Vincent Bouad

Marc Guerre

Cédric Totée

Gilles Silly

Olinda Gimello

Bruno Améduri

Jean- François Tahon

Rinaldo Poli

Sophie Barrau

Vincent Ladmiral

RAFT Polymerisation of Trifluoroethylene: The importance of understanding reverse additions

This article is the first report of the RAFT polymerisation of trifluoroethylene (TrFE). Trifluoroethylene is a rare but very important fluoromonomer, as it allows the preparation of materials endowed with unique electroactivity via copolymerisation with vinylidene fluoride (VDF) and other fluoromonomers. RAFT polymerisations carried out using O-ethyl-S-(1-methoxycarbonyl) ethyldithiocarbonate as chain transfer agent and a thermal initiator were carefully examined. The polymerisation, its kinetics and the chain-end evolution were investigated by GPC, 1 H{ 19 F} and 19 F{ 1 H} NMR spectroscopies as well as MALDI-TOF mass spectrometry. Similarly to the RAFT polymerisation of VDF, irreversible transfer reactions and reverse additions significantly affect the control of the polymerisation as well as the chain-end functionality. However, in contrast to VDF, unusual reverse propagation of TrFE, although limited to a few monomer units, was evidenced thanks to a combined NMR spectroscopy and DFT calculations approach. RAFT polymerisation afforded relatively well-defined PTrFE with crystalline structure consistent with previous reports.

A. Introduction

Fluoropolymers possess a set of remarkable properties such as a high thermal stability, chemical resistance, low permittivity but also good weathering durability, hydro-and oleophobicity as well as, in certain cases, electroactivity (ferro-, piezoelectricity). 1,2 Polyvinylidene fluoride (PVDF) is one of the most used fluoropolymer (second after polytetrafluoroethylene, PTFE). 3,4 It has found application in filtration membranes, architectural coating and, with its electroactive properties, is a potential candidate for use in high technology devices. 5,6 The copolymer of vinylidene fluoride (VDF) and trifluoroethylene (TrFE) is a very interesting polymer as it combines the high ferro-and piezoelectric properties of PVDF with a much better processability. In contrast, compared to PVDF, PTrFE has been much less studied. This is likely due to its high price, relative rarity and inferior electroactive properties compared to PVDF or P(VDFco-TrFE). 7,8 Although TrFE has been widely used and studied as a comonomer (in particular with VDF), only few studies have been dedicated to the homopolymerisation of TrFE. Just like VDF, TrFE only polymerises via the conventional radical polymerisation mechanism and is prone to chain defects caused by reverse additions (i.e. head-to-head (HH) additions and tail-to-tail (TT) additions, where the head and the tail are the CF2 and CHF groups, respectively; see Figure 10 for the different possible additions). The first studies dedicated to the TrFE radical polymerisation and PTrFE structure by Naylor and Lasoski identified the two main 19 F NMR resonance regions for the CFH and CF2 groups and highlighted the spectrum complexity, resulting from the PTrFE multiple stereochemical centres, but did not provide a detailed assignment of the different resonances. 9 In later work, Yagi described for the first time the PTrFE microstructure and estimated, using the Monte Carlo simulation method, that 50% of the monomers are backward-added. 10 Note that Yagi's definition of a backward-added monomer is not identical to the definition of reverse additions used by other authors. A backward-added monomer is a monomer that has been incorporated in the polymer chain via HH or via a HH-TH addition (i.e. via an addition which generates a -CHF • propagating radical). Importantly, a monomer incorporated in the chain via a TT addition is not considered backward-added. In 1982, Cais et al. provided a different estimation of the fraction of reverse additions in PTrFE based on 19 F NMR spectroscopy and rotational isomeric state (RIS) models. 11 In this article, the authors only considered reverse additions (HH and TT additions). However, they assumed that these reverse additions occurred, as in PVDF, only as the HH-TT addition sequence. They did not consider the possibility of "reverse propagation" (i.e. sequence of tail-to-head (TH) additions). They estimated these HH-TT additions to amount to 11.6 % of the total monomer additions. 11 In a previous paper using [START_REF] Tonelli | Advances in Chemistry Series[END_REF] C and RIS modelling, which they afterwards deemed erroneous, the same authors had evaluated the fraction of HH-TT defects as 50%. 12 An update, based on 19 F{ 1 H} NMR spectra, estimated the HH-TT additions to be 20% of the overall additions. [START_REF] Tonelli | Advances in Chemistry Series[END_REF] More recently, using the definition of reverse additions given by Cais (HH-TT additions sequence), and more modern NMR techniques (higher frequency and more efficient 1 H decoupling), Harris reached an estimation of around 13.5%. [START_REF] Reinsberg | [END_REF] To this day, apart from the work of Yagi, the addition defects in PTrFE have only been considered as HH-TT sequence, as in PVDF. Cais and Kometani, for example, described the synthesis of isoregic (HT additions only) PTrFE by dechlorination or debromination of the precursor poly(chlorotrifluoroethylene) (PCTFE) or poly(bromotrifluoroethylene). 15 They also described the synthesis of pure HH-TT PTrFE by the alternating copolymerisation of 1,2-difluoroethylene and tetrafluoroethylene. 16 Reversible Deactivation Radical Polymerisation techniques, have not yet been employed to polymerise TrFE. 17 Several articles report the telomerisation of TrFE using various transfer agents such as hydrogen disulfide 18 ,mercaptan and trifluoromethanethiol, 19 bistrifluoromethyl disulphide, 20 iodine monochloride, 21 or dibromodifluoromethane 22,23 for example. Balagué et al. studied the telomerisation of TrFE using fluoroalkyl iodides as transfer agents. 24 They found that TrFE was less reactive than VDF and that the monoadducts were composed of regularly added and reversely added TrFE unit. 24 More recently, Gosset et al. used telomerisation to prepare TrFE-dimethyl phosphite monoadducts and obtained the tail (CF2HCHFPO3(CH3)2) and head (CFH2CF2PO3(CH3)2) regioisomers in a 3:1 ratio. 25 Finally, Colpaert 26 et al. prepared PTrFE using the thermal degradation of perfluoro-3-ethyl-2,4-dimethyl-3-pentyl (leading to CF3˙ radical) as initiator.

The electroactive properties of PTrFE are highly dependent on the crystal structure. PTrFE is a semi-crystalline polymer with different crystal phases and chain conformations as described by Tashiro et al. 27 and Lovinger et al. 16 Oka et al. 8 reported two crystal phases related to the cooling rate from the melt : the non-polar phase (also called S-phase because of a single peak observed by X-ray diffraction) obtained by quenching, and the polar phase (also called D-phase because of a double peak) obtained by slow cooling from the melt. 28 The first evidence of ferroelectricity in PTrFE was reported for a polymer cooled at a rate of approximatively 2°C.min -1 . 8 This article is the first study of the TrFE RAFT polymerisation. It is a follow-up to the work of Guerre et al. on the RAFT polymerisation of VDF. [29][30][31] Those comprehensive studies showed that the HH reverse additions lead to the formation of less reactivatable -CH2-xanthate-terminated PVDF chains, which impairs the control of the polymerisation. This article examines the behaviour of TrFE under RAFT polymerisation conditions and focusses on the evolution of the PTrFE chain-ends during the polymerisation.

B. Experimental Section

B.1. Materials

All reagents were used as received unless stated otherwise. Trifluoroethylene (TrFE) was kindly supplied by Arkema (Pierre-Bénite, France). O-Ethyl-S-(1-methoxycarbonyl) ethyldithiocarbonate (CTAXA) was synthesized according to the method described by Liu et al. 32 tert-Amyl peroxy-2ethylhexanoate (Trigonox 121, purity 95%) was purchased from AkzoNobel (Chalons-en-Champagne, France). Dimethylcarbonate (DMC) was purchased from Sigma Aldrich and used as received. Acetone-d6 was purchased from Eurisotop and used as received.

B.2. Instrumental methods

Size-Exclusion Chromatography (SEC).

Size-exclusion chromatograms (SEC) were recorded using a triple-detection GPC from Agilent Technologies with its corresponding Agilent software, dedicated to multidetector GPC calculation. The system used two PL1113-6300 ResiPore 300 × 7.5 mm columns with THF as the eluent with a flow rate of 1 mL•min -1 . The detector was a 390-LC PL0390-0601 refractive index detector. The entire SEC-HPLC system was thermostated at 35 °C. PMMA standards were used for calibration. Typical sample concentration was 20 mg/mL. MALDI-TOF Spectrometry. MALDI-TOF mass spectra were recorded using a Bruker Ultraflex III time-of-flight mass spectrometer using a nitrogen laser for MALDI (λ 337 nm). The measurements in positive ion were performed with a voltage and reflector lens potential of 25 and 26.3 kV, respectively. For negative ion mode, the measurements were performed with ion source and reflector lens potential of 20 and 21.5 kV respectively. Mixtures of peptides were used for external calibration.

B.3. NMR SPectroscopy

The polymer NMR spectra were collected at 25 °C on a Bruker Avance III 400-MHz spectrometer equipped with two independent broadband ( 15 N-31 P and 15 N- 19 F, 300 W) and a high band ( 1 H, 100W) rf channels. A 5 mm 1 H/ 19 F/ [START_REF] Tonelli | Advances in Chemistry Series[END_REF] C TXO triple resonance pulsed field gradient probe for which [START_REF] Tonelli | Advances in Chemistry Series[END_REF] C and 19 F are on the inner coil and 1 H on the outer coil is used for three channels experiments. This probe has a lower background 19 F signals compared to standard dual-channel probes. This triple resonance 1 H/ 19 F/ [START_REF] Tonelli | Advances in Chemistry Series[END_REF] C probe is capable of producing short 90° pulses of 6.5 μs width for 19 F, 9.5 μs for 13 C and 9.2 μs for 1 H channels. In all experiments, 1 H decoupling is realized with waltz16. 19 F decoupling was performed with nested loops using 0.5 ms and 1 ms chirped adiabatic pulses with 80 kHz bandwidth in order to desynchronize and minimize decoupling artifacts. C 1D NMR with 1 H, 19 F and 1 H+ 19 F Decoupling. A one pulse 90° pulse sequence was used with 1.1 s acquisition time, 30 kHz spectral window 4100 transient and 1 s recycle delay.

19

F 2D NMR COSY with 1 H Decoupling. The cosygp pulse sequence from Bruker catalog was modified in order to include 1 H decoupling over the whole pulse sequence. The acquisition parameters were 1 s acquisition time, 75 kHz spectral windows in F2 and in F1, 4 transients and recycle delay of 1 s. Processing involved a magnitude calculation phase correction in the F1 dimension. 1 H{ [START_REF] Tonelli | Advances in Chemistry Series[END_REF] C} 2D NMR HSQC with 1 H Decoupling. The hsqcetgpsi2 HSQC pulse sequence from the Bruker catalog was modified in order to apply 19 F decoupling over the whole pulse sequence. Acquisition parameters were 0.3 s acquisition time, 7.5 kHz spectral window in F2, 25 ms acquisition time, 30.2 kHz spectral window in F1, 1 JCH = 152 Hz, garp decoupling for 13 C, 8 transients and recycle delay of 1 s. Processing involved an exponential window multiplication in both dimensions.

19

F{ [START_REF] Tonelli | Advances in Chemistry Series[END_REF] C} 2D NMR HSQC with 1 H Decoupling. The pulse sequence described by Li et al. 33 (2D NMR studies of a model for Krytox® perfluoropolyethers) was written from scratch for a Bruker system, the only modifications being 1 H decoupling over the whole pulse sequence, [START_REF] Tonelli | Advances in Chemistry Series[END_REF] C decoupling performed with nested loops using 0.5 ms and 1 ms chirped adiabatic pulses with 30 kHz bandwidth in order to desynchronize and minimize decoupling artefacts and echo-antiecho quadrature detection in F1. Acquisition parameters were 83 ms acquisition time and 75 kHz spectral window in F2, 99 ms acquisition time and 10 kHz spectral window in F1, 1 JCF = 260 Hz, 16 transients and recycle delay of 1 s.

Processing involved linear prediction of an exponential window multiplication in both dimensions and a magnitude calculation phase correction in the F1 dimension. Differential Scanning Calorimetry (DSC). DSC experiments were performed on a DSC Q20 apparatus (TA Instruments). PTrFE samples of around 10 mg were analysed in the temperature range [-80, 220°C] at a heating rate of 10°C.min - 1 under nitrogen gas flow.

X-ray scattering. Wide Angle X-ray Scattering (WAXS) experiments were carried out using a Xeuss 2.0 (XENOCS) with a GeniX3D microsource (λ=1.54 Å) operating at 0.6 mA and 50 kV. The sample-to-detector distance was 120 mm.

B.4. Syntheses

Pressure reactor procedure. The PTrFE used for NMR analyses was synthesized by RAFT polymerisation in a 50 mL Hastelloy Parr autoclave system (HC 276), equipped with a mechanical Hastelloy stirring system, a 3000-psi rupture disk, inlet and outlet valves, and a Parr electronic controller to regulate the stirring speed and heating. Prior to reaction, the autoclave was pressurised with 30 bars of nitrogen to check for leaks. The autoclave was then kept under vacuum (20 10 -3 bar) for 30 minutes to remove any trace of oxygen. A degassed solution of tert-amyl peroxy-2-ethylhexanoate, the initiator (0.281 g, 1.22 10 -3 mol), and CTAXA (1.27 g, 6.09 10 -3 mol) were introduced via a funnel under vacuum. The reactor was then cooled using a liquid nitrogen bath and 10 g of TrFE was transferred by double weighing (i.e. mass difference before and after filling the autoclave with TrFE). After warming to ambient temperature, the autoclave was heated to the target temperature under mechanical stirring. The reaction was stopped after 30 min. The autoclave was cooled to room temperature (ca. 20 °C), purged from the residual monomers, and the dimethylcarbonate was removed under vacuum. The crude product was dissolved in 10 mL of acetone and left under vigorous stirring for 10 min. This polymer solution was then precipitated from 100 mL of chilled hexane. The precipitated polymer (yellow wax) was filtered through a filter funnel and dried under vacuum (15 10 -3 mbar) for 2 h at 40 °C. The polymerisation yield (2.5 %) was determined gravimetrically (mass of dried precipitated polymers/mass of monomer introduced in the pressure reactor).

Carius tube procedure. The RAFT polymerisation of TrFE was carried out in thick 8 mL Carius tubes in which a solution of the initiator tert-amyl peroxy-2-ethylhexanoate (Trigonox® 121) and CTAXA in DMC (5 mL), was added and then degassed by performing at least three freeze-pump-thaw cycles. The gaseous monomer was introduced into the Carius tube at the liquid nitrogen temperature (TrFE, 1.5 g, 1.83 10 -2 mol, 0.8 ΔP) using a custom-made manifold that enables accurate measurement of the gas amounts (using "pressure drop vs. mass of monomer" calibration curves). The tube was then sealed under dynamic vacuum at the liquid nitrogen temperature, before being placed horizontally in a shaking water bath thermostated at 73 °C. At the desired polymerisation time (1h, 2h, 4h, 6h, 8h and 15h), the tubes were placed into liquid nitrogen, opened, and then the dimethyl carbonate was evaporated at 40 °C under reduced pressure. Conversions were determined gravimetrically after drying under vacuum until constant weight.

B.5. Computational details

The computational work was carried out using the Gaussian09 suite of programs. 34 The geometry optimizations were performed in the gas phase without any symmetry constraint using the B3PW91 functional in combination with the 6-31G(d,p) basis functions for all atoms. The unrestricted formulation was used for all radicals, yielding negligible spin contamination in all cases. The ZPVE, PV, and TS corrections at 298.15 and at 343.15 K were obtained with Gaussian09 from the solution of the nuclear equation using the standard ideal gas and harmonic approximations, which also verified the nature of all optimized geometries as local minima or firstorder saddle points. A correction of 1.95 kcal/mol was applied to all G values to change the standard state from the gas phase (1 atm) to solution (1 M). 35

C. Results and discussion

The conditions for the TrFE RAFT polymerisation were adapted from those used by Guerre et al. 31 for the RAFT polymerisation of VDF (Figure 1). DMC was chosen over other solvents as it provides higher polymerisation rates and is relatively less prone to H-abstraction than other solvents. 36 TrFE was expected to behave similarly to VDF. Indeed, TrFE is, like VDF, prone to chain defects as shown in previous studies. 10,[START_REF] Reinsberg | [END_REF]37,38 The HH reverse additions proved to play a crucial role in the RAFT polymerisation of VDF as they were shown to be responsible for the slowdown of the RAFT equilibrium and ultimately for the loss of control. 23,30 The chain ends formed during the RAFT polymerisation of TrFE and their evolution with conversion were thus examined.

The NMR study of the RAFT polymerisation of TrFE revealed the presence of four characteristic xanthate-terminated chains: i) in the 7.1-7.2 ppm region of the 1 H NMR spectrum and in the -171/-175 ppm region of the 19 F NMR spectrum assigned to -CFH-XA-terminated chains and ii) in the -80/-90 ppm region of the 19 F NMR spectrum assigned to -CF2-XAterminated chains (Figure 2). In addition, the presence of monoadducts (i.e. molecule formed by insertion of only one TrFE unit in a CTA) was also detected by 19 F NMR (Figure S1). Two monoadducts were observed (Figure S1): the H-adduct formed by addition of the CTA R-group to the TrFE CFH (tail) moiety and the T-adduct resulting from the corresponding addition to the CF2 (head) moiety. A detailed description of these monoadducts and the determination of the chain ends is provided elsewhere. 39 In the course of the polymerisation the H-adduct disappears very quickly from the reaction medium; it can only be observed at low conversions (up to 15%). In contrast, the amount of T-adduct decreases more gradually with conversion and still persistent at the end of the reaction. It is possible to see the T-adducts as functional chain ends, as they are terminated by a xanthate moiety, or as a xanthate trapping agent, as T-adducts seemed to be poorly reactivated. Table S1 and S2 and Figure S2 and S3 details the evolution of the chain end functionality with and without considering the T-adducts. Here, T-adducts are considered as -CFH-XA terminated chains and are taken into account in the calculation of chain end functionnality. Dead chains, produced by hydrogen abstraction from solvent (DMC), monomer or polymer were also observed at -130 to -134/6.5 ppm (-CF2H termini) and -244.6/5.0 ppm (-CFH2 termini) in the 19 F/ 1 H NMR spectra (Figure 2), respectively.

The TrFE RAFT polymerisation kinetics and the molar mass, dispersity and chain-end resonance evolutions were monitored for two targeted degrees of polymerisation: DP50 and DP100. Table 1 gathers the data related to these experiments.

Figure 3 shows the PTrFE ω chain-end evolution (xanthate chain-ends), monitored and quantified using 19 F NMR spectroscopy. As in the case of PVDF, the regularly-terminated chains (terminated by a -CF2-XA moiety and noted PTrFEH-XA) quickly disappeared from the reaction medium. Indeed, even at low TrFE conversion (around 5%), these PTrFEH-XA chains represented already only 45% or less of the total number of chains (for both targeted DP). The precise TrFE conversion for which the PTrFEH-XA chains completely disappear is however difficult to determine. The data points shown in Figure 3 suggest that this likely occurs between 5 and 20% conversion for the DP50 experiment and between 5 and 30 % conversion for the DP100 polymerization. As in the case of VDF, 29 the proportion of regularly-terminated chains decreased faster with conversion when higher degrees of polymerisation were targeted (compare entries 2 and 7 in Table 1 and Figure 3). The complete disappearance of the PTrFEH-XA chains thus likely occurs below 20% conversions in the two polymerizations studied. However, additional datapoints would be required to accurately determine these specific conversion values. Conversely, the proportion of reversely terminated PTrFE-XA chains (-CFH-XA termini, noted PTrFET-XA and formed by transfer to a xanthate of a CHF • radical) reached its maximum at low conversion with about 75 % for the DP100 and as high as 95 % for the DP50 experiments. The formation of these two types of xanthate-terminated chains leads to a competition between the non-degenerate and degenerate processes of chain transfer to xanthate (Figure 4). Here, as in the case of VDF, this competition induces the slowdown of the overall RAFT chain equilibrium which impairs the control of the polymerisation resulting in broader molar mass distributions. 29 These PTrFET-XA chains accumulate in the reaction medium because they are less reactivatable than their PTrFEH-XA counterparts. Indeed, according to DFT calculations (vide infra), the barrier for the reactivation of PTrFEH-XA (degenerative exchange) is lower than for PTrFET-XA. Again, as in the RAFT polymerisation of VDF, the overall PTrFE chain-end functionality (i.e. the proportion of xanthateterminated chains) decreased rapidly in the course of the polymerisation due to irreversible transfer reactions, and barely reached 55% or 30 % after 15 h of polymerisation for the DP50 and DP100 experiments, respectively. Note that this functionality values also consider T-adducts; the real functionality values are actually slightly lower (50 and 25 %, see Tables S1 andS2 and Figures S2 andS3). This loss of In order to substantiate the above mechanistic interpretation, DFT calculations were carried out on a model system, at the same level of theory used for the previously reported PVDFH-XA and PVDFT-XA reactivation investigation. 29 The PTrFEH-XA and PTrFET-XA macro-CTAs were modelled by the H-CHFCF2-XA and H-CF2CHF-XA molecules, in which the polymer chain beyond the xanthate-linked monomer was replaced by an H atom. In addition, the OEt group in the xanthate group was replaced by an OMe group. These simplifications reduce the computational cost and are not expected to introduce any major electronic change (polarity, homolytic strength) or steric effect in the bonds that are involved in the computed processes. Therefore, the calculated energy differences should not be significantly affected. The relative barriers for the additions and fragmentations relative to thedegenerate (H/H, T/T) and non-degenerative (H/T) exchange processes are summarised in Figure 5. The addition barrier is the lowest for the head radical addition to the head CTA (10.6 kcal/mol, blue curve), leading to thedegenerate H/H exchange through an intermediate adduct that is nearly isoergic with the separate CTA and free radical. The tail radical addition to the head CTA (magenta curve) has an intermediate barrier of 11.9 kcal/mol and leads to a non-degenerative exchange, producing the head radical and the tail CTA in a slightly exoergic process (-1.3 kcal/mol). The intermediate adduct is again essentially isoergic with the separate fragments. Finally, the degenerate T/T exchange (red curve) has a higher addition barrier of 12.9 kcal/mol. The reverse non-degenerative exchange, corresponding to the reactivation of the tail CTA by the more abundant head free radical (magenta energy profile from right to left), has an overall activation free energy of 13.2 kcal/mol and is thus the most difficult process. These values slightly increase when calculated at 70 °C, because of a negative activation entropy (see data in the SI, Figure S4).

These trends are qualitatively similar to those previously obtained for the PVDF CTA models. 30 Therefore, the calculations confirm that the inverted TrFE monomer additions lead to accumulation of less easily reactivable tail CTA (PTrFET-XA), as in the case of the VDF system. There are, however, two relevant differences. The calculations gave greater addition barriers for the TrFE system than for the VDF system (the latter were calculated as 9.0, 9.0 and 9.5 kcal/mol for the H/H, H/T and T/T processes, respectively), suggesting that the controlled polymerisation of TrFE with this RAFT chain transfer agent should be slightly less efficient. The second difference is the smaller thermodynamic gain for the non-degenerative exchange (-1.3 kcal/mol, vs. -6.1 kcal/mol for the VDF system). Consequently, the reverse T/H exchange has a greater barrier for the VDF system (15.1 kcal/mol). The reason for this lower thermodynamic gain is clearly related to the greater similarity of the two chain ends (CF2 vs. CHF) relative to VDF (CF2 vs. CH2). The smaller difference between the tail and head CTA reactivation barriers for TrFE may lead one to anticipate that the inverted monomer additions should have a lower impact, relative to VDF, on the loss of polymerisation control. However, this loss of control does not only depend on how fast the tail CTA is reactivated, but also on how much faster it accumulates as a consequence of the head-to-head additions. This point will be further discussed below. In light of the observed evolution of the chain end functionalities and of the DFT calculations, the kinetics (Figure 6) and the molar mass and dispersity evolutions with conversion (Figure 7) for the RAFT polymerisation of TrFE can be better interpreted.

Figure 6 shows several phases in the TrFE RAFT polymerisation. First, there is an inhibition period of around 90 min, in which the polymerisation seemed to proceed at low speed or not at all. Inhibition periods are not unusual in RAFT polymerisation and this phenomenon was already observed in the RAFT polymerisation of VDF. 31 This inhibition period might be due to the selective formation of monoadducts at the beginning of the reaction. This phenomenon has been demonstrated by Pound et al. 40 when the monoadduct possesses a poor leaving group compared to the R group of the RAFT agent. In the present case the formation of T-adduct which are not easily reactivated by the majority CF2 • radicals likely contribute to this inhibition. The polymerisation rate observed for the DP50 experiment seemed slower than that of the DP100 experiment. This rate retardation is also relatively common in RAFT polymerisation. 41 For the DP50 experiment, the polymerisation seemed to accelerate after 4 h. The 4 h mark is also the time when the PTrFEH-XA disappeared from the reaction medium. In the case of VDF, an acceleration of the polymerisation was also noted when the molar fraction of PVDFH-XA reached zero. This rate acceleration was not observed in the DP100 experiment. This is presumably because of the lack of datapoints in this experiment. The rate acceleration likely happened in the 2 h -4 h interval (when PTrFEH-XA disappeared). In both experiments, a decrease of the polymerisation rate was observed towards the end of the polymerisations. This decrease of the polymerisation rate, also seen for the polymerisation of VDF, remains unclear. 31 It may be caused by the decrease of the monomer concentration in solution due to the decrease of its partial pressure as the polymerisation proceeds.

At a first glance, Figure 7 seems to show a relatively linear evolution of the molar masses with conversion for both the DP50 and the DP100 polymerisations, suggesting that they are relatively well controlled. However, a slight decrease of the slope may be seen after the 4 h (15.3%, 2,900 g mol -1 ) data point of the DP50 experiment. This point almost coincides with the total disappearance of the PTrFEH-XA chains. The same phenomenon was also observed for the RAFT polymerisation of VDF. 30 This change of slope cannot be observed for the DP100 experiment presumably because of the lack of data points between the 4% and 33% TrFE conversion values as mentioned above. Nevertheless, the dispersity (Ɖ), which remained low (< 1.2) in the first few hours of polymerisation, increased as the polymerisations proceeded and reached almost 1.6 at higher conversion (55-70%). This Đ increase is believed to result from two causes: 1) irreversible transfer reactions to monomer, polymer or solvent, leading to dead chains; and 2) to the slowdown of the RAFT chain transfer process due to reverse additions. These phenomena were shown to be responsible for a similar increase of Ɖ in the RAFT polymerisation of VDF. [29][30][31] Further characterisation of the structure of the PTrFE synthesized by RAFT polymerisation (Entry 12, Table 1) was performed via matrix-assisted laser desorption/ionization coupled time-of-flight mass spectrometry (MALDI-TOF), using both positive and negative ion modes (Figure 8 The positive ion mode showed a very weak signal and only displays the polymer chain terminated via irreversible transfer (i.e. chains terminated by -CF2H or -CFH2). In contrast, the negative ion mode displayed a distribution of xanthateterminated PTrFE chains centered on a DP = 13 chain at m/z = 1229. Note that these distributions showed a fragmentation of the relatively weak C-O bond of the xanthate moiety presumably occuring during the analysis. Surprisingly, only the R group initiated population were observed, contrary to what had been observed by Guerre et al. on PVDF made by RAFT polymerisation. 31 As mentioned above, the RAFT polymerisation of TrFE faces the same problem as that of VDF: the quick formation of less reactivatable PTrFET-XA chains, which are produced by headto-head (HH) additions followed by transfer to the xanthate group. For PVDF, the HH additions are systematically followed by tail-to-tail (TT) additions and amount to about 4-4.5% of the total number of VDF additions (hence the HH-TT sequence amounts to about 8-9 % of the additions). For PTrFE, the chain defects were also believed to be constituted by HH-TT sequences (see Introduction) and these defects were estimated by several authors to be about 13.5 %. [START_REF] Reinsberg | [END_REF] To better estimate the occurrence of these chain defects and their impact on the TrFE RAFT polymerisation, a DFT approach of the radical addition onto TrFE as well as a thorough NMR study PTrFE chain-ends were carried out. The standard activation barriers associated to the four possible monomer addition modes, obtained by the DFT calculations at 25 °C, are shown in Figure 9. These barriers were also calculated at 70 °C (see SI, Figure S5). The general reactivity trend is the same as that previously determined for the polymerisation of VDF (both the head and the tail radicals have lower addition barriers to the monomer tail end yielding preferential HT and TT additions, respectively). 29 The energetic barrier of a HT addition is slightly lower for TrFE than for VDF, respectively 11.4 kcal/mol and 11.9 kcal/mol. These values suggest a slightly faster polymerisation rate of TrFE compared to that of VDF. For the TrFE system, the activation energy difference between the HT and HH additions is 1.4 kcal/mol at 25 °C. From this Δ(ΔG), it is possible to derive a probability of occurrence of 8.6% for the HH additions (and 91.4% for the HT addition). This HH probability value is 3.6 times higher than that calculated for VDF (only 2.4%). 30 Thus, the tail dormant species, which is more difficult to reactivate as discussed above, accumulates faster for the TrFE polymerisation than for the VDF polymerisation, in agreement with the observed rapid loss of control. The activation energy difference between the TT and TH additions is only 1 kcal/mol for TrFE, vs. 4 kcal/mol at 25°C for VDF. This smaller difference translates into a much higher probability (15.6%, vs. 0.12% for VDF) of occurrence for the TH addition in PTrFE. At 70 °C, although all Gibbs energy barriers increase, the Δ(ΔG) values remain unchanged relative to 25 °C (Figure S5), but the temperature effect in the Eyring relationship slightly modifies the relative addition probabilities: 84.4% of HT and 15.6% of HH additions for the main head radical chains; 81.2% of TT and 18.8% of TH additions for the minor tail radical chains. These calculations thus suggest that, in addition to a much greater impact of the monomer addition errors for the TrFE polymerisation, the HH additions are not systematically followed by a TT addition. A non-negligible fraction of HH-TH (-CHFCF2-CF2CHF-CF2CHF-) sequences may be generated. To the best of our knowledge, only Yagi reported the possibility of TH addition in PTrFE, 10 whereas all the subsequent studies on PTrFE considered that HH additions were systematically followed by a TT addition. However, the probability of two consecutive TH additions, as calculated from the probabilities of HH and TH additions under the terminal model approximation, is only 1.3 % at 25°C or 2.1% at 70°C. The conclusion is that two main types of chain defects should exist in PTrFE: the most probable one derived from HH-TT addition sequence (7.3% of the triads at 25 °C, 9.25% at 70 °C) and a less important one derived from the HH-TH addition sequence. The experimental proof of the existence of these TH additions was so far missing since NMR spectroscopy is not directional and cannot distinguish between HT and TH additions (leading to -CHFCF2-CHFCF2-or to -CF2CHF-CF2-CHF motifs respectively). A thorough examination of the 1 H and 19 F NMR 1) and PTrFE-XA (Entry 5, Table 1). spectra of PTrFE prepared by RAFT polymerisation was thus carried out. This study, detailed in a different article 39 , has evidenced the existence of a -CFH-XA chain end where the CFH group 19 F NMR resonance at -174.9 ppm is correlated in 3 JF-F with the resonance of a CF2 group and in 4 JF-F with that of a CFH group, proving the existence of a CFH • radical resulting from a TH addition or in other words, to a reversed propagation (Figure S9).

Finally, two samples of PTrFE (made by RAFT and by conventional free radical polymerisation and indicated as PTrFE-XA and PTrFE, respectively, see Table S4) were heated to 210°C and cooled down from the melt either by fast cooling (around 1000°C/min) or by slow cooling (< 2°C/min), trying to reproduce the work of Oka et al. 8 The DSC thermograms of the samples presented in Figure 11 do not show any significant differences for the different cooling rates. The heat capacity jump attributed to the glass transition temperature (Tg) is located around -23°C for both PTrFE samples and the endothermic peak characteristic of the melting of the primary crystals (Tm) is observed at around 168°C for PTrFE-XA and at around 181°C for PTrFE. It is interesting to note the higher melting temperature for the PTrFE sample prepared by conventional polymerisation. This can be explained the presence of longer macromolecular chainswhich induce the formation of crystalline domains. Because of the relatively low molar mass of both polymers, the melt temperatures are much lower than the values of 213°C or between 186°C and 194°C reported in the literature. 26,42,43 In order to identify the PTrFE crystal phase, the structure was investigated by WAXS at ambient temperature. A representative scattering pattern of PTrFE and the associated integrated intensity profiles are depicted in F. For all samples, the WAXS pattern is a ring characteristic of an isotropic distribution of the macromolecular chains. The corresponding intensity profile presents a main peak at 2 = 18.2° and two smaller peaks at 2 = 31.8 and 36.7° associated to the (100), (110) and (200) planes, respectively, and characteristic to the non-polar crystal phase of PTrFE. 27,44 No difference was established between the two different cooling rates and the crystal phase does not show any trace of the polar phase responsible for the ferroelectric behaviour of those polymers. This behaviour may also be caused by the relative low molar masses of the PTrFE examined.

D. Conclusions

This work shows that xanthates are suitable chain transfer agent for the RAFT polymerisation of trifluotroethylene (TrFE). However, as in the case of vinylidene fluoride (VDF), the RAFT mechanism is adversely affected by: 1) irreversible transfer caused by the strong propensity of PTrFE radicals for H-abstraction, and 2) reverse monomer additions (head-tohead, HH), leading to the formation of less reactive chain ends that slow down the RAFT chain equilibrium. These effects, which lead to a loss of polymerisation control, are even more pronounced with TrFE than with VDF, because TrFE is much more prone to reverse additions. In consequence, welldefined PTrFE may only be prepared, in solution, at low conversions before irreversible transfer becomes dominant. Polymerisation in dispersed aqueous media may partially alleviate this problem. However, the RAFT equilibrium slowdown problem, which is inherent to the HH addition probability, may not have a solution. Organometallicmediated radical polymerisation and specifically cobaltmediated radical polymerisation would probably reactivate both chain-ends and afford higher molar masses and better control over the polymerisation. 45,46 Nevertheless, thanks to the ability of the RAFT polymerisation to temporarily trap endgroups, the chain-end evolution monitoring during the RAFT polymerisation, combined with DFT calculations, led to the discovery of limited reverse (tail-to-head) propagation. This behaviour seems, so far, unique to TrFE; the HH additions are systematically followed by TT additions in the VDF polymerisation. This work provides a much better understanding of the polymerisation behaviour of TrFE, paves the way to the study of the RAFT copolymerisation of VDF and TrFE, and provides important information to decipher the microstructure of important TrFE-containing electroactive fluoropolymers.

1H

  1D NMR. A one pulse 90° (9.25 μs) pulse sequence was used with 6 s acquisition time, 3 kHz spectral window, 1 transient and 1s recycle delay.

19 F

 19 1D NMR. A one pulse 90° (6.5 μs) pulse sequence was used with 0.08 s acquisition time, 75 kHz spectral window 1 transient and 1s recycle delay.

2JCF = 30

 30 Hz was used for the 2 J 19 F {[START_REF] Tonelli | Advances in Chemistry Series[END_REF] C} 2D NMR HSQC with 1 H Decoupling.

Figure 2 : 19 F

 219 Figure 2 :19F NMR spectrum of PTrFE made by RAFT (entry 5, Table 1) recorded in (CD3)2CO.

Figure 3 :

 3 Figure 3: Evolution of the different chain ends for the RAFT polymerisation of TrFE: DPtarget = 50 (top) and DPtarget = 100 (bottom) calculated using Equation S1 for -CFH-XA, Equation S2 for -CF2-XA and Equation S3 for Irreversible transfer.

Figure 2 :

 2 Figure 2: Energy profiles for the degenerate (H/H and T/T) and non-degenerate (H/T) radical exchange with the dormant chain. The reported values are ΔG°298.15 in kcal/mol.

CH

Figure 6 .

 6 Figure 6. First-order kinetic plots of the RAFT homopolymerisation of TrFE with DPtarget = 50 (black squares) and DPtarget = 100 (red circles). Reaction conditions: [TrFE]0:[CTA]0:[I]0 = [DPtarget]:1:0.2. Reaction time = 15h, T = 73°C, CTA = O-ethyl-S-(1-methoxycarbonyl)ethyldithiocarbonate, I = tert-amyl peroxy-2ethylhexanoate.

Figure 7 .

 7 Figure 7. Evolution of the molar mass (Mn) and the dispersity (Đ) vs conversion for the RAFT polymerisation of TrFE DPtarget = 50 (black squares) and DPtarget = 100 (red circles). Reaction conditions: [TrFE]:[CTA]0:[I]0 = [DPtarget]:1:0.2. Reaction time = 15h, T = 73°C, CTA = O-ethyl-S-(1-methoxycarbonyl)ethyldithiocarbonate, I = tert-amyl peroxy-2-ethylhexanoate.

Figure 3 .

 3 Figure 3. Scheme of the twodegenerate chain transfers and of the non-degenerative chain transfer at work during the RAFT polymerisation of TrFE.

  and S10).

Figure 4 .

 4 Figure 4. MALDI-TOF mass spectra in negative and positive ion mode of PTrFE prepared by RAFT polymerisation (Entry 11, Table 1). Reaction conditions: [M]0:[CTA]0:[I]0 = 100:1:0.2, reaction time = 15 h at 73 °C, conversion = 67%.

Figure 9 .

 9 Figure 9. Detail of the four different XA chain end and Energy profiles for (a) the HT and HH additions of the PTrFEH • model radical and (b) the TH and TT additions of the PTrFET • model radical to TrFE. The reported values are ΔG°298.15 (1M standard state) in kcal/mol.

Figure 5 .

 5 Figure 5. DSC thermogram of PTrFE-XA (made by RAFT; entry 5 table1) and PTrFE (made by conventional radical polymerisation, entry 12 table 1)) fast-cooled (fc) and slow-cooled (sc) from the melt.

Figure 12 .

 12 Figure 12. WAXS patterns and intensity profiles of slow-cooled (sc) and fast-cooled (fc) PTrFE (Entry 12, Table1) and PTrFE-XA (Entry 5, Table1).

Table 1 :

 1 Summary of the results for the RAFT syntheses of PTrFE.

Entry [TrFE]0:[CTA]0:[I]0 Time (h) Conversion (%) Mn (theo) Mn f (exp) Ɖ -CFH-XA (%) c -CF2-XA (%) d Irreversible transfer (g/mol) (g/mol) (%) e

	1 a	50:1:0.2	1	/	/	/	/	/	/	/
	2	50:1:0.2	2	4.6	400	1,700	1.01	54.5	45.5	0
	3	50:1:0.2	4	15.3	840	2,900	1.19	94.5	3.4	2.1
	4	50:1:0.2	6	40.6	1,900	4,600	1.32	80.1	0	19.9
	5	50:1:0.2	15	56.6	2,500	5,700	1.58	56.3	0	43.7
	6 a	100:1:0.2	1	/	/	/	/	/	/	/
	7	100:1:0.2	2	4.3	560	2,100	1.06	74	24	2
	8	100:1:0.2	4	33.6	2,900	6,300	1.37	76.7	0	23.3
	10	100:1:0.2	8	48.3	4,200	8,000	1.39	54.9	0	45.1
	11	100:1:0.2	15	67	5,700	9,900	1.58	30.7	0	69.3
	12	50:0:0.2	15	79	3,500	5,700	4.30	/	/	/
	13 b	20:1:0.2	0.5	2.5	/	/	/	/	/	/

a No polymer was obtained for this experiment. b Synthesis carried out in autoclave, no GPC analysis as the polymer obtained was used entirely for NMR inverstigations. c Proportions calculated considering both the H-and T-adducts (regular and reverse monoadducts respectively). Calculated using equation S1. d Calculated using equation S2. e Calculated using equation S3. The chain-end proportions take the monoadducts into account. f GPC traces presented Figure

S6

and S7.

Figure 1. Scheme of the RAFT polymerisation of TrFE, using tert-amyl peroxy-2-ethylhexanoate as initiator, O-ethyl-S-(1-methoxycarbonylethyl)dithiocarbonate as CTA and dimethylcarbonate as solvent.

Acknowledgements

This work was partly supported through grant ANR-16-CE08-0025, ANR NanoPiC from the French National Research Agency. The authors thank Arkema (Pierre Bénite, France) for providing TrFE.

Author Contributions

Conflicts of interest

There are no conflicts to declare

Notes and references


Jean-Luc Garden

Hervé Guillou

M A Ramos

Thermodynamics of Glasses

Introduction

Thermodynamics states that the properties of a system in equilibrium depend neither on time nor on past history. Glasses clearly violate this postulate. Not only do their properties depend on history but they also vary with time at temperatures at which relaxation toward internal thermodynamic equilibrium does occur, but at a rate slow enough to be observable at the timescale of the experiment performed. To deal with glasses, thermodynamics must thus consider nonequilibrium states and their actual cause, namely the irreversibility of the transition that occurs when relaxation times eventually become much longer than experimental timescales such that the material freezes in as a glass.

Much attention is currently paid to the processes driving the glass transition at a microscopic scale and also to their implications for the macroscopic properties of glasses. Because this topic is extensively discussed in this chapter, we will deal here with a second fundamental issue, namely that of the phenomenological approaches followed to understand the observable macroscopic properties of glasses and, thus, to design new applications. To quote a single example, density gradients in tempered glasses are the key to thermal strengthening, which is achieved irreversibly upon cooling (Chapter 3.12).

In this chapter, the basic concepts of macroscopic nonequilibrium thermodynamics will first be summarized and illustrated with experimental heat capacities for a model system, PolyVinylAcetate [PVAc, (C 4 H 6 O 2 ) n ]). The basic concepts of equilibrium and nonequilibrium will then be introduced to point out why glasses challenge the laws of thermodynamics. Next, properties of the supercooled liquid state above T g will be presented and the phenomenology of the glass transition examined in the light of calorimetric data, in particular in terms of configurational properties. The basics of nonequilibrium thermodynamics in the glass transition range will finally be reviewed along with the issue of aging below the glass transition range.

Basics of Nonequilibrium Thermodynamics

In thermodynamics one investigates the changes occurring when a system passes from a state A to another state B. At constant chemical composition, the system is in internal equilibrium if its state is defined by only two macroscopic variables such as temperature (T), pressure (P), volume (V), enthalpy (H), internal energy (U), or Gibbs free energy (G). Their values are not only constant but independent of the pathway actually followed between any two states A and B. As stated by the First Law of thermodynamics, between A and B the internal energy varies as:

ΔU A B = Q A B + W A B 1
where Q A B and W A B are the heat and work exchanged by the system with its surroundings, respectively. Likewise, the entropy is decomposed into two parts,

ΔS A B = Q A B T + Δ i S A B , 2
where the first represents the heat exchanged with the surroundings and the second the entropy created within the system itself during the transformation.

U N C O R R E C T E D P R O O F S
If two equilibrium states are connected by a reversible process, then Δ i S A B = 0. If the system undergoes instead an irreversible process through which it falls out of equilibrium, then Δ i S A B > 0 since a spontaneous process is always associated with an entropy increase of the system. Upon glass formation by cooling, pressure increase, or other means, the equilibrium liquid is continuously losing internal equilibrium. As will be discussed here, the question arises as to whether there is any finite production of entropy andif sowhether this quantity is of importance regarding the other terms involved in the process.

The Third Law of thermodynamics postulates that the entropy of a perfectly ordered system is zero at 0 K. In contradiction with it, however, calorimetric measurements indicate that glasses not only possess nonzero entropies at 0 K but that this residual entropy depends on thermal history as illustrated by a simple entropy cycle calculated from measured heat-capacities and entropy of fusion (ΔS f ). Beginning with a perfectly ordered crystal, whose entropy thus is 0 at 0 K, one derives the entropy of the crystal at its congruent temperature of fusion T f , then that of the melt from this temperature down to the glass transition, and finally that of the glass down to 0 K (Figure 1). The difference S 0 between this entropy and that of the crystal at 0 K is the residual entropy (Table 2), which increases with higher glass transition temperatures and, thus, with higher cooling rates, reflecting the increasingly wide distribution of configurational states obtaining with increasing temperatures.

A finite residual entropy at 0 K might seem to contradict the Third Law of thermodynamics. As justified by Jones and Simon [START_REF] Jones | Qu'est-ce qu'un verre?[END_REF], however, there is no contradiction because this law applies only to crystals and other systems in internal equilibrium, which are necessarily ordered at 0 K to minimize their Gibbs free energies. This is not the case of glasses, which do obey the Nernst theorem [START_REF] Nernst | The New Heat Theorem[END_REF] since they cannot pass from one entropy state to another at 0 K (ΔS = 0 for two neighboring glassy states at 0 K).

Although such determinations also made for partially disordered crystals like ice I h or CO have long been explained by simple statistical mechanical models, the very concept of residual entropy has recently been debated [START_REF] Takada | [END_REF]. On the assumption that ergodicity must hold for the entropy to be defined, the proponents of a kinetic view have claimed that the configurational entropy undergoes an abrupt jump at the glass transition in order to reach the zero value of the crystal entropy at 0 K. In contrast, the proponents of the conventional view have stressed that what matters is not time averages but spatial averages of configurational microstates [START_REF] Takada | [END_REF], which is the reason why the measured residual entropies do make sense physically and correlate with the specific structural features of glasses and disordered crystals.

By definition, equilibrium thermodynamics cannot alone account for fundamental questions raised when relaxation is too slow with respect to experimental timescales. Owing to the kinetic nature of the problem, use has been made of the formalism originally developed for the kinetics of chemical reactions by De Donder and his school [START_REF] De Donder | Thermodynamic Theory of Affinity, A Book of Principles[END_REF]. With values increasing as the reaction proceeds, a new variable, the advancement of reaction, ξ(t) is defined to characterize the state of the system as a function of time, t, such that the reaction rate is simply dξ(t)/dt. This extensive variable, expressed in mol, accounts for the distribution of matter (local mass or density variation), or the molecular structure, within the system at any time. A new state function, the affinity, A, is then introduced to relate ξ(t) to the driving force of the reaction, its Gibbs free energy (at constant T and P):

A P t , T t , ξ t = -∂G ∂ξ P,T 3
The affinity A, expressed in J/mol, is the intensive conjugate variable of ξ. All time dependences are thus embedded into the time variations of the internal parameter ξ, or A, and of the other variables that are controlled experimentally (e.g. T, P).

For a relaxing system, the instantaneous entropy production was simply written by De Donder as the product of the thermodynamic force and the corresponding flux [START_REF] De Donder | Thermodynamic Theory of Affinity, A Book of Principles[END_REF], S Glass at 0 K Glass with a higher fictive temperature Supercooled liquid

T K T g T m T
Liquid at equilibrium Crystal at 0 K S = 0 J/K

U N C O R R E C T E D P R O O F S σ i = d i S dt = A T × dξ dt , 4
where the thermodynamic force actually is A/T, for the sake of dimensional analysis (the entropy production being in W/K).

Regarding the glass transition, the problem boils down to know A and ξ (or dξ/dt) and how they evolve with time. Depending on the values of both parameters, however, at this point several cases must be distinguished because not all of them are relevant (Table 1). The first and simplest case is that for which both A and dξ/dt are zero. It is that of the equilibrium liquid, which will thus be first considered in its metastable, supercooled extension.

Supercooled Liquids

Although the liquid state is generally far from simple, it can be considered as an equilibrium reference at viscosities (η) low enough that flow is easy, i.e. at high-enough temperatures at the pressure considered. In that case, the diffusion of microscopic entities, be they molecules or atoms, obeys the Stoke-Einstein relation, which relates the diffusivity D to the temperature and viscosity with:

D = k B T Cη , 5
where the coefficient C is a geometrical factor fixed by the boundary condition of the flow.

From its position at time t 0 , a diffusing entity travels a kind of random walk over an average distance d D tt 0 as a function of time. For low-viscosity liquids and high temperatures, D is high so that entities explore a great many different positions and configurations in a time shorter than that needed to perform a physical measurement. They do it through degrees of freedom that include not only thermal motions of translation, rotation, and vibration but also the complex kinds of atomic motions collectively termed configurational, which are governed by strong short-range repulsions and long-range attractions in molecular liquids. The measurement then averages out all these configurations.

Picturing these motions at a microscopic scale is difficult, however, especially for complex liquids or melts with various interacting entities. In various types of glass-forming liquids [START_REF] Wondraczek | Towards ultrastrong glasses[END_REF], local order can nonetheless be described in terms of degree of polymerization, formation of channels or sublattices, or formation of interpenetrating networks. Like the advancement of a chemical reaction, such structural features may be described in terms of the aforementioned parameter ξ. In internal thermodynamic equilibrium, i.e. in the liquid state, ξ is equal to ξ eq (T,P), but not in the glass transition range where ξ(t) becomes a function of T(t), P(t), and A(t), revealing its nonequilibrium nature. Below the glass transition range, where the relaxation time of the configurational degrees of freedom exceeds the experimental timescale, they cease to contribute to the measured property. At temperature low enough, the structure then eventually freezes in for good in one state defined by one particular value of ξ(t), which becomes independent of the external parameters T and P.

From a practical standpoint, the timescale defined by the viscosity of the material is important to determine the temperature at which the system will fall out of equilibrium when observed at the timescale of a particular experiment. There is not yet a unique model for describing relaxation phenomena in all glass-forming liquids (Chapter 3.7), whether strong or fragile with Arrhenian or non-Arrhenian viscosities, respectively [START_REF] Angell | Formation of glasses from liquids and biopolymers[END_REF]. In measurements of macroscopic properties, one nonetheless considers generally that experimental timescales τ exp are of the order of τ exp ~10 2 -10 3 seconds. The viscosity should then be of the order of 10 12 Pa.s or 10 13 P for structural relaxation to be complete under these conditions. To stress the usually tremendous variations of viscosity

U N C O R R E C T E D P R O O F S
down to the glass transition, it will suffice to note that the viscosities of stable liquids (i.e. above the melting or liquidus temperature) range from 10 -3 to 10 2 Pa.s depending on chemical composition and structural type.

Glass as a Nonequilibrium Substance

Time-dependent effects appearing at the glass transition are clearly observed in the heat capacities measured for PVAc (Figure 2), which is a model polymeric system extensively studied because of its excellent glass-forming ability and standard T g close to room temperature. The observed hysteresis loop between cooling and heating demonstrates that the heat capacity does not only depend on T and P but also on time. Moreover, upon heating, the heat capacity shows a typical overshoot, i.e. an endothermic event, named structural recovery process. To come back to the initial liquid state, the system needs to recover the amount of internal enthalpy that has previously been lost. From such measurements, it is possible to determine the configurational contribution to the heat capacity

ΔC conf P .
Here, it is defined by the difference at every temperature between the heat capacities actually measured and estimated for the glass phase:

ΔC conf P = C P -C g P 6
This type of definition also applies to other thermodynamic variables such as the thermal expansion coefficient α P , or the isothermal compressibility κ T . A configurational contribution consequently represents the thermodynamic contribution that originates in configurational changes in the liquid.

The glassy state then is defined as that for which the configurational movements have been frozen-in, i.e.

ΔC conf P = 0 . In this state, only the vibrational motions, i.e. the fast degrees of freedom (faster than the experimental timescale), contribute. To define this contribution over the entire temperature interval of interest, an extrapolation of the glass heat capacity from low to high temperatures is needed (Figure 2). The heat capacity of the supercooled liquid can also be extrapolated toward low temperatures (Figure 2). The difference between these values for the supercooled liquid and the glass,

ΔC conf,eq P = C l P -C g P 7
then yields the equilibrium configurational contribution, which keeps increasing below T g even though the actually observed values do vanish (Figure 3). From the equilibrium and actual configurational contributions, the variation of the configurational enthalpy ΔH conf and entropy ΔS conf , taken between two temperatures, are calculated with:

ΔH conf T = T T 1 ΔC conf P dT and ΔS conf T = T T 1 ΔC conf P T dT 8
where T 1 = 360 K is in Figure 2 an arbitrarily selected reference temperature. Absolute values of both state functions could be obtained from the enthalpy and entropy of an isochemical crystalline compound through the crystallization values of these functions (see Figure 1). For lack of such a compound for PVAc, only relative values are thus presented (Figure 4) in such a way that both the actual and equilibrium values are equal from 360 K to the temperature of about 315 K at which internal equilibrium is lost. Since these variations are similar for the configurational enthalpy and entropy, only the former is shown in Figure 4.

N C O R R E C T E D P R O O F S
Contrary to their equilibrium counterparts, which continue to decrease upon cooling, both the actual configurational enthalpy and configurational entropy level off in the amorphous state (Figure 4). Owing to the large width of the glass transition range, the heat capacity variations at the glass transition are much too smooth to be interpreted as reflecting the discontinuity of a second-order phase transition. Such a discontinuity can nonetheless be identified at a temperature T M defined by the intersection of the extrapolated glass and supercooled liquid (Figure 4, inset). Both configurational enthalpy and entropy are thus continuous at that temperature, which separates the glass from the supercooled liquid. The same applies to other properties such as volume. Because entropy and volume are the first derivatives of the Gibbs free energy with respect to temperature and pressure, respectively, the following relations initially derived by Ehrenfest should hold when second-order derivatives of the free energy vary discontinuously at this point M:

dP dT M = Δα P Δκ T 9a dP dT M = ΔC P T M V M Δα P 9b
To express these equations in terms of discontinuities of equilibrium configurational contributions at T M , e.g. ΔC conf,eq P of Eq. ( 7), Prigogine and Defay [START_REF] Prigogine | Thermodynamique Chimique, Nouvelle Rédaction[END_REF] assumed that the supercooled liquid is in internal equilibrium down to T M (i.e. A = 0 and dA = 0) whereas the glass below T M is defined by dξ = 0. These two equalities can then be grouped to yield the so-called Prigogine-Defay (PD) ratio [START_REF] Prigogine | Thermodynamique Chimique, Nouvelle Rédaction[END_REF]: Although considering an internal parameter ξ, this approach assumes that the glass transition occurs continuously at T M where ξ g = ξ l . If so, it would follow from Eq. ( 9) that the PD ratio should be unity. As indicated by the values listed for widely different glass-forming liquids (Table 2), however, calculated PD ratios are higher or even much higher than unity. One can explain such values by taking into account the kinetic nature of the glass transition [START_REF] Garden | Non-equilibrium configurational Prigogine-Defay ratio[END_REF]. Physically, it is making sense to assume that isobaric temperature derivatives such as ΔC P or Δα P are not measured under the same kinetic conditions as an isothermal pressure derivative like Δκ T . Whereas this inconsistency may be removed if more than one internal order parameters ξ are involved in the thermodynamics of the glass transition [START_REF] Davies | Thermodynamic and kinetic properties of glasses[END_REF], the problem may in contrast be compounded by the uncertainties arising from the extrapolation procedures used for deriving the relevant parameters at the temperature T M .

Π = ΔC P Δκ T T M V M Δα P 2 10 0 -2 -4 -6 -8 -5 -6 -7 -8 -9
Another puzzling fact has been long ago pointed out by Kauzmann [START_REF] Kauzmann | The nature of the glassy state and the behavior of liquids at low temperature[END_REF] who wondered what would happen if the entropy of a supercooled liquid were extrapolated down to temperatures much lower than the experimentally observed T g . The conclusion was that it would become lower than that of the isochemical crystal at a temperature T K , thus termed the Kauzmann temperature (Table 2), which could suggest that the liquid undergoes a continuous phase transition toward the crystalline phase at T K analogous to the critical point of fluids.

One way out of the paradox implies kinetic arguments and assumes that the viscosity of the supercooled liquid diverges at a temperature close to T K . This assumption may be represented by the Vogel-Fulcher-Tammann (VFT) equation (Chapter 4.1):

η = A exp B T -T 0 11
where the temperature T 0 of the viscosity divergence is actually close to the Kauzmann temperature (Table 2) even though they may depend on the specific sample and the method of measurement. Another way out is to take with great caution the extrapolations of the heat capacity and other thermodynamic functions of the supercooled liquid. As long pointed out [e.g. 11], there is no current theory for these properties in liquid state analogous to the Einstein or Debye models that provide functional forms at all temperatures for heat capacities of crystals.

As derived from strikingly old questions in glass science, these counterintuitive features indicate that glasses cannot be described by equilibrium thermodynamic states only. Nonequilibrium thermodynamics is, therefore, likely to be useful to characterize glasses and the glass transition.

Nonequilibrium Thermodynamics of the Glass Transition

The questions raised by the Kauzmann paradox or the PD ratio clearly illustrate the need for a more fundamental thermodynamic description of the glass transition. Following the pioneering work of Tool [START_REF] Tool | Relaxation of stresses in annealing glass[END_REF][START_REF] Tool | Relation between inelastic deformability and thermal expansion of glass in its annealing range[END_REF] and Davies and Jones [START_REF] Davies | Thermodynamic and kinetic properties of glasses[END_REF], different approaches and phenomenological models have been developed to deal with the glass transition range itself, many within the framework of classical nonequilibrium thermodynamics [START_REF] De Donder | Thermodynamic Theory of Affinity, A Book of Principles[END_REF][START_REF] Nemilov | Thermodynamic and Kinetic Aspects of the Vitreous State[END_REF].

The starting point has been the phenomenological concept of fictive temperature (T f ) propounded by Tool [START_REF] Tool | Relaxation of stresses in annealing glass[END_REF][START_REF] Tool | Relation between inelastic deformability and thermal expansion of glass in its annealing range[END_REF] to characterize the state of a relaxing system at any time. This temperature is similar to an order parameter ξ. It thus overcomes the limitations of the fixed limiting temperature T M , which characterizes only the point at which internal equilibrium is suddenly lost in a quenched state. On an analogous basis, a more detailed description is made in terms of two-temperature thermodynamics [START_REF] Leuzzi | Thermodynamics of the Glassy State[END_REF] whereby the vibrational and configurational degrees

U N C O R R E C T E D P R O O F S
of freedom are distinguished by a "classical" temperature for fast modes (phonons bath), and an effective temperature for the slow modes, respectively. The first physical models have then relied on two different approaches. In free-volume theories, one generally considers that the dynamics of the system is determined by the free space present around its atoms, which makes configurational rearrangements more or less easy. In entropy theories, among which that of Adam-Gibbs is the best known [START_REF] Adam | On the temperature dependence of cooperative relaxation properties in glassforming liquids[END_REF], the same determining role is attributed to configurational entropy. In other words, these theories assign the strong increase of relaxation times with decreasing temperatures and the eventual structural freezing in to decreases of either free volume or configurational entropy. Other more recent theories of the glass transition rely on mode coupling, random first-order transitions or energy-landscape descriptions [e.g. 16]. These different approaches have the common goal of finding the exact expression for the structural relaxation time, or its distribution, as a function of controlling parameters such as temperature or pressure, or structural order parameter.

For the sake of simplicity, let us consider here conditions of constant pressure. If the additional parameter ξ is taken into account, the total differential of the enthalpy of a system can be written as the sum of two contributions (considering pressure, the generalization to three contributions would be obvious):

dH P = ∂H ∂T P,ξ dT + ∂H ∂ξ P,T dξ 12
The isobaric heat capacity is written as:

C p = dH dT P = ∂H ∂T P,ξ + ∂H ∂ξ P,T dξ dT P 13
The first term on the right-hand side is the heat capacity at constant ξ, i.e. C glass P , and the second, the configurational contribution ΔC conf P as defined by Eq. ( 6). To account for the kinetic nature of the glass transition, it is then necessary to rewrite Eq. ( 13) as:

C P = ∂H ∂T P,ξ + ∂H ∂ξ P,T
dξ dt dT dt P 14

When the rate of change of ξ becomes much smaller than the rate of change of temperature, (dξ/dt) P (dT/ dt) P , the configurational contribution is negligible.

Hence, it is the ratio between these two rates that is controlling the relative value of the experimentally recorded configurational heat capacity. This ratio is maximum in the supercooled liquid state, and decreases throughout the glass transition range to become negligible in the glassy state (cf. Figure 3). There, only the first right-hand side term in Eq. ( 14) contributes:

C P,ξ = C glass P < C P < C eq P = C liquid P or 0 < ΔC conf P < ΔC conf,eq P 15
The next step thus consists in taking into account the time dependence of ξ at every temperature through the temperature dependence of the relaxation time τ. The simplest way to do this is to assume a simple exponential decay for ξ at fixed temperature and pressure:

dξ dt = - ξ -ξ eq τ 16
where ξ eq (P,T) is the equilibrium value of the order parameter, i.e. a variable characterizing the liquid structure that depends only on P and T. Although the relaxation time itself has been given different temperature dependences with Arrhenius, VFT, or others laws (Chapter 3.7), the important point is that they are all of an exponential nature with respect to T or P to ensure the structural freezing-in of the system. Interesting applications of these concepts have been made with the lattice-hole model of liquids, which has the advantage of lending itself to an evaluation of the order parameter ξ. Schematically, this model considers a liquid as a lattice in which disorder is represented by unoccupied sites whose fraction x depends on both temperature and pressure [START_REF] Simha | On the statistical thermodynamics of spherical and chain molecule fluids[END_REF]. From the equilibrium value of the order parameter, it is thus possible to solve the linear differential Eq. ( 16) to find its temperature dependence and, then, to calculate the variations of the heat capacity within the glass transition range under varied conditions [START_REF] Möller | Freezing-in and production of entropy in vitrification[END_REF]. Likewise, the configurational Gibbs free energy may also be computed analytically as a function of temperature, pressure, and order parameter. A similar approach has been followed to incorporate the effects of pressure in the expression of the structural relaxation time for determining also how the heat capacity, thermal expansion coefficient, and isothermal compressibility vary under different conditions [START_REF] Garden | Affinity and its derivatives in the glass transition process[END_REF].

From the configurational Gibbs free energy calculated for the lattice-hole model, one readily simulates with the definition (3) of the affinity its variations upon vitrification (cooling) and structural recovery (heating) [START_REF] Garden | Affinity and its derivatives in the glass transition process[END_REF]. Thermodynamic data measured on o-terphenyl may be used to simulate the corresponding affinities during temperature ramps (Figure 5): cooling at 0.3 K/min from an initial temperature of 255 K is followed by heating at the same rate, and then by further cooling at 0.5 K/min preceding final heating at 20 K/min. That the supercooled liquid begins to lose internal equilibrium from 248 K is indicated by the departure at this temperature of the affinity curve from the zero line, which represents the

U N C O R R E C T E D P R O O F S
maximum (equilibrium) value of the affinity during cooling. The affinity then linearly decreases with temperature below 240 K in the glassy state, with higher values for slower cooling as a result of lower glass transition temperatures. Upon heating, the affinity begins to increase linearly according to the same line pathway before crossing the equilibrium line. It then exhibits a peak whose position shifts toward higher temperatures and whose magnitude and width increase with the heating rate in ways such that the configurational heat capacity and the other thermodynamic coefficients can be simulated [START_REF] Garden | Affinity and its derivatives in the glass transition process[END_REF].

The entropy production can also be calculated from Eq. (4) (inset in Figure 5). In agreement with previous results [START_REF] Möller | Freezing-in and production of entropy in vitrification[END_REF], it shows a single peak upon cooling but two peaks upon heating. With respect to the experimental data, the advantage of the calculation is thus to distinguish clearly two contributions to the entropy produced when heat is brought to the material. The first peak is associated with a decrease of the configurational energy of the system, which is taking place because of the delay introduced by the relaxation time, even though heat is being supplied. As to the second peak, it is in contrast associated with the configurational energy necessary to recover internal equilibrium in the supercooled liquid state.

Here the wording "is associated" instead of "represents" is necessary because the entropy produced and configurational entropy changes necessarily differ as a result of the irreversible nature of the glass transition. Whereas the entropy production is the product of the thermodynamic force and flux (see Eq. ( 4)), the variation of the configurational entropy is written as, see Eqs. ( 8) and ( 13):

dS conf dt = ΔC conf P T × dT dt = ∂H ∂ξ p,T dξ dt P T 17
The rate of entropy production thus reflects the spontaneous or irreversible microscopic processes that take place within the system during relaxation. As dictated by the Second Law of thermodynamics, it is always positive whether upon cooling or heating (Figure 5, inset). Physically, it can be thought of the heat irreversibly generated by friction at a microscopic scale. The resulting thermal power P i = Tσ i , where σ i is the entropy creation in Eq. ( 4), is produced much too quickly to be compensated instantaneously by an exchange of heat with the surrounding heat bath. Under this circumstance, this is why an effective or fictive temperature can be defined. This surrounding heat bath is sometimes called the phonon bath since it is characterized by fast or vibrational modes.

On the contrary, the change in configurational entropy is a reversible process related to the heat exchanged with the surrounding heat bath whose relevant thermal power is:

P th = T dS conf dt = ΔC conf P × dT dt 18
Because the configurational entropy becomes constant upon vitrification, its variations have vanished (i.e. the configurational heat capacity) below the glass transition range. Above this range, in the supercooled liquid state, they of course differ from zero as indicated by dS conf,eq dt = ΔC eq P T × dT dt = ∂H ∂ξ eq P,T dξ eq dT T × dT dt 19

In the transition range, the variations of the configurational entropy of the system are consequently positive or negative upon heating and cooling, respectively. As already evaluated long ago either theoretically [START_REF] Davies | Thermodynamic and kinetic properties of glasses[END_REF] or experimentally [START_REF] Bestul | Limits on calorimetric residual entropies of glasses[END_REF][START_REF] Tombari | Change in entropy in thermal hysteresis of liquid-glass-liquid transition and consequences of violating the Clausius theorem[END_REF], the entropy produced is generally negligible with respect to the configurational entropy changes. The integration of the heat capacity curves measured by calorimetry is thus a pertinent way to access to the absolute value of the residual entropy at 0 K [START_REF] Takada | [END_REF]. As seen from a direct comparison of Eqs. ( 4) and ( 17), arriving at this conclusion is tantamount to neglecting the affinity with respect to the enthalpy of advancement of the configurational change at every temperature:

Entropy production negligible ⟺ A ∂H ∂ξ P,T 20

Physical Aging

Relaxation times that depend only on temperature and pressure have been considered. Nevertheless, the complexity of microscopic structures in glasses implies the existence of a distribution of relaxation times. Relaxation processes then also depend on the instantaneous state of the system itself and, thus, on its history as, for instance, described by the Tool-Narayanaswamy-Moynihan model (Chapter 3.7, [START_REF] Moynihan | Structural relaxation in vitreous materials[END_REF][START_REF] Narayanaswamy | A model of structural relaxation in glass[END_REF]). A consequence is that some nontrivial relaxation processes can take place well below the glass transition range. Hence, it is interesting to study such a time-dependent process termed physical aging, which has a practical relevance through its possible effects on glass properties. The process can be illustrated with DSC scans for PVAc (Figures 2 and6). If the sample is cooled down at some rate to 297 K, i.e. about 10 K below the calorimetric T g , and its temperature is kept constant for a time interval t a , then its enthalpy (and entropy) relax to their lowest equilibrium values for the temperature (and pressure) considered. Experimentally, physical aging manifests itself as differences between the areas of DSC scans upon heating recorded for samples annealed (72 hours at 297 K) and not annealed. For the experiment without annealing, the lowest temperature of 278 K has been directly reached (see cooling curve in Figure 6). The amount of enthalpy relaxed during aging is equal to the difference between the light gray and dark gray areas in Figure 6. Of course, for a given annealing temperature, such a difference increases with the aging duration. As recently carried out on polymeric glass-formers annealed at relatively low aging temperatures for a very long time, calorimetry (DSC) can bring to light two different timescales for glass equilibration, revealing the complexity and richness of relaxation processes well below T g [START_REF] Cangialosi | Dynamics and thermodynamics of polymer glasses[END_REF].

Phenomenologically, however, in simple cases aging can be accounted for with the same approach as developed in previous sections. It is related to the relaxation of the order parameter ξ toward its equilibrium value ξ eq (P,T) whereas the affinity A is relaxing at the same time toward zero. When applicable, the lattice-hole theory can be used to solve Eq. ( 16) at constant P and T to reproduce the observed process. As done for o-terphenyl [START_REF] Garden | Affinity and its derivatives in the glass transition process[END_REF], the order parameter is calculated at constant pressure and aging temperature T = 229.5 K with Eq. ( 16) and a temperature-and pressure-dependent relaxation time [START_REF] Garden | Affinity and its derivatives in the glass transition process[END_REF]. The affinity has been calculated upon heating at 60 K/ min, either after cooling at 6 K/min without aging, or after cooling at the same rate but with an aging process at 229.5 K (Figure 7). Upon isothermal aging, the affinity increases markedly (see the arrow in Figure 6) and then increases at the same rate as without aging. The difference is that the zero line is crossed at a lower temperature so that a much bigger peak is observed when the affinity finally recovers its zero equilibrium value at high temperatures. Since the affinity is an integrated measure of the heat capacity, the large peaks either calculated or

U N C O R R E C T E D P R O O F S
observed for these properties are clear signatures of aging [START_REF] Garden | Affinity and its derivatives in the glass transition process[END_REF]. More complex calculations can of course be made to deal with at least two separate timescales [START_REF] Cangialosi | Dynamics and thermodynamics of polymer glasses[END_REF], or a more realistic distribution of relaxation times.

Perspectives

Whether in the form of affinity, fictive temperature, or structural order parameter, additional variables must be introduced to deal with the nonequilibrium thermodynamics of glass-forming systems and, in particular, with the time dependence of their properties in relaxation regimes. Phenomenological advances now make it possible to predict these properties as a function of time and temperature or to determine accurately the entropy irreversibly produced, but the mechanisms involved at the atomic or molecular level generally remain to be deciphered. The physical nature of the glass transition is a case in point, as are the origins of Kauzmann catastrophe, of the strong variations of the PD ratio, of the diversity of relaxation timescales or of, as illustrated by the wellknown memory effects, the complex nonlinear coupling of the parameters of the differential equations with which these processes are described. Not only could highly sensitive calorimetric experiments yield valuable original data in this respect but coupling of different techniques such as dielectric spectroscopy and temperature-modulated calorimetry should bring new insights on the dynamics and thermodynamics of the glass transition. Recent experiments on ultra-stable organic glasses obtained by vapor deposition techniques are, for instance, promising [START_REF] Swallen | Organic glasses with exceptional thermodynamic and kinetic stability[END_REF][START_REF] Rodríguez-Tinoco | Ultrastable glasses portray similar behaviour to ordinary glasses at high pressure[END_REF]. And whereas very long aging performed well below T g should also give new clues on the laws driving complex relaxation processes in the glassy state [START_REF] Cangialosi | Dynamics and thermodynamics of polymer glasses[END_REF], experiments made at extremely rapid timescales (e.g. spectroscopy) are in contrast needed to investigate relaxation in supercooled liquids where equilibrium is quickly achieved. To give a single example, ultrastable organic glasses obtained by vacuum-deposition techniques should be of special interest in view of their internal stability that is equivalent of that of hyper-aged glasses (with aging time of millions of years) obtained by conventional melt cooling [START_REF] Swallen | Organic glasses with exceptional thermodynamic and kinetic stability[END_REF]. For this particular class of glasses, the aforedefined T M values are so much lower (by a few tens of degrees) than the standard glass transition temperatures that T M and T g cannot be indiscriminately used in Eq. (9b) [START_REF] Swallen | Organic glasses with exceptional thermodynamic and kinetic stability[END_REF][START_REF] Rodríguez-Tinoco | Ultrastable glasses portray similar behaviour to ordinary glasses at high pressure[END_REF]. Among other consequences, new insights should then be gained on the non-unity of the PD ratio. Finally, such experiments should of course be firmly complemented by fundamental work. Microscopic theories and atomistic simulations must be developed and, as stringent tests of their value, their predictions checked in terms of macroscopic physical properties.

Figure 2

 2 Figure 2 Heat capacity of PVAc measured across the glass transition range by differential scanning calorimetry at the same rate of 1.2 C/min first upon cooling (solid circle) and then upon heating (empty circle). Dashed lines: fits made from the heat capacities measured for the glass and supercooled liquid.

3. 2

 2 Thermodynamics of Glasses 4 0004539173.3D 4 21/11/2019 3:24:30 PM

U

Figure 4 Figure 3 5 0004539173

 435 Figure 4 Difference between the configurational enthalpy of PVAc and a zero reference-value taken at 360 K. Actual value (solid circle) and equilibrium value (empty circle). Inset: magnification of Figure 4 showing extrapolated values of the glass and supercooled liquid of this differential configurational enthalpy intersecting at the point M, which defines the limiting fictive temperature T M = T f .

Figure 5

 5 Figure 5 Simulated affinities of o-terphenyl in the glass transition range upon cooling and heating as calculated from the lattice-hole model. Solid circle for -0.3 K/min and solid square for -0.5 K/min; empty circle for +0.3 K/min and empty square for +20 K/min. The horizontal line represents equilibrium (A = 0). Inset: entropy production rates calculated from the previous affinities. Solid circle upon cooling and empty circle upon heating.

Figure 6

 6 Figure 6 Effect of aging on the heat capacities of PVAc recorded upon heating at the same rate. Heat capacity after a cooling (solid circle with line) and heat capacity after cooling and 72-hour annealing at 297 K (empty circle with line). The enthalpy released during aging estimated by the difference between the two areas included between these two curves. Heat capacities upon continuous cooling shown as solid circles.

Figure 7

 7 Figure 7 Effect of aging on affinities calculated with the lattice-hole model upon heating at the same rate of 60 K/min, first after a continuous cooling at 6 K/min (black line with solid circle), and second after annealing at 229.5 K (black line with empty circle). The black arrow simulating the relaxation of affinity upon aging at 229.5 K. Affinity upon continuous cooling shown as solid circles.

Table 1

 1 Thermodynamic states in terms of affinity and its derivatives and in terms of rate of advancement of the process.Liquid, glass, and relaxing liquid states are indicated by gray cells. The other cells indicate particular states that can be encountered or not during the glass transition. The value of the rate of production of entropy is indicated in each cell.

			3 Supercooled Liquids 3
	Rate of advancement dξ/dt (extensive, mol/s)
	Affinity A (intensive, J/mol)	dξ/dt = 0	dξ/dt 0
	A = 0 and dA = 0	True equilibrium;	Unphysical
		liquid state; σ i = 0
	A = 0 and dA 0	Isomassic state; σ i = 0	False equilibrium; nonequilibrium state; σ i = 0
	A 0 and dA = 0	Isomassic, isoaffine state;	Isoaffine state; σ i 0
		σ i = 0
	A 0 and dA 0	Nonequilibrium;	Nonequilibrium;
		glassy state; σ i = 0	viscous state; σ i 0

0004539173.3D 3 21/11/2019 3:24:30 PM

Table 2

 2 Thermodynamic parameters measured from five different glass-formers.

	6	3.2 Thermodynamics of Glasses
		Material	T g (K)	ΔS 0 (J/K/mol)	PD ratio	T K (K)	T 0 (K)
		SiO 2	1480	5.1	>10 3	1150	NA
							Arrhenius relaxation
		CaAl 2 Si 2 O 8	1109	36.2	1.5-22	815	805
		Glucose	305	1.7	3.7	241	242
		PVAc	301	NA	2.2	239	250
				No crystal
		Glycerol	183	19.4	3.7	134	123
		Se	295	3.6	2.4	207	226
		The values are taken from the literature.

5

  Nonequilibrium Thermodynamics of the Glass Transition 7

	0004539173.3D 7	21/11/2019 3:24:32 PM

Acknowledgments

The authors thank J. Richard for data treatments and simulations carried out with the lattice-hole model, G. McKenna for PVAc samples, and the reviewers for their time, remarks, and useful corrections.