters[END_REF], 141 [START_REF] Illers | Heat of fusion and specific volume of poly(ethylene terephthalate) and poly(butylene terephthalate[END_REF] Poly(L-lactic acid) PLLA 135 [START_REF] Miyata | Crystallization behaviour of poly(l-lactide)[END_REF], 91 [START_REF] Pyda | Heat capacity of poly(lactic acid)[END_REF], 146 [START_REF] Badrinarayanan | A comparison of crystallization behavior for melt and cold crystallized poly (l-Lactide) using rapid scanning rate calorimetry[END_REF], 96 [START_REF] Kalish | Spectroscopic and thermal analyses of α′ and α crystalline forms of poly(L-lactic acid)[END_REF], 143 (α-crystals) [START_REF] Righetti | Enthalpy of melting of α′and α-crystals of poly(l-lactic acid)[END_REF], 107 (α'-crystals) [START_REF] Righetti | Enthalpy of melting of α′and α-crystals of poly(l-lactic acid)[END_REF], ∆ = 20.9 + 0.74 -0.0011

• for the α'-form and ∆ = 45.7 + 0.74 -0.0011

• for the α-form [START_REF] Righetti | Enthalpy of melting of α′and α-crystals of poly(l-lactic acid)[END_REF] Poly(phenylene sulfide) PPS 80 [START_REF] Cheng | Glass transition and melting behavior of poly(thio-1,4-phenylene)[END_REF], 146 [START_REF] Maemura | Characterization of Crystallinity and Orientation in Poly-p-Phenylene Sulfide[END_REF], 112 [START_REF] Huo | Effects of thermal history on the rigid amorphous phase in poly(phenylene sulfide)[END_REF] Poly(ethylene 2,5-furandicarboxylate) PEF 137 [START_REF] Papageorgiou | Synthesis of poly(ethylene furandicarboxylate) polyester using monomers derived from renewable resources: thermal behavior comparison with PET and PEN[END_REF], 140 [START_REF] Stoclet | Isothermal crystallization and structural characterization of poly(ethylene-2,5-furanoate)[END_REF], 185 [START_REF] Van Berkel | Isothermal Crystallization Kinetics of Poly (Ethylene 2,5-Furandicarboxylate): Isothermal Crystallization Kinetics of Poly (Ethylene 2,5-Furandicarboxylate)[END_REF] Poly(butylene 2,5-furandicarboxylate) PBF 129 [START_REF] Papageorgiou | Evaluation of polyesters from renewable resources as alternatives to the current fossil-based polymers. Phase transitions of poly(butylene 2,5-furandicarboxylate)[END_REF] Even if some authors estimated the equilibrium enthalpy of melting through methods such as the Flory equation [36] or density measurement [START_REF] Danusso | Fusion enthalpy and entropy of isotactic polypropylene[END_REF][START_REF] Van Berkel | Isothermal Crystallization Kinetics of Poly (Ethylene 2,5-Furandicarboxylate): Isothermal Crystallization Kinetics of Poly (Ethylene 2,5-Furandicarboxylate)[END_REF], the value of ∆Hm 0 for most crystallizable polymers has been more traditionally determined by generating different microstructures with an increasing degree of crystallinity, and then cross-comparing the results of microstructural characterizations performed by at least two techniques, such as XRD and DSC [START_REF] Righetti | Enthalpy of melting of α′and α-crystals of poly(l-lactic acid)[END_REF][START_REF] Badrinarayanan | A comparison of crystallization behavior for melt and cold crystallized poly (l-Lactide) using rapid scanning rate calorimetry[END_REF][START_REF] Kalish | Spectroscopic and thermal analyses of α′ and α crystalline forms of poly(L-lactic acid)[END_REF][START_REF] Cheng | Glass transition and melting behavior of poly(thio-1,4-phenylene)[END_REF][START_REF] Maemura | Characterization of Crystallinity and Orientation in Poly-p-Phenylene Sulfide[END_REF][START_REF] Papageorgiou | Synthesis of poly(ethylene furandicarboxylate) polyester using monomers derived from renewable resources: thermal behavior comparison with PET and PEN[END_REF][START_REF] Stoclet | Isothermal crystallization and structural characterization of poly(ethylene-2,5-furanoate)[END_REF][START_REF] Van Berkel | Isothermal Crystallization Kinetics of Poly (Ethylene 2,5-Furandicarboxylate): Isothermal Crystallization Kinetics of Poly (Ethylene 2,5-Furandicarboxylate)[END_REF][START_REF] Papageorgiou | Evaluation of polyesters from renewable resources as alternatives to the current fossil-based polymers. Phase transitions of poly(butylene 2,5-furandicarboxylate)[END_REF]. On one side, XRD provides an overall apparent degree of crystallinity that is calculated as a ratio of areas, i.e. the area corresponding to the sharper (crystalline) peaks divided by the total area of the pattern (crystalline peaks plus amorphous halo). On the other side, DSC provides the enthalpy [J g -1 ] associated to the melting process of the percentage of polymer crystals previously quantified by XRD. A linear regression of several experimental points collected by this method on samples with different crystallinities allows extrapolating the values of enthalpy of melting to the ideal case of a 100% crystalline polymer. Most of the time, the degree of crystallinity is obtained from XRD patterns, the enthalpy of melting ∆ is estimated from DSC curves, and the equilibrium enthalpy of melting ∆ is calculated according to equation [START_REF] Righetti | Enthalpy of melting of α′and α-crystals of poly(l-lactic acid)[END_REF].

∆ = ∆ / "# (1)
Sometimes other techniques, such as infrared [START_REF] Cebe | Heat of fusion of polymer crystals by fast scanning calorimetry[END_REF][36][START_REF] Hu | Determining Beta-Sheet Crystallinity in Fibrous Proteins by Thermal Analysis and Infrared Spectroscopy[END_REF][START_REF] Miller | Determination of Crystallinity and Morphology of Fibrous and Bulk Poly(ethylene terephthalate) by Near-Infrared Diffuse Reflectance Spectroscopy[END_REF][START_REF] Yao | Investigation of Structure and Crystallization Behavior of Poly(butylene succinate) by Fourier Transform Infrared Spectroscopy[END_REF] or Raman spectroscopy [START_REF] Stuart | Polymer crystallinity studied using Raman spectroscopy[END_REF][START_REF] Nielsen | Estimation of crystallinity of isotactic polypropylene using Raman spectroscopy[END_REF][START_REF] Rull | Estimation of crystallinity in polyethylene by Raman spectroscopy[END_REF][START_REF] Taylor | The Quantitative Analysis of Crystallinity Using FT-Raman Spectroscopy[END_REF][START_REF] Strobl | Raman spectroscopic method for determining the crystallinity of polyethylene[END_REF][START_REF] Schenzel | New Method for Determining the Degree of Cellulose I Crystallinity by Means of FT Raman Spectroscopy[END_REF], are used to quantify the overall percentage of crystallinity to be compared to the value of enthalpy of melting obtained by DSC, but the use of XRD cross-compared to DSC is by far the most common. In a recent study about PLA, Righetti et al. [START_REF] Righetti | Enthalpy of melting of α′and α-crystals of poly(l-lactic acid)[END_REF] used XRD patterns cross-compared to conventional DSC to obtain the equilibrium enthalpy of melting as a function of temperature rather than a single extrapolated value. Recently, Cebe et al. [START_REF] Cebe | Heat of fusion of polymer crystals by fast scanning calorimetry[END_REF] proposed a method for determining the equilibrium enthalpy of melting from Fast Scanning Calorimetry (FSC). This method requires the measurement of the enthalpy of melting plotted against the product of the sample mass times its crystallinity for several samples having variable masses and/or crystallinities; in this case, ∆Hm 0 is deduced from the slope of the plot, expected to be linear, representing ∆Hm [J] vs. $ * . This method is interesting for it has two major advantages: (1) it allows determining the equilibrium enthalpy of melting for samples that degrade right after melting and therefore can be hardly characterized by conventional DSC [START_REF] Cebe | Beating the Heat -Fast Scanning Melts Silk Beta Sheet Crystals[END_REF][START_REF] Corvis | Vitreous State Characterization of Pharmaceutical Compounds Degrading upon Melting by Using Fast Scanning Calorimetry[END_REF][START_REF] Puente | Effects of Size and Specific Surface Area of Boron Nitride Particles on the Crystallization of Bacterial Poly(3-hydroxybutyrateco -3-hydroxyvalerate)[END_REF][START_REF] Puente | Effect of boron nitride as a nucleating agent on the crystallization of bacterial poly(3-hydroxybutyrate)[END_REF], and (2) it reduces the uncertainties that could be reproached to conventional DSC, because melting/recrystallization processes are observed at standard heating conditions [START_REF] Puente | Effects of Size and Specific Surface Area of Boron Nitride Particles on the Crystallization of Bacterial Poly(3-hydroxybutyrateco -3-hydroxyvalerate)[END_REF][START_REF] Puente | Effect of boron nitride as a nucleating agent on the crystallization of bacterial poly(3-hydroxybutyrate)[END_REF][START_REF] Kong | The measurement of the crystallinity of polymers by DSC[END_REF] but largely suppressed at higher heating rates [START_REF] Toda | Insights into polymer crystallization and melting from fast scanning chip calorimetry[END_REF][START_REF] Minakov | Melting and reorganization of poly(ethylene terephthalate) on fast heating (1000 K/s)[END_REF][START_REF] Adamovsky | Scanning microcalorimetry at high cooling rate[END_REF][START_REF] Santis | Scanning Nanocalorimetry at High Cooling Rate of Isotactic Polypropylene[END_REF][START_REF] Esposito | From a Three-Phase Model to a Continuous Description of Molecular Mobility in Semicrystalline Poly(hydroxybutyrate-co-hydroxyvalerate)[END_REF]. According to the authors, this method would reduce errors and can be applied to all types of polymers, copolymers, and blends regardless of their degree of crystallinity. In a paper dealing with the heat capacity of poly(trimethylene terephthalate), Pyda et al. [START_REF] Pyda | Heat capacity of poly(trimethylene terephthalate)[END_REF] raised the question about the possibility of correlating the heat capacity at the glass transition ∆& ' ( )*) + [J g -1 mol -1 ] to the heat of fusion ∆Hf [kJ mol -1 ] obtained by calorimetry on the same sample subjected to different thermal treatments (as received, partially quenched, quenched, crystallized isothermally, annealed, after cooling with 10 K min -1 ) and therefore with different microstructures and crystallinities.

They compared the experimental results to the values predicted on the basis of the ATHAS Data Bank [START_REF] Pyda | ATHAS Data Bank[END_REF] and observed that the extrapolated value of the heat of fusion, which was used as a control over the measured values of heat capacity, was in accord with an extrapolation of the experimental points only when the corresponding microstructure contained little or no rigid amorphous content. The thing is, when it comes to the microstructural description of semi-crystalline polymers, in most cases the two-phase model is better replaced by a more complex three-phase model [START_REF] Wunderlich | Reversible crystallization and the rigid amorphous phase in semicrystalline macromolecules[END_REF][START_REF] Chen | Vitrification and Devitrification of Rigid Amorphous Fraction of PET during Quasi-Isothermal Cooling and Heating[END_REF], which involves an additional "phase" to explain the connection between the ordered and disordered domains, the so-called rigid amorphous fraction (RAF). As such, the RAF should be considered as an interphase, rather than a phase, for it usually forms within the amorphous phase and involves all the entangled macromolecular segments that are too disordered to be part of a crystalline domain, yet too constrained by the nearby crystals to relax as freely as the mobile amorphous fraction (MAF).

The appearance of a RAF is highly probable, for it is due to the length of the polymer chains, which is intrinsically much larger than the lamellar thickness [START_REF] Wunderlich | Macromolecular nucleation[END_REF][START_REF] Androsch | The link between rigid amorphous fraction and crystal perfection in cold-crystallized poly(ethylene terephthalate)[END_REF]. The amount of RAF can vary from a polymer to another, sometimes reaching up to 40% [START_REF] Arnoult | Mobile amorphous phase fragility in semi-crystalline polymers: Comparison of PET and PLLA[END_REF]. Quite intuitively, the amount of RAF is expected to decrease if crystals grow bigger and the macromolecular segments get less entangled [START_REF] Chen | Vitrification and Devitrification of Rigid Amorphous Fraction of PET during Quasi-Isothermal Cooling and Heating[END_REF][START_REF] Androsch | The link between rigid amorphous fraction and crystal perfection in cold-crystallized poly(ethylene terephthalate)[END_REF][START_REF] Schick | Formation and disappearance of the rigid amorphous fraction in semicrystalline polymers revealed from frequency dependent heat capacity[END_REF][START_REF] Righetti | Enthalpy-based determination of crystalline, mobile amorphous and rigid amorphous fractions in semicrystalline polymers[END_REF]. Several studies have shown how the RAF can be used to explain some macroscopic properties of semi-crystalline materials [START_REF] Guinault | Impact of crystallinity of poly(lactide) on helium and oxygen barrier properties[END_REF][START_REF] Nguyen | Mechanical investigation of confined amorphous phase in semicrystalline polymers: Case of PET and PLA[END_REF][START_REF] Martín | Microstructure and properties of branched polyethylene: Application of a three-phase structural model[END_REF][START_REF] Olson | Positron Annihilation Lifetime Spectroscopy of Poly(ethylene terephthalate): Contributions from Rigid and Mobile Amorphous Fractions[END_REF][START_REF] Drieskens | Structure versus properties relationship of poly(lactic acid). I. Effect of crystallinity on barrier properties[END_REF][START_REF] Kolesov | The rigid amorphous fraction of cold-crystallized polyamide 6[END_REF][START_REF] Fernandes Nassar | Multi-scale analysis of the impact of polylactide morphology on gas barrier properties[END_REF] and understand their behavior [START_REF] Esposito | From a Three-Phase Model to a Continuous Description of Molecular Mobility in Semicrystalline Poly(hydroxybutyrate-co-hydroxyvalerate)[END_REF], which in turns means that controlling the amount of RAF is a key parameter for tailoring the performance of semi-crystalline polymers -almost as much as controlling the global amount, the size and the regularity of the crystalline domains. As pointed out by previous works on the establishment of the RAF [START_REF] Chen | Vitrification and Devitrification of Rigid Amorphous Fraction of PET during Quasi-Isothermal Cooling and Heating[END_REF][START_REF] Androsch | The link between rigid amorphous fraction and crystal perfection in cold-crystallized poly(ethylene terephthalate)[END_REF][START_REF] Schick | Formation and disappearance of the rigid amorphous fraction in semicrystalline polymers revealed from frequency dependent heat capacity[END_REF][START_REF] Righetti | Enthalpy-based determination of crystalline, mobile amorphous and rigid amorphous fractions in semicrystalline polymers[END_REF], the connection between the amorphous and the crystalline domains can be evidenced either during [START_REF] Wunderlich | Reversible crystallization and the rigid amorphous phase in semicrystalline macromolecules[END_REF][START_REF] Schick | Formation and disappearance of the rigid amorphous fraction in semicrystalline polymers revealed from frequency dependent heat capacity[END_REF][START_REF] Schick | Vitrification and devitrification of the rigid amorphous fraction of semicrystalline polymers revealed from frequency-dependent heat capacity[END_REF][START_REF] Righetti | Time and Temperature Evolution of the Rigid Amorphous Fraction and Differently Constrained Amorphous Fractions in PLLA[END_REF] or after the crystallization process [START_REF] Wunderlich | Reversible crystallization and the rigid amorphous phase in semicrystalline macromolecules[END_REF][START_REF] Chen | Vitrification and Devitrification of Rigid Amorphous Fraction of PET during Quasi-Isothermal Cooling and Heating[END_REF][START_REF] Schick | Formation and disappearance of the rigid amorphous fraction in semicrystalline polymers revealed from frequency dependent heat capacity[END_REF][START_REF] Righetti | Enthalpy-based determination of crystalline, mobile amorphous and rigid amorphous fractions in semicrystalline polymers[END_REF][START_REF] Righetti | Time and Temperature Evolution of the Rigid Amorphous Fraction and Differently Constrained Amorphous Fractions in PLLA[END_REF]. Moreover, the amount of RAF depends on the crystallization conditions (thermal treatment [START_REF] Pyda | Heat capacity of poly(trimethylene terephthalate)[END_REF][START_REF] Righetti | Crystalline, mobile amorphous and rigid amorphous fractions in poly(L-lactic acid) by TMDSC[END_REF], time and temperature of crystallization [START_REF] Esposito | From a Three-Phase Model to a Continuous Description of Molecular Mobility in Semicrystalline Poly(hydroxybutyrate-co-hydroxyvalerate)[END_REF], thickness [START_REF] Nassar | Structural and Dynamic Heterogeneity in the Amorphous Phase of Poly(L,L-lactide) Confined at the Nanoscale by the Coextrusion Process[END_REF], mechanical treatments [START_REF] Hamonic | Rigid amorphous fraction versus oriented amorphous fraction in uniaxially drawn polyesters[END_REF][START_REF] Miranda | Fluoropolymer microstructure and dynamics: Influence of molecular orientation induced by uniaxial drawing[END_REF]). If it is possible to control the amount of RAF by adjusting and carefully controlling the experimental conditions for crystallization, with a suitable set of crystallization parameters it should be possible to eventually create semi-crystalline microstructures in which almost no RAF is formed.

As previously mentioned, the equilibrium enthalpy of melting ∆ may be calculated according to equation [START_REF] Righetti | Enthalpy of melting of α′and α-crystals of poly(l-lactic acid)[END_REF] where is an estimation of the crystallinity degree obtained from XRD patterns, and the enthalpy of melting ∆ is calculated from DSC curves. On one side, being part of the amorphous phase, the RAF is expected to contribute to the amorphous halo [START_REF] Huo | Effects of thermal history on the rigid amorphous phase in poly (phenylene sulfide)[END_REF], which should be carefully subtracted to obtain "# . On the other side, the presence of RAF is known to be responsible for a progressive change in the baseline of the DSC curves [START_REF] Wunderlich | Reversible crystallization and the rigid amorphous phase in semicrystalline macromolecules[END_REF][START_REF] Minakov | Superheating in linear polymers studied by ultrafast nanocalorimetry[END_REF], which also leads to either overestimate or underestimate ∆ , depending on the choice of the baseline and of the temperature range selected for integration. As a consequence, the cross-comparison of "# with ∆ is subjected to large uncertainties on the estimation of ∆ . Besides, and most importantly, as for any other cross-comparison method, the assumption has to be made that two samples prepared in different ways, having different dimensions, and measured with two different techniques, actually represent the same system. From an experimental point of view, the RAF is distinguished from the MAF as it does not contribute to the heat capacity change at the glass transition [START_REF] Menczel | Heat capacity hysteresis of semicrystalline macromolecular glasses[END_REF][START_REF] Ma | Constraints in semicrystalline polymers: Using quasiisothermal analysis to investigate the mechanisms of formation and loss of the rigid amorphous fraction[END_REF]. Due to its highly constrained behavior, the "relaxation" of the RAF requires more energy to be accomplished with respect to the relaxation of the MAF, therefore its contribution to the increase in the heat capacity occurs at higher temperatures, contributing to a slight but continuous change in the DSC baseline over the entire temperature range of devitrification [START_REF] Wunderlich | Reversible crystallization and the rigid amorphous phase in semicrystalline macromolecules[END_REF][START_REF] Minakov | Superheating in linear polymers studied by ultrafast nanocalorimetry[END_REF]. In general, the development of a significant amount of RAF is associated to the growth of irregular crystalline domains [START_REF] Esposito | From a Three-Phase Model to a Continuous Description of Molecular Mobility in Semicrystalline Poly(hydroxybutyrate-co-hydroxyvalerate)[END_REF][START_REF] Righetti | Time and Temperature Evolution of the Rigid Amorphous Fraction and Differently Constrained Amorphous Fractions in PLLA[END_REF][START_REF] Kanchanasopa | Solid-State Microstructure of Poly(L-lactide) and L-Lactide/meso-Lactide Random Copolymers by Atomic Force Microscopy (AFM)[END_REF][START_REF] Cruz | Novel Aspects in the Structure of Polyethylene terephthalate) As Revealed by Means of Small-Angle X-ray Scattering[END_REF] that are potentially subjected to crystalline reorganization. As a consequence, most of the time the error made on the estimation of ∆ is not only related to the devitrification of the RAF and the choice of a good baseline used to integrate the melting peak, but also to the eventual reorganization (melting/recrystallization) of small and imperfect crystal boundaries [START_REF] Wunderlich | Reversible crystallization and the rigid amorphous phase in semicrystalline macromolecules[END_REF]. In this case, performing DSC measurements at conventional heating rates (up to 20 K min -1 ) exposes to the risk of introducing an additional source of error in the estimation of ∆ . FSC is the only experimental technique able to reduce and eventually suppress any contribution due to melting/recrystallization, as shown by Prof. Schick's pioneer work [START_REF] Minakov | Melting and reorganization of poly(ethylene terephthalate) on fast heating (1000 K/s)[END_REF][START_REF] Adamovsky | Scanning microcalorimetry at high cooling rate[END_REF][START_REF] Santis | Scanning Nanocalorimetry at High Cooling Rate of Isotactic Polypropylene[END_REF]. FSC measurements can only be performed if the size of the samples is decreased to nanoscale, within a range of thickness 1-10 µm [START_REF] Toda | An evaluation of thermal lags of fast-scan microchip DSC with polymer film samples[END_REF]; Nassar et al. [START_REF] Nassar | Structural and Dynamic Heterogeneity in the Amorphous Phase of Poly(L,L-lactide) Confined at the Nanoscale by the Coextrusion Process[END_REF] recently observed that crystallizing nanoscale samples in which the polymer thickness is reduced to a few nanometers dramatically reduces the chances of developing RAF, even when the temperature selected for isothermal crystallization does not necessarily favor phase decoupling.

This work aims at (1) using FSC to estimate the equilibrium enthalpy of melting of several crystallizable polymers, such as poly(ethylene terephthalate) (PET), poly(L-lactic acid) (PLLA), poly(phenylene sulfide) (PPS), poly(ethylene 2,5-furandicarboxylate) (PEF) and poly(butylene 2,5-furandicarboxylate) (PBF), and (2) discussing the results in comparison with the literature and the most common method that uses DSC cross-compared with XRD.

MATERIALS AND METHODS

Materials

Commercial grades of poly(ethylene terephthalate) (PET), poly(L-lactic acid) (PLLA) and poly(phenylene sulfide) (PPS) were purchased in the form of pellets, with the exception of PET that was purchased as a film. Poly(ethylene 2,5-furandicarboxylate) (PEF) and poly(butylene 2,5-furandicarboxylate) (PBF) were synthesized in the laboratories of the Wageningen Food and Biobased Research (FBR), The Netherlands. PEF was synthesized according to the procedure reported in [START_REF] Thiyagarajan | Biobased furandicarboxylic acids (FDCAs): effects of isomeric substitution on polyester synthesis and properties[END_REF]. PBF was obtained by melt polymerization of dimethyl-furan-dicarboxylic acid (DMFDCA) with 1,4-butanediol using Ti(O-i-Pr)4 as a catalyst. The synthesized polymer was subjected to solid-state post-condensation (SSPC) and then used without further purification. All the samples were dried prior to measurement: PET and PLLA were dried at Tg + 10°C for at least 4h, whereas PPS, PEF and PBF were stored in a desiccator with P2O5 for at least 24h. The list of samples used in this study is reported in Table 2. Nanoscale samples were crystallized in situ on Fast Scanning Calorimetry (FSC) sensors and subsequently characterized by FSC. Bulk samples were crystallized in an oven and then characterized by wide-angle X-Ray Diffractometry (XRD), Differential Scanning Calorimetry (DSC), Modulated-Temperature DSC (MT-DSC) and FSC.

Table 2 List of samples along with their number-average molecular weight (, -), weightaverage molecular weight (, . ), polydispersity index (, . , - / ), grade and source.

Sample 0 1 [g mol -1 ] 0 2 [g mol -1 ] 0 2 0 1 / Grade,

Fast Scanning Calorimetry (FSC)

FSC measurements were performed using a Flash-DSC 1 calorimeter (Mettler-Toledo) equipped with a HUBER TC100 intracooler. Prior to use, each MultiSTAR UFS 1 MEMS empty chip was conditioned and corrected according to the manufacturer's procedure.

Temperature calibration was performed according to the procedures reported in the literature [START_REF]Fast Scanning Calorimetry[END_REF]. The dynamic thermal lag corresponds to half the distance between the glass transition temperatures measured at the same heating and cooling rates; the static thermal lag corresponds to a third of the distance between the onset melting temperatures of two indium samples, one placed on top of the polymer sample and the other directly placed on the reference [START_REF] Schawe | Measurement of the thermal glass transition of polystyrene in a cooling rate range of more than six decades[END_REF]. The dynamic thermal lag ∆TD (which depends on the selected heating and cooling rates) was measured at β + = | β -| = 1000 K s -1 and found to be less than 4K. As for the static thermal lag ∆TS (which mostly depends on the sample thickness), values less than 2K

were ensured by preparing samples with thickness of 10 ± 3 µm, as recommended by Toda et al. [START_REF] Toda | An evaluation of thermal lags of fast-scan microchip DSC with polymer film samples[END_REF] to prevent thermal gradients, and in agreement with the findings previously reported in the literature for different polymers [START_REF] Schawe | Description of thermal relaxation of polystyrene close to the thermal glass transition[END_REF][START_REF] Dhotel | Combining Flash DSC, DSC and broadband dielectric spectroscopy to determine fragility[END_REF][START_REF] Monnier | Vitrification of PLA by fast scanning calorimetry: Towards unique glass above critical cooling rate?[END_REF]. The thickness of the samples was estimated from the determined mass, the literature value of density, and the area measured by optical microscopy, as previously done by Toda et al. [START_REF] Toda | An evaluation of thermal lags of fast-scan microchip DSC with polymer film samples[END_REF]. A constant nitrogen flow of 20 mL min -1 was used to purge the measurement cell. Prior to in situ crystallization, the samples were quenched to their reference amorphous state by performing five heating/cooling steps at a rate of 1000 K s -1 over a temperature range going from -60°C to the melt in order to ensure that any previous thermo-mechanical history was erased. The mass of the FSC samples used for this study ranged between 27 and 251 ng. The mass values were estimated using equation ( 2):

$ = ∆& ' 3 4 5 /∆& ' 3 6) 5 (2)
Where ∆& ' 3 4 5 is the heat capacity step at the glass transition estimated from FSC curves (β + = 1000 K s -1 ) and ∆& ' 3

6) 5
is the heat capacity step at the glass transition obtained by a MT-DSC scan of the quenched reference bulk samples (heating rate

β + = 2 K min -1 ≈ 0.033 K s -1 ).

Modulated-Temperature Differential Scanning Calorimetry (MT-DSC)

MT-DSC measurements were carried out on a DSC Q100 (TA Instruments) using the Tzero technology. Temperature, energy and heat capacity calibrations were performed with indium and sapphire standards. All the thermal treatments and characterizations were done under a constant nitrogen flow of 50 mL min -1 to prevent any oxidative degradation of the samples.

The mass of MT-DSC bulk samples ranged between 5 and 10 mg. The modulatedtemperature heating ramps were designed using a heat-only protocol, starting from -70°C and reaching a temperature that ensures the complete melting of each sample, with a heating rate of 2 K min -1 , a modulating amplitude of ± 0.318K and a period of 60s, as recommended in [START_REF] Lacey | Theory and Practice of Modulated Temperature Differential Scanning Calorimetry[END_REF], to prevent reversible crystallization during melting. MT-DSC was used to determine the heat capacity step at the glass transition for the quenched bulk samples

∆& ' 3 6)
5 , which is further needed to estimate the mass of FSC nanoscale samples according to equation (2).

Differential Scanning Calorimetry (DSC)

DSC measurements were made on a DSC TA2920 (TA Instruments). Energy and temperature calibrations were carried out using an indium standard. All the characterizations were done after ballistic cooling to -20°C, with a heating rate of 20 K min -1 under a constant nitrogen flow of 50 mL min -1 to prevent any oxidative degradation of the samples. The mass of DSC bulk samples ranged between 5 and 10 mg. DSC was used to estimate the enthalpy of melting ∆ of bulk samples to be used in equation (1).

Wide-angle X-Ray Diffractometry (XRD)

XRD spectra were recorded at room temperature on (16 mm x 16 mm) samples by a Bruker D8 X-ray diffractometer in the angular range 27 = 5-40°, with a step of 0.05° and a counting time of 1s/step, using a Co Kα radiation (λ = 2.29Å). The patterns were obtained by subtracting the background signal and averaging at least three spectra recorded on the same spot of the sample. XRD patterns were then exploited with different fitting methods (multipeak fitting with Gaussian or Pearson functions) by two different operators to have an estimation of the uncertainties on the calculation of the apparent degree of crystallinity "# .

This value was determined as a ratio of areas, i.e. the area corresponding to the crystalline peaks divided by the total area of the pattern (crystalline peaks plus amorphous halo), and used in equation ( 1).

Sample preparation and thermal treatments

In order to create a microstructure with a reduced amount of RAF, nanoscale samples were crystallized in situ on FSC sensors using selected conditions of temperature and time.

Choice of the crystallization temperature Tc

The samples were melted, cooled down to a temperature within the crystallization temperature range, held in isothermal conditions for a constant crystallization time (60 minutes), cooled down to -60°C with a cooling rate of 1000 K s -1 (the same rate used for measurements), and then heated again to observe the melting endotherm. This protocol was repeated at progressively increasing temperatures (the increase was done by steps of 5°C) and the crystallization temperature Tc was selected as the highest temperature at which the crystallization induction time was shorter than the isothermal time (i.e. crystallization started within a time ≤60 minutes). The choice of 60 minutes results from a compromise between a reasonable crystallization time and a sufficiently high crystallization temperature for crystal perfection.

Preparation of maximum crystallized nanoscale samples

A cycle of isothermal crystallizations was performed on each sample at the selected temperature Tc for progressively increasing durations until a microstructure was formed that showed no further increase in the melting peak, i.e. until the crystallinity degree reached its maximum. The corresponding time was taken as tc max .

Preparation of maximum crystallized bulk samples to be compared to nanoscale samples

The results obtained by FSC on nanoscale samples were compared to the results obtained by more conventional techniques (XRD, DSC and MT-DSC), which require bigger samples. To this purpose, semi-crystalline bulk samples were also prepared by melting the polymer samples between two Teflon sheets, then quickly transferring the assembly to an oven previously set at the selected crystallization temperature Tc and holding the isothermal conditions for a crystallization time at least equal to tc max . Maximum crystallized bulk samples were obtained by holding the selected crystallization temperature Tc for a crystallization time varying between 2 and 5h. The fully amorphous counterparts were obtained by melting the samples between two Teflon sheets, followed by quenching in cold water.

Determination of the equilibrium enthalpy of melting (∆8

= ∆ /∆ ( 3
)
Where ∆ is the enthalpy of melting obtained by integration of the melting endothermic peak. Cebe et al. [START_REF] Cebe | Heat of fusion of polymer crystals by fast scanning calorimetry[END_REF] recently applied equation ( 1) to FSC experiences and rewrote it as follows:

∆ 4 = ∆ • * $;<< * (4)
With obtained either by combining FSC results with information found in the literature (i.e. the heat of fusion of 100% crystal at Tm and the specific heat capacity at a given temperature, judiciously chosen) [START_REF] Cebe | Heat of fusion of polymer crystals by fast scanning calorimetry[END_REF] and/or by a different experimental technique (through independent measurements performed on similarly treated samples -under the assumption that uniform thermal treatment would result in uniform crystallinity, whatever the size of the sample and the experimental technique).

According to equation ( 4), the value of ∆ can be determined if the sample mass, degree of crystallinity and enthalpy of melting are known, i.e. ∆ can be directly deduced from the slope of a plot representing the enthalpy of melting ∆ 4 as a function of $;<< * .

When the conditions for a two-phase model are fulfilled, the degree of crystallinity can be directly estimated from the ratio between the heat capacity change at the glass transition of a crystallized sample (∆& ' =>?@ ) and the value previously found for its fully amorphous counterpart (∆& ' 3 ) as follows:

'A = 1 -∆& ' =>?@ /∆& ' 3 [START_REF] Chiang | Equilibrium between Crystalline and Amorphous Phases in Polyethylene 1[END_REF] If the works of Cebe et al. [START_REF] Cebe | Heat of fusion of polymer crystals by fast scanning calorimetry[END_REF] (using FSC) and Pyda et al. [START_REF] Pyda | Heat capacity of poly(lactic acid)[END_REF][START_REF] Pyda | Heat capacity of poly(trimethylene terephthalate)[END_REF] (correlating the enthalpy of melting and the heat capacity change at the glass transition in two-phase microstructures) are merged, equations ( 1), ( 4) and ( 5) can be combined as:

∆ 4 • = -∆ • /∆& ' 3 4 • • 5 * ∆& ' =>?@ 4 • • 5 + ∆ • (6)

RESULTS AND DISCUSSION

Figure 1 shows the FSC curves recorded on nanoscale samples upon heating at β + = 1000 K s - 1 after in situ isothermal crystallization from the melt at the crystallization temperature Tc during different crystallization times tc followed by cooling down to -60°C at β -= 1000 K s -1 .

Some of the samples investigated in this study are known to be potentially subjected to the special care was given to the selection of Tc so that only the most stable crystalline phase was formed. In the case of PLLA, for instance, the crystallization from the melt at temperatures higher than 130°C leads to the formation of α crystals (helical chain segments aligned in an orthorhombic unit cell), whereas at temperatures lower than 100°C the formation of conformationally disordered α' crystals is observed [START_REF] Righetti | Enthalpy of melting of α′and α-crystals of poly(l-lactic acid)[END_REF]. The presence of metastable crystalline phases increases the risk of crystalline reorganization, which in turn affects the calculation of the enthalpy of melting. For this reason, PLLA samples were crystallized at Tc = 145°C. For similar reasons, Tc was set at 175°C in the case of PEF and 135°C in the case of PBF (whose thresholds between different crystalline phases were found at 170°C [START_REF] Martino | Nucleation and Self-Nucleation of Bio-Based Poly(ethylene 2,5-furandicarboxylate) Probed by Fast Scanning Calorimetry[END_REF] and 130°C [START_REF] Ma | Synthesis and crystallinity of poly(butylene 2,5furandicarboxylate)[END_REF], respectively). The first encouraging evidence is that in Figure 1 all the samples show a single and relatively sharp melting peak, excluding the possibility of a melting/recrystallization process during heating, which would have been typical of polymorphism or crystalline reorganization and is usually revealed by an exothermic signal partially or entirely overlapping the melting endotherm (as observed, for instance, in PLA [START_REF] Di Lorenzo | Calorimetric analysis of the multiple melting behavior of poly(L-lactic acid)[END_REF][START_REF] Pan | Polymorphous Crystallization and Multiple Melting Behavior of Poly(L-lactide): Molecular Weight Dependence[END_REF][START_REF] Yasuniwa | Thermal analysis of the double-melting behavior of poly(L-lactic acid)[END_REF][START_REF] Melnikov | Re-exploring the double-melting behavior of semirigid-chain polymers with an in-situ combination of synchrotron nano-focus X-ray scattering and nanocalorimetry[END_REF]). As the crystallization time increases, the intensity of the melting endotherm increases. Since no cold crystallization occurs during the heating ramp, the values of ∆ 4 obtained by integrating the endothermic peaks in Figure 1 can be reliably and exclusively associated to the melting of the crystalline domains progressively created during the previous isothermal treatments. For some polymers (e.g. for PET and PPS), the maximum of the melting endotherm slightly shifts to higher temperatures as the crystallization time increases, suggesting that the crystallization progresses and the crystalline lamellae grow thicker, in agreement with the Gibbs-Thomson equation [START_REF] Höhne | Another approach to the Gibbs-Thomson equation and the melting point of polymers and oligomers[END_REF].

One may also note that the baseline of the FSC curves in Figure 1 slightly changes as the crystallinity of the samples increases. This dependence is observed in the solid state (from the glassy state up to the onset of melting) but never observed in the molten state, and is particularly visible when the sample mass is relatively large (e.g. for PLLA and PPS). These changes may be due to some interfacial effects associated with stress transfers between the polymer sample and the SiN membrane of the sensor. Stress transfers could be the consequence of a mismatch in the thermal conductivity of the polymer sample and the SiN membrane of the sensor, but also to the shrinkage related to crystallization. The second hypothesis is most likely because the shrinkage of a polymer sample is expected to be proportional to the extent of crystallization, proportional to the sample mass and only visible when the sample is semi-crystalline (in the molten state the polymer is supposed to relax).

These effects are not due to a sample mass change, because precautions were taken to make sure that the mass of polymer in contact with the sensor remained constant during the whole experimental protocol. XRD scans were recorded on maximum crystallized bulk samples to confirm that the crystalline phases grown during the isothermal crystallization at Tc are the most stable ones (Figure 2), which is consistent with the information previously reported in the literature about PLA [START_REF] Righetti | Enthalpy of melting of α′and α-crystals of poly(l-lactic acid)[END_REF][START_REF] Zhang | Crystal Modifications and Thermal Behavior of Poly(L-lactic acid) Revealed by Infrared Spectroscopy[END_REF], PEF [START_REF] Van Berkel | Isothermal Crystallization Kinetics of Poly (Ethylene 2,5-Furandicarboxylate): Isothermal Crystallization Kinetics of Poly (Ethylene 2,5-Furandicarboxylate)[END_REF] and PBF [START_REF] Ma | Synthesis and crystallinity of poly(butylene 2,5furandicarboxylate)[END_REF]. The XRD patterns in Figure 2 were also used to estimate the apparent degree of crystallinity [START_REF] Seidel | Characterization and Analysis of Polymers[END_REF], a method that requires a careful subtraction of the amorphous halo. In this work, all the samples maximum crystallized in the bulk showed a complex amorphous halo that required two Gaussian peaks for fitting. Huo et al. [START_REF] Huo | Effects of thermal history on the rigid amorphous phase in poly (phenylene sulfide)[END_REF] have already pointed out that the presence of RAF affects XRD patterns; being part of the amorphous phase, the RAF is expected to be included in the amorphous halo along with the MAF. The literature reports that the amorphous halo of PET can be deconvoluted in two

Gaussian contributions [START_REF] Murthy | Structure of the amorphous phase in crystallizable polymers: poly(ethylene terephthalate)[END_REF]: the first one attributed to the interchain distances perpendicular to the plans of aligned aromatic rings, the second one attributed to the interchain distances within the plane of the aligned aromatics rings. Similarly to PET, two Gaussian contributions were used to deconvolute the amorphous halos of PLLA and PPS according to information reported in the literature [START_REF] Mano | Structural evolution of the amorphous phase during crystallization of poly (llactic acid): A synchrotron wide-angle X-ray scattering study[END_REF][START_REF] Murthy | General procedure for evaluating amorphous scattering and crystallinity from X-ray diffraction scans of semicrystalline polymers[END_REF]. As for PEF and PBF, two Gaussian contributions were also used, even if no information is reported in the literature.

Figure 3 shows the values of the enthalpy of melting ∆ 4 plotted against the values of the heat capacity step at the glass transition in the semi-crystalline samples ∆& ' =>?@ 4 , both measured on the FSC curves reported in Figure 1. that can be found in the literature (Table 1) plus the values obtained in this work.

According to its definition, ∆ could be straightforwardly calculated with the datasets reported in Figure 3, provided that the hypothesis of a two-phase microstructure is verified.

Indeed, the comparison between the values of ∆ (with respect to its reference ∆ ) and ∆& ' =>?@ (with respect to its reference ∆& ' 3 ) obtained from the same DSC curve is generally used to discuss the pertinence of a two-phase model to describe the microstructure of semicrystalline polymers. Whenever a discrepancy is revealed in the information provided by ∆ (crystalline fraction calculated according to equation ( 3)) and ∆& ' =>?@ (residual amorphous fraction relaxing at the glass transition), a three-phase model is adopted in the place of equation ( 5) and the RAF is introduced to solve the discrepancy.

1 = O'A + ∆& ' =>?@ /∆& ' 3 + #P4 (7)
Nassar et al. [START_REF] Nassar | Structural and Dynamic Heterogeneity in the Amorphous Phase of Poly(L,L-lactide) Confined at the Nanoscale by the Coextrusion Process[END_REF] recently reported that growing crystals in a nanoscale confined environment reduces the development of RAF, even when the temperature conditions are supposed to favor the connections between the crystalline domains and the surrounding amorphous phase. The samples used in FSC experiments are intrinsically nanoscale (the biggest weighs 250 ng), which may eventually help limiting the formation of RAF for similar reasons, i.e. for a sort of finite-size effect. With this being said, the choice of a two-phase model to determine the degree of crystallinity using equation ( 5) applied to FSC curves sounds encouraging. Besides, the experimental conditions used in this study for crystallization were designed to minimize the development of RAF, and the fact that the baselines of the FSC curves in Figure 1 look quite straight in the temperature range between the glass transition and the melt, suggests that a two-phase model could actually apply to all the semi-crystalline microstructures obtained by in situ isothermal crystallization on nanoscale samples. The literature reports that the amount of RAF formed during the crystallization process typically decreases as the size and thermal stability of the crystalline phase increases [START_REF] Androsch | The link between rigid amorphous fraction and crystal perfection in cold-crystallized poly(ethylene terephthalate)[END_REF], and it is well known that the size and thermal stability of the crystalline lamellae increases with the crystallization temperature [START_REF] Hoffman | Regime III crystallization in melt-crystallized polymers: The variable cluster model of chain folding[END_REF]. The literature also reports that the development of RAF is typically associated to an increase in the glass transition temperature (revealing a mobility restriction of the RAF on the MAF) [START_REF] Okazaki | Modulated differential scanning calorimetry in the glass transition region, V. activation energies and relaxation times of poly(ethylene terephthalate)s[END_REF],

as well as a modification of its shape (due to a different distribution of the relaxation times, revealing a stronger coupling between phases) [START_REF] Esposito | From a Three-Phase Model to a Continuous Description of Molecular Mobility in Semicrystalline Poly(hydroxybutyrate-co-hydroxyvalerate)[END_REF]. These modifications of the glass transition, especially when associated to the development of irregular crystalline domains melting over an extended temperature range, would make it impossible to distinguish the microstructural information conveyed by the heat capacity change as a function of temperature (baseline) and the melting endotherm, respectively. None of these modifications were observed in the FSC curves recorded for this study.

When equation ( 6) is used to fit the data in Figure 3, the linear regression gives a slope that

corresponds to ∆ • /∆& ' 3 4 • • 5
and an intercept that directly provides the value of ∆ . The dispersion of the experimental values around the linear fit (grey hatched areas in Figure 3) corresponds to ± 5 % uncertainty on the estimation of the degree of crystallinity. If this uncertainty were introduced in the calculation, the value of ∆ would be affected by an uncertainty of ± 10 J g -1 (which is acceptable when compared to the error introduced by a wrong estimation of the enthalpy of melting, e.g. in the case of crystalline reorganization during DSC measurement ramps at conventional heating rates).

Indeed, there are several possible sources of uncertainties, and the scattering in the values of ∆ found in the literature is quite explicit, as illustrated by the grey horizontal areas in Figure 3 covering all the values previously reported in Table 1. One of the main sources of uncertainty in determining the equilibrium enthalpy of melting by this method, is due to the fact that this is an extrapolative method. It is quite difficult to obtain a polymer with a highly crystalline microstructure for which the two-phase assumption can be assuredly made. Most of the polymers investigated in this paper have a degree of crystallinity that barely reaches 30%. For these polymers, the uncertainty associated with the slope of the ∆ vs ∆& ' plots is much relevant. This being said, the values of ∆ obtained in this study fall in the same range as the values previously reported in the literature by several authors, as illustrated by the grey areas in Figure 3. In the case of PET, for instance, for which no substantial divergences are found in the literature, the FSC protocol provides a value of 138 J g -1 , which is in agreement with the value previously obtained by Wunderlich and Androsch (140 J g -1 ) [START_REF] Pyda | ATHAS Data Bank[END_REF][START_REF] Androsch | The link between rigid amorphous fraction and crystal perfection in cold-crystallized poly(ethylene terephthalate)[END_REF]. As a consequence, when it comes to relatively new polymers for which no data are found in the literature, such as PEF and PBF, one may assume that this method provides at least a preliminary estimation of ∆ that is fairly reliable, yet debatable if different values are successively found by other experimental techniques. Table 3 summarizes the experimental conditions used for the in situ isothermal crystallization performed on the FSC sensors, as well as the main outcomes of the crystallization process measured by FSC on the maximum crystallized nanoscale samples (exploitation of Figure 1) and the corresponding values of ∆ . . The value obtained for PET is 37 ± 5%. Ruland method (based on the conservation of the total scattered intensity by a set of atoms, independent on their structural order) could have been used to obtain the absolute degree of crystallinity [START_REF] Seidel | Characterization and Analysis of Polymers[END_REF], but most of the works reported in the literature use the method based on the ratio of areas. When equation ( 1) is used to crosscompare 3Q "# with the enthalpy of melting measured by conventional DSC, a value of 162 J g -1 is obtained. When FSC results are used instead of DSC in equation ( 1), the crosscomparison with 3Q "# provides a value of 165 J g -1 . These values are quite different with respect to the values found in the literature (140 J g -1 [START_REF] Pyda | ATHAS Data Bank[END_REF], 125 J g -1 [22], Table 1), independently on the technique used to measure the enthalpy of melting. The value obtained by the FSC method is much closer to the values of the literature (138 J g -1 ). Comparing different values of ∆ is definitely a multifactor problem that requires a deeper understanding of both the advantages and drawbacks of the experimental procedures used to obtain them. The differences may be acceptable or very large, depending on the nature of the sample (some polymers crystallize faster, easier and more regularly than other polymers, generating different amounts of RAF), on the design of the experimental protocol used for crystallization (which includes the size of the sample and the crystallization conditions, i.e.

the crystallization temperature and time), as well as on the calculation procedure. Since ∆ is estimated by extrapolating the experimental data (crystallinity degree Xc vs. enthalpy of melting ∆ obtained on samples with different crystallinity degrees, it is mandatory to ensure the accuracy on the calculation of these quantities. In general, techniques such as XRD and DSC are used to cross-compare the results obtained on bulk samples crystallized in controlled conditions. Bulk samples are subjected to bigger thermal gradients, resulting in less controlled crystallization conditions and less regular microstructures. Besides, using conventional DSC to determine the enthalpy of melting of semi-crystalline polymers expands the level of uncertainty because of crystalline reorganization that is sometimes observed during the heating ramps, and that is hardly distinguished from the baseline drift related to the devitrification of the RAF. Using FSC improves the accuracy of measurement thanks to the extremely fast heating rates, which exclude any possible crystalline reorganization and suppress the effects eventually due to polymorphism. Last but not least, in suitable and controlled crystallization conditions, FSC allows the concomitant measurement of 6P4 = 1 -and ∆ , considerably reducing the uncertainties due to the cross-comparison of samples whose microstructures are not necessarily the same because of sampling heterogeneities. The main source of uncertainty for the FSC method proposed here is related to the fact that ∆ is estimated under the assumption that a two-phase microstructure is obtained during the in situ crystallization protocol. For this reason, an accurate selection of the crystallization temperature Tc should be done with the purpose of growing regular crystalline domains with reduced connections between phases, i.e. with a negligible amount of RAF.

Among the samples shown in Figure 1, PLLA and PPS are the ones with the largest masses (∼250 ng). Sample mass and crystallization temperature are two parameters that possibly affect the results of FSC measurements and the following data treatment. However, it is quite difficult to distinguish how these parameters respectively contribute to the combined uncertainty on ∆ because it is very hard to prepare several FSC sensors with samples having exactly the same mass and shape, and placed exactly in the same position on the membrane. Figure 4 shows the results obtained on smaller samples of PLLA and PPS (58 ng and 94 ng respectively) at higher crystallization temperatures (149°C and 243°C respectively) with respect to Figure 1. 138 J g -1 and 133 J g -1 respectively) are quite similar and in perfect agreement with the literature (140 J g -1 [START_REF] Pyda | ATHAS Data Bank[END_REF] and 125 J g -1 [22]) (Figure 5). provides a values of ∆ = 137 ± 4 J g -1 . The grey horizontal area represents the domain covered by the values of ∆ found in the literature (Table 1).

Other samples may be more sensitive to the choice of Tc, especially if they have peculiar crystallization features such as polymorphism. PLLA, for instance, exhibits several types of crystal modifications (α, α′, β, and γ phases) that are sometimes difficult to isolate, even if the crystallization temperature is accurately selected. The ∆ vs ∆& ' plots reported in Figure 4 (bottom) suggest that PLLA is extremely sensitive to the choice of Tc. Table 3 reported a value of ∆ = 96 ± 10 J g -1 for PLLA crystallized at Tc = 145°C, but a much different value is obtained for Tc = 149°C (∆ = 134 ± 10 J g -1 ). None of these values is in disagreement with the literature, because Pyda et al. [START_REF] Pyda | Heat capacity of poly(lactic acid)[END_REF] and Kalish et al. [START_REF] Kalish | Spectroscopic and thermal analyses of α′ and α crystalline forms of poly(L-lactic acid)[END_REF] reported values of 91 J g -1 (at Tc = 145°C) and 96 J g -1 (at Tc = 150°C) respectively, whereas Miyata et al. [START_REF] Miyata | Crystallization behaviour of poly(l-lactide)[END_REF] and Badrinarayanan et al. [START_REF] Badrinarayanan | A comparison of crystallization behavior for melt and cold crystallized poly (l-Lactide) using rapid scanning rate calorimetry[END_REF] reported values of 135 J g -1 and 146 J g -1 respectively; the most accurate approach seems to be the one proposed by Righetti et al. [START_REF] Righetti | Enthalpy of melting of α′and α-crystals of poly(l-lactic acid)[END_REF], which suggests to take into account the temperature dependence of ∆ .

These preliminary results indicate that polymers are not equally sensitive to parameters such as the sample mass and the crystallization temperature. As potential sources of uncertainty on the extrapolated value of the equilibrium enthalpy of melting, these parameters definitely deserve further investigations. As a perspective, it would also be interesting to measure the crystallinity degree by recording XRD patterns directly on the FSC sensor, according to the technical solutions proposed by Ivanov, Cavallo, Vlassak and their-coworkers [START_REF]Fast Scanning Calorimetry[END_REF][START_REF] Melnikov | Re-exploring the double-melting behavior of semirigid-chain polymers with an in-situ combination of synchrotron nano-focus X-ray scattering and nanocalorimetry[END_REF][START_REF] Van Drongelen | Microfocus wide-angle X-ray scattering of polymers crystallized in a fast scanning chip calorimeter[END_REF][START_REF] Baeten | Simultaneous Synchrotron WAXD and Fast Scanning (Chip) Calorimetry: On the (Isothermal) Crystallization of HDPE and PA11 at High Supercoolings and Cooling Rates up to 200 °C s -1[END_REF], who worked out some feasible solutions to combine XRD with FSC.

CONCLUSIONS

This work shows that the equilibrium enthalpy of melting ∆ can be determined by fast scanning calorimetry (FSC), provided that suitable crystallization conditions are selected to reduce the connection between phases. The metrological concept relies, as usual, on the characterization of samples having microstructures with different degrees of crystallinity and the extrapolation of the experimental data to the theoretical situation of a 100% crystalline polymer. Contrarily to the methods commonly found in the literature, which are based on the cross-comparison of different experimental techniques (such as XRD and DSC) performed on different samples, this method is only calorimetric and significantly reduces the uncertainties related to (1) thermal gradients (FSC measurements are performed on nanoscale samples), [START_REF] Thiyagarajan | Biobased furandicarboxylic acids (FDCAs): effects of isomeric substitution on polyester synthesis and properties[END_REF] polymorphism and/or crystalline reorganization (FSC ramps are recorded at very high heating rates), and (3) sample heterogeneities (all the information required to calculate ∆ is obtained from the same curve, recorded on the same sample). This method applies to any crystallizable polymer that is quenchable and for which a two-phase microstructure with a negligible amount of rigid amorphous fraction and reduced connections between phases is obtained under controlled crystallization conditions. The protocol of isothermal crystallization is performed in situ on nanoscale samples placed on the FSC sensor. The optimization of the crystallization parameters (temperature and time) should be preliminarily done according to the selected samples, especially for polymers that are subjected to polymorphism and/or particularly sensitive to the choice of the crystallization temperature. This method was used to estimate the equilibrium enthalpy of melting ∆ of well-known polymers, such as PET and PPS, as well as of more recent biopolymers, such as PLLA, PEF and PBF.

∆ 4 •• 5 .

 45 can therefore be calculated from the intercept of a plot representing the enthalpy of melting ∆4 • versus the heat capacity step at the glass transition ∆& ' =>?@ ∆ 4 is obtained by integrating the endothermic peak observed on the FSC curves normalized to the sample mass (according to equation (2)) with a linear baseline going from the end of the glass transition up to the melt. ∆& ' =>?@ 4 is obtained by extrapolating the baselines of the FSC curves in the glassy/solid state (T<Tg) and in the liquid state (T>Tg) through the glass transition Tg (read as the midpoint of the heat capacity step), and calculating the difference between B;<CDEFC GHIJHK 4 ( )*)L and B;<CDEFC ?MGHK 4 ( )*)L . Values equivalent to a change in the heat capacity are obtained from the values of heat flow [mW] by conversion to [J] and normalization to the sample mass [g].

Figure 1 FSC

 1 Figure 1 FSC curves recorded upon heating at β + = 1000 K s -1 on nanoscale samples of PET, PLLA, PPS, PEF and PBF after in situ isothermal crystallization from the melt at the crystallization temperature Tc during different crystallization times tc.

Figure 2

 2 Figure 2 XRD patterns recorded at room temperature on maximum crystallized bulk samples of PET, PLLA, PPS, PEF and PBF after isothermal crystallization from the melt at the crystallization temperature Tc (solid lines). The raw data are in grey. The thick solid lines represent the fitting result. Thinner solid and dashed lines are also reported to represent the crystalline and amorphous contributions to each pattern (multi-peak fitting with Gaussian functions). The amorphous halos were fitted by two Gaussian peaks; the dashed lines

Figure 3

 3 Figure 3 Enthalpy of melting vs. heat capacity change at the glass transition obtained from FSC curves normalized to the sample mass and the heating rate (Figure 1) for nanoscale samples of PET, PLLA, PPS, PEF and PBF crystallized in situ at the crystallization temperature Tc for different crystallization times tc. The linear regression of the experimental data (solid lines) extrapolated to = 100% and ∆& ' = 0 J g -1 K -1 reveals the equilibrium enthalpy of melting ∆ under the assumption of a two-phase model. The grey hatched areas around the solid lines represent the uncertainty of ± 5% on the estimation of the apparent

Figure 4 (

 4 Figure 4 (top) FSC curves recorded upon heating at β + = 1000 K s -1 on nanoscale samples of PLLA and PPS after in situ isothermal crystallization from the melt at two different crystallization temperatures (145°C and 149°C for PLLA, 235°C and 243°C for PPS) during different crystallization times. Smaller samples were used with respect to Figure 1 (58 ng vs. 250 ng for PLLA, 94 ng vs. 251 ng for PPS). (bottom) Enthalpy of melting vs. heat capacity change at the glass transition obtained from the FSC curves normalized to the sample mass

a

  different crystallization temperature. The same PET sample was crystallized at three different temperatures (190°C, 200°C, and 210°C) and the values of ∆ obtained (141 J g -1 ,

Figure 5

 5 Figure 5 Extrapolation based on the FSC measurements on a nanoscale sample of PET (92 ng) isothermally crystallized from the melt at three different crystallization temperatures (190°C, 200°C and 210°C) during different crystallization times. The linear regression (solid lines) of the experimental data (crosses) extrapolated to = 100% and ∆& ' = 0 J g -1 K -1

  # ) are available for cross-comparison. With the value of ∆ , the degree of crystallinity of a semi-crystalline polymer can be determined by performing a single DSC run and applying the following equation[START_REF] Mathot | Temperature dependence of some thermodynamic functions for amorphous and semi-crystalline polymers[END_REF][START_REF] Mathot | Thermal Characterization of States of Matter[END_REF]:

		9 : )
	Equation (1) gives a good estimate of ∆	only if the same microstructure can be obtained by
	two different techniques (e.g. DSC and XRD) and the corresponding results (i.e. ∆	and

"

Table 3

 3 Crystallization temperature (Tc) and crystallization time (tc max ) selected to reach the maximum crystallinity degree during in situ isothermal crystallization aiming to reduce coupling between phases. Tg and Tm are the glass transition temperature and the melting temperature of the maximum crystallized nanoscale samples measured by FSC. The

	equilibrium enthalpy of melting ∆	was obtained according to equation (6) based
	exclusively on FSC results. For comparison's purposes, the crystallinities of maximum
	crystallized bulk and nanoscale samples, which were obtained by XRD and FSC respectively,
	are also reported (	"#	3Q	and	4	3Q	).

Table 1

 1 plots reveal that, for these polymers, the extrapolated value of ∆ is indeed quite dependent on the sample mass and/or on the choice of Tc. Further studies are required to evaluate the influence of these parameters. It should be reminded that all extrapolative methods have intrinsic limitations that are sources of uncertainties, and this is particularly true when the extrapolation is done on a large range of values starting with few points unevenly distributed (the case of PPS crystallized at 243°C is a good example). From this point of view, the extrapolation done on the basis of FSC measurements is neither better nor worse than any other extrapolative method, but provides values in agreement with the literature. Besides, there are polymers (such as PET) that are quite insensitive to the choice of

	) plus the

ACKNOWLEGMENTS

The authors thank the Re ǵion Normandie and the European FEDER for their financial support through the SCAMPI project (Aurélie Bourdet's PhD program) and the FARM Rin Recherche project (Estève Ernault's post-doc position), as well as the French Ministère de l'Enseignement Supérieur et de la Recherche for the financial support for Clément Fosse's PhD program. The authors also thank Dr Benoît Vieille for providing PPS pellets.