Olivier De Almeida

Jean-Franc ¸ois Ferrero
email: jean-francois.ferrero@univ-tlse3.fr

Laurent Escale

Ge ´rard Bernhart

Charpy test investigation of the influence of fabric weave and fibre nature on impact properties of PEEK-reinforced composites

Keywords: Thermoplastic composites, impact, Charpy test, fibre nature, yarn crimp effect

The aim of the work is to use Charpy impact test for quick evaluations of different Polyether-ether-ketone (PEEK)-reinforced composites to be used for impact protection. In the first part, the influence of weave pattern was first analysed by comparing the impact behaviour of three PEEK composites reinforced with plies of unidirectional (UD) tapes, 5H satin fabrics and 2 Â 2 twill fabrics made of high-strength carbon fibres. In the second part, the influence of fibre nature was investigated for the same weave pattern. The impact behaviour of five 2 Â 2 twill fabrics made from inorganic fibre (carbon, glass and basalt) and organic fibre (aramid and poly(p-phenylene-2,6-benzobisoxazole) (PBO)) has been compared. Two main types of failure modes were identified: a brittle behaviour mode with high failure strength and a highly deformable behaviour mode in which energy absorption is more important. The balance between brittle behaviour and highly deformable behaviour results from competition between the yarn crimp, weave pattern and fibre properties of the composite. Slight yarn crimp and small ply thickness increase the stiffness of the composite and induce brittle behaviour characterized by fibre failure in tension and a steep peak on the loading curves. This behaviour is observed in UD and 5H satin carbon-reinforced composites or 2 Â 2 twill glass and basalt fabric-reinforced composites. In contrast, aramid and PBO 2 Â 2 twill fabric composites exhibit high shear strength. The highly deformable behaviour of the specimens during the Charpy impact

Introduction

Composites are extensively used in aerospace applications due to their high strength to weight ratio. However, one of the major concerns is their susceptibility to impact damage, generally due to assembly or maintenance activities (handling, storage and accidental tool drops). The effect of foreign object impact can severely degrade the strength of composite structures while being undetectable by visual inspection. [START_REF] Abrate | Impact on composite structures[END_REF] Many researchers have investigated the brittle behaviour of unidirectional (UD) carbon epoxy plates. It is strongly dependent on the matrix cracking, which plays an important role in its progressive failure mode. Among the many studies that have tried to explain the different types of damage developing in a composite laminate plate during a low velocity/low energy impact, [START_REF] Abrate | Impact on composite structures[END_REF][START_REF] Richardson | Review of low-velocity impact properties of composite materials[END_REF][START_REF] Bouvet | Low velocity impact modelling in laminate composite panels with discrete interface elements[END_REF] many works propose a scenario of impact damage, which gives matrix cracks a precursor role in the development of the delamination in the laminates. This scenario begins with the development of matrix cracks in the impact zone below the impactor. These cracks increase during loading following the fibre direction and can induce delamination in the interply area. This behaviour is particularly observed for continuous fibre-reinforced thermosetting composites that have been intensively used over the past three decades in aeronautical applications. [START_REF] Abrate | Impact on composite structures[END_REF] High-performance thermoplastic resin (e.g. PEEK) offers a promising alternative to thermosetting resins because of its better resistance to impact damage. [START_REF] Bessard | Mate´riaux composites structuraux à base PEEK e´labore´s par thermocompression dynamique: relation proce´de´-proprie´te´s[END_REF][START_REF] Bismarck | Effect of hot water immersion on the performance of carbon reinforced unidirectional poly(ether ether ketone) (PEEK) composites: stress rupture under end-loaded bending[END_REF][START_REF] Tennyson | Hypervelocity impact damage to composites[END_REF] Many efforts have been made to understand why such materials are often more damage tolerant. The most studied configuration remains PEEK resin-reinforced carbon fibre, such as AS4/APC2 or AS4/APC1 (for the older studies) UD prepreg. [START_REF] Vieille | Investigations on stamping of C/PEEK laminates: influence on meso-structure and macroscopic mechanical properties under severe environmental conditions[END_REF] Drop weight impact produced less extensive damage in carbon/PEEK fibre laminates. [START_REF] Bishop | The mechanical performance and impact behaviour of carbon-fibre reinforced PEEK[END_REF][START_REF] Dorey | On the impact performance of carbon fibre laminates with epoxy and PEEK matrices[END_REF][START_REF] Vieille | Influence of matrix toughness and ductility on the compression-after-impact behavior of woven-ply thermoplastic-and thermosettingcomposites: a comparative study[END_REF] Because of the less extensive damage and greater delamination fracture energy, the residual compressive strengths were significantly greater with a PEEK matrix. For carbon/PEEK fibre woven ply laminates, a recent contribution can be found in the study by Vieille et al. 11 where the impact behaviour of a thermosetting-based (epoxy) and two thermoplastic-based (PEEK and polyphenylene sulfide [PPS]) laminates is compared. From this work, it seems that the matrix plastic yielding plays an important role by locally promoting permanent deformation and, therefore, reducing the extent of damage.

All these failure modes under low-velocity impact loading conditions are strongly dependent on the fibre type, resin type, fibre-resin interface, loading velocity and projectile type. For low-velocity impact events, the use of pendulums like those employed in the Charpy test, drop towers or drop weights has become standard. [START_REF] Abrate | Impact on composite structures[END_REF] Nevertheless, these two test methods have different objectives that are related to the specificity of the specimen used. The drop tower is typically used to impact plates. [START_REF] Abdulhamid | Influence of internally dropped-off plies on the impact damage of asymmetrically tapered laminated CFRP[END_REF][START_REF] Hongkarnjanakul | Validation of low velocity impact modelling on different stacking sequences of CFRP laminates and influence of fibre failure[END_REF][START_REF] Mohmmed | Finite element analyses of low-velocity impact damage of foam sandwiched composites with different ply angles face sheets[END_REF] This allows observations of the evolution of the damage in the plane of the specimen (matrix cracking, debonding at the fibre-matrix interface and particularly ply delamination) created by the drop weight according to the stacking sequence studied. The damage is then equivalent to that created by the fall of a tool or a slight impact on a composite structure.

Many researchers have used the Charpy test to investigate the impact behaviour of various composite material structures. It allows the energy absorbing capability and the strength of the laminates to be estimated over the entire phase of loading and subsequent failure of the specimen. [START_REF] Adams | Instrumented Charpy impact tests of several unidirectional composite materials[END_REF][START_REF] Adams | Low level Charpy impact of graphite epoxy hybrid composites[END_REF][START_REF] Arifin | Investigation of the behaviour of a chopped strand mat/woven roving/foam-Klegecell composite lamination structure during Charpy testing[END_REF][START_REF] Li | Charpy impact properties and failure mechanism of 3D MWK composites at room and cryogenic temperatures[END_REF][START_REF] Martin | Analysis of impact properties of 3D-carbon-carbon composite materials by the instrumented Charpy test[END_REF] For this, it is necessary to use measurement chains of sufficiently high frequency response and striker tups of sufficiently high sensitivity. [START_REF] Kalthoff | Characterization of the dynamic failure behaviour of a glass-fiber/vinyl-ester at different temperatures by means of instrumented Charpy impact testing[END_REF] Hufenbach et al. [START_REF] Hufenbach | Charpy impact tests on composite structures -an experimental and numerical investigation[END_REF][START_REF] Hufenbach | Multidisciplinary damage analysis of textilereinforced composites for impact and crash applications[END_REF] have experimentally investigated the influence of major parameters, such as fibre type and architecture, hybrid composite and fibre orientation, on the impact damage resistance of composite structures. Ghasemnejad et al. [START_REF] Ghasemnejad | Charpy impact damage behaviour of single and multi-delaminated hybrid composite beam structures[END_REF][START_REF] Ghasemnejad | To improve impact damage response of single and multi-delaminated FRP composites using natural Flax yarn[END_REF] have studied the energy absorbing capability of a delaminated hybrid composite beam (glass/epoxy and carbon epoxy) and delaminated fibre-reinforced polymer pinned using natural flax yarn. Fernandez-Canteli et al. [START_REF] Fernandez-Canteli | Dynamic fracture toughness measurements in composites by instrumented testing: influence of aging[END_REF] have investigated the effect of ageing on the dynamic fracture toughness of three different fibre fabric composite laminates. They showed that ageing significantly affects the dynamic fracture toughness but had less effect on the fracture toughness.

The purpose of this study is to experimentally investigate the influence of different parameters like fibre nature or weave pattern on the impact properties of PEEK matrix composites by means of Charpy impact tests. In the first part, special emphasis is placed on the comparison between 1D and 2D carbon fibrous architectures and the second part aims at comparing the impact behaviour of inorganic-based (carbon, glass and basalt) and organic-based (aramid and poly(p-phenylene-2,6-benzobisoxazole) (PBO)) reinforcements of the same weave.

Experiments Materials

Three different quasi-isotropic carbon/PEEK laminates were first investigated. These composite laminates essentially differ in the reinforcement weave pattern. The first material was prepared with 40 plies of AS4/APC2 tapes, provided by Cytec, using a quasi-isotropic lay-up [90/45/0/À45] 5s . The fibre volume fraction and nominal thickness of the UD prepreg tapes were, respectively, 60% and 0.137 mm. After consolidation in a hot press at 395 C for 25 min, the laminate thickness was 5.4 mm. The second material was a carbon/PEEK laminate prepared with 16 plies of Pi-Preg ® semi-preg provided by Porcher Composites. This semi-preg is a two-face powdered carbon fabric (high tenacity carbon fibres [HTA] carbon fibres, 5H satin weave, Reference 3106-P17) having a nominal ply thickness after consolidation of 0.31 mm and a fibre volume fraction of 50%. The consolidation of this [90/45/0/À45] 2s lay-up laminate was performed in a fast heating and cooling press for 10 min at 400 C. The laminate thickness was 5.0 mm. The third material was prepared by a film stacking technique using 38 mm thick Victrex Aptiv 2000 films (grade 150) and a 2 Â 2 twill carbon fabric. The dry carbon fabric had an areal weight of 280 g/m 2 and was provided by Hexcel under the reference HexForce 46285. The [90/45/0/À45] 3s laminate lay-up was made by alternating 64 films of PEEK and 16 plies of carbon fabric, which corresponded to a nominal fibre volume fraction of 50%. As for the second composite, a fast heating and cooling press was used to perform the composite consolidation. A final thickness of 4.9 mm was obtained after processing the stacked sequence at 400 C for 20 min.

In order to analyse the influence of fibre nature on the impact resistance of PEEK matrix composites, the 2 Â 2 twill carbon-reinforced laminate was compared with four other composites prepared with 2 Â 2 twill fabrics made of glass, basalt, aramid and PBO fibres. The characteristics of the fabrics are summarized in Table 1: although all the fabrics had the same weave, they exhibited different nominal thicknesses because of different areal weights.

Therefore, different numbers of plies were used to manufacture 5 mm thick plates with a fibre volume fraction of 50%. In particular, glass-and basalt-reinforced composites were composed of 16 layers of 2 Â 2 twill fabrics, while carbon, aramid and PBO composites were made with 32 layers of 2 Â 2 twill fabrics (Table 2).

All composites were consolidated using the same methodology and processing conditions as for the carbon-reinforced composite described previously, that is, consolidation was performed in a fast heating and cooling mould at 400 C for 20 min.

Charpy test

The interest of Charpy tests is that they only require test specimens of small dimension, such as beams, and, as it involves high energy impacts, they generate much more damage than those obtained with the other low-velocity impact tests. Although this test does not give data that can be used for absolute and quantitative strength calculations, they allow valuable comparative data that are determined in a simple and effective manner.

A Zwick Roell Charpy test device was used to investigate the impact damage resistance of the composite laminate structures. All the Charpy tests were carried out with a mass of pendulum of 21.9 kg, a swing length of 748.6 mm and a height of 1400 mm, which resulted in an impact speed of 5.2 m/s and an impact energy of 300 J. All the specimens were 70 mm long and 10 mm wide and were simply supported by anvils with a 40 mm support span (Figure 1).

No notches were machined on the specimens, contrary to what is conventionally done for the characterization of metallic Charpy samples. Indeed, in the present study, the Charpy impact tests performed do not aim at characterizing the fracture toughness but are only used as a reproducible experiment for comparison of impact behaviour of various composite materials.

The loading force F(t) acting on the tip of the striking hammer during the failure process was measured by a force transducer located on the striker of the Charpy hammer. Energy losses due to bearing friction and air resistance were disregarded because of their small contribution to the energy balance.

At least five specimens of each type were tested. The force-displacement curves of five specimens of AS4/APC2 are shown in Figure 2. It can be seen that the impact behaviour is perfectly reproducible and, for clarity of curve comparisons, only one representative curve will be presented for each type of specimen.

After Charpy tests, a post-mortem analysis of the damaged specimens was carried out using optical microscope and/or visual observations.

Results and discussion

Charpy data analysis

Figure 3 shows a representative set of data obtained for the AS4/APC2 specimen. The energy absorbed over time (respectively hammer displacement) was obtained by a double integration of the loading force over time. The entire loading and failure process of the specimen, that is, up to the displacement (respectively time) at which the loading force had decreased to zero force value again, represents the total energy for damaging and for breaking the specimen. This energy agrees with the one that would be determined from the difference in the heights of the striker before and after the test. The characteristic force-displacement curve exhibits a first loading phase corresponding to an energy W init absorbed by the specimen until the load has reached its maximum value F max . W init represents a quantity of the energy associated with the initiation of failure, through different damage mechanisms.

The second part of the curve (W prop ) is the quantity of energy that is required to totally or partially break the specimen until it is ejected out of the anvils. This partial energy usually characterizes the energy used for crack propagation, generally related to the main mechanisms, that is, matrix microcracking, delamination and fibre fracture.

Influence of weave pattern

Results. The influence of weave pattern was first analysed by comparing the impact behaviour of three PEEK composites reinforced with plies of UD tapes, 5H satin fabrics and 2 Â 2 twill fabrics made of high-strength carbon fibres. The measured Charpy impact data (force and energy over hammer displacement), photographs of broken specimens and schematic representation of the fibres breakage are shown in Figure 4.

For all three materials, Charpy impact tests led to complete fracture of the specimens, that is, the fracture of all carbon fibres through the specimen depth.

The force-displacement and energy-displacement curves reveal significant differences in the impact behaviour of the different weave patterns. The initial slope of the curves shows that the material stiffness of APC2/AS4 composite (Figure 4(a)) is higher than 5H satin (Figure 4(b)) and 2 Â 2 twill composites (Figure 4(c)).

The load curves of UD-and satin-reinforced laminates are both characterized by a steep rise to a loading peak followed by an instantaneous drop of the applied force corresponding to the failure of the sample.

For both materials, F max corresponds to a critical damage accumulation leading to the instantaneous failure of the fibres of the specimens. For the 2 Â 2 twill laminate (Figure 4(c)), the force-displacement curve still has a main loading peak at F max but its level was only a quarter of that of the AS4/APC2 laminate. Moreover, the rupture of the specimens was not as brittle as for the other two materials.

The analysis of the tested samples shows differences in the damage mechanisms and fracture type. Ply delamination occurred once or twice in each sample of AS4/APC2 laminate on all or part of the surface of the specimens. Overall, the two other carbon fabric-reinforced composites do not show delamination. Only the last interface opposite to the impacted side is slightly delaminated over a very short length. The shape of the fibres failure shows a bending failure for the UD-reinforced laminate specimens (fibres breaks in the axis of the impact) and shear failure for the woven specimens (fibres breaks oriented at 45 in the shearing cone).

As shown in Figure 5, due to the brittle behaviour, the loading phase largely contributes to energy absorption for both UD and satin fabric-reinforced composites: W init represents 45% of the total absorbed energy. However, the F max value of the satin fabric composite is 30% lower than that of UD composite. This results in a total absorbed energy W total for the AS4/APC2 composite that is twice as high as for the satin composite. For the 2 Â 2 twill laminate, the absorbed energy W init during the first loading phase only represented 17% of the total absorbed energy. As a consequence, although they exhibited very different behaviours, the total energy absorbed by the composites made with fabrics was of the same order (5 J, Figure 6).

Discussion on influence of weave pattern. According to the previous results, both the load-displacement behaviours and the failure modes during Charpy impact tests can be related on one hand to the plies interpenetration and on the other to the crimp ratio.

Delamination is a damaging mechanism observed in UD laminates, but is limited in fabric-reinforced laminates because of the interpenetration of fabric plies: the chaotic interface between any two plies limits the propagation of delamination. [START_REF] Abrate | Impact on composite structures[END_REF] This explains why delamination is not observable in fabric-reinforced composites while it appeared as a significant damaging mechanism in AS4/APC2 laminate. The crimp ratio is the ratio of the yarn length to the length of fabric produced from that yarn. [START_REF] Adanur | Handbook of weaving[END_REF] In balanced fabrics, this ratio is all the higher when warp and weft interlace per unit length of fabric. It is known to largely influence the mechanical properties of fabrics, particularly the tensile modulus, [START_REF] Abot | Effect of woven fabric architecture on interlaminar mechanical response of composite materials: an experimental study[END_REF][START_REF] Scida | A micromechanics model for 3D elasticity and failure of woven-fibre composite materials[END_REF] but also the resistance of the yarns in tension. [START_REF] Osada | Initial fracture behavior of satin woven fabric composites[END_REF][START_REF] Kergomard | Intralaminar and interlaminar damage in quasi-unidirectional stratified composite structures: experimental analysis[END_REF] In AS4/APC2 composite, all the carbon yarns are straight and aligned within the plies which give a crimp ratio equal to 1. The tensile modulus and strength of the AS4/APC2 is therefore higher than in 5H satin fabric and even higher than that of 2 Â 2 twill fabric and the F max value is governed by the failure of carbon yarns in tension. As the warp tows and the weft tows interlace twice every five yarns and twice every two yarns, this ratio increases in 5H satin and even more in 2 Â 2 twill fabrics (Figure 6).

The F max value is governed by the failure of carbon yarns. This failure is due to the tension of carbon yarns but also of their shear. The distribution between tension loading and shear loading depends on the crimp level of the fabrics in the laminates. In fabrics, the higher the crimp ratio, the higher is the shear loading of the yarns and the lower is F max . For the 2 Â 2 twill laminate, even if F max is weak, the flexibility of both the plies and the structure induces a diffuse fracture. This progressive failure still allows the material to absorb a significant quantity of energy before total fracture and ejection from the anvils. This behaviour (low F max and high energy absorption) can be interesting in armour applications. From these observations, it may be interesting to study the influence of the nature of the fibres to optimize this process.

Effect of fibre nature

Results. The influence of fibre nature on the impact behaviour of 5 mm thick quasiisotropic composites was analysed using the same methodology as for the comparison of the carbon-reinforced composites. Figure 7 shows the Charpy impact data and a photograph of a representative impacted sample for each type of fibre: carbon (Figure 7(a)), glass (Figure 7(b)), basalt (Figure 7(c)), aramid (Figure 7(d)) and PBO (Figure 7(e)). These five fibres cover the main families of reinforcement: non-siliceous inorganic fibre (carbon), siliceous inorganic (mineral) fibres (glass and basalt) and organic fibres (aramid and PBO). Figure 8 shows micrographic observations of the damaged areas.

As shown in Figure 7(b) and (c), the 2 Â 2 twill glass and basalt composites exhibited a loading curve similar to the carbon satin and carbon UD laminates: a distinct peak load followed by prompt failure. In both cases, the specimens broke under the Charpy hammer and the fractures were principally characterized by fibre breakage. The effect of fabric weave observed for carbon-reinforced laminates was therefore different for mineral fibres. The unstable propagation of the fracture was smoother for carbon than for the mineral fibres.

The impacted samples of glass and basalt fibres-reinforced composites exhibit a more localized fracture area than the carbon material (Figure 8(a) and (b)). No delamination is observable which agrees with the previous analysis of the influence of weave on Charpy impact behaviour.

Unlike the inorganic fibre-reinforced composites, the composites with organic fibres did not break during the Charpy impact test (Figure 8(c) and(d)). As shown on the load/ displacement curves of aramid and PBO, the resistance to impact and the energy absorption occurred during 30 mm of displacement. This displacement corresponds to The load curves obtained for these materials also contrast with those of composites with inorganic fibres by their shape. Although the maximum load F max can be identified, the Charpy test curves are flattened, with a load constantly below 1 kN.

Even if the samples were ejected before fracture, damage occurred during the impact (Figure 8(c) and(d)). In both aramid-and PBO-reinforced materials, the impact induced matrix microcracking. In the case of aramid, it led to the apparition of shear bands starting from the impacted side and to a few fibre failures on the non-impacted side (Figure 8(c)).

Discussion on influence of fibre nature. The chart in Figure 9 compares the different types of fibres in regard to the Charpy impact characteristic properties, that is, the total absorbed energy W total , the maximum force F max and the absorbed energy at peak W init . Three behaviours can be distinguished, corresponding to the three different families of fibres: the organic population, which is characterized by high but progressive energy absorption (low F max ), the mineral population, which has brittle behaviour but a reasonable absorbed energy due to a high F max and the non-siliceous inorganic population, which shows limited performance (low W total and F max ).

Many factors can influence the Charpy impact properties of composite materials but, in the present case, as it corresponds to the main families of fibres, this classification into three main populations must be correlated with the intrinsic properties of the fibres. The tensile properties of the fibres are summarized in Table 3 and, for the sake of clarity, Figure 10 proposes a graphical representation of the mechanical behaviour of the different types of fibres. The ply thickness was also considered in the comparison.

Carbon fibres, which have the highest stiffness, differ from the other fibres by their low deformability. In contrast, glass and basalt fibres exhibit poor stiffness and strength but a strain at rupture that is twice as high as that of carbon. Organic fibres, aramid and PBO, show an interesting compromise of stiffness, strength and strain at rupture. In particular, PBO fibres have a modulus only 23% lower than carbon and the highest strength and deformability.

The ply thickness of carbon, aramid and PBO composites is similar. The 5 mm thick laminates were fabricated with the same number of plies, and it can be considered that the specimens and fabric plies carried the same loading forces during Charpy impact. The difference of impact behaviour and composite resistance was thus only due to the difference of the mechanical properties of the fibres. The absence of a loading peak for organic fibre-reinforced composite may then be due to the lower stiffness of these laminates because of the lower Young modulus of the fibres. The totally different energy absorption process, progressive carbon fibre failure versus specimen bending with matrix cracking for organic fibres, may be the consequence of the differences in deformability and strength of the organic fibres.

Since the mineral fibre-reinforced composites comprise twice as many plies of fabric as the other composites, the analysis of their mechanical response to Charpy impact cannot be only based on the fibre performances. Thinner plies induce a lower crimp ratio and therefore a higher stiffness of the specimen's structure. The competition between the structural effect, governed by the ply thickness and yarn crimp in the fabric, and the intrinsic mechanical performance of the fibres led to an absorbed energy that was significantly higher in the case of mineral reinforcements than the energy absorbed by the carbon composite. The brittle behaviour observed for these composites and the moderate mechanical properties of glass and basalt fibres then suggest that the structure has a dominant effect on the composite reaction to Charpy impact loading.

Conclusions

The comparison of different carbon weaves and fibre types by means of Charpy impact tests has highlighted two main behaviours that result from the complementarity of yarn crimp, delamination onset, ply thickness and properties of the fibre.

In the case of AS4/APC2, the low yarn crimp, the small ply thickness and the rigidity of carbon produce a very stiff composite that leads to a tensile loading of the reinforced fibres during impact. This loading mode induces brittle fracture of the specimens due to the apparition of delamination and fibre failure.

The slight increase of yarn waviness in 5H satin fabric and the decrease of ply thickness in glass and basalt fabrics lead to the same brittle behaviour. The effect of crimp seems to be a dominant parameter controlling the failure mode of the composites when delamination cannot occur.

On the contrary, specimens with low stiffness, that is, composites made of thick 2 Â 2 twill fabrics, can bend. For the carbon specimen, the load applied during the impact leads to a progressive but total fracture of the specimen. In the same configuration, organic fibres demonstrated their ability to deform without rupture and to absorb energy during impact. The fibres, and therefore the specimens, did not break neither in shear nor in bending. The ability to deform without shear sensitivity appeared to be more efficient for energy absorption than the stiff resistance to impact.

In particular, PBO absorbed 50% more energy than AS4/APC2 during Charpy impact testing. This organic fibre is thus an interesting alternative to aramid for composite armours, when deflection is allowed for projectile protection.

This comparative study has pointed out the effect of crimp and ply thickness on the deformation mechanisms of composites. However, no relation could be found between the local loading applied to the fibres and their multiaxial intrinsic properties. Further investigations are thus required to better understand the origin of impact resistance. In particular, the dynamic behaviour and the shear strength of the fibres remain poorly known.

Figure 1 .

 1 Figure 1. Schematic drawing of the Charpy test device used in this study.

Figure 2 .

 2 Figure 2. Load curves of Charpy testing on five specimens of the UD carbon/PEEK composite. UD: unidirectional.

Figure 3 .

 3 Figure 3. Typical set of instrumented Charpy impact data measured on APC2/AS4 specimen: measured force-displacement and calculated energy-displacement.

Figure 4 . 2 Â 2

 422 Figure 4. Charpy test load curves, macroscopic observations and schematic representation of representative damaged specimens of carbon/PEEK composites: (a) UD tape, (b) 5H satin fabric and (c) 2 Â 2 twill fabric. UD: unidirectional.

Figure 5 .

 5 Figure 5. Partial and total absorbed energies W init and W total (left) and maximum force at peak F max (right) measured on the carbon/PEEK composites.

Figure 6 .

 6 Figure 6. Schematic representation of a ply cross section of each reinforcement highlighting the difference of tow crimp in the reinforcement weaves investigated: (a) UD, (b) 5H satin and (c) 2 Â 2 twill. UD: unidirectional.

Figure 7 .

 7 Figure 7. Charpy test load curves on representative PEEK composite specimens fabricated with 2 Â 2 twill fabrics: (a) carbon, (b) glass, (c) basalt, (d) aramid and (e) PBO. PBO: poly(p-phenylene-2,6-benzobisoxazole).

Figure 8 .

 8 Figure 8. Microscopic observations of the damaged area of Charpy specimens reinforced with glass (a), basalt (b), aramid (c) and PBO (d) fibres. PBO: poly(p-phenylene-2,6-benzobisoxazole).

Figure 9 .

 9 Figure 9. Comparison of characteristic properties of Charpy impacted specimens.

Figure 10 .

 10 Figure 10. Mechanical properties of the five types of fibres and ply thickness in composite samples.

Table 1 .

 1 Characteristics of the fabrics.

		Carbon	Glass	Basalt	Aramid	PBO
	Supplier	Hexcel	Brochier	Flocart	Teijin	Toyobo
	Areal weight (g/m 2 )	2 8 0	1 9 5	2 2 0	2 2 0	2 3 0
	Thickness (mm)	0.28	0.20	0.16	0.28	0.27

PBO: poly(p-phenylene-2,6-benzobisoxazole).

Table 2 .

 2 Film stacking configurations and laminate characteristics.

		Carbon	Glass	Basalt	Aramid	PBO
	Number of fabrics	16	32	32	16	16
	Number of films	64	64	64	64	64
	Thickness (mm)	5.1	5.2	5.2	5.0	5.0

Table 3 .

 3 Properties of inorganic and organic fibres.

		Carbon	Glass	Basalt	Aramid	PBO
	Elastic modulus (GPa)	235	73	86	120	180
	Strength (GPa)	4.4	2.2	2.8	3.6	5.8
	Strain at break (%)	1.7	3.5	3.1	2.8	3.5
	Density	1.78	2.60	2.67	1.45	1.54

PBO: poly(p-phenylene-2,6-benzobisoxazole).

 2 twill fabrics: (a) carbon, (b) glass, (c) basalt, (d) aramid and (e) PBO. PBO: poly(p-phenylene-2,6-benzobisoxazole).

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.


Céline Vicard

Olivier De Almeida
email: olivier.dealmeida@mines-albi.fr

Arthur Cantarel

Gérard Bernhart

6 synthesized from ϵ-caprolactam

Keywords:

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

Modeling of the polymerization and crystallization kinetic coupling of polyamide

Introduction

Reactive thermoplastic systems are promising alternatives to conventional thermoplastic matrices for the manufacturing of composite laminates. Contrary to high molecular weight thermoplastics, which high viscosity limits fiber impregnation rate and requires high pressure processes, the low viscosity of reactive systems make them compatible with liquid processes used for thermoset matrix composite manufacturing (Resin Transfer Molding, Liquid Resin Infusion) [START_REF] Van Rijswijk | Reactive processing of textile fiber-reinforced thermoplastic composites -an overview[END_REF]. Thermoplastic composites made from reactive systems combine the advantages of thermoset composite processes, that offer the possibility of manufacturing large and non-uniform 3D composite parts, and thermoplastic matrix properties, which enable welding, reshaping and recycling composite parts afterwards.

Among the available reactive thermoplastic systems, polyamide 6 (PA6) synthesized by anionic ring-opening polymerization (AROP) of εcaprolactam is particularly attractive for transportation and energy applications. It offers a suitable balance between mechanical properties and material cost [START_REF] Van Rijswijk | Reactive processing of textile fiber-reinforced thermoplastic composites -an overview[END_REF] and does not required extensive modifications of standard liquid process equipment. PA6 polymerization is achieved in situ during and after impregnation of composite reinforcements. The synthesis can be parametrized using various pair and ratio of catalyst and activator and controlled by temperature [START_REF] Van Rijswijk | Reactive processing of textile fiber-reinforced thermoplastic composites -an overview[END_REF][START_REF] Khodabakhshi | Monitoring of small-scale anionic polymerization of caprolactam; a method to be used in an additive manufacturing process[END_REF][START_REF] Khodabakhshi | Anionic polymerisation of caprolactam at the small-scale via DSC investigations[END_REF].

The particularity of the PA6 synthesis is that the chains polymerization and their crystallization can occur simultaneously depending on the temperature applied [START_REF] Wichterle | The anionic polymerization of caprolactam[END_REF][START_REF] Komoto | Formation of spherulites during polymerization of lactams[END_REF][START_REF] Wichterle | Adiabatic crystallization of amorphous polycaprolactame[END_REF] and both affect the viscosity of the system. These phenomena are even coupled as the crystallization kinetics depend on the amount of polymer chains available. The understanding of the polymerization/crystallization interaction is even more critical given that their temperature dependencies are contrary. A high processing temperature accelerates the polymerization kinetics while disenabling crystallization (which will occur during cooling). Conversely, a low temperature slows down the polymerization but facilitates crystallization; though issues due to the coupling between phenomena may appear.

Different hypotheses have been reported in the literature to explain how phenomena interact. Specifically, crystallization could not proceed right from the beginning of the reaction as a minimum concentration or a minimal length of polymerized oligomers are required for nucleation [START_REF] Wunderlich | Macromolecular Physics -Crystal Nucleation Growth Annealing[END_REF][START_REF] Korshak | Kinetics of activated anionic polymerization of εcaprolactam under non-isothermal conditions[END_REF][START_REF] Davtyan | fson, Problems of non-isothermal character in polymerisation processes[END_REF][START_REF] Vicard | Experimental study of polymerization and crystallization kinetics of polyamide 6 obtained by anionic ring opening polymerization of ε-caprolactam[END_REF]. The crystallization could also change the concentration of reactants in the mixture, either by reducing the volume available for polymerization (increases the concentration) or by entrapping reactive species in crystals (decreases the concentration), which would have an impact on the synthesis kinetics [START_REF] Komoto | Formation of spherulites during polymerization of lactams[END_REF][START_REF] Wunderlich | Macromolecular Physics -Crystal Nucleation Growth Annealing[END_REF][START_REF] Van Rijswijk | Thermoplastic Composite Wind Turbine Blades -Vacuum Infusion Technology for Anionic Polyamide-6 Composites[END_REF][START_REF] Davé | Polyamides from lactams via anionic ring-opening polymerization: 2[END_REF][START_REF] Lee | Reaction-induced crystallization kinetics during the anionic polymerization of ε-caprolactam[END_REF].

As the temperature range of PA6 synthesis coincides with that of crystallization, a proper control of the reaction advancement and a prediction of final composite properties are complex yet crucial. To avoid expensive trial-and-error empirical optimization, a process simulation can be developed in order to optimize the cycle time, composite properties and dimensional stability after demolding. In particular, this requires a modeling of the synthesis kinetics including both polymerization and crystallization. This modeling also allows construction of a Time-Temperature-Transformation (TTT) diagram, which summarizes the events occurring over time at different temperatures [START_REF] Luisier | Time-Temperature-Transformation diagram for reactive processing of polyamide 12[END_REF]. Such diagrams represent a convenient tool for assessing a cure path and final properties of composites involving reactive matrices. From a unique diagram, different processing routes can be defined for any process while avoiding use of complex simulations.

The global heat flow φ [W g -1 ] recorded during the PA6 AROP from ε-caprolactam is the sum of the polymerization heat flow φ p and the crystallization heat flow φ c (eq. ( 1)).

= + φ t φ t φ t ( ) ( ) ( ) p c
(1)

To model the evolution of φ, the relative degree of conversion α [-] and the relative degree of crystallinity β [-] are generally preferred as they allow using kinetic differential equations. α is defined by equation [START_REF] Khodabakhshi | Monitoring of small-scale anionic polymerization of caprolactam; a method to be used in an additive manufacturing process[END_REF], where X p [wt.%] is the mass fraction of polymerization and

∞ H ∆ p [J g -1 ] is the final enthalpy of polymerization. ∫ = = ∞ ∞ α t X t X H φ τ τ ( ) ( ) 1 ∆ ( ) d p p p t p 0 (2)
In the case of the PA6 synthesis, the polymerization kinetics are modeled either through a mechanistic differential equation based on chemical reaction kinetics and monomer concentration [START_REF] Sittler | Alkaline polymerization of 6-caprolactam. XXXII. The kinetics of polymerization activated by N,N,N',N'-tetraacetylhexamethylenediamine[END_REF][START_REF] Greenley | The kinetic equation for the initiated anionic polymerization of ε-caprolactam[END_REF][START_REF] Reimschuessel | Ring-Opening Polymerization[END_REF][START_REF] Rigo | Kinetic study of anionic polymerization of 6-caprolactam by differential calorimetry[END_REF][START_REF] Lin | A kinetic study of the activated anionic polymerization of ε-caprolactam[END_REF][START_REF] Cimini | A mechanistic kinetic model for the initiated anionic polymerization of ε-caprolactam[END_REF], or by using a semi-empirical equation [START_REF] Lin | A kinetic study of the activated anionic polymerization of ε-caprolactam[END_REF][START_REF] Wittmer | Über die anionische schnellpolymerisation von caprolactam[END_REF][START_REF] Ya Malkin | The nonisothermal anionic polymerization of caprolactam[END_REF][START_REF] Bolgov | Role of the functionality of activators during isothermal crystallization accompanying the activated anionic polymerization of ε-caprolactam[END_REF][START_REF] Ya Malkin | Kinetics of anionic polymerization of lactams. (Solution of non-isothermal kinetic problems by the inverse method)[END_REF][START_REF] Ya Malkin | The exothermal effects of superimposed processes of activated anionic polymerization of ε-caprolactam and crystallization of the polymer formed[END_REF][START_REF] Kamal | Kinetics and thermal characterization of thermoset cure[END_REF][START_REF] Sourour | Differential scanning calorimetry of epoxy cure: isothermal cure kinetics[END_REF]. Wittmer et al. early proposed a first-order equation with an Arrhenius temperature dependence [START_REF] Wittmer | Über die anionische schnellpolymerisation von caprolactam[END_REF]. Malkin et al. [START_REF] Ya Malkin | The nonisothermal anionic polymerization of caprolactam[END_REF][START_REF] Ya Malkin | Kinetics of anionic polymerization of lactams. (Solution of non-isothermal kinetic problems by the inverse method)[END_REF][START_REF] Ya | Rheology in polymerization processes[END_REF] improved this polymerization equation by introducing an autocatalytic reaction rate term and Camargo et al. [START_REF] Camargo | Bulk polymerization kinetics by the adiabatic reactor method[END_REF] proposed to use an n th -order reaction. This model has since been extensively used in the literature [START_REF] Lee | Reaction-induced crystallization kinetics during the anionic polymerization of ε-caprolactam[END_REF][START_REF] Yeh | Adiabatic anionic polymerization of caprolactam in the presence of N-acylated caprolactam macroactivator: kinetic study[END_REF][START_REF] Kim | Mechanism and kinetics of adiabatic anionic polymerization of ε-caprolactam in the presence of various activators[END_REF][START_REF] Kim | Kinetics of adiabatic anionic copolymerization of ε-caprolactam in the presence of various activators[END_REF][START_REF] Teuwen | Novel reaction kinetic model for anionic polyamide-6[END_REF][START_REF] Russo | Fast-activated anionic polymerization of ε-caprolactam in the bulk under quasi-adiabatic conditions: comparison of different kinetic models[END_REF][START_REF] Nagy | Modeling the influence of flow phenomena on the polymerization of ε-caprolactam[END_REF] and remains the reference.

In a similar manner, the crystallization heat flow is defined through the relative degree of crystallinity β [-] by equation [START_REF] Khodabakhshi | Anionic polymerisation of caprolactam at the small-scale via DSC investigations[END_REF], where X c [wt. %] is the mass fraction crystallinity.

∫ = = ∞ ∞ β t X t X H φ τ τ ( ) ( ) 1 ∆ ( ) d c c c t c 0 (3)
In order to model the crystallization during the PA6 synthesis, the literature proposed to use models describing bulk polymer crystallization, such as the Avrami-Kolmogorov model [START_REF] Bolgov | Role of the functionality of activators during isothermal crystallization accompanying the activated anionic polymerization of ε-caprolactam[END_REF][START_REF] Teuwen | Novel reaction kinetic model for anionic polyamide-6[END_REF]. Malkin et al. [START_REF] Ya Malkin | Kinetics of anionic polymerization of lactams. (Solution of non-isothermal kinetic problems by the inverse method)[END_REF][START_REF] Ya Malkin | Rheokinetics of anionic polymerization of caprolactam[END_REF][START_REF] Ya Malkin | Macrokinetics of polymer crystallization[END_REF][START_REF] Ya Malkin | General treatment of polymer crystallization kinetics -Part 1. A new macrokinetic equation and its experimental verification[END_REF][START_REF] Ya Malkin | General treatment of polymer crystallization kinetics -Part 2. The kinetics of nonisothermal crystallization[END_REF] have proposed a new differential equation to describe either bulk polymer crystallization and PA6 crystallization during synthesis. This model includes an autocatalytic term improving the kinetic prediction at the end of crystallization.

To model the coupling behavior between polymerization and crystallization, different levels of interaction have been considered in the literature. A recent study [START_REF] Taki | A kinetic model of viscosity development for in situ ring-opening anionic polymerization of ε-caprolactam[END_REF] proposed to use a simple sum (eq. ( 4)), considering that both transformations are independent. The total heat flow of reaction φ is calculated from ∞ φ p and ∞ φ c [W g -1 ], that refer respectively to the final heat of polymerization and crystallization of the system at the end of the reaction, and from the degree of conversion α and the degree of crystallinity β. This approach is however not valid from a physical point of view as crystallization necessarily depends on the availability of polymerized chains.

= + ∞ ∞ φ t φ α t φ β t ( ) d d d d p c (4)
In the literature [START_REF] Lee | Reaction-induced crystallization kinetics during the anionic polymerization of ε-caprolactam[END_REF][START_REF] Teuwen | Large Thermoplastic Composite Wind Turbine Blades -Process Optimization for Vacuum Infusion of Large Blades[END_REF], the most used coupling equation has been proposed by Malkin and Bolgov (eq. ( 5)). The term B [-] has been defined in this study as the degree of crystallinity within the reactive mixture. It differs from the relative degree of crystallinity of equation ( 3) that corresponds to the crystallization of the bulk polymer. Although if no coupling with polymerization is considered, then = B β. In equation [START_REF] Komoto | Formation of spherulites during polymerization of lactams[END_REF], the crystallization kinetics β ˙are proportional to the degree of conversion α at any time and both phenomena are initiated simultaneously with temperature. It thus assumes that crystallization can occur right from the beginning of the reaction, even when the system only consists of monomers and oligomers. Moreover, the crystallization kinetics of a newly synthesized chain is assumed to be only dependent on the time elapsed since the beginning of the reaction, with no consideration to the moment it was actually formed. These assumptions induce issues with the applicability of the Malkin coupling equation.

= + ∞ ∞ φ t φ α t φ β t α t ( ) d d d d ( ) p c B t d d (5)
The aim of this study is to investigate the limitations of the Malkin coupling model and to overcome them by proposing a new coupling equation. Models are based on a previous experimental study of the PA6 AROP from ε-caprolactame [START_REF] Vicard | Experimental study of polymerization and crystallization kinetics of polyamide 6 obtained by anionic ring opening polymerization of ε-caprolactam[END_REF]. The reaction kinetics were directly identified using calometric method (DSC), contrary to most of the literature dealing with this synthesis, which are based on temperature measurements in an adiabatic reactor. DSC results could be directly used for the identification of kinetic models.

The polymerization kinetic model was identified on this experimental database and complementary experiments of crystallization from the melt were achieved in order to define an appropriate crystallization model. The results obtained with the Malkin coupling model are then compared and a new coupling equation is presented. It was adapted from a kinetic coupling developed by Hillier to model the crystallization of bulk polymers when they exhibit a secondary crystallization [START_REF] Hillier | Modified Avrami equation for the bulk crystallization kinetics of spherulitic polymers[END_REF]. After presenting the coupling formulation, the accuracy of the model to predict the total heat flow over time and its coherence with physical phenomena are discussed. Finally, an isothermal TTT diagram of the PA6 synthesis was built based on the model proposed, gathering both polymerization and crystallization evolutions.

Experimental

Synthesis kinetics and coupling behavior

The modeling of PA6 synthesis was based on experimental data detailed in Ref. [START_REF] Vicard | Experimental study of polymerization and crystallization kinetics of polyamide 6 obtained by anionic ring opening polymerization of ε-caprolactam[END_REF]. In this previous study, PA6 was synthesized by anionic ring-opening polymerization from ε-caprolactam with products supplied by Brüggemann Chemical, Germany. The products used for PA6 polymerization were the ε-caprolactam monomer "AP-Nylon®", caprolactam magnesium bromide as catalyst (MgBrCL -"Nyrim C1®" 1.4 mol kg -1 in caprolactam) and bifunctional hexamethylene-1,6-dicarbamoylactam as activator (HDCL -"Brüggolen C20P®" 2.0 mol kg -1 in caprolactam). Syntheses were achieved with a reactive mixture which MgBrCL/HDCL ratio was 0.79/1.10 mol.% of CL.

The global heat flow φ [W g -1 ] released during synthesis was recorded by DSC at various temperatures between 403 and 473 K (Fig. 1) and heating rates. The degree of global transformation γ [-] was defined by equation [START_REF] Wichterle | Adiabatic crystallization of amorphous polycaprolactame[END_REF], where Q tot [J g -1 ] is the total heat of reaction (both polymerization and crystallization). The global synthesis kinetics γ˙[s -1 ] are obtained from the temporal derivative of the degree of global transformation γ.

∫ = γ t Q φ τ τ ( ) 1 ( ) d tot t 0 (6)
The DSC curves and data recorded by TGA, DSC and WAXS on polymerized samples were used to quantify the total heat of reaction Q tot [J g -1 ], the degree of conversion ∞ X p [wt.%] and the crystallization enthalpy H

∆ c [J g -1 ] for each isothermal synthesis temperature (Table 1). The theoretical total enthalpy of polymerization H ∆ p 100% was also calculated and estimated as ± 123. [START_REF] Komoto | Formation of spherulites during polymerization of lactams[END_REF] 4.1 J g -1 . From the analysis of this experimental dataset, specific behaviors of the coupling between polymerization and crystallization have been identified. When the PA6 synthesis is achieved at high temperature, i.e. above 473 K, the low supercooling prevents the crystallization to occur and the synthesis only depends on the polymerization kinetics. The crystalline structure is formed afterward during the cooling and the final PA6 properties are similar to a standard PA6. At lower synthesis temperature, the polymerization kinetics are reduced and the higher degree of supercooling activates the crystallization in a second stage of the synthesis. Below 433 K, the crystallization kinetics strongly depend on the kinetics of chain extension and the polymerization thus controls the overall synthesis time.

The previous results also highlighted that a high degree of crystallinity and a thick lamellar morphology are obtained when crystallization occurs simultaneously with polymerization, probably due to a higher mobility of polymer chains within the monomer medium. Finally, at low temperature, it was observed that crystallization does not start at the early stage of the synthesis. This was explained by the fact that nucleation requires a minimal chain length [START_REF] Wunderlich | Macromolecular Physics -Crystal Nucleation Growth Annealing[END_REF][START_REF] Korshak | Kinetics of activated anionic polymerization of εcaprolactam under non-isothermal conditions[END_REF][START_REF] Hoffman | Theoretical aspects of polymer crystallization with chain folds: bulk polymers[END_REF][START_REF] Lindenmeyer | Imperfections in polymer crystals[END_REF] or a minimal concentration of oligomers [START_REF] Wunderlich | Macromolecular Physics -Crystal Nucleation Growth Annealing[END_REF][START_REF] Davtyan | fson, Problems of non-isothermal character in polymerisation processes[END_REF] to proceed.

Isothermal crystallization from the melt

The experimental study in Ref. [START_REF] Vicard | Experimental study of polymerization and crystallization kinetics of polyamide 6 obtained by anionic ring opening polymerization of ε-caprolactam[END_REF] was completed with a DSC crystallization from the melt campaign performed on a PA6 sample synthesized at T iso = 423 K. Tests were conducted with a power-com- pensated PerkinElmer 8500 DSC under nitrogen on 9-12 mg samples sealed in aluminum pans. The sample was first heated at 543 K, i.e. above the equilibrium melting temperature of an infinite crystal T m 0 (533 K for PA6). A dwell time of 8 min was used in order to erase the thermal history. The sample was then cooled down at -150 K min -1 to different isothermal temperature T iso c

, between 453 and 463 K (every 2 K). Crystallization control tests at -10 K min -1 were repeated before and after the campaign and confirmed the sample stability along the thermal cycles.

The heat flow curves of crystallization are presented in Fig. 2. The higher the applied temperature T iso c , , the slower the kinetics and longer induction time, which is expected in this temperature range for PA6 [START_REF] Magill | Crystallization kinetics study of nylon 6[END_REF] since the supercooling is lower. The crystallization is also characterized by unsymmetric peaks that reflect a kinetic slowdown at the end of the crystallization and suggests the existence of a secondary crystallization [START_REF] Cebe | Crystallization behaviour of poly(ether-ether-ketone)[END_REF]. Fig. 3 shows the superposition of the crystallization curve obtained from the melt at T iso c , = 453 and 463 K with the second peak (corresponding to the crystallization) of the synthesis curve at the same temperature. The good agreement between curves indicates that when crystallization occurs separately from the polymerization, its kinetics correspond to the one obtained from the melt. This result suggests that the crystallization process is similar during synthesis and from the melt when polymerization and crystallization are almost decoupled.

Numerical methods

Synthesis kinetic formulation

By using a normalized kinetic formulation, the synthesis kinetics are expressed irrespective of the enthalpies associated to polymerization and crystallization. A weight factor of polymerization w p [-] has been Fig. 1. PA6 global synthesis kinetics γ˙measured in isothermal conditions with DSC.

Table 1

Total heat of reaction, mass fraction of polymerization and crystallization enthalpy measured during the PA6 synthesis at different isothermal temperatures.

Synthesis temperature

Total heat of reaction

Mass fraction of polymerization

Crystallization enthalpy defined as the ratio of the final enthalpy of polymerization ∞ H ∆ p to the total heat of reaction Q tot according to equation [START_REF] Wunderlich | Macromolecular Physics -Crystal Nucleation Growth Annealing[END_REF]. The weight factor of crystallization w c [-] is defined in a similar manner, involving

T iso [K] Q tot [J g -1 ] ∞ X p [wt.%] H ∆ c [J g -
+ = w w 1 p c . = = + ∞ ∞ ∞ ∞ w T H T Q T H H H T ( ) ∆ ( ) ( ) ∆ ∆ ∆ ( ) p p tot p p c ( 7
)
As shown in Table 1, the total heat of reaction Q tot decreases with the increase in the synthesis temperature, while the mass fraction of polymerization ∞ X p remains globally constant whatever the synthesis condition, with an average of ∞ X ‾ p = 94.2 ± 1.4 wt%. The weight factors w p and w c were therefore mainly dependent on the variation of crystallization enthalpy H ∆ c with temperature. For sake of simplicity, a constant value was thus considered for the final enthalpy of polymerization ∞ H ∆ p . It was calculated from the average mass fraction of polymerization ∞ X ‾ p and the theoritical total enthalpy of polymerization H ∆ p 100% .

As for crystallization, a second-order polynomial equation (eq. ( 8)) was used to model the temperature dependence of

∞ H T ∆ ( ) c
(Fig. 4). Beyond the roots of the polynomial curve, i.e. below 365.4 K and above 472.4 K, the crystallization was assumed impossible which was in good agreement with experimental observations reported in Ref. [START_REF] Vicard | Experimental study of polymerization and crystallization kinetics of polyamide 6 obtained by anionic ring opening polymerization of ε-caprolactam[END_REF].

= - + - ∞ H T T T ∆ ( ) 0.0354 29.651 6107.5 c 2 (8)
Using eqs. ( 7) and ( 8), the degree of global transformation γ (eq. ( 6)) can be used to define the coupling equation without using enthalpies of phenomena. The Malkin coupling model (eq. ( 5)) can be thus rearranged as equation [START_REF] Davtyan | fson, Problems of non-isothermal character in polymerisation processes[END_REF].

∫ = + - γ t w T α t w T α θ θ β θ θ ( ) ( ) ( ) (1 ( )) ( ) d d ( ) d p p t B t 0 ( ) (9)

Model parameters identification procedure

The polymerization, crystallization and coupling models were identified sequentially on the experimental data of the degree of conversion α, the degree of crystallinity β and the degree of global transformation γ respectively. A least squares method involving an interiorpoint optimization algorithm developed in Matlab® was used for determining model parameters. The identification was achieved on all datasets together, meaning that temperature dependent parameters and constant parameters could be distinguished.

The objective function was normalized by the number of data points on curves N m (eq. ( 10)) and by the number of curves N c (eq. ( 11)). The identification was run several times with different sets of initial parameters in order to avoid the convergence of the optimization algorithm into a local minimum set of parameters.

∑ = ⎛ ⎝ ⎜ - ⎞ ⎠ ⎟ = f x N Y Y x Y ( ) 1 ( ) max(
)

j m i N j i exp j i sim j exp 1 , , 2 m (10)
∑ = = f x N f x ( ) 1 ( ) tot c j N j 1 c (11)

Polymerization and crystallization models

Isothermal polymerization modeling

The polymerization kinetics α ˙were described by the differential equation [START_REF] Davé | Polyamides from lactams via anionic ring-opening polymerization: 2[END_REF] proposed by Camargo et al. [START_REF] Camargo | Bulk polymerization kinetics by the adiabatic reactor method[END_REF] based on Malkin's work [START_REF] Ya Malkin | The nonisothermal anionic polymerization of caprolactam[END_REF][START_REF] Ya Malkin | Kinetics of anionic polymerization of lactams. (Solution of non-isothermal kinetic problems by the inverse method)[END_REF][START_REF] Ya | Rheology in polymerization processes[END_REF]. This model presents an Arrhenius temperature dependence in which A 0 [s -1 ] is a constant representative of the number of colli- sions between molecules, B 0 [-] is an autocatalytic factor, n p [-] corresponds to the reaction order, E a [J mol -1 ] is the activation energy of the polymerization and R [J K -1 mol -1 ] is the universal gas constant. )exp

= - + ⎛ ⎝ - ⎞ ⎠ α t A α B α E RT d d (1 ) (1
n a 0 0 p ( 12
)
Between 463 and 473 K, the polymerization peak is completely separated from the crystallization, which allowed the assessment of the four parameters of the Malkin-Camargo model by optimization. Fig. 5 compares the experimental polymerization kinetics α ˙with the model prediction. The good agreement between results demonstrates the relevance of the model. Final parameters of the model were optimized with the coupling model and the full temperature range of isothermal synthesis.

Isothermal crystallization modeling

As previously mentioned, crystallization during synthesis is similar to crystallization from the melt (Fig. 3) when it is decoupled from the polymerization ( = T iso 453 and 463 K). It was thus decided to identify the crystallization model from results recorded on the synthesized PA6 sample and presented in Fig. 2. This strategy to model crystallization from bulk crystallization was postulated by Malkin et al. [START_REF] Bolgov | Role of the functionality of activators during isothermal crystallization accompanying the activated anionic polymerization of ε-caprolactam[END_REF][START_REF] Ya Malkin | The exothermal effects of superimposed processes of activated anionic polymerization of ε-caprolactam and crystallization of the polymer formed[END_REF][START_REF] Ya Malkin | Macrokinetics of polymer crystallization[END_REF][START_REF] Ya Malkin | General treatment of polymer crystallization kinetics -Part 1. A new macrokinetic equation and its experimental verification[END_REF][START_REF] Ya Malkin | General treatment of polymer crystallization kinetics -Part 2. The kinetics of nonisothermal crystallization[END_REF]. It implies that the interaction with the polymerization progress is only considered through the coupling equation.

Among available models describing the crystallization from the melt of thermoplastic polymer, the Nakamura model [START_REF] Nakamura | Some aspects of nonisothermal crystallization of polymers. I. Relationship between crystallization temperature, crystallinity, and cooling conditions[END_REF][START_REF] Nakamura | Some aspects of nonisothermal crystallization of polymers. II. Consideration of the isokinetic condition[END_REF] was selected as it enables modeling isothermal and non-isothermal crystallization with a unique simple equation. The relative degree of crystallinity β was expressed with equation ( 13), in which K N [s -1 ] is the Nakamura global kinetic constant and n [-] the Avrami exponent distinctive of the type of nucleation (instantaneous or sporadic) and geometry of crystal growth.

∫ ⎜ ⎟ = - ⎛ ⎝ ⎜ -⎛ ⎝ ⎞ ⎠ ⎞ ⎠ ⎟ β t KT θ θ ( ) 1 exp ( ( )) d t N n 0 (13)
The temperature dependent kinetic constant K T ( )

N
was modeled using the Hoffman-Lauritzen equation (eq. ( 14)) [START_REF] Hoffman | Treatise on Solid State Chemistry[END_REF]. In equation ( 14), U* refers to the activation energy of macromolecular motion in the molten state, that was set to 6300 J mol -1 , K g is a constant related to crystallization growth, the temperature ∞ T corresponds to the differ- ence -T 30 g K, T m 0 refers to the equilibrium melting temperature of PA6 crystals and f defines a corrective factor introduced by Hoffman for high supercooling This factor is given by the relation

= + f T T T 2 ( ) m 0 .
The parameters n, K 0 and K g were determined by optimization while ∞ T and T m 0 were set to 293 K and 533 K respectively [START_REF] Illers | Struktur und Kristallinität von 6-Polyamid[END_REF].

⎜ ⎟ ⎜ ⎟ = ⎛ ⎝ - - ⎞ ⎠ ⎛ ⎝ - - ⎞ ⎠ ∞ K T K U R T T K T T T f ( ) exp ( ) exp ( ) N g m 0 * 0 (14)
The main advantage of this model is that it considers the temperature dependence of macromolecular diffusion and nucleation in a temperature range lasting from the glass transition temperature up to the equilibrium melting temperature of PA6 crystal T m 0 . It allows representing the maximum crystallization kinetics around 413 K and the reduction of crystallization kinetics close to T g and T m 0 , that have been observed for PA6 by MaGill on spherulite growth rate [START_REF] Magill | Crystallization kinetics study of nylon 6[END_REF] or by fast scan DSC measurements more recently [START_REF] Kolesov | Structure formation of polyamide 6 from the glassy state by fast scanning chip calorimetry[END_REF][START_REF] Schick | Fast Scanning Calorimetry[END_REF].

Besides the crystallization kinetic equation, an induction time t T ( )

c,0
was introduced in the model in order to consider the time required for the nucleus to form before crystal growth can start. This induction time was modeled with an Arrhenius equation related to the equilibrium melting temperature T m 0 (eq. ( 15)). The induction time introduced two parameters to be identified by optimization: a pre-exponential factor A t and an activation energy E t .

⎜ ⎟ = ⎛ ⎝ - ⎞ ⎠ t T A E R T T ( ) exp ( ) c t t m ,0 0 (15)
Fig. 6 compares the measured crystallization kinetics β ˙and the model predictions (eqs. ( 13) and ( 14)); here the time is normalized with the induction time, i.e. -t t c,0 . The latter observed experimentally (Fig. 2) is compared with the Arrhenius model (eq. ( 15)) in Fig. 7.

Results in Fig. 6 are in good agreement except by the end of crystallization where a possible secondary crystallization can occur. In order to take into account this phenomenon, a coupling equation between two crystallization models could be used, like the couplings proposed by Hillier [START_REF] Hillier | Modified Avrami equation for the bulk crystallization kinetics of spherulitic polymers[END_REF], Velisaris and Seferis [START_REF] Velisaris | Crystallization kinetics of polyetheretherketone (PEEK) matrices[END_REF] or Cebe [START_REF] Cebe | Application of the parallel avrami model to crystallization of poly(etheretherketone)[END_REF]. However it was decided to limit the complexity of the crystallization model to a single Nakamura model. It is indeed aimed at modeling PA6 crystallization during synthesis between 403 and 473 K but it was identified on experimental curves recorded only between 453 and 463 K. The extrapolation of the crystallization model was therefore considered critical with regards to the secondary crystallization process and a reduced number of parameters was preferred.

Table 2 presents the crystallization parameters obtained by the global optimization. It appears that n is relatively small and is not consistent with the spherulitic geometry usually observed for PA6 crystallized from the melt. This result could be a consequence of using a unique Nakamura model to represent the entire transformation without considering the secondary crystallization. Parameters obtained are thus an average of the ones attributed to each mechanism. The crystallization parameters identified from the melt will be fixed for the coupling modeling, being therefore excluded to the global optimization. Fig. 5. Simulation of decoupled isothermal polymerization with Malkin-Camargo model; (-) exp and (-⋅ -).sim Fig. 6. Simulation of the relative degree of crystallinity during the bulk crystallization of a synthesized PA6 sample; (-) β exp and (-⋅ -).β sim Fig. 8 shows the crystallization curves simulated with Nakamura-Hauffman-Lauritzen model and the induction time over the temperature range used for PA6 synthesis from ε-caprolactam, i.e. from 403 to 473 K. At low temperature, where K N is maximal, crystallization only lasts 10 s. Conversely, 5 h are required to complete crystallization at 473 K.

Coupling modeling

Malkin coupling

In order to take into account the coupling between polymerization and crystallization, Malkin et al. [START_REF] Bolgov | Role of the functionality of activators during isothermal crystallization accompanying the activated anionic polymerization of ε-caprolactam[END_REF][START_REF] Ya Malkin | The exothermal effects of superimposed processes of activated anionic polymerization of ε-caprolactam and crystallization of the polymer formed[END_REF] proposed that the term due to crystallization depends on the degree of conversion α achieved at time t as shown in equation ( 5) and rearranged as equation ( 9) using equations ( 1)-( 7). This coupling has been used in the literature to model the temperature rising during adiabatic syntheses [START_REF] Lee | Reaction-induced crystallization kinetics during the anionic polymerization of ε-caprolactam[END_REF][START_REF] Bolgov | Role of the functionality of activators during isothermal crystallization accompanying the activated anionic polymerization of ε-caprolactam[END_REF][START_REF] Ya Malkin | The exothermal effects of superimposed processes of activated anionic polymerization of ε-caprolactam and crystallization of the polymer formed[END_REF][START_REF] Teuwen | Thermoplastic Composite Wind Turbine Blades -Kinetics and Processability[END_REF].

The introduction of the induction time of crystallization t c,0 gives equation ( 16), which distinguishes two cases depending on the sign of θ t T ( )

c,0
described with the Heaviside step function H.

∫ = + - × - - γ t w T α t w T H θ t T α θ β θ t T θ ( ) ( ) ( ) (1 ( )) ( ( ) ) ( ) ( ( ) )d p p t c θ c B t 0 ,0 d d ,0 ( ) (16)
This coupling assumes that both polymerization and crystallization start at the beginning of the reaction and occur at the same time. It means that at each new increment of polymerization, the formed phase is involved in the global crystallization in progress. Moreover, it does not consider the possible influence of the crystallization on the polymerization at low synthesis temperature, even if this possibility has been mentioned in literature [START_REF] Komoto | Formation of spherulites during polymerization of lactams[END_REF][START_REF] Wunderlich | Macromolecular Physics -Crystal Nucleation Growth Annealing[END_REF][START_REF] Vicard | Experimental study of polymerization and crystallization kinetics of polyamide 6 obtained by anionic ring opening polymerization of ε-caprolactam[END_REF][START_REF] Davé | Polyamides from lactams via anionic ring-opening polymerization: 2[END_REF][START_REF] Lee | Reaction-induced crystallization kinetics during the anionic polymerization of ε-caprolactam[END_REF].

Fig. 9 presents results at T iso = 413 and 443 K obtained with Malkin coupling (eq. ( 16)), the Malkin-Camargo polymerization model (eq. ( 12)) and the Nakamura-Hoffman-Lauritzen crystallization model (eqs. ( 13)-( 15)) compared to experimental data from DSC. Fig. A in Supplementary data gathers the synthesis curves simulated from T iso = 403-473 K. It appears that this coupling equation completely inhibits crystallization.

These results are a direct consequence of the form of the term B in the Malkin coupling (eq. ( 16)), which could reached 1 only if phenomena are perfectly decoupled, i.e. if the polymerization is completed (α = 1) prior to crystallization. However, the crystallization model extrapolation, especially at low temperature (403-423 K), implies very high kinetics (Fig. 8) within a time scale of tens of seconds while the polymerization takes one to 2 h. The product of the two components in term B is thus almost zero and leads to a reaction only described by the polymerization kinetics.

This incoherence is inherent of the hypothesis of a global crystallization starting at the beginning of the synthesis. At low temperature, it means both that the crystallization starts even though chains are not yet polymerized, and that chains formed after the end of the global crystallization cannot crystallize despite the thermal conditions being favorable for a crystalline organization.

Hillier coupling

In order to counterbalance the lack of physical meaning involved in the Malkin coupling, the Hillier equation [START_REF] Hillier | Modified Avrami equation for the bulk crystallization kinetics of spherulitic polymers[END_REF] has been adapted to the case of the PA6 polymerization and crystallization coupling (eq. ( 17)). The difference with the Malkin coupling (eq. ( 9)) is in the time dependence of the local decoupled crystallization term β. Indeed the term B, which represents the coupled crystallization, corresponds here to the degree of crystallinity at time t of an element that was polymerized at time θ. In other words, each new polymerized part (new increment of α) starts its crystallization with its own initiation time and its own kinetics.

Hillier coupling thus represents the influence of polymerization on crystallization in terms of the availability of chain segments, which corresponds to the experimental observations. Nevertheless, as with the Malkin coupling, this adaptation of Hillier coupling assumes that crystallization does not influence polymerization, although this possibility has been mentioned previously [START_REF] Van Rijswijk | Reactive processing of textile fiber-reinforced thermoplastic composites -an overview[END_REF].

∫ = -- - - γ t w T α t w T α θ θ β t θ θ ( ) ( ) ( ) (1 ( )) ( ) d d ( )d p p t B t 0 ( ) (17)
By introducing the crystallization induction time t c,0 (eq. ( 15)) into equation ( 17), the coupling is described according to equation [START_REF] Rigo | Kinetic study of anionic polymerization of 6-caprolactam by differential calorimetry[END_REF], Fig. 7. Simulation of the induction time t c,0 appearing prior crystallization from the melt of a synthesized PA6 sample.

Table 2

Parameters of Nakamura-Hoffman-Lauritzen model identified by optimization on curves of crystallization from the melt.

Parameter

Value Unit

K 0 × 7.59 10 8 [s -n ] K g × 6.63 10 5 [K 2 ] N 1.59 [-] A t × - 5.17 10 2 [s -1 ] E t 4.45 [kJ mol -1 ]
Fig. 8. Simulation of the degree of crystallinity β through the Nakamura-Hoffman-Lauritzen model including the induction time t c,0 over the temperature range of PA6 isothermal synthesis.

where H is the Heaviside step function.

∫ = -- × -- -- γ t w T α t w T H t θ t T α θ β t θ t T θ ( ) ( ) ( ) (1 ( )) ( () ) () ( () ) d
p p t c θ c 0 ,0 d d ,0 (18)
with:

∫ ∫ -- = - - -- -- - β t θ t T K T τ nK T K T τ ( () ) exp( ( ( ) d ) ) ( )( ( ) d )
θ c t θ t T N n N t θ t T N n d d , 0 0 ( ) 0 ( ) 1 c c ,0 ,0
The identification of the four parameters of the Malkin-Camargo polymerization model (eq. ( 12)) was performed on the γ sigmoid obtained for the isothermal syntheses between 403 and 473 K (every 10 K). Fig. 10 details the kinetic curves simulated at 413 and 443 K compared to experimental data from DSC and Fig. B in Supplementary data gathers the synthesis curves simulated from T iso = 403-473 K. Unlike the Malkin coupling, the crystallization is predicted for all temperatures. At low temperature, the crystallization is limited by the polymerization and then adopts its kinetics, since each new increment crystallizes almost instantaneously. This is consistent with the interpretation of the kinetic results obtained in Ref. [START_REF] Vicard | Experimental study of polymerization and crystallization kinetics of polyamide 6 obtained by anionic ring opening polymerization of ε-caprolactam[END_REF]. At 463 K, the coupling slightly shifts the crystallization compared to the Malkin coupling, which improves the prediction (see Fig. B).

However, since the crystallization starts at the beginning of the polymerization, the model does not reproduce the crystallization peaks in the early stages of the polymerization between 403 and 413 K and the shoulders observed between 423 and 443 K. As previously mentioned, these peaks are due to the germination that cannot start in the first moments of the polymerization. Thus at 413 K (Fig. 10a), the germination seems to start from about 10 min and at 443 K (Fig. 10b) after 4 min. The induction time of the crystallization t c,0 only translates the influence of the temperature on the crystallization initiation and does not consider the effect of molar masses or the concentrations reached.

If the use of the Hillier coupling with the Malkin-Camargo and Nakamura-Hoffman-Lauritzen crystallization models improves the simulation of PA6 synthesis by correcting the absence of crystallization at low temperature predicted with the Malkin coupling, all the physical phenomena are not taken into account. The introduction of a corrective factor in the crystallization term B was therefore proposed in order to improve the prediction of synthesis kinetics.

Corrective factor

An empirical corrective factor was added to the coupling equation to correct the particular crystallization kinetics observed during early Fig. 9. Simulation of PA6 synthesis kinetics γ˙, polymerization kinetics α ˙and crystallization kinetics B ˙with Malkin coupling (eq. ( 16)) versus DSC data. Fig. 10. Simulation of PA6 synthesis kinetics γ˙, polymerization kinetics α ˙and crystallization kinetics B ˙with Hillier coupling (eq. ( 18)) versus DSC data.

stages of polymerization at the low temperature, i.e. between 403 and 443 K. It was inspired by the diffusion factor introduced in thermosetting curing models to take into account the drastic decrease of the crosslinking kinetics due to vitrification. The corrective factor f d in equation [START_REF] Cimini | A mechanistic kinetic model for the initiated anionic polymerization of ε-caprolactam[END_REF] depends on the degree of conversion α and the temperature T. f d displays a sigmoid shape defined between 0 and 1 and parametrized by C and D [-], which govern the half-height slope and the shift of f d sigmoid on α axis respectively.

= - + -+ f α T C T α D T ( , ) 1 1 1 exp( ( )( 1) ( )) d ( 19
)
adding the corrective factor f d into the coupling equation [START_REF] Rigo | Kinetic study of anionic polymerization of 6-caprolactam by differential calorimetry[END_REF] gives equation [START_REF] Wittmer | Über die anionische schnellpolymerisation von caprolactam[END_REF].

∫ = -- × -- -- γ t w T α t w T H t θ t T f α θ T α θ β t θ t T θ ( ) ( ) ( ) (1 ( )) ( () ) ( () , ) () ( () ) d p p t c d θ c 0 ,0 d d ,0 (20)
An overall optimization was performed on sigmoidal γ curves of isothermal syntheses in order to refine the four parameters of the Malkin-Camargo polymerization model (eq. ( 12)) and to identify parameters C and D for each T iso since their temperature dependence is not known a priori. For their identification, constraints were imposed on these parameters: D was set higher than 6.9 in order to get f T (0.999, ) Fig. 12 presents the evolution of the corrective factor f d as a func- tion of the degree of conversion α for each temperature. The higher the synthesis temperature, the higher the degree of conversion α for which the sigmoid f d proceeded. This is consistent with the fact that the higher the temperature, the higher the critical nucleus size to reach and therefore the higher the molar mass or the concentration. At higher temperature (T iso between 453 and 463 K) f d has little effect since the induction time of crystallization t c,0 is enough to take into account the late crystallization observed. It is consistent with the experimental observations presented previously (Fig. 1): the crystallization is decoupled from the polymerization and its kinetics is similar to that taking place from the molten state. At 473 K crystallization does not occur, meaning that f d has no effect at this temperature. Fig. 13 shows the evolution of the parameters C and D as a function of the synthesis temperature. No proper model was found to correctly represent their temperature dependence due to the high sensitivity to parameters C and D. The values necessary for the simulation of the synthesis at other isothermal temperatures or under non-isothermal conditions were thus interpolated from the parameters C and D identified at the different T iso .

Table 3 summarizes the polymerization parameters obtained by a global optimization with the Hillier coupling and the corrective factor f d . The activation energy of the reaction E a reaches more than 90 kJ mol -1 , whereas the values reported in the literature with the same catalyst (MgBrCL) are between 53 and 84 kJ mol -1 [START_REF] Davé | Polyamides from lactams via anionic ring-opening polymerization: 2[END_REF][START_REF] Teuwen | Thermoplastic Composite Wind Turbine Blades -Kinetics and Processability[END_REF][START_REF] Merna | Polymerization of lactams: 97. Anionic polymerization of ε-caprolactam activated by esters[END_REF]. However, it should be noted that the results of Davé et al. [START_REF] Davé | Polyamides from lactams via anionic ring-opening polymerization: 2[END_REF] and Teuwen [START_REF] Teuwen | Thermoplastic Composite Wind Turbine Blades -Kinetics and Processability[END_REF] were obtained by modeling the temperature rise of the medium in an adiabatic reactor, which limits the measurement sensitivity for an accurate characterization of the temperature dependence of phenomena contrary to the DSC characterization used in this study. Moreover, results of Merna et al. [START_REF] Merna | Polymerization of lactams: 97. Anionic polymerization of ε-caprolactam activated by esters[END_REF] correspond to copolymerization with the ε-caprolactone monomer (forming polycaprolactone), which may explain the different activation energies E a obtained. In the lit- erature, most of the modeling studies have focused on the sodium caprolactamate (NaCL) catalyst, which has a reaction activation energy E a comprised between 62.8 and 104.3 kJ mol -1 [START_REF] Lee | Reaction-induced crystallization kinetics during the anionic polymerization of ε-caprolactam[END_REF][START_REF] Ya Malkin | The nonisothermal anionic polymerization of caprolactam[END_REF][START_REF] Ya Malkin | Kinetics of anionic polymerization of lactams. (Solution of non-isothermal kinetic problems by the inverse method)[END_REF][START_REF] Ya Malkin | The exothermal effects of superimposed processes of activated anionic polymerization of ε-caprolactam and crystallization of the polymer formed[END_REF][START_REF] Camargo | Bulk polymerization kinetics by the adiabatic reactor method[END_REF][START_REF] Kim | Mechanism and kinetics of adiabatic anionic polymerization of ε-caprolactam in the presence of various activators[END_REF][START_REF] Kim | Kinetics of adiabatic anionic copolymerization of ε-caprolactam in the presence of various activators[END_REF][START_REF] Sibal | Designing nylon-6 polymerization systems for RIM[END_REF] for adiabatic synthesis. The reaction order n p identified is in good agreement with literature for the MgBrCL catalyst [START_REF] Davé | Polyamides from lactams via anionic ring-opening polymerization: 2[END_REF][START_REF] Teuwen | Thermoplastic Composite Wind Turbine Blades -Kinetics and Processability[END_REF], which reports values between 0.93 Fig. 11. Simulation of PA6 synthesis kinetics γ˙, polymerization kinetics α ˙and crystallization kinetics B ˙with Hillier coupling including the corrective factor f d (eq. ( 21)) versus DSC data. Fig. 12. Evolution of the corrective factor f d with the degree of conversion α for the different synthesis temperatures.T iso and 1.2. The autocatalytic factor B 0 is greater than the range of 7-21 reported by literature [START_REF] Davé | Polyamides from lactams via anionic ring-opening polymerization: 2[END_REF][START_REF] Teuwen | Thermoplastic Composite Wind Turbine Blades -Kinetics and Processability[END_REF]. Finally, the parameter A 0 is hardly comparable because its value depends directly on the identified activation energy E a (Arrhenius law).

This empirical corrective factor reveals current limits of the coupling equation proposed in this study. The influence of the polymerization on crystallization at the early stages of synthesis could be achieved by introducing polymerization-dependent parameters into the crystallization model, instead of adding a corrective factor dependent on the degree of conversion α. The crystallization kinetic parameters would be either temperature dependent and functions of the molecular weight, as already evoked in the literature [START_REF] Okui | Temperature and Molecular Weight Dependencies of Polymer Crystallization[END_REF][START_REF] Crist | Polymer spherulites: a critical review[END_REF], or the chain concentration within the mixture. However, this would increase the level of complexity of the identification of parameters as it can no longer be separately identified on crystallization from the melt of a synthesized PA6.

In addition, the influence of crystallization on the polymerization should be considered in a further improvement of the synthesis kinetic model as it affects the degree of conversion ∞ X p for T iso < 443 K [START_REF] Vicard | Experimental study of polymerization and crystallization kinetics of polyamide 6 obtained by anionic ring opening polymerization of ε-caprolactam[END_REF].

Time-Temperature-Transformation diagram

Isothermal TTT diagram of the PA6 synthesis was developed due to the new model proposed in this study and is shown on Fig. 14. This diagram represents iso-polymerization curves (X p from 1 to 94 wt% in green) and iso-crystallinity curves (X c from 1 to 43 wt% in blue). This diagram must be horizontally read at a constant temperature. Readers should keep in mind that the studied experimental range is between 403 and 473 K, thus simulations outside of these boundaries have not been characterized.

The strong coupling between polymerization and crystallization appears clearly on Fig. 14 for low synthesis temperature (383-433 K), since the crystallization is restrained by the polymerization progress. At higher temperature, phenomena progressively separate due to the polymerization acceleration and crystallization deceleration, until a limit temperature of 468 K, above which the newly formed chains do not crystallize.

Conclusion

The Hillier's formalism demonstrated its ability in predicting the overall synthesis of polyamide 6 from its monomer ε-caprolactam. It accurately describes the influence of polymerization on the crystallization kinetics when synthesis is achieved at high temperature, i.e. when the coupling level of both phenomena is limited. This coupling equation resolves the lack of physical meaning of previous models from the literature as the level of coupling does not affect the final crystallization enthalpy.

Although the sole coupling fails in predicting the synthesis kinetics when crystallization is faster than polymerization at low temperature, it allows consideration that a PA6 molecule crystallizes as in bulk case with its own kinetics from the moment it is polymerized, but it does not consider the case of short molecules crystallizing in a dilute medium of reactive mixture.

This can be solved by introducing a corrective factor into the coupling without modifying the bulk crystallization model. This empirical factor, which depends on the degree of conversion α and temperature, provides a convenient and simple solution for that purpose. A better understanding of the crystallization of low molecular weight PA6 in the reactive mixture is however required to obtain a predictive modeling of PA6 synthesis that is suitable for non-isothermal modeling as well. An alternative to that corrective factor could be a crystallization model involving parameters that directly depend on the degree of conversion α.

Fig. 3 .

 3 Fig.3. Superposition of isothermal crystallization curves obtained from the melt with the crystallization peak recorded during synthesis at the same temperature.

Fig. 4 .

 4 Fig. 4. Temperature dependence of the crystallization enthalpy

d = 1

 1 and -C D > 6.9 to get f T (0.001, ) d = 0. Fig.11details the kinetic curves simulated at T iso = 413 and 443 K compared to experimental data from DSC. Results of all synthesis temperatures are presented in Figs. C to F in Supplementary data. The results clearly show the contribution of the corrective factor f d , as the model reproduces the particular crystallization kinetic behavior observed at the beginning of polymerization by delaying its initiation.

Fig. 13 .

 13 Fig. 13. Parameters C and D of the corrective factor f d versus synthesis temperature.T iso

Fig. 14 .

 14 Fig. 14. Simulated isothermal TTT diagram of PA6 synthesis.

Table 3

 3 Parameters of Malkin-Camargo polymerization model identified by optimization on synthesis curves with Hillier coupling, Nakamura-Hoffman-Lauritzent c,0 crystallization model and corrective factor. f d

	Parameter	Value	Unit
	A 0 E a B 0 n p	1.86 10 7 × 91.2 47.4 1.14	[s -1 ] [kJ mol -1 ] [-] [-]

Appendix A. Supplementary data

Supplementary data to this article can be found online at https:// doi.org/10.1016/j.polymer.2019.121681.