Alisson Magnan

Léo Georges

Alexandre Corbin

Sandrine Rossi

Christiane Prebet

Sylvie Ghiringhelli

René Rossin

Claudine Barbe

Henri Sabbadin

Frédéric Le Guyader

Laetitia Tribolet

Pauline Jessica Aurélien

Enfin

Celeste Obrigada

Keywords: Rheology, polyamide, glass transition, dynamic fragility, dielectric spectroscopy, physical aging

Je voudrais remercier tout d'abord Florence Clément et Didier Long, directrice et directeur adjoint du Laboratoire des Polymères et Matériaux Avancés pour m'avoir donné l'opportunité de réaliser ce projet de thèse au sein du laboratoire, mais également pour les nombreux et riches échanges tout au long de ces trois années.

Mes plus profonds remerciements vont à mon directeur de thèse, Paul Sotta et à mon encadrant

v

Résumé en français

Les polyamides sont des thermoplastiques techniques qui présentent de bonnes propriétés mécaniques et barrières. Le comportement viscoélastique complet des polyamides est rarement décrit dans la littérature à cause de la présence d'une phase cristalline et à l'instabilité thermique de ces polymères.

Dans une première partie, le comportement rhéologique de polyamides amorphes PA 6I avec différentes masses molaires a été étudié. L'augmentation de la masse molaire se traduit par l'apparition d'un plateau caoutchoutique et le décalage du temps de relaxation terminal vers les plus basses fréquences, en accord avec les modèles de Rouse et de reptation.

Des interactions ioniques ont été ajoutées au PA 6I en copolymérisant les mêmes monomères avec différents acides isophtaliques substitués, avec des taux de groupements ioniques de 5 à 20 mol%, ce qui entraîne une augmentation de la température de transition vitreuse de 10 à 40°C. Les courbes maîtresses des PA 6I et des copolyamides substitués non enchevêtrés se superposent dans toute la gamme de fréquences en utilisant une température de référence appropriée, proche mais non strictement identique à Tg. Les groupements ioniques ont un effet sur la fragilité dynamique, c'est-à-dire sur la variation en température de la réponse rhéologique près de Tg. Le modèle de Rouse décrit correctement la réponse rhéologique des polyamides non enchevêtrés, montrant que les interactions ioniques n'ont pas d'effet sur la viscoélasticité de ces matériaux. Les polyamides enchevêtrés suivent le modèle de la reptation. La masse entre enchevêtrements augmente avec l'ajout de groupements ioniques dû à une augmentation de la rigidité de la chaîne. La diffusion des rayons-X aux petits angles montre qu'il n'y a pas de ségrégation de domaines ioniques dans les copolyamides non enchevêtrés et une faible ségrégation dans les copolyamides enchevêtrés.

L'ajout de liaisons hydrogènes plus fortes se traduit par l'augmentation de la masse entre enchevêtrements et de la Tg sans aucune modification significative dans l'allure des courbes maîtresses qui restent bien décrites par le modèle de Rouse.

L'effet des groupements ioniques sur la dynamique à l'état solide (au-dessous de Tg) a été étudié par spectroscopie diélectrique. Les groupements ioniques n'ont pas d'effet sur les relaxations secondaires des polyamides, tandis que la relaxation est décalée comme la Tg mesurée par DSC. Aucune différence n'a été observée entre les copolyamides ioniques et le PA 6I lors de tests de vieillissement physique, démontrant que les groupements ioniques n'introduisent pas d'hétérogénéités dynamiques.

Introduction

This PhD thesis was carried out in the Laboratoire Polymères et Matériaux Avancés UMR5268, a joint research unit between CNRS and Solvay. This work was done in the framework of a Solvay project which deals with the effects of adding strong interactions in polymers, with the aim of developing new supramolecular materials. In this work, the fundamental understanding of the dynamics of differently modified polyamides in both solid and molten states was performed.

Polyamides belong to the family of thermoplastic polymers widely used in textile, automotive and food packaging industries, due to their excellent mechanical and barrier properties [START_REF] Kohan | Nylon Plastics Handbook[END_REF]. Most known polyamides are aliphatic and semi-crystalline such as PA 6, PA 66 and PA 6,10. However, other structures have been developed in the last years for specific applications, such as semi-aromatic or aromatic polyamides, as well as amorphous polyamides [START_REF] Kohan | Nylon Plastics Handbook[END_REF].

The demanded solid state properties of high performance polymers are often difficult to optimize and these materials are frequently difficult to process due to high viscosities and high processing temperatures, often close to degradation. The aim of this PhD project is to develop a new class of polyamides with reversible interactions and to understand the mechanisms that control molten and solid state properties.

The objective is to combine the good mechanical properties of typical covalent polymers with those of low molecular weight molecules, i.e. the mildness of the processing conditions.

Strong intermolecular interactions, such as hydrogen bonds, ionic interactions or metal complexation [START_REF] Stadler | Thermoplastic elastomers by hydrogen bonding 1. Rheological properties of modified polybutadiene[END_REF]- [START_REF] Goldansaz | Controlling the melt rheology of linear entangled metallosupramolecular polymers[END_REF], have a strong impact on the rheology of non-polar or low Tg polymers, as it has been demonstrated by numerous studies and modeled by so-called 'sticky' Rouse or reptation models. This was demonstrated in model cases in which strongly interacting groups are diluted and provide a time scale which is well separated from main chain relaxation time. The case of polyamides is more complex. In systems with a high density of hydrogen bonds, cooperativity of bonds and overlap of time scales raise fundamental issues which have not been described yet.

Furthermore, the complete picture of the viscoelastic behavior of polyamides has not been extensively reported, due to the lack of stability (post condensation and hydrolysis reactions may take place during experiments) and to crystallization that limits the range of temperatures at which rheological measurements can be performed [START_REF] Pezzin | Capillary Viscometry of Molten Polycaprolactam[END_REF], [START_REF] Van Ruymbeke | Structure and rheology of branched polyamide 6 polymers from their reaction recipe[END_REF]. This study has been conducted very extensively and full rheological master curves were established, using end-blocked and amorphous modified polyamides.

The aim of this investigation is to understand the role of the strength and density of interactions in polyamides. A new class of polyamides was designed for this study, in order to avoid limitations inherent to polyamide's rheological study as already mentioned. Since rheological properties depend on the chemical structure of the polymer, special attention was given to the introduction of the interactions on the polyamide backbone. Different interacting groups were used, with different interaction strengths, while keeping the polymer backbone constant. First, it will be important to understand the rheological behavior of amorphous polyamides with 2 H-bonds per monomer and to understand if theoretical models for unentangled and entangled non-entropic polymers are still valid in the case of polyamides. The effect of strong intermolecular interactions in the rheology of polyamides will then be studied, introducing different interacting groups (such as H-bonds and ionic groups), with the aim of increasing the strength of the interactions between polymer chains as well as their density. The effect of these interactions will be looked into in both molten and solid state dynamics, using rheological and dielectric spectroscopy measurements.

The first chapter of this manuscript presents a background review on the influence of strong interactions, followed by the state of the art on polyamide chemistry and structure. The second chapter deals with the presentation of the studied polyamides as well as the description of the experimental techniques used that allowed the study of polyamide mechanical, structural and thermal properties.

The third chapter consists on the complete study of the molten state dynamics of polyamides without further introduction of interactions. The effect of molecular weight will be studied as well as fitted using known models. Then, the effect of different strength and density of interaction on the viscoelasticity of polyamides will be presented in fourth and fifth chapters. Finally, the sixth chapter deals with the dynamics in the solid state of polyamides with different interacting groups. A comparison between two different techniques, dielectric spectroscopy and dynamic mechanical analysis, will also be presented.

Chapter 1 -State of the art

Polyamides are generally semi-crystalline materials with excellent barrier and mechanical properties, widely used in both textile and automobile industries, though their rheological study has so far not been developed in details. This chapter consists in a review of polymer dynamics in both molten and solid states, followed by a review of polyamide properties and structure to better emphasize the interest of studying this specific class of polymers.

Polymer dynamics in the molten state

Rheology can be defined as the science of the flow of liquids and the deformation of solids (Eugene C. Bingham and Markus Reiner, in 1920) [START_REF] Barnes | An Introduction to Rheology[END_REF], [START_REF] Macosko | Rheology: principles, measurements, and applications[END_REF]. One of its main goals is to develop quantitative relationships called constitutive relations, between force and deformation in materials, primarily liquids. This section starts with an introduction to rheology and its constitutive equations, followed by the rheology of polymer melts in the linear viscoelastic regime, with special focus on the influence of molecular weight, polydispersity, chemical structure, etc. The specific case of strong interactions will be discussed in the last section.

Solids, liquids and viscoelastic materials

One of the most common rheological tests is to submit a thin polymer sample to small amplitude sinusoidal shear oscillations, as in equation 1.1 (where γ0 is the strain amplitude, ω is the angular frequency and t is the time) and measure its stress response [START_REF] Malkin | Rheology fundamentals[END_REF].

Small amplitude sinusoidal shear oscillation is required in order to stay in the linear regime as shown in Figure 1. In the linear region, the stress is sinusoidal, proportional to the strain or strain rate, and the dynamic moduli is independent of the applied strain amplitude at a fixed frequency. However, in the nonlinear region, the dynamic moduli become a function of the strain amplitude and the resulting stress waveforms may be distorted from sinusoidal waves.

An ideal elastic solid behaves according to Hooke's law in the linear regime [START_REF] Barnes | An Introduction to Rheology[END_REF]. The stress measured is proportional to the applied strain, as represented in equation 1.2, where is the shear strain and G is the shear modulus. Thus, for an oscillatory shear, the stress response is in phase with the deformation (δ = 0°). This model is commonly represented by a spring as in Figure 2-a, i.e. an element in which the force is proportional to the extension.

On the other side, an ideal viscous liquid behaves according to Newton's law [START_REF] Barnes | An Introduction to Rheology[END_REF], as presented in equation 1. [START_REF] Chen | Ionomer dynamics and the sticky Rouse model[END_REF], where is the viscosity and is the shear rate. In this case, the stress is out of phase with the strain (δ = 90° and ). Newton's model is commonly represented by a dashpot as in Figure 2-b, i.e. an element where the force is proportional to the rate of extension. A material that exhibits combined elastic and viscous behavior is called viscoelastic. Several models have been developed in order to fully understand the viscoelastic response of materials. The most known model in the representation of viscoelastic materials is the Maxwell model. In this case, viscoelastic materials are represented as a combination of both elastic (spring) and viscous (dashpot) elements in series, as schematized in Figure 2-c, which are capable of both storage and dissipation of energy, respectively [START_REF] Barnes | An Introduction to Rheology[END_REF], [START_REF] Macosko | Rheology: principles, measurements, and applications[END_REF], [11]. The stress response of a viscoelastic material to an oscillatory shear is still sinusoidal but shifted with a phase angle, δ, between 0 and 90°. The stress response can be separated into two components, the elastic "in-phase" contribution and the viscous "out of phase" contribution. G' represents the elastic storage modulus which is in phase with the strain and G'' the viscous loss moduli which is in phase with the shear rate (and out of phase with the strain). In equation 1.4, represents the complex modulus, which can also be written as , resulting in equation 1.5. The ratio of the storage and loss modulus corresponds to the tangent of the phase angle, also called the loss trangent (equation 1.6). Both G' and G'' can be calculated from the Maxwell model and their equations are given by equation 1.7 and 1.8. The relaxation time, , is given by the ratio and determines the liquid/solid behavior transition. For slow motions (when compared to the relaxation time) the Newtonian behavior dominates (when 0, G'' , liquid-like behavior), while at short times, the elastic behavior dominates (when , G' G0, solid-like behavior).

The viscoelastic moduli, G' and G'', as a function of the angular frequency (in a log-log scale) are presented in Figure 3. The crossover point between G' and G'' corresponds . The relaxation time, , is a useful measure of the viscoelastic response time of the material. At times longer than the relaxation time, G' and G'' vary with a slope of 2 and 1, respectively. Alternatively, the response of a viscoelastic material can be analyzed as a function of time, as shown in Figure 4. Figure 3 illustrates the behavior of an ideal system with a single relaxation time. Polymer melts are viscoelastic materials, however, it is not possible to describe their rheological behavior using a single relaxation time (Figure 5-a) [START_REF] Rubinstein | Polymer physics[END_REF]. In reality, polymer melts have a set of relaxation times, that is often described as a continuous spectrum of relaxation times, H(t), which if often described using the generalized Maxwell model, composed of many Maxwell elements in parallel as observed relaxation times, as schematized in Figure 5-b.

Polymer rheology is strongly dependent on molecular weight, polydispersity, chemical structure, architecture and temperature. In the next sections, we will further discuss the impact of each parameter [11], [START_REF] Rubinstein | Polymer physics[END_REF], [START_REF] Bird | [END_REF], [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF].

Polymer melts

A polymer chain consists in an array of atoms joined by chemical bonds and it is often represented as a bead-spring chain. The most simple chain representation is the freely-jointed chain, in which each segment can orient freely. In the case where each configuration is equally probable, the mean squared end-to-end distance follows a Gaussian distribution. Polymer molecules exhibit Brownian motion, being able to diffuse through other chains and adopt multiple conformations due to their length and flexibility.

When a deformation is applied to a polymer melt, the macroscopic deformation is just the reflection of the microscopic deformations in the chain configurations of the material. Viscoelastic behavior arises from the different intra or intermolecular motions that depend on temperature and frequency (or time). For example, in a relaxation experiment at constant temperature, for observation times shorter than molecular motions, polymer chains are effectively frozen and molecular motion does not happen, thus the polymer behaves like a solid; while, for long observation times, these motions will start and the polymer will eventually relax, as schematized in Figure 4.

Influence of the molecular weight

The molecular weight is one of the main structural parameters which affect polymer flow behavior. The general behavior of the viscoelastic response of a polymer with increasing molecular weight is illustrated by the example shown in Figure 6.

Conversely, it was observed that the molecular weight has a strong influence on the zero shear viscosity (i.e. viscosity at the limit of low shear rate). Focusing on linear monodisperse polymers, at a given temperature, the zero shear viscosity, 0, varies linearly with molecular weight below the critical molecular weight, Mc, while above it, the viscosity varies with M 3,4. . as represented in Figure 7 [11], [START_REF] Fetters | Packing Length Influence in Linear Polymer Melts on the Entanglement, Critical, and Reptation Molecular Weights[END_REF].

The critical molecular weight is directly linked with the molecular weight between entanglements and the packing length, a characteristic parameter that controls many of the fundamental physical properties of polymers. The critical molecular weight can be described as the minimum molecular weight required for a temporary network structure to exist due to the topological constraints created by surrounding chains. The molecular weight between entanglements, Me, is defined as the average molecular weight between topological constraints and calculated from the value of the plateau modulus, which appears for sufficiently high molecular weight samples, as observed in Figure 6. The appearance of the plateau modulus in the dynamics moduli of polymer melts will be discussed further in this manuscript. The packing length will be discussed in details in section 1.1.2.4 [START_REF] Fetters | Packing Length Influence in Linear Polymer Melts on the Entanglement, Critical, and Reptation Molecular Weights[END_REF]- [START_REF] Fetters | Chain dimensions and entanglement spacings[END_REF].

The dynamic moduli, G' and G'', are also strongly affected by the molecular weight of the polymer. As an example, master curves of the dynamic elastic modulus of a linear monodisperse poly(cyclohexylethylene) (PCHE) obtained by time-temperature superposition are presented in Figure 6.

A qualitative change of the dynamic moduli with increasing molecular weight is observed: a plateau appears in G', when molecular weight is above the molecular weight between entanglements. Note that the molecular weight between entanglements of PCHE is equal to 40200 g•mol -1 [START_REF] Zhao | Thermal and Viscoelastic Behavior of Hydrogenated Polystyrene[END_REF]. At high frequencies, the large scale motions of the chain are frozen and the only mechanism for relaxation is the stretching and bending of bonds. Polymer is in its glassy state and behaves like a solid, with a modulus around 1 GPa. As a basic rule, two distinct regions in the G', G'' curves can be observed as the frequency decreases. The region in which G varies in between roughly 10 9 down to 10 6 Pa corresponds to the so-called glassy regime. The region in which G becomes smaller than roughly 10 6 Pa up to terminal relaxation corresponds to the rubber-like regime.

Several models have been developed to understand the evolution of the dynamic moduli with increasing polymer molecular weight, such as the Rouse model for unentangled polymers and the reptation model for entangled polymers. These two models will be described in details in the next two sections.

Theoretical models: Rouse Model

As it was mentioned before, the molecular weight is an important factor in polymer rheology, especially when compared to the molecular weight between entanglements or to the critical molecular weight, Me and Mc, respectively. A polymer with a molecular weight below the critical molecular weight is called unentangled and its dynamics in the melt state are described by the Rouse model, developed for short ideal chains [START_REF] Rouse | A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers[END_REF]. On the other hand, the dynamics of entangled polymers is given by the "tube" model (also known as reptation model [START_REF] De Gennes | Entangled polymers[END_REF] or M. Doi and S. Edwards model [START_REF] Doi | The Theory of Polymer Dynamics[END_REF]).

As in the generalized Maxwell model mentioned before, the linear viscoelastic response of a polymer measures a complete spectrum of relaxation times and dynamic moduli cannot be described using a single relaxation time. Rouse introduced the concept of the relaxation modes, since smaller sections of a polymer chain with m monomers can relax as a whole chain with m monomers.

The polymer chain in the Rouse model [START_REF] Rouse | A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers[END_REF] is represented as N beads connected by N-1 springs of root mean-square size l0, as shown in Figure 8. Each bead and spring element represent a small portion of the chain made of a few elementary segment of monomers. There is no interaction between the beads, beside that through the connecting springs. Each bead is characterized by its own independent friction, with friction coefficient . The total Rouse chain friction coefficient, , is given by the sum of the contributions of each of the N beads.

In the Rouse model, each polymer chain with N beads has N relaxation modes (p=N) and each relaxation mode p has a characteristic relaxation time, , as given by equation 1.9 where M corresponds to the molecular weight of the polymer, and correspond to the molecular weight and length of an elementary sub-chain, respectively, R is the ideal gas constant and T is the temperature. The Rouse relaxation time, , is given for p=1 and it corresponds to the longest relaxation time of the chain, i.e. the time over which the polymer diffuses a distance about its own size.

Dynamic moduli are given by the sum of all relaxation modes and are calculated using the relaxation times given by equation 1.9, as shown in equations 1.10 and 1.11, where is the polymer density. corresponds to the number of chains per volume unit.

In the terminal regime, for frequencies smaller than the longest relaxation time (1/ < ), equations 1.10 and 1.11 predict a slope of 2 and 1, for G' and G'' respectively, on a log-log plot. At higher frequencies, both equations predict a slope of 1/2. This behavior was experimentally observed for several entropic (non-polar) polymers, as exemplified in Figure 9. The Rouse model describes the viscoelastic behavior of unentangled polymers well above their glass transition. However, this model can be complemented by a high frequency term which takes into account more local motions of the Rouse sub-chains. As in fact, the glassy modulus ( Pa) has a strong impact on the transition zone and terminal region, as pointed out by Majesté et al. [START_REF] Majesté | Viscoelasticity of low molecular weight polymers and the transition to the entangled regime[END_REF].

Majesté et al. studied the viscoelastic behavior of unentangled polystyrene samples which was correctly fitted using the Rouse model and introducing a high frequency contribution, an empirical Davidson and Cole expression [START_REF] Gray | Dynamic Viscoelastic Behaviour of Low-Molecular-Mass Polystyrene Melts[END_REF].

In Figure 10, it is shown that simple Rouse model underestimates the dynamic moduli in the transition region (dashed line). By adding a glassy contribution, the modified Rouse model, represented by the solid lines, correctly describe the viscoelastic data of polystyrene samples.

Theoretical models: Reptation model

Entangled polymers, which are polymers with a molecular weight above the critical molecular weight, are materials in which the entanglements play a role on the viscoelastic behavior. In reality, a clear plateau appears over a certain range of frequency between glassy and terminal region, causing the slow relaxations to shift progressively to longer times while the fast relaxations remain unaffected. The typical dynamic moduli of an entangled polymer are shown in Figure 11.

At high frequencies (short times), the polymer is in its glassy state and behaves like a solid, with a modulus around 1 GPa. The relaxation time associated with the transition from the glassy state to the transition region is called the Kuhn monomer relaxation and it is the shortest relaxation time, given for p=N and it is given by equation 1.12 [START_REF] Rubinstein | Polymer physics[END_REF]. At times longer than the Kuhn relaxation time, local motions come into play and there is a transition zone during which modulus drops about 1 to 3 decades. At , as in Figure 11, polymer chains cannot completely relax since their motion is hindered by surrounding chains (entanglements act as a temporary network), which prevents further relaxation by the normal Rouse process and leads to a plateau. The value of the modulus in the plateau region is given by equation 1. [START_REF]16th European School on Rheology[END_REF]. Focusing on one single chain (see Figure 12-a), surrounding chains create complex topological constraints and create a tube around this one chain [START_REF] Rubinstein | Polymer physics[END_REF], [START_REF] Doi | The Theory of Polymer Dynamics[END_REF], [START_REF] Van Ruymbeke | Quantitative tube model predictions for the linear viscoelasticity of linear polymers[END_REF], [START_REF] Larson | Definitions of entanglement spacing and time constants in the tube model[END_REF]. Chain relaxation will happen when the chain steps out of the tube where it was confined, by chain motions which are parallel to the tube. This movement along the tube is called reptation. At times longer than the reptation time, , polymer chains can escape constraints created by surrounding chains and relax. When this happens, G' and G'' have the same slopes of 2 and 1, respectively, as expected by the Rouse model. The reptation model predicts that the relaxation time of the chain (τ) is proportional to the cube of the molecular weight ( ). The actual observed relationship is , which makes clear that additional relaxation mechanisms are required. In reality, in addition to the pure reptation of the chain, there are also contour length fluctuations (Figure 12-b) [START_REF] Doi | The Theory of Polymer Dynamics[END_REF], [START_REF] Van Ruymbeke | Quantitative tube model predictions for the linear viscoelasticity of linear polymers[END_REF] and constraint release relaxations (Figure 12-c) [START_REF] Van Ruymbeke | Quantitative tube model predictions for the linear viscoelasticity of linear polymers[END_REF]. Contour length fluctuations correspond to the movement of the chain ends that are not contained in the tube, which can be modeled using a Rouse-like relaxation of the chain ends, not requiring the relaxation of the center of mass of the polymer chain. The "tube" represents the topological constraint on a given chain and it is itself surrounded by other chains. According to the motion of the surrounding chains, the tube itself is able to move allowing lateral motions of internal segments of the chain and to an increase of the effective tube diameter, accelerating relaxation.

The length of the rubbery plateau, as defined by the ratio between and , depends on the molecular weight of the polymer and on its molecular weight between entanglements, as given by equation 1.14, obtained using the M. Doi and S. Edwards model [START_REF] Doi | The Theory of Polymer Dynamics[END_REF], [START_REF] Larson | Definitions of entanglement spacing and time constants in the tube model[END_REF]. The exponent can vary between 3 and 3.4 as some polymers follow a strict Reptation behavior, while in others constrain release and contour length fluctuations play an important role in polymer relaxation.

Parameters controlling rheological behavior

In 1969, Flory wrote "Comprehension of the conformational statistics of chain molecules is indispensable for a rational interpretation and understanding of their properties" [START_REF] Flory | Statistical mechanics of chain molecules[END_REF]. This recognition led to developments in polymer physics in attempt to correlate viscoelastic properties, such as the plateau modulus, , with the chain dimensions [START_REF] Graessley | The Entanglement Concept In Polymer Rheology[END_REF]. Rheological properties and other physical properties of polymers depend on various length scales related to the overall size of the polymer or to the local structure at the scale of the monomer.

The Kuhn segment is related to the local rigidity of the chain backbone, as schematized in Figure 13. The length of the Kuhn segment (Kuhn length, lk) is defined as the ratio of the actual mean square end-to-end distance,

, to the fully extended size, Rmax, as given in equation 1.15.

The characteristic ratio, , is defined as the ratio of to the mean square end-to-end distance, , if the chain was infinitely flexible and given by equation 1.16. where l0 is the length of one backbone unit bond and n0 is the number of unit bonds along the polymer chain. Thus the chain dimensions may be written as in equation 1.17, taking into account the bond angles as in a polyethylene chain.

To better understand the relationship between chain dimensions and its properties, Witten et al. [START_REF] Witten | Theory of stress distribution in block copolymer microdomains[END_REF] defined a new quantity called the packing length, p. The packing length is related to the ratio of the overall volume explored by a chain, Vsp, to the own volume occupied by all monomers of a chain, Vch, as given in equation 1. [START_REF] Fetters | Chain dimensions and entanglement spacings[END_REF]. It is closely related to the local "bulkiness" of the chain monomer and it is defined in this way, in order to be independent of the chain length.

One chain occupies the volume, Vsp, defined as the volume of a sphere with radius , as given in equation 1. [START_REF] Zhao | Thermal and Viscoelastic Behavior of Hydrogenated Polystyrene[END_REF]. The gyration radius, , is equal to 1/6 times the mean square end-to-end distance, for Gaussian chains. The number of chains, Nsp, of molecular weight M that would completely fill a volume Vsp, is defined in equation 1. [START_REF] Rouse | A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers[END_REF], where corresponds to the volume of one chain.

Nsp may be considered as the number of chains with which a given chain is entangled. Making the hypothesis that the molecular weight between entanglements, Me, is defined when Nsp = 2, as given in equation 1. The packing length reflects polymer architecture and is found to be a characteristic parameter that controls many of the fundamental physical properties of polymer, such as the molecular weight between entanglements, plateau modulus, etc [START_REF] Fetters | Packing Length Influence in Linear Polymer Melts on the Entanglement, Critical, and Reptation Molecular Weights[END_REF]- [START_REF] Fetters | Chain dimensions and entanglement spacings[END_REF].

Experimentally, when the polymer molecular weight is greater that the critical molecular weight, Mc, a temporary network structure exists due to the topological constraints (entanglements) created by surrounding chains and this creates an elastic response. The plateau modulus can be used to determine Me, the molecular weight between entanglements points, as given in equation 1.13.

Several approaches to calculate the value of the plateau modulus have been proposed. One of them is to take as the value of G' corresponding to the minimum of the loss trangent or the minimum of G'' [START_REF] Liu | Evaluation of different methods for the determination of the plateau modulus and the entanglement molecular weight[END_REF], [START_REF] Wu | Chain entanglement in homopolymers, copolymers and terpolymers of methyl methacrylate, styrene and N-phenylmaleimide[END_REF], as in Figure 14-a. Another approach is to calculate as the value of the complex modulus, G*, corresponding to the minimum of the phase angle, , in the Van Gurp-Palmen plot [START_REF] Trinkle | Van Gurp-Palmen-plot: a way to characterize polydispersity of linear polymers[END_REF], as shown in Figure 14-b. On the other hand, the critical molecular weight, Mc, can be calculated experimentally as the point in the melt viscosity as a function of the molecular weight plot where the exponents change from 1 to 3.4. The ratio Mc/Me was long taken to be between 2 or 3 [START_REF] Aharoni | On entanglements of flexible and rodlike polymers[END_REF]. However, a recent empirical compilation has shown that this ratio is dependent on the packing length of the polymer. For example, polyethylene has a packing length of 1.69 Å and Mc/Me=3.5, while poly(cyclohexylethylene) has a packing length of 5.59 and a ratio of molecular weight of 1.4. Based upon experimental studies including tens of polymers, it was proposed that the ratio Mc/Me could be written as follows:

where Mc=Me at p=p* 10 Å. At least four polymers exist with a packing length close to 10 Å. Nylon 6, for example, has a packing length of 1.98 Å and a molecular weight between entanglements of 2470 g•mol -1 [START_REF] Fetters | Chain dimensions and entanglement spacings[END_REF].

Time-temperature superposition and the WLF equation

The relaxation times depend on the ability of the material to reorganize. For solids, large scale molecular reorganization is very slow, while for liquids the relaxation times are fast. If molecular motion cannot happen due to the lack of mobility of the chains, relaxation time will increase. In polymer melts, it is usual to perform experiments at different temperatures in order to achieve longer or smaller relaxation times. For example, when temperature increases, polymer chains start to move faster and when a deformation is imposed, polymer can relax during the time of the experimental test.

Polymer relaxation moduli often vary over many decades in frequency, from the glassy state to the terminal region (see for example Figure 11). However, a rheometer can usually provide dynamic information over a relatively small range of frequency (generally 3 decades), which is insufficient to obtain the complete range of polymer relaxation. To obtain the complete picture of polymer relaxation, several experiments are done at different temperatures, usually between Tg+5 and Tg+100°C and the time-temperature superposition principle is applied to build a master curve. This process can be applied only to thermorheological simple materials [START_REF] Plazek | The Temperature Dependence of the Viscoelastic Behavior of Poly(vinyl acetate)[END_REF]- [START_REF] Ngai | Identification of different modes of molecular motion in polymers that cause thermorheological complexity[END_REF] (see section 1.1.2.6).

It relies on the very strong hypothesis that all relaxation processes in the polymer are scaled with a single elementary time which has a definite variation with temperature.

The time-temperature superposition is a result of a vertical and a horizontal shift, as it can be observed in Figure 15. The vertical shift considers differences in the density and modulus between each temperature. The vertical shift factors, , are given by equation 1.26 for each temperature and are often close to 1. To apply this shift, the temperature dependency of the density, , needs to be known. Though for some polymers the density does not vary significantly with temperature and in these cases it can be neglected. The factor Tref/T is used to compensate the variation of the pre-factor in equation 1.10 and 1.11.

The horizontal shift factors, , are determined using an error minimization method. Their temperature dependency is complex and can be described using the Williams-Landel-Ferry (WLF) equation as given in equation 1.27 [START_REF] Williams | The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-forming Liquids[END_REF]. The shift factors are often applied to the tan curves, as this variable is independent of the vertical shift factors and measurement conditions, but it can also be applied to the modulus or viscosity curves.

where C1 and C2 are the WLF constants and Tref is the arbitrary chosen reference temperature. The values of the two WLF constants can be reported to Tg using equation 1.28 and 1.29 [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF]. The constants and were initially considered universal with the values 17.4 and 51.6 K, respectively [START_REF] Williams | The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-forming Liquids[END_REF]- [START_REF] Angell | Formation of glasses from liquids and biopolymers[END_REF]. Later, it was shown that they vary from one polymer to another. Note that = . Since the only characteristic temperature in amorphous polymers is the glass transition temperature, a solution to compare different polymers is to choose Tg as the reference temperature. The possibility of using Tg as reference temperature for each master curve allows comparing various polymers at isofrictional conditions. This is supported by the fact that several authors state that "the difference in the glass transition temperature for X samples varied by only some degrees °C, so using a common reference temperature for each introduces only a small error" [START_REF] Osterwinter | Rheological Consequences of Hydrogen Bonding: Linear Viscoelastic Response of Linear Polyglycerol and Its Permethylated Analogues as a General Model for Hydroxyl-Functional Polymers[END_REF], [START_REF] Huang | Rheological Behavior of Partially Neutralized Oligomeric Sulfonated Polystyrene Ionomers[END_REF]. This might be acceptable for small glass transition variations, such as in cases demonstrated in the literature [START_REF] Chen | Ionomer dynamics and the sticky Rouse model[END_REF], [START_REF] Chen | Linear Viscoelastic and Dielectric Properties of Phosphonium Siloxane Ionomers[END_REF]- [START_REF] Staropoli | Melt dynamics of supramolecular comb polymers: Viscoelastic and dielectric response[END_REF]. The comparison of master curves of different polymers with different glass transition temperatures should be done at the Tg of each polymer. If master curves of different polymers are compared at Tg, thus master curves may be compared at iso-frictional conditions.

Thermorheological complexity

Time-temperature superposition sometimes cannot be used to generate a complete response of the viscoelastic behavior of polyamides. In reality, it is often difficult to describe the transition from the glassy state to the molten state using one single temperature dependence, causing time-temperature superposition to fail. This happens because there are two different time dependencies, subsequently it is believed that in this transition zone both segmental α-relaxation responsible for glass transition and the generalized sub-Rouse modes for undiluted polymers are present, which have different temperature dependencies.

The thermorheological complexity was observed for several amorphous polymers, starting in 1965 by Plazek et al. using polystyrene, followed by several other authors [START_REF] Plazek | The Temperature Dependence of the Viscoelastic Behavior of Poly(vinyl acetate)[END_REF]- [START_REF] Ngai | Identification of different modes of molecular motion in polymers that cause thermorheological complexity[END_REF], using polyvinyl acetate, atactic polypropylene or poly(methylmethacrylate), while polyisobutylene seems to be thermorheologically simple. An example of the thermorheological complexity effect is shown in Figure 16. In conclusion, thermorheological complexity was so far observed in several polymers and it is not particular to polyamides or interacting polymers. Using the WLF parameters, and , it is possible to calculate a parameter that characterizes the sensitivity of molecular dynamics to temperature, the so-called dynamic fragility index or fragility of chain relaxation, m (not to be confused with fragility in the mechanical meaning, as opposed to toughness). Furthermore, it characterizes the degree of deviation of the temperature dependence from Arrhenius behavior. This is defined as the derivative at Tg of the log of the shift factor η(T)/η(Tg) vs T/ Tg (the so-called Angell plot) as given by equation 1.30. There are two general types of glasses defined by the fragility parameter: so-called fragile systems (m > 50) and so-called strong ones [START_REF] Angell | Relaxation in liquids, polymers and plastic crystals-strong/fragile patterns and problems[END_REF], which have a nearly Arrhenius-like behavior (m < 30), as it can be observed in Figure 18.

Non-polymeric liquids were extensively studied by many authors [START_REF] Sokolov | Why many polymers are so fragile[END_REF]- [START_REF] Scopigno | Is the Fragility of a Liquid Embedded in the Properties of Its Glass?[END_REF]. The fragility of almost all small molecule systems is lower than 100, with few exceptions. However, many polymers are quite fragile, with m > 150. For small molecules, several approaches have been proposed to correlate fragility with thermodynamical [START_REF] Ito | Thermodynamic determination of fragility in liquids and a fragile-to-strong liquid transition in water[END_REF], [START_REF] Martinez | A thermodynamic connection to the fragility of glassforming liquids[END_REF], vibrational [START_REF] Scopigno | Is the Fragility of a Liquid Embedded in the Properties of Its Glass?[END_REF] and mechanical properties [START_REF] Novikov | Poisson's ratio and the fragility of glass-forming liquids[END_REF]. For example, for small molecules, it is possible to relate the dynamic fragility to the heat capacity jump during the glass transition, as in Figure 18. However, the inverse trend is often observed in polymers [START_REF] Sokolov | Why many polymers are so fragile[END_REF], [START_REF] Huang | Dynamic fragility in polymers: A comparison in isobaric and isochoric conditions[END_REF].

The effect of molecular weight and chain rigidity on the polymer dynamic fragility was studied [START_REF] Sokolov | Why many polymers are so fragile[END_REF], [START_REF] Kunal | Role of Chemical Structure in Fragility of Polymers: A Qualitative Picture[END_REF], [START_REF] Roland | Temperature dependence of local segmental motion in polystyrene and its variation with molecular weight[END_REF]. Polymers such as poly(dimethyl siloxane), poly(propylene oxide) or poly(isoprene) show no significant dependence of fragility on molecular weight (about 10% variation for an increase in molecular weight from 1000 to 20000 g•mol -1 ). Conversely, the fragility of polymers like polystyrene or poly(methyl methacrytalate) show a strong dependence on molecular weight. For example, polystyrene fragility almost doubles between 500 and 100 000 g•mol -1 . Several correlations have been reported, to understand the variation of dynamic fragility in polymers [START_REF] Novikov | Poisson's ratio and the fragility of glass-forming liquids[END_REF], [START_REF] Cangialosi | A thermodynamic approach to the fragility of glass-forming polymers[END_REF]. Non-polymeric liquids as well as polymer whose fragility does not depend on the molecular weight follow the correlations discussed above, while the other group of polymers (PS, PMMA, PC, etc) deviates from this behavior at high molecular weights.

Sokolov et al. [START_REF] Kunal | Role of Chemical Structure in Fragility of Polymers: A Qualitative Picture[END_REF] studied the relationship between the dynamic fragility and the chemical structure of the polymer. One could consider that Tg and fragility are related, since both depend on molecular weight and chemical structure (rigidity of the chain and of side groups). In reality, it is the flexibility of the polymer backbone and side groups that play a decisive role in polymer fragility, since it depends on the chain packing ability. Dudowicz et al. [START_REF] Dudowicz | Fragility of Glass-Forming Polymer Liquids[END_REF], [START_REF] Dudowicz | The Glass Transition Temperature of Polymer Melts[END_REF] explained that fragility is related to the packing ability of each polymer. Rigid chains and/or bulky side groups induce difficulties in packing and leave excess free volume, resulting in higher fragility values. For example, the packing lengths of poly(isoprene) (PI) and polystyrene (PS) are 2.95 Å and 3.99 Å [START_REF] Fetters | Chain dimensions and entanglement spacings[END_REF], respectively. The packing length is significantly higher for polystyrene samples, which explains the strong increase in fragility, not observed in the case of poly(isoprene). Sokolov et al. [START_REF] Kunal | Role of Chemical Structure in Fragility of Polymers: A Qualitative Picture[END_REF] also proposed that the fragility of a polymer does not depend simply on the stiffness of the backbone or that of the side groups, but in the contrast between the two. For example, both PI and polydimethylsiloxane (PDMS) have flexible backbones and identical flexible side groups, but quite different Tg's (~ -130°C for PDMS and ~ -70°C for PI) and quite different fragilities (80-100 for PDMS and ~ 46 for PI). The Si-O backbone is more flexible (with torsional energy barriers: E = 2 kJ•mol -1 ) than a C-C backbone (E = 18-25 kJ•mol -1 ), nevertheless side groups also play a role in the dynamic fragility. Comparing the energy barrier for conformational changes of the side groups, PI side groups motions have similar activation energy to that of the chain, meaning that motions of both backbone and side groups will happen at similar temperatures, resulting in a very efficient packing. For PDMS, backbone and side groups have different activation energies, meaning that the side groups act as rigid groups in comparison to the polymer backbone, resulting in poor packing and higher dynamic fragility.

Sokolov et al. [START_REF] Agapov | Effect of Polar Interactions on Polymer Dynamics[END_REF] also studied the effect of the introduction of polar interactions on the polymer backbone in terms of dynamic fragility. They observed that segmental dynamics slow down (Tg increases) and have steeper temperature dependence (higher fragility index) when a polar group is attached directly to the polymer backbone, as in polypropylene or poly(isoprene). On the other hand, when a polar atom is separated from the polymer backbone by a benzene ring, both the glass transition temperatures and fragility index vary without a systematic trend, depending on the position of the polar atom. For example, for the samples of poly(4-chlorostyrene) (P4ClS) and poly(4-methylstyrene) (P4MS) (Figure 17) they observed a decrease in the fragility index, while the glass transition temperature increased with the introduction of polar groups. However, when the lateral group was in the orto-or meta-positions, they observed a decrease in the glass transition temperature. Unfortunately, the fragility values for the polymers in which the lateral group was in the orto-or meta-positions are not available in the literature. In the case of the orto-and meta-substituted polymers presented in that work [START_REF] Agapov | Effect of Polar Interactions on Polymer Dynamics[END_REF], (P2ClS-P2MS and P3ClS-P3MS -Figure 17), the larger volume of the -CH3 side group creates a steric hindrance which makes the chain more rigid, contrary to the -Cl substituent, which may possibly explain the decrease in the glass temperature with the introduction of polar interactions. In poly(4-chlorostyrene), the chlorine atom is far from the polymer backbone and may have a negligible contribution to intramolecular rotational energy barriers. Thus, for P4ClS and P4MS the main difference in segmental relaxation must be coming from intermolecular interactions, resulting in a decrease of the fragility index. The same effect was observed by Osterwinter et al. [START_REF] Osterwinter | Rheological Consequences of Hydrogen Bonding: Linear Viscoelastic Response of Linear Polyglycerol and Its Permethylated Analogues as a General Model for Hydroxyl-Functional Polymers[END_REF] who studied the influence of hydrogen bonds on the fragility of polyglycerol and its methylated analogue. They observed a decrease in fragility with increasing intermolecular interactions (from linPG-OMe to linPG-OH). The same behavior is observed in the case of molecular glass forming samples with the introduction of polar groups, as observed in the Angell plot, shown in Figure 18, glass formers show different behavior when compared to glycerol. For example, o-terphenyl shows higher fragility than glycerol, which may be due to the interactions present in the case of glycerol. Thus, the same effects could then be present in polyamide materials.

In conclusion, the dynamic fragility value depends strongly on molecular weight and polymer structure (relative flexibility of backbone and side groups), which influences the packing length, as well as the type of interactions between the polymer backbone.

Influence of polydispersity and branching

Models used so far can be applied to monodisperse and linear polymers. However, industrial polymers are rarely monodisperse or linear, due to secondary reactions, branching reactions, etc. For instance, polyamides have a polydispersity of about 2 and may have branched chains leading to higher polydispersity (as it will be explained in section 1.2, see Figure 28). Moreover, several studies have been done using highly branched polymers, in order to better understand this topic.

Figure 19 shows the differences observed in the dynamic moduli when the molecular weight distribution is broader. For polydispersed samples, the decrease of the storage modulus during the relaxation time is slower and non-exponential. In fact, the broadening of the molecular weight distribution causes a broadening in the spectrum of relaxation times. The rubbery plateau has the same value but often less clear, since when approaching lower frequencies, smaller chains have already relaxed [START_REF] Trinkle | Van Gurp-Palmen-plot: a way to characterize polydispersity of linear polymers[END_REF], [START_REF] Wasserman | Prediction of linear viscoelastic response for entangled polyolefin melts from molecular weight distribution[END_REF], [START_REF] Cross | Polymer rheology: Influence of molecular weight and polydispersity[END_REF]. For unentangled polymers, it is possible to take into account the effect of polydispersity in the prediction of the dynamic moduli. For the Rouse model, to consider polydispersity, a distribution of molecular weights is implemented in equations 1.11 and 1.12 as given in equations 1.31 and 1.32.

where is the weight fraction of chains of average molecular weight (obtained experimentally from GPC) and is the relaxation time of the p mode of the polymer with molecular weight . In entangled polymers melts, the development of a model to take into account polydispersity is more difficult and led to the development of more complex models, for example the double reptation model [START_REF] Van Ruymbeke | Prediction of linear viscoelastic properties for polydisperse mixtures of entangled star and linear polymers: Modified tube-based model and comparison with experimental results[END_REF].

The effect of branching on rheological properties cannot be easily generalized, since it depends on the type of branching: stars [START_REF] Snijkers | Effects of Core Microstructure on Structure and Dynamics of Star Polymer Melts: From Polymeric to Colloidal Response[END_REF], combs, short or long-side chains [START_REF] Wood-Adams | Thermorheological Behavior of Polyethylene: Effects of Microstructure and Long Chain Branching[END_REF], [START_REF] Majesté | From linear viscoelasticity to the architecture of highly branched polyethylene[END_REF], hyperbranched structures [START_REF] Schubert | Can Hyperbranched Polymers Entangle? Effect of Hydrogen Bonding on Entanglement Transition and Thermorheological Properties of Hyperbranched Polyglycerol Melts[END_REF]. Generally, the terminal regime is delayed to lower frequencies and several inflection points may appear in the plateau region due to the existence of several distinct relaxation mechanisms/times. When branching is not controlled, it can also affect the molecular weight distribution. In this case, it is quite difficult to separate the influences of branching and polydispersity.

Effect of intermolecular interactions

The rheological behavior of purely entropic (i.e. non polar) polymers, such as polybutadiene or polystyrene, is well described by classical models like Rouse or reptation models [START_REF] Rouse | A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers[END_REF]- [START_REF] Doi | The Theory of Polymer Dynamics[END_REF], described in details in sections 1.1.2.2 and 1.1.2.3. Conversely, it is not fully clear how intermolecular interactions (e.g. ionic associations or hydrogen bonds) influence the rheological properties of polymer chains. This aspect has been studied in the literature for the past decades using mainly known polymers in which polar groups are introduced [START_REF] Van Ruymbeke | Preface: Special Issue on Associating Polymers[END_REF]. In this section, some of the more remarkable works on this subject will be discussed in details. In the first part, the introduction of hydrogen bonding groups will be presented followed by a review on the effect of the introduction of ionic groups on polymer dynamics. Other types of interactions may be used, such as metal-ligand complexation or π-π interactions. However, as it is not the main subject of this PhD, these will not be mentioned.

The introduction of these interactions results in supramolecular polymers. Unlike conventional bonded polymers, supramolecular polymers present a variety of non-covalent interactions that define their properties. Due to the presence of these reversible noncovalent interactions, supramolecular materials may exhibit different dynamic properties, such as self-healing [START_REF] Li | A highly stretchable autonomous self-healing elastomer[END_REF].

Hydrogen bonding

Although hydrogen bonds are not amongst the strongest non-covalent interactions, they are widely used in supramolecular chemistry due to their directionality and versatility [START_REF] Brunsveld | Supramolecular Polymers[END_REF]. The strength of a single hydrogen bond depends on the nature of the donor and acceptor. It increases if multiple hydrogen bonding happens simultaneously. First studies on this subject used systems containing monodisperse linear polymers in which a small amount of lateral groups were introduced, the so-called stickers capable of interacting between each other. The strong interactions between stickers create a thermoreversible network which modifies the rheological behavior of polymers.

A important work in this domain was initially done by Stadler's group [START_REF] Stadler | Thermoplastic elastomers by hydrogen bonding 1. Rheological properties of modified polybutadiene[END_REF], [START_REF] Stadler | Thermoplastic Elastomers by Hydrogen Bonding 2. IR-Spectroscopic Characterization of the Hydrogen Bonding[END_REF]- [START_REF] Müller | Influence of hydrogen bonding on the viscoelastic properties of thermoreversible networks: analysis of the local complex dynamics[END_REF]. The authors studied polybutadienes of narrow molecular weight distribution modified by the addition of small amounts of 4phenyl-1,2,4-triazoline-3,5-dione (urazole) groups (Figure 20-a). These highly polar groups can interact by two hydrogen bonds, building a thermoreversible network. The first effect which was noticed was the increase in the glass transition temperature (Tg) upon the insertion of the stickers. Rheological measurements confirmed the existence of such a hydrogen bonding network. The value of the elastic plateau was increased with increasing concentration of stickers and the longest relaxation time was shifted to lower frequencies (and higher temperatures) with increasing concentration of stickers as it can be seen in Figure 20-b andc. The slopes of G' (storage modulus) and G'' (loss modulus) in the terminal zone decreased with increasing concentration of stickers, which could be attributed to an increase in the effective molecular weight and an apparently broader molecular weight distribution. An additional relaxation was observed in the plateau region for the loss modulus, related to the association and dissociation dynamics of the stickers.

The sticker association/dissociation times were investigated by Müller et al. by dielectric spectroscopy [START_REF] Müller | Influence of hydrogen bonding on the viscoelastic properties of thermoreversible networks: analysis of the local complex dynamics[END_REF]. Figure 21 presents the relaxation map of polybutadiene/stickers systems. The characteristic relaxation time of the stickers was found to be at temperatures above the glass transition temperature (Tg < -90°C), namely, *-relaxation process was found to take place at temperatures above the -relaxation, normally associated with Tg. Both *-and -relaxation times were shifted to higher temperatures with increasing concentration of urazole groups.

Several other studies were performed using hydrogen bonding groups [START_REF] Osterwinter | Rheological Consequences of Hydrogen Bonding: Linear Viscoelastic Response of Linear Polyglycerol and Its Permethylated Analogues as a General Model for Hydroxyl-Functional Polymers[END_REF], [START_REF] Brunsveld | Supramolecular Polymers[END_REF], [START_REF] Lewis | The Influence of Hydrogen Bonding Side-Groups on Viscoelastic Behavior of Linear and Network Polymers[END_REF]- [START_REF] Callies | Linear Rheology of Supramolecular Polymers Center-Functionalized with Strong Stickers[END_REF]. Their conclusions, namely an increase in the elastic plateau and a shift of the relaxation times to lower frequencies and/or higher temperatures, confirmed the effects observed by Stadler et al. A thorough overview of these works is shown in Table 1. Different studies are compared in terms of: chemical structure of the polymer and the interacting groups, shift in Tg due to the introduction of H-bonding groups, whether polymers are entangled or unentangled, bond density (how many H-bonds per sticker) and the main results observed. As a general trend, time-temperature superposition works, the longest relaxation times are shifted to lower frequencies and the plateau modulus increases with increasing sticker content.

Shabbir et al. [START_REF] Shabbir | Effect of Hydrogen Bonding on Linear and Nonlinear Rheology of Entangled Polymer Melts[END_REF] observed an increase in the plateau modulus related to the increase in the content of interacting groups, in this case equal to the fraction of hydrolyzed polymer. They suggested that the increase in the plateau modulus is related to a decrease of the packing length induced by polymer hydrolysis. Thus, hydrogen bonding contributes only to the viscous terminal regime and the differences in the plateau modulus are mainly due to the differences in the packing length (and therefore in the molecular weight between entanglements). More recently, a new class of transient networks has been studied, with hydrogen bonds being replaced by ionic interactions. Ionomers are ion-containing polymers, typically up to 15% of ionized units covalently bonded to the polymer chain. Ionomers can be used in several applications, such as in membranes, in adhesives and in the packaging industry. Contrary to hydrogen bonding polymers, ionomers can have high glass transition temperatures and show a nanophase separation of ion-rich aggregates, typically with a characteristic scale of 2-4 nm [START_REF] Weiss | Viscoelastic behavior of lightly sulfonated polystyrene ionomers[END_REF]. At relatively low temperatures, ionic aggregates act as physical crosslinks, while at higher temperatures, the association/dissociation dynamics of ion pairs becomes fast, a phenomena called ionic hopping, which allows melt flow [START_REF] Weiss | Rheological behavior of oligomeric ionomers[END_REF], [START_REF] Kim | Introduction to Ionomers[END_REF].

The effect of ionomers on rheology was studied by Weiss et al. [START_REF] Weiss | Viscoelastic behavior of lightly sulfonated polystyrene ionomers[END_REF], [START_REF] Weiss | Rheological behavior of oligomeric ionomers[END_REF], [START_REF] Weiss | Viscoelastic Behavior of Very Lightly Sulfonated Polystyrene Ionomers[END_REF]. They studied the rheological behavior of alkali metal salts of oligomeric sulfonated polystyrene (PS-SO3X) ionomers with a narrow molecular weight distribution, using two sulfonation levels, 2.5 and 4.8 mol%, which corresponded to 1 and 2 sulfonate groups per chain on average. Ionomers exhibited nanophase separation of an ion-rich phase. The results of Small-Angle X-ray Scattering (SAXS) measurements exhibited a scattering peak around q=1-2 nm -1 as observed in Figure 23-a, which corresponds to a characteristic distance between ionic clusters of 4-6 nm. Rheological results showed once again a shift of the terminal relaxation times to lower frequencies and the appearance of a rubbery plateau, due to the constraints created by the ionic clusters that act as physical crosslinks. The magnitude of these results depend on the alkali metal used (Li, Na, K, Rb, Cs), as shown in Figure 23-b andc. Both PS and HSPS (acid derivative of sulfonated polystyrene) samples behaved as Rouse chains. This result was expected for PS oligomer samples, but not for HSPS oligomers, since hydrogen bonding is strong enough to increase Tg about 5°C. However, it seemed that hydrogen bonding was weakened at experimental temperature, therefore it has no influence in the dynamic moduli. At these same temperatures, ionomers displayed a plateau modulus, revealing a stronger interaction in the case of ionic clusters. In reality, polystyrene has low ion solvation abilities, leading to a high-energy for ionic dissociation and a long association lifetime.

The viscoelastic behavior of polysiloxanes with phosphonium fraction up to 26 mol% and oligo(ethyleneoxide) phosphonium random copolymer ionomers (Figure 24-a) was studied by Chen et al. [START_REF] Chen | Linear Viscoelastic and Dielectric Properties of Phosphonium Siloxane Ionomers[END_REF]. They also observed a delay in polymer relaxation, which increased with increasing ion content. Dynamic moduli of substituted polymers were described by the Sticky-Rouse model up to 11 mol% phosphonium fraction, while for higher fractions, one single broad relaxation was observed from glassy to terminal regions (Figure 24-b). This broad relaxation can be described by a single Kohlrausch-Williams-Watt (KWW) equation. Contrarily to Weiss' work shown in Figure 23, polysiloxane-based phosphonium ionomers did not present a plateau modulus and time-temperature superposition worked well, in a wide frequency range, suggesting ionic association is weak. This argument is supported by the lack of ionic aggregation peak in X-ray scattering.

The rheology of polyester ionomers with a high molar fraction, f, of ionic substituted monomers (up to 100%) was also studied by Chen et al. [START_REF] Chen | Ionomer dynamics and the sticky Rouse model[END_REF]. In this study, two different polyesters were used, poly(ethylene oxide) -PEO -and poly(tetramethylene oxide) -PTMO. Their chemical structures are presented in Figure 25-a and b. The dynamic moduli of PEO-f Na ionomer samples is presented in Figure 25-c. The glass transition of these systems increased from -45°C, for f=0, to -2°C, for f=100, which explains that master curves did not overlap in the glassy regime when compared at the same temperature. The introduction of ionic groups in the PEO samples does not lead to a clear plateau, contrarily to what was observed in Weiss' work (Figure 23). It may be speculated that the more polar chemical structure of PEO facilitates ion pair dissociation from ionic clusters, lowering the energy for dissociation.

In the case of PTMO, the dynamic moduli of the fully substituted polymer, PTMO650-100%Na, were well described by the Sticky-Rouse model, as shown in Figure 25-d. In contrast to the PEO-f Na ionomers shown in Figure 25-c, the PTMO650-100%Na sample exhibits a plateau covering almost 9 decades, while its neutral analogue presents a Rouse-like behavior, typical of unentangled polymers. In fact, PTMO has a lower dielectric constant (lower polarity than PEO), leading to an higher energy for the ion pair to move outside the ion clusters [START_REF] Wheatle | Effect of Polymer Polarity on Ion Transport: A Competition between Ion Aggregation and Polymer Segmental Dynamics[END_REF]. Chen et al. [START_REF] Huang | Reversible Gelation Model Predictions of the Linear Viscoelasticity of Oligomeric Sulfonated Polystyrene Ionomer Blends[END_REF]- [START_REF] Zhang | Association energy in strongly associative polymers[END_REF] studied the linear viscoelastic response of lightly sulfonated unentangled polystyrene as a function of the number of ionic groups per chain when compared to the gel point, which corresponds to an average of one cross-link per chain. The rheological response of polystyrene drastically changes in the vicinity of the gel point, as observed for the NaSPS-0.76 in Figure 26. Above the gel point, samples show typical reversible gel behavior characterized by a clear plateau and longest relaxation time shifted to lower frequencies as it is governed by the ionic association lifetime.

Several other studies were performed using ionic interactions [START_REF] Chen | Linear Viscoelastic and Dielectric Properties of Phosphonium Siloxane Ionomers[END_REF], [START_REF] Chen | Viscoelasticity of Reversible Gelation for Ionomers[END_REF], [START_REF] Kim | Filler and percolation behavior of ionic aggregates in styrene-sodium methacrylate ionomers[END_REF]- [START_REF] Song | Small-Angle X-ray Scattering Study on Poly(ethyl acrylate-coitaconate) Ionomers Neutralized with Various Cations[END_REF]. Their conclusions namely an increase in the elastic plateau and a shift of the relaxation times to lower frequencies and/or higher temperatures, confirmed the effects observed by Stadler et al. An overview of these works is done in Table 2. Different studies are compared in terms of: chemical structure of the polymer and the interacting groups, shift in Tg due to the introduction of ionic groups, whether polymers are entangled or unentangled, whether there is cluster formation or not, number of ionic groups per chain and the main results observed. tTS worked well for a great majority of ionomers, with the longest relaxation times being shifted to lower frequencies, the appearance of a rubbery plateau in the case of unentangled polymers, due to the creation of ionic aggregates that will act as topological constraint, inhibiting polymer relaxation as expected by the Rouse model.

In conclusion, it is important to fully understand the rheological behavior of polymers with interactions. Most literature studies have been done in low Tg polymers, up to 100°C, mainly non-polar, with a limited amount of H-bonding groups or ionomers. Several common observations have been made but some points are yet to be clarified: which is the role of Tg, what about entangled polymers, how does polymer polarity affect the rheological response of ionomers, which parameters are affected by the introduction of H-bond interacting groups or ionic groups.

Polyamides

Polyamides (or nylons, as named by DuPont) were first synthesized by W. Carothers, in 1938, by a condensation reaction (diamine and diacid), leading to a -CONH-as repeating unit [START_REF] Kohan | Nylon Plastics Handbook[END_REF], [START_REF] Guérin | Polyamides PA[END_REF]. Nylon was the first semi-crystalline plastic, thus when comparing to polystyrene or poly(methyl methacrylate), it meant a sharp transition from solid to melt state and a much higher service temperature. Polyamides belong to the family of the thermoplastic engineering plastics and are widely used due to their excellent mechanical and barrier properties. Most known polyamides are aliphatic semi-crystalline such as PA 6, PA 66 and PA 6,10 [START_REF] Deopura | Polyesters and polyamides[END_REF]. Other structures have been developed in the last years for specific applications, such as semi-aromatic or aromatic polyamides, as well as amorphous polyamides.

Polyamide synthesis

Polyamide are generally denoted as PA n, where n indicates the number of carbons in the amino acid or lactam, or PA n,m, where the first number, n , indicates the number of carbons in the diamine and the second, m, the number of carbons in the diacid. For example, nylon-6 is obtained from caprolactam (C6H11NO) and PA 6,6 is obtained from the reaction between hexa(methylene)diamine (C6H16N2) and adipic acid (C6H10O4). PA n,m polyamides are generally obtained via a polycondensation reaction of a diacid and a diamine (or their corresponding salts) as shown in Figure 27. During synthesis, an acid catalyst can be used in order to increase the monomer conversion and to hinder secondary reactions, such as branching reactions. The condensation reaction relies on the equilibrium between reactants and products, thus its equilibrium constant can be defined as the product of the concentration of the produced groups (NHCO and H2O) divided by the product of the concentration of the reactants, as express in Equation 1.33.

Water removal by vaporization is the key point in polyamide synthesis. The condensation reaction is promoted by the extraction of water vapor, which shifts the equilibrium of the reaction to the products side, resulting in an increase of the molecular weight. In the opposite way, if water is not removed, the hydrolysis reaction is favored and the molecular weight of the resulting polyamide decreases. Another important factor to control, in order to control the molecular weight of the polyamide, is the loss of diamine because of its relative volatility.

The polyamide manufacturing process consists of 4 main steps [START_REF] Helft | Internal report on polyamide and its synthesis[END_REF]: concentration, prepolycondensation, decompression and polycondensation. During the first step, the initial aqueous salt solution is concentrated, at low or medium pressure, in order to minimize diamine loss. At the end of the concentration step, temperature and pressure are quickly increased (up to 220°C and 18 bars) and the prepolycondensation begins by the formation of oligomeric chains. This permits to minimize diamine loss, since volatility of oligomeric chains is lower than that of the diamine. This step is stopped when the remaining water concentration is lower than 5%. In the decompression step, pressure is released to nearly atmospheric pressure and temperature is increased to 270-280°C (above melting temperature of semicrystalline polyamides, e.g. PA 6,6 has a melting temperature of about 265°C [START_REF] Guérin | Polyamides PA[END_REF]). Lastly, the polycondensation takes place at atmospheric or reduced pressure at a temperature between 270-280°C and the polymer molecular weight increases significantly as a result of low water content in the melt.

Based on statistical considerations, Flory developed a theory, which is valid only in the case of linear polycondensation [START_REF] Flory | Principles of Polymer Chemistry[END_REF]. This theory relies on the fact that the probability that a chain-end reacts to form an amide group is equal to the extent of reaction, p, for balanced polymer and the probability that a chainend has not reacted is 1-p. p is close to 1 for high molecular weights. Hence the probability that a chain is composed of x repeat units is given by and the number of chains, Nx, with x repeat units is given by equation 1.34.

N, the total number of chains of all sizes is given by equation 1. [START_REF] Aharoni | On entanglements of flexible and rodlike polymers[END_REF], where N0 represents the total number of units (monomers):

The weight fraction, wx, of chains of x repeat units is given by . Combining equations 1.34 and 1.35, the Flory distribution equation is obtained as in equation 1.36.

The expected polydispersity index (Ip) expressed as the ratio between the weight-average molecular weight and the number-average molecular weight, is then given by Equation 1.37, for high degree of polymerization.

As mentioned above, during polyamide synthesis, secondary reactions can occur, possibly leading to branched polymer chains which strongly affects the molecular weight distribution and rheological properties. Branching can occur, for example, when a reaction between two diamines or two amide chain ends happens, as presented in Figure 28. The bis(n-alkyl)triamine has three amine sites where polymerization can take place. The amount of branching can be estimated by analytical methods, such as gel permeation chromatography (GPC) or nuclear magnetic resonance (NMR) that will be presented in chapter 2.

Temperature stability

At high temperature, if the water concentration does not correspond to that in equilibrium of a certain molecular weight, the polyamide will continue to evolve, in terms of molecular weight and thus viscosity. It is therefore very important to control water content in polyamides prior to rheological measurements at high temperature: if the water concentration is too low, the molecular weight will increase by post-condensation, and the viscosity will increase during tests; on the other hand, if the water concentration is too high, the hydrolysis reaction is favored and the molecular weight will decrease [START_REF] Acierno | Rheological behavior of polyamide 11 with varying initial moisture content[END_REF].

To prevent this to happen, it is possible to block the chain ends by decreasing the quantity of available amine or acid functions, precluding post condensation reaction. Nevertheless, it is crucial to control the water content and the time during which samples stay in contact with ambient humidity in order to perform reproducible measurements.

Structure of Polyamides

Polyamides chemical structure consists in an aliphatic, semi-aromatic or aromatic backbone separated by amide groups -NHCO. These amide groups are formed by an amine (NH) that behaves as a hydrogen bond donor and a carbonyl that behaves as a hydrogen bond acceptor, capable of forming strong hydrogen bonds between polymer chains, as presented in Figure 29. The interaction strength is estimated at 25 kJ•mol -1 [START_REF] Jorgensen | Optimized intermolecular potential functions for amides and peptides. Structure and properties of liquid amides[END_REF], [START_REF] Klotz | Hydrogen Bonds between Model Peptide Groups in Solution[END_REF] (estimated from molecular simulations in vacuum). Hydrogen bonds are directional interactions, meaning that chains need to be parallel in order to form hydrogen bonds, which favors polyamide crystallization. Polyamides exhibit crystalline polymorphism, i.e. several crystalline structures can be obtained depending on the type of polyamide and on processing conditions. The various degrees of crystallinity of polyamides are the result of the ability of polymer chains to pack in a regular crystal lattice and of hydrogen bonds that act as links between polyamide chains [START_REF] Kohan | Nylon Plastics Handbook[END_REF]. Some monomers can inhibit crystallization in polyamides, by disrupting the organization of polymer chains into crystalline lattices. One possibility to do so is to introduce alkyl substituents along a linear aliphatic chain or more than one methyl group to aliphatic monomer backbone. Other monomers can be used to inhibit crystallization such as isophorodiamine [START_REF] Dolden | Structure-property relationships in amorphous polyamides[END_REF], whose structure inhibits crystallization or isophthalic acid [START_REF] Dolden | Structure-property relationships in amorphous polyamides[END_REF]- [START_REF] Bocahut | Modeling Molecular Relaxation Mechanisms in Amorphous Polymers: Application to Polyamides[END_REF], that introduces an inflection point into the polymer chain by the stereochemical configuration of the carbonyl groups next to the benzene ring, hindering regular chain packing and thus crystallization. The structure of PA 6I (hexamethylene diamine and isophthalic acid) presented in Figure 30, contrasts with that of PA 66 or PA 6T (teraphthalic acid), which have linear structures.

Viscoelasticity of polyamides

Although polyamides are industrial polymers, their rheology has not been extensively reported, due to serious issues which make its study difficult. The major difficulty in the characterization of the rheological behavior of polyamides is due to their hygroscopic nature and also to possible evolution of molecular weight if water concentration is different from that at equilibrium [START_REF] Dijkstra | Guidelines for rheological characterization of polyamide melts (IUPAC Technical Report)[END_REF]. Some authors [START_REF] Pezzin | Capillary Viscometry of Molten Polycaprolactam[END_REF], [START_REF] Acierno | Rheological behavior of polyamide 11 with varying initial moisture content[END_REF], [START_REF] Khanna | New developments in the melt rheology of nylons. I: Effect of moisture and molecular weight[END_REF] discuss this molecular weight evolution during rheological analysis, due to post-condensation or hydrolysis reaction, if water content is to low or too high, respectively. Pezzin et al. [START_REF] Pezzin | Capillary Viscometry of Molten Polycaprolactam[END_REF] observed that the melt viscosity increased with time for PA 6 samples with a low water content (<0.24%), due to post condensation, while it decreased for higher humidity levels (>0.31%), due to hydrolysis reaction (see Figure 27 for post condensation and hydrolysis reaction).

In 1996, Khanna et al. [START_REF] Khanna | New developments in the melt rheology of nylons. I: Effect of moisture and molecular weight[END_REF] studied the effect of moisture and molecular weight in the melt rheology of PA 6, PA 66 and PA 12. They observed that the zero shear viscosity decreased with the moisture content in the range of 0.02 to 0.2%, demonstrating the effect of water as a plasticizer. The activation energies for flow were between 40 and 70 kJ•mol -1 . A plasticizer is a substance that, when introduced and dispersed in a polymer, decreases its viscosity and glass transition temperature. In polyamides, H-bonding molecules, which decrease the interactions between chains, are generally used as plasticizers. Classical polyamide plasticizers are benzene sulfonamides [START_REF] De Groote | Effect of benzenesulfonamide plasticizers on the glass-transition temperature of semicrystalline polydodecamide[END_REF]- [START_REF] Groote | The effect of benzenesulfonamide plasticizers on the glass transition temperature of an amorphous aliphatic polyamide[END_REF] or phenolic resins [START_REF] Hartikainen | Structure and Morphology of Polyamide 66 and Oligomeric Phenolic Resin Blends: Molecular Modeling and Experimental Investigations[END_REF]. Water and alcohols acts as well as a plasticizer.

Acierno et al. [START_REF] Acierno | Rheological behavior of polyamide 11 with varying initial moisture content[END_REF] studied the effect of humidity content on the rheological behavior of PA 11. They observed a large increase in its viscosity (Figure 31) and attributed this evolution to the post condensation reaction. Indeed, the exponential increase of the viscosity could be fitted with second-order reaction kinetics.

Another major cause for the lack of data on polyamide rheology is that most polyamides are semicrystalline. The shear modulus of polymer increases during crystallization and is difficult to measure with common rheometers. Moreover, the measured response would be mainly due to the crystalline structure that has a mechanical effect dominant over polymer structure or interactions in the remaining molten fraction. Therefore, all literature studies cited in this section discuss the rheological behavior above melting temperature, which is justified when studying processing conditions. However, for a more fundamental study on the influence of molecular weight, interaction between chains, etc, the access to the complete master curve from the glassy state to the terminal region is needed. To the best of our knowledge, there is no study where the complete master curve of polyamides was presented.

Van Ruymbeke et al. [START_REF] Van Ruymbeke | Structure and rheology of branched polyamide 6 polymers from their reaction recipe[END_REF] studied the viscoelasticity of linear and branched polyamide 6. To prevent molecular weight evolution, chain ends were blocked using a mono-functional benzoic acid as a chain stopper. The viscoelastic response was fitted using the tube model, for both linear and branched polyamides, taking into account the architecture of each sample.

In conclusion, polyamides investigated in the literature are water-sensitive and semi-crystalline, impeding the full understanding of the rheological behavior of polyamides and the accomplishment of a complete master curve. It is interesting to study polyamide rheology due to the high density of hydrogen bonds (more than 1 H-bond per monomer), but so far there is no information on its rheological behavior, plateau modulus, etc.

Polymer dynamics in the solid state

In amorphous polymers, the main relaxation process is the glass transition, at which the polymer undergoes a change from the glassy (solid) state to a rubbery state. This process corresponds to the onset of collective large-scale motions, involving various monomers of the polymer backbone. Below the glass transition temperature, the large scale motion of the polymer backbone is essentially frozen. Therefore, most viscoelastic properties in the glassy state result from local molecular motions, corresponding to subglassy or secondary relaxations. Many polymeric materials present more than one structural relaxations when investigated over a wide temperature range [START_REF] Beaume | Secondary transitions of aryl-aliphatic polyamides. I. Broadband dielectric investigation[END_REF]- [START_REF] Taniguichi | Dielectric relaxation of thin films of polyamide random copolymers[END_REF]. In addition to the glass transition, other secondary transitions can appear in the glassy state, associated to local motions of molecular groups and segmental relaxations, that strongly affect mechanical properties of polymers, specifically the -relaxation [START_REF] Bocahut | Modeling Molecular Relaxation Mechanisms in Amorphous Polymers: Application to Polyamides[END_REF], [121].

The polymer relaxation times in the glassy state, including the glass transition, can be measured by Dynamic Mechanical Analysis (DMA) or by Broadband Dielectric Spectroscopy (BDS). The glass transition temperature can also be measured using Differential Scanning Calorimetry (DSC). The variations of the characteristic relaxation times (or frequencies) with temperature can be plotted in a relaxation map (log frequency vs. 1/T), as represented in Figure 33. Secondary and relaxations have an Arrhenius-like dependence on temperature, as in equation 1. [START_REF] Plazek | Viscoelastic properties of polymers. 4. Thermorheological complexity of the softening dispersion in polyisobutylene[END_REF], where 0 is the relaxation time at infinite temperature, EA is the activation energy, R is the ideal gas constant and T is the temperature. The temperature variation of the relaxation time of the -relaxation is generally fitted using a Vogel-Fulcher-Tammann (VFT) equation (equation 1.39) and it depends on the VFT constant, AVFT, and on the Vogel temperature, that varies in between 30-70K below the glass transition [START_REF] Kremer | Broadband Dielectric Spectroscopy[END_REF] . Equivalently, it can also be fitted using the WLF equation (equation 1.27).

Solid-state dynamics of polyamide

Three main relaxation processes have been identified in polyamides [START_REF] Beaume | Secondary transitions of aryl-aliphatic polyamides. I. Broadband dielectric investigation[END_REF], [START_REF] Beaume | Secondary transitions of aryl-aliphatic polyamides IV. Dynamic mechanical analysis[END_REF], [START_REF] Laredo | Water Absorption Effect on the Dynamic Properties of Nylon-6 by Dielectric Spectroscopy[END_REF]- [START_REF] Laurati | Dynamics of Water Absorbed in Polyamides[END_REF], as schematized in Figure 32. At the lowest temperature (and higher frequencies), the -process is generally attributed to local motions of short methylene segments, which are internal to the monomer. Therelaxation results from the rotation of the amide groups, inciting changes in conformation and hydrogen bond dissociation/association. The -relaxation is associated with large scale segmental motions, involving disruption of hydrogen bonds. It is normally associated with the glass transition. The resulting relaxation map of a PA 6 sample is shown in Figure 33-b. Both secondary relaxations were fitted using Arrhenius equations while the -relaxation was fitted using VFT equation.

In Figure 33, it is clear that the presence of water drastically change the relaxation map of polyamides. The -relaxation was modified since water acts as a plasticizer and decreases the segmental relaxation temperature. Also both secondary relaxation were affected as it can be observed in this figures. As mentioned before, polyamides are hygroscopic polymers [START_REF] Kohan | Nylon Plastics Handbook[END_REF], [START_REF] Acierno | Rheological behavior of polyamide 11 with varying initial moisture content[END_REF]. The presence of water in polyamide causes the disruption of inter-or intra-chain hydrogen bonds, leading to a decrease of the glass transition, a phenomenon called plasticization. While amide groups are considered hydrophilic absorption sites, the rest of the polymer is strongly hydrophobic. This results in an odd structural organization of water that can be aggregated in cavities as in hydrophobic materials. The presence of water strongly affects the relaxation map of polyamide, as studied by Laurati et al. [START_REF] Laurati | Dynamics of Water Absorbed in Polyamides[END_REF], as demonstrated in Figure 34. In the dry state (full symbols), two relaxation processes can be observed, corresponding to and relaxations, with activation energies of 37.6 and 63.7 kJ•mol -1 , respectively. In the wet state (open symbols), three relaxations can be distinguished, with intensities higher than those of the dry state. A relaxation process, named process A (EA = 56.0 kJ•mol -1 ) was observed in the same temperature range where -relaxation is observed, in the dry state. However, the overall intensity of this relaxation is higher than that of the -relaxation (~0.45 for wet state compared to 0.12 in the dry state) and it can be associated to water motions. The large dipole moment of water molecules could explain the larger strength of the process, hiding the real -relaxation of the polymer that cannot be observed in this case due to its comparatively lower intensity. The B process is attributed to the water molecules associated with amide groups by hydrogen bonding and has an activation energy around 55 kJ•mol -1 . Since water dynamics is faster when compared to amide groups, it is possible that the motions of amide-water complexes are faster than those of amide-amide groups. Finally, the process C has a temperature dependence and dielectric strength comparable with that of the -relaxation in the dry state and it is attributed to the amide groups that are not bonded to water molecules, even if showing a different activation energy (around 57 kJ•mol -1 compared to the activation energy of the -relaxation in the dry state which is 64 kJ•mol -1 ).

The glass transition temperature

Physical aging experiments

A glass is a disordered system for which the time scales (or relaxation times) of molecular motions are longer than the experimental time scale. Glasses are non-equilibrium solids, resulting from the fast cooling of a liquid that can evolve toward equilibrium in a process known as structural recovery, as schematized in Figure 35 [125]-[132].

In the liquid state, the enthalpy (or specific volume) varies nearly linearly with temperature, from which it is possible to calculate the heat capacity of the liquid, given by the derivative of the enthalpy curve. As the temperature approaches Tg (A B in Figure 35), molecular motions become slower and polymer chain do not have enough time to rearrange into the equilibrium state. At Tg, the slope of the volume or enthalpy curves as a function of temperature deviates from equilibrium, resulting in an apparent change of heat capacity, ΔCp. This results in the usual definition of Tg, which is the temperature at which there is a jump in the heat capacity. However, it is important to keep in mind that the glass transition temperature depends on the cooling rate. If the cooling rate is smaller, the material will stay in equilibrium down to a lower temperature (A D) thus the measured glass transition temperature will be lower. The polymer in its glassy state is not at equilibrium. If the glass is kept at a constant temperature Ta < Tg close to Tg, as time is given, the material will evolve toward equilibrium, resulting in a decrease in the specific volume or enthalpy (B C). This process, often referred to as aging or structural recovery, may potentially continue until the equilibrium state is established for a given aging temperature. It is important to emphasize that ageing can occur only at temperatures not too far below from Tg, such that the glass retain some mobility.

The fictive temperature is given by the crossover point between the liquid equilibrium and glassy state lines for a point in between the glassy and equilibrium lines, point D, as represented in Figure 35. For an unaged glass the fictive temperature is equal to the glass transition temperature. In the other way, for a material at equilibrium the fictive temperature is equal to the aging temperature.

For instance, in the study of blends of two polymers with very distinct glass transition temperatures, aging experiments can be of great use. This technique permits to characterize the presence of dynamical heterogeneities in the blend, i.e., zones where the local dynamics are heterogeneous due to different local compositions of the blend, leading to a broadening of the glass transition temperature. For example, in the polybutadiene/styrene-butadiene random copolymer blend studied by Shi et al. [START_REF] Shi | Glass Transition Distribution in Miscible Polymer Blends: From Calorimetry to Rheology[END_REF], [START_REF] Shi | PhD thesis: Linear and non linear mechanical properties of miscible polymer blends near glass transition[END_REF], both polymers have quite different glass transitions (-100°C for PB and -20°C for SBR). Figure 36 represents a polymer blend of these two polymers where it is possible to observe some regions rich in polymer A and some regions rich in polymer B. This results in different glass transition temperatures, and . Ionomers generally show nanophase separation, with a non-polar continuous phase and ionic nanodomains (or clusters), which may exhibit two different glass transitions, that of the polymer and that of the ionic clusters. Several studies focused on the possibility that the ionomer melt in between this two temperatures, and , is likely to be a non-equilibrium state, due to the restricted mobility of the polymer chains bounded to the ions within the clusters.

For instance, Weiss et al. [START_REF] Chun | The development of the ionic microphase in sulfonated poly(ethylene-co-propylene-co-ethylidene norbornene) ionomers during physical aging above Tg[END_REF] studied the evolution of the ionic microphase in zinc salt of sulfonated poly(ethylene-co-propylene-co-ethylidene norbene) ionomers during physical aging. They observed that phase separation was enhanced during physical aging above the polymer glass transition and below the glass transition of the ionic clusters. They confirmed this by both DMA and SAXS experiments, where a clear increase in the scattering peak during aging was observed, as shown in Figure 37.

Fractional Free Volume

The glass transition is often described as the point where the thermal properties of polymers undergo a discontinuity. For instance, there is a jump in both thermal expansion coefficient and heat capacity when going through the glass transition. This jump in thermal properties corresponds to a change in the slope of the specific volume or enthalpy against temperature, as shown in Figure 35 and Figure 38.

Many properties of liquids prove the presence of a substantial proportion of free volume, present as holes of the order of monomeric dimensions or smaller, associated to packing irregularities. The fractional free volume may be defined as an intrinsic property of the polymer matrix created by the voids left between polymer chains. In the last decades, several works have been established with the aim of better understanding the effect of the polymer backbone and packing efficiency in the free volume at Tg.

The original theory on the fractional free volume, fg, at the glass transition temperature dates from 1950 and was developed by Fox and Flory [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF], [START_REF] Fox | Influence of molecular weight and degree of crosslinking on the specific volume and glass temperature of polymers[END_REF]. In fact, they considered Tg as an "iso-free volume state", independent of molecular weight and polymer backbone. At high enough temperature, above the glass transition of amorphous polymers, a lowering of temperature is accompanied by a decrease of the free volume, as the molecular adjustments take place freely within a normal experimental time scale. With decreasing temperature, molecular adjustments are slower, until reaching a temperature at which a decrease in free volume does not happen at all within experimental time scales. The fractional free volume in the glassy state is presumed constant, as shown in Figure 38.

In free volume theories, it is assumed that the local relaxation time or equivalently the viscosity depends on the available free volume in the material. This relationship in generally described by the Doolittle equation as shown in equation 1.41, where A and B are empirical constants, v is the total volume of the system and vf is the free volume of the system [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF], [START_REF] Doolittle | Studies in Newtonian Flow. II. The Dependence of the Viscosity of Liquids on Free Space[END_REF], [START_REF] White | Polymer Free Volume and Its Connection to the Glass Transition[END_REF]. Note that the quantity represents the "close packing volume".

Neglecting variations in

, it is straightforward to deduce equation 1.41, which links Doolittle's equation to the WLF equation [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF], [START_REF] Doolittle | Studies in Newtonian Flow. II. The Dependence of the Viscosity of Liquids on Free Space[END_REF], [START_REF] White | Polymer Free Volume and Its Connection to the Glass Transition[END_REF]. Note that is the shift factors for .

where f = is the fraction of free volume, and fg is the fraction of free volume at Tg. If the free volume fraction increases linearly with temperature in accordance with the equation , where is the thermal expansion coefficient of the free volume, thus equation 1.41 becomes equation 1.42, identical in form with the WLF equation [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF], [START_REF] Doolittle | Studies in Newtonian Flow. II. The Dependence of the Viscosity of Liquids on Free Space[END_REF], [START_REF] White | Polymer Free Volume and Its Connection to the Glass Transition[END_REF].

The fractional free volume at Tg can thus be expressed in terms of the WLF parameters, as B is arbitrarily set at unity, as given in equation 1.43 [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF], [START_REF] Doolittle | Studies in Newtonian Flow. II. The Dependence of the Viscosity of Liquids on Free Space[END_REF], [START_REF] White | Polymer Free Volume and Its Connection to the Glass Transition[END_REF].

The WLF constants are calculated by fitting the experimental horizontal shift factors using the WLF equation (equation 1.27). Then, the value of is calculated by rescaling the value of C1 to Tg using equation 1.28.

If the value of would be considered universal and equal to 17.4, then, using equation 1.43, the fractional free volume would be equal to 0.025, which is considered a universal value as well. Some exceptions to this value were already observed. For the solutions of cellulose derivatives, for instance, the fractional free volume is slightly smaller.

Experimental results by Positron annihilation spectroscopy allowed better comprehension on the effect chemical structure which determines chain flexibility on the resulting fractional free volume at Tg. For instance, diene elastomers with double bonds in the chain backbone show a more effective packing resulting in low level of overall free volume. Conversely, vinyl elastomers are less flexible which leads to both larger voids and larger fractional free volume. Aromatic polymers are more flexible than the polydienes, however the presence of phenyl rings in the main chain prevents the structural units from being more effectively packed. Consequently, a higher level of fractional free volume is observed, around 0.035 instead of 0.025, for example for polycarbonate samples [START_REF] Bartoš | Free volume microstructure of amorphous polymers at glass transition temperatures from positron annihilation spectroscopy data[END_REF].

Approach of the current study

The targeted solid state properties of high performance polymers, such as strength, impact strength, barrier properties are often difficult to optimize. Their processability is frequently problematic due to high viscosities, high processing temperatures, often close to degradation. The aim of this PhD project is to study a new class of systems with reversible interactions and to understand the mechanisms that control molten and solid state properties. The reversible character of these interaction allows, at low temperature, that the polymer behave as a high molecular weight material, due to strong interactions between polymer chains, while at high temperatures, interactions dissociate and polymer processing is easier (lower processing temperatures, lower risk of polymer degradation, less energy consumed, etc). The objective is to combine the properties of typical covalent polymers or of polymeric network, in other words, the good mechanical properties of high molecular weight materials, with those of low molecular weight molecules, i.e. less severe processing conditions (high temperatures, etc).

Strong intermolecular interactions (such as hydrogen bonds, ionic interactions or metal complexation) have a strong impact on the rheology of non-polar, such as polystyrene or polybutadiene, or low Tg polymers, such as poly(ethyleneoxide), as it has been demonstrated by numerous studies (by Stadler et al. [START_REF] Stadler | Thermoplastic elastomers by hydrogen bonding 1. Rheological properties of modified polybutadiene[END_REF], Weiss et al. [START_REF] Weiss | Rheological behavior of oligomeric ionomers[END_REF], etc -see section 1.1.3 above), and modeled by so-called 'sticky' Rouse or reptation models. This was demonstrated in model cases in which strongly interacting groups are diluted and provide a new time scale which is well separated from main chain relaxation time. In this context, it is important to study the effect of intermolecular interactions on the viscoelastic behavior of polymers with a high density of intermolecular hydrogen bonds, such as polyamides. The case of polyamides is more complex, since in systems with a high density of hydrogen bonds, cooperativity of bonds and overlap of time scales raise fundamental issues which have not been described yet. Thus, it is important to clarify whether ionic groups can provide a new time scale distinct from chain relaxation or with a distinct temperature variation (due to different activation energy of ionic interactions). Furthermore, complete studies of polyamide rheology, by construction of master curves, are scarce in the literature, due to stability problems (post-condensation and hydrolysis) and the presence of a crystalline phase that limits the range of temperatures at which the dynamic moduli can be measured.

Therefore, the aim of this investigation is to understand the role of the strength and density of interactions in polyamides, by introducing interactions stronger than amide-amide H-bonds. A new class of polyamides was designed for this study, in order to avoid limitations inherent to polyamide's rheological study as discussed in Chapter 1. Polyamide stability was achieved by blocking chain ends using a monofunctional benzoic acid unit as a chain stopper. Limitations related to the presence of a crystalline phase can be avoided by using an amorphous polyamide. There has been important progress in synthesizing amorphous polyamide and the use of an isophthalic acid as co-monomer can ensure the amorphous character [START_REF] Dolden | Structure-property relationships in amorphous polyamides[END_REF], [START_REF] Gaymans | Nylon 4,I: an amorphous polyamide[END_REF]. Since rheological properties depend on the chemical structure of the polymer, special attention was given to the introduction of the interactions on the polyamide backbone. Different interacting groups were used, with different interaction strengths, while keeping constant the polymer backbone.

First, it will be important to understand the rheological behavior of polyamides (2 H-bonds per monomer) and to understand if theoretical models for unentangled and entangled non-entropic polymers are still valid in the case of polyamides. The effect of strong intermolecular interactions in the rheology of polyamides will be studied, introducing different interacting groups (such as H-bonds and ionic groups), with the aim of increasing the strength of the interaction between polymer chains as well as their density. The effect of these interactions will be observed in both solid state and molten state dynamics, using dielectric spectroscopy measurements and rheological measurements.

The second chapter deals with the presentation of the studied polyamides as well as the description of the experimental techniques used that allowed the study of polyamide mechanical, structural and thermal properties. Then, in the third chapter the complete study of the molten state dynamics of polyamides with increasing molecular weight is presented. The investigation of the role of interaction on the viscoelasticity of polyamides, using different interacting groups and different molecular weight distributions, will be presented in the fourth and fifth chapters. Finally, the last part deals with the dynamics in the solid state of polyamides with different interacting groups. A comparison between two different techniques (dielectric spectroscopy and dynamic mechanical analysis) will also be presented.

Chapter 2 -Materials and Methods

In this chapter, the different polyamides used in this work will be presented, as well as the synthesis routes used to increase the density and strength of intermolecular interactions. The experimental techniques used to characterize the viscoelastic behavior in both molten and solid state, as well as the thermal and structural characterization techniques will be presented.

Materials

A new class of polyamides was designed for this study, based on a PA 6I basis, which is an amorphous polyamide obtained by the condensation reaction of hexamethylene diamine (HMD) with isophthalic acid (IA). Contrary to most common polyamides like PA 6 or PA 66, PA 6I did not crystallize even at very small cooling rates during DSC experiments or during isothermal rheology tests.

To prevent any evolution of the molecular weight, and thus of the viscosity, during rheological measurements, namely by post-condensation, chain ends were blocked during synthesis step, with a monofunctional benzoic acid. The monoacid introduced reacts with the amine chain ends during polymer synthesis, therefore the quantity of amine chain ends is close to zero, meaning that post condensation reaction cannot take place. The introduction of this blocker helps as well targeting a certain molecular weight. Prior to dynamical tests, polyamide samples were dried in vacuum for several days, at a temperature close to Tg, until reaching a water content smaller than 500 ppm, as measured by Karl Fischer titration. This permits to avoid hydrolysis reaction during polymer processing, rheological tests or dielectric spectroscopy measurements, as well as, avoid any plasticization effect (mainly a Tg decrease) during thermal analysis.

Intermolecular interactions were introduced directly on the PA 6I polymer backbone, in order to introduce as less changes as possible in the polymer backbone (influencing many polymer properties such as packing length, free volume, molecular weight between entanglements, etc). The effect of interactions on the dynamics of polyamides was studied by introducing hydrogen bonding groups or ionic functional groups on the PA 6I backbone, as presented in Figure 39. To that end, isophthalic acid substituted with different groups -OH and -SO3Li or -SO3Na was used (5-hydroxyisophthalic acid and 5-sulfoisophthalic acid lithium or sodium salts). This option allows as well keeping all polyamides amorphous.

Hydrogen bonding polyamides

PA 6HIA is synthesized by condensation reaction of HMD and 5-hydroxyisophthalic acid. Compared to PA 6I, it has an additional -OH group capable of forming stronger H-bonds with amide groups. While an amide-amide standard H-bond (as in Figure 29) has energy of interaction around 25 kJ•mol -1 [START_REF] Jorgensen | Optimized intermolecular potential functions for amides and peptides. Structure and properties of liquid amides[END_REF], [START_REF] Klotz | Hydrogen Bonds between Model Peptide Groups in Solution[END_REF], the interaction of an amide group with the -OH function from the phenol group has 40 kJ•mol -1 (energy values obtained by simulation in vacuum) [START_REF] Hartikainen | Structure and Morphology of Polyamide 66 and Oligomeric Phenolic Resin Blends: Molecular Modeling and Experimental Investigations[END_REF].

Ionic copolyamides

Ionic copolyamides, PA 6LiSIPA or NaSIPA, are the result of the condensation of HMD and 5-sulfoisophthalic acid lithium or sodium salts. PA 6LiSIPA was synthesized as a copolymer with PA 6I in a molar proportion of 95/5, 90/10, 85/15 and 80/20, while PA NaSIPA was synthesized in a single molar proportion of 90/10. The lithium-amide interaction has already been thoroughly studied in the literature in several polymer blend miscibility studies [START_REF] Molnar | Miscibility of polyamide-6 with lithium or sodium sulfonated polystyrene ionomers[END_REF]- [START_REF] Tucker | Small-Angle Neutron Scattering Analysis of Blends with Very Strong Intermolecular Interactions: Polyamide/Ionomer Blends[END_REF] by FTIR (Fourier-transform infrared spectroscopy), NIR (Near Infrared), DMA (Dynamic mechanical analysis) and solid-state 15 N NMR (Nuclear Magnetic Spectroscopy). In these studies, it was observed that an ionomer-amide complex is formed, involving the coordination of the carbonyl group of the polyamide backbone by the metal cation. This ionomer-amide complex presumably goes through different stages depending on the molar ratio between carbonyl groups and metal cations as schematized in Figure 40 [139]:

A. The complex corresponds to the structure of the lithium sulfonate group in polystyrene. Other entities, namely X, are represented as being in the region to fill space around the lithium cation. They can be for example, other sulfonate groups to create ionic clusters, or water molecules, since it is known that these ionic groups are highly hygroscopic. B. When the amide/lithium molar ratio equals one, the lithium cation can associate with one carbonyl group while the sulfonate group SO3 -can interact with the free NH from the amide group, through a hydrogen bond. The energy of this complex is around 10-15 kJ•mol -1 [START_REF] Rajagopalan | Molecular interpretation of miscibility in polyamide-6 blends with alkali ionomers of sulfonated polystyrene[END_REF]. C. When the amide/lithium molar ratio equals or exceeds 4, Li + can be surrounded by four carbonyl groups as the sphere of coordination of the lithium cation is 4. On the other side, the sulfonate groups can also interact with the free NH, creating a hydrogen bond. The energy of this complex is around 50 kJ•mol -1 [START_REF] Rajagopalan | Molecular interpretation of miscibility in polyamide-6 blends with alkali ionomers of sulfonated polystyrene[END_REF], which is the double of the interaction energy between two amide groups.

The influence of the nature of the counter-ion was also studied by synthesizing a copolyamide using a sodium 5-sulfoisophthalic salt. The choice of the counter-ion was based on literature results and will be further explained in section 4.4. Copolyamides with a lower glass transition temperature were synthesized, while maintaining the amorphous character of polyamides. To do so, the diamine chain segment was increased, by replacing the hexamethylene diamine by decamethylene diamine, as shown in Figure 41.

In this study, we report the study of different series of polyamides: PA 6I, PA 6HIA and copolymers of PA 6I/6LiSIPA and PA 6I/6NaSIPA. The PA 6I series will permit to develop knowledge in the behavior of non-modified amorphous and stable polyamides, a topic which has not been addressed in the literature. Then, introducing interacting groups will allow the study of the influence of interactions on the rheological properties of unentangled and entangled polyamides, as well as on the dynamics in the solid state.

Polyamide synthesis

None of the studied polyamides are commercially available. All polyamides were synthesized in the Polymers Design and Development Laboratory in Solvay (V. Bossenec, M.-L. Michon). The polyamides were prepared by polycondensation in a 300 mL stirred pressure autoclave.

The needed quantities of diacid and diamine to synthesize up to 130 g of polyamide were introduced in the reactor. Hexamethylene diamine was used for the majority of polyamides, except for PA 10I/10NaSIPA, while the diacid was changed in order to synthesize polyamides with different interactions. For PA 6I/6LiSIPA or PA 6I/6NaSIPA, the quantities of acid monomer were calculated in order to introduce the desired lithium/sodium sulfonate contents. For instance, to synthesize PA 6I, isophthalic acid and hexamethylenediamine were poured with demineralized water and antifoam in a stainless steel clave. The atmosphere was purged with nitrogen, and the temperature was increased progressively to 220°C, with continuous stirring, letting pressure increase up to 17.5 bar. The temperature was increased progressively up to 250°C, maintaining the same level of pressure. In the decompression step, the pressure was then progressively released while the temperature was increased to about 272°C. Vacuum was sometimes applied to reach higher conversion rates for about 20 to 40 min at the same temperature under continuous stirring. The vacuum was broken with nitrogen and the polyamide was removed from the reactor, using a bottom drain valve.

The complete list of materials, as well as the characteristics of each series of polyamides (later described in details) is presented in Table 3. The full characterization in terms of branching and molecular weight distribution will be presented in the third and fourth chapter, along with the rheological measurements, since both are tightly related. Each polymer will be named after its chemical structure (PA 6I, PA 6HIA or ionic copolymers PA 6I/6LiSIPA or PA 6I/6NaSIPA, followed by the non-ionic/ionic ratio) and the approximate molecular weight.

Methods

Chemical structure characterization

The chemical structure of polyamides was studied using End group titration (EGT) and Gel permeation chromatography (GPC) to know the molecular weight and molecular weight distribution. The quantity of branching created during synthesis was also estimated using a proprietary dosage protocol as well as by solution-state NMR (nuclear magnetic resonance).

Molecular weight distribution

The number-and/or weight-average molecular weight of the studied polyamides was analyzed by End group titration (EGT) and by Size exclusion chromatography (or gel permeation chromatography -GPC).

End group titration was performed by dissolving the polyamide in a trifluroethanol/chloroform mixture and both amine -NH2 and acid -COOH groups were titrated by potentiometric back titration (Solvay Analytical Laboratory, K. Longé, N. Delon-Anik). It permitted to quantify the number of acid (CEG) and amine (AEG) end groups (in mmol•kg -1 ) while the quantity of blocked (BEG) end groups is known from the reaction recipe, assuming total reaction with the amine chain ends. These values allow calculating a number-average molecular weight, Mn, as shown in equation 2.1, where concentrations are in mmol•kg -1 .

The Mn calculated by EGT analysis was not always in full quantitative agreement with the value obtained by GPC, especially for low molecular weights, due to the truncation of oligomers in the GPC analysis.

GPC measurements depend on the hydrodynamic volume of each polymer chain. To perform this measurement, a polymer sample is dissolved in a solvent and the solution is passed over a column, filled with a porous media. Depending on the chain size, the eluted molecules can enter the pores, while long chains flow with the solvent. The weight concentration of the polymer in the eluting solvent is continuously measured. The curve of polymer concentration as a function of the elution time can thus be compared to the one of a standard polystyrene and the molecular weight in PS equivalents is obtained [START_REF] Lesec | Masses molaires moyennes[END_REF]. The advantage of GPC over other techniques is that it provides both number-and weight-average molecular weight as well as the polydispersity index, which is an essential parameter when studying the rheological behavior of polymers. GPC curves can also be used to check if the polycondensation reaction led to statistical, linear chains, by comparing the measured molecular weight distribution to the Flory prediction for linear statistical chains, as shown in equation 1.36. In the case where GPC data cannot be correctly fitted using Flory distribution, secondary reactions may have occurred during polyamide synthesis, leading to branched polymers [START_REF] Flory | Principles of Polymer Chemistry[END_REF].

The determination of the molecular weight distribution by GPC is submitted to a certain error, related to the truncation of the oligomers. In fact, in all GPC chromatograms, some peaks due to the presence of oligomers or unreacted species are observed in the low molecular weight range, as shown in Figure 42-a.

Each peak represented by a letter from A to F can be associated to one definite species, based on the corresponding molecular weight: A = Diacid unreacted; B = Monomer 6I; C = Oligomer formed by two diacids and one diamine; D = Oligomer with 2 monomers; E = Oligomer with 3 monomers; F = Oligomer with 4 monomers. As a general rule, the molecular weight distribution of unentangled polyamides was truncated after the second oligomer (peak D -5000 g•mol -1 ), while for high molecular weight samples, the truncation was performed after the forth oligomer (peak F -1000 g•mol -1 ).

In GPC, the molecular weight distribution is given as dw/dlogM where w is the weight fraction of chains with x repeat units, as a function of M, the molecular weight. To fit the Flory distribution (equation 1.36) and determine the extent of reaction, p, the data has to be presented as the weight fraction of chains bearing x repeat units as a function of the number of repeat units ( ), where M0 represents the molecular weight of the monomer. Thus, equation 2.2 shows how to transform GPC raw data to the data needed in the Flory distribution, presented in Figure 42-b.

Molecular weight distributions were analyzed by size exclusion chromatography (GPC), using a GPC PL Lab 120 device with double detection (viscometer and UV detector) in the Solvay Analytical Laboratory (K. Longé, N. Delon-Anik). Since polyamide is not soluble in most common solvents at room temperature, specials solvents needed to be used, following an internal protocol developed by Solvay Analytical Laboratory.

Branching quantification

As explained in section 1.2.1, a branching reaction can occur during polyamide synthesis by the formation of bis-hexamethylene triamine (BHT), as shown in Figure 28. The concentration of BHT can be determined by dosage after hydrolyzation of polyamide, following an internal protocol developed by Solvay Analytical Laboratory. The polyamide is introduced in a test tube, as well as the solvent responsible for the polymer hydrolysis. The mix is then neutralized and transferred to a separation funnel (liquid-liquid extraction). The organic phase is recovered as it contains all the amines and also the BHT. To be detected by the UV detector, a previous derivatization is performed. A calibration set with different BHT solutions is available to make a calibration curve for the BHT content. The determined BHT concentration is given in ppm (parts per million) referred to the initial polyamide weight.

Another branching reaction may specifically occur during the synthesis of PA 6HIA. During the synthesis of this polyamide, the molecules of the diacid monomer, 5-hydroxyisophtalic acid, can react with each other. In fact, the alcohol function (from the phenol) and the carboxylic acid function (-COOH) can react, forming an ester, as shown in Figure 43-a. The ester continues having three carboxylic acid functions that can react with the diamine monomers and be inserted in the polymer chain. The presence of the phenolic ester can be accessed by Nuclear Magnetic Resonance, bearing in mind that also the mono-functional acid introduced to end-cap amine chain ends can react with -OH groups (Figure 43-b). PA 6HIA polymer samples were analyzed in a Bruker Av400 (Solvay Analytical Lab, F. Le Guyader, L. Tribolet), using as solvent deuterated hexafluoroisopropanol (HFIP-D2) at ambient temperature. Spectra of 13 C and 1 H NMR are presented in Figure 44 where the chemical shifts observed in PA 6HIA in agreement with those of phenolic ester molecules are visible. This technique permits to calculate the quantity of phenolic ester with a 10% error bar.

Thermal and structural characterization

Structural characterization is necessary to better understand the dynamic behavior of polymers. In the case of polyamides, it is essential when characterizing the rheological behavior of polymers, for which amorphous polymers are needed to obtain the complete dynamic behavior. To do so, it is necessary to confirm the amorphous character of polyamides, by differential scanning calorimetry (DSC) and by wide angle X-ray scattering (WAXS). Small angle X-ray scattering (SAXS) measurements were also performed with the aim of understanding whether ionomers (PA6I/6LiSIPA or PA 6I/6NaSIPA copolymers) presented nanophase separation, as in other works reported in the literature (see chapter 1) [START_REF] Weiss | Rheological behavior of oligomeric ionomers[END_REF].

Differential Scanning Calorimetry

The principle of this technique consists in applying a temperature ramp on two pans, one with the polymer and one empty considered as the reference and recording the difference of energy that needs to be provided to keep both pans at the same temperature. This difference of energy, called heat flow, and its variation with temperature, give information on the polymer thermal properties, such as the glass transition temperature, melting temperature, crystallization, etc. As mentioned before, the presence of a crystalline phase is one of the main reasons for the lack of literature on polyamide rheology, since it limits the range of temperatures at which the dynamic moduli can be measured to the terminal region. In this study, DSC measurements were performed to identify the glass transition temperatures of polyamides and to verify the amorphous character of polyamides, on account of the absence of melting or crystallization signals [START_REF] Teyssèdre | Caractérisation des polymères par analyse thermique[END_REF].

The glass transition of amorphous polymers can be easily identified as a step in the heat flow or heat capacity from DSC measurements, as presented in Figure 45-b, or equivalently by a peak in the derivative of the heat capacity, as in Figure 46. Glass transition is sometimes a broad phenomenon making its analysis difficult using a standard procedure. A better precision can be obtained using modulated DSC (mDSC), which allows to separate time dependent and independent processes, into the reversing and nonreversing heat flows, as presented in Figure 45-b. Compared to the standard DSC mode in which a temperature ramp is applied linearly, in mDSC the temperature is modulated by a few degrees around an average ramp, as shown in Figure 45-a.

The mDSC mode allows distinguishing the glass transition process from other kinetic processes, such as chemical reactions, solvent evaporation, etc, meaning that the total response of heat flow over time, dH/dt, is separated in two components: one that depends on the heat capacity, Cp, of the sample and the heating rate, and another term depending on time and absolute temperature, , as given in equation 2.3.

Differential scanning calorimetry data were acquired on a TA Instruments Q2000 modulated DSC (mDSC) at a heating rate of 3°C•min -1 with a temperature modulation of ±2°C and a period of 60s.A rapid cooling system (RCS) with a nitrogen flow of 50 mL•min -1 was used. Non-hermetic pans were filled with 5-15 mg of polymer sample. Only the second heating run was considered for the determination of glass transition temperatures (-30 to 200°C), as the maximum of the derivative of the reversible heat capacity jump, or equivalently, at the inflexion point of the heat capacity curve at the glass transition. To check the repeatability of the measurements, two to three measurements were done for each polymer and all curves superpose almost perfectly.

From the thermograms obtained from mDSC experiments, it is possible to calculate as well the heat capacity jump during glass transition. This can be done by calculating the difference between the liquid state and glassy state Cp or by plotting the derivative with time of the reversing heat capacity signal measured by DSC and by integrating this peak, as presented in Figure 46.

Differential Scanning Calorimetry for Physical Aging Experiments

In addition to classical glass transition measurements described in the previous section, aging experiments were performed for ionic copolymer samples (PA 6I/6LiSIPA and PA 6I/6NaSIPA), with the objective of understanding whether ionic groups act as dynamic heterogeneities i.e. zones where the local dynamics are heterogeneous due to different local compositions of the ionic groups in the copolymers. This experiment was implemented following a first heating cycle to remove thermal history and by performing an isothermal step at Ta<Tg and observing how the heat flow in the heating cycle (from B) is affected, when compared to the unaged sample (from A), as presented in Figure 47. During the isothermal aging step, the glassy sample will spontaneously evolve toward equilibrium, a process called structural recovery or physical aging. The enthalpy gain from A to B corresponds to the enthalpy gained during aging, or in other words to the gain in density (lower volume) that corresponds to a state closer to or at equilibrium, if enough time is given to the sample to reach its equilibrium. The enthalpy gain between points A and B is calculated as the area in between the two heat capacity curves corresponding to the heating cycles from points A and B. For the unaged sample, the enthalpy variation in the glass transition is given by equation 2.4, where HA corresponds to the enthalpy of the glass at the point A at the aging temperature Ta and HM corresponds to the enthalpy at a given temperature TM > Tg. The same equation can be written for the enthalpy variation in the glass transition for the unaged sample, as presented in equation 2.5, where represents the enthalpy of the aged sample at point B, after an aging step at temperature Ta.

Since the difference of the integrals between two curves is equal to the integral of the difference of two curves, it is possible to calculate the enthalpy gain as in equation 2.6.

To perform this experiment, non-hermetic pans were filled with 5-15 mg of polymer sample and a detailed sequence was used for all experiments, as presented in Figure 48. Aging step was performed for 5h for all aging temperatures ranging from Tg -5 to Tg -40°C. The first heating cycle (step 1) was performed to erase the thermal history of the sample, after which an isothermal step was performed at Ta (step 2). After the aging step, sample is cooled until -30°C and then heated to measure the glass transition of the aged sample (step 3). To avoid errors from sample-to-sample variation, DSC calibration, etc., the glass transition of the unaged sample was measured in the same sequence. After the third process, sample was kept for 5 minutes at high temperature, to erase thermal history followed by a classical coolingheating cycle to measure the glass transition (step 4). Finally, it is possible to define the overshoot as the difference between the two heat capacity curves of the aged and unaged samples, calculated by the integral presented in equation 2.6. The overshoot is then plotted as a function of the aging temperature, which allows comparing different polymers, independently of each glass transition.

Wide Angle X-ray Scattering

Wide angle X-ray scattering (WAXS) is a non-destructive technique of characterization that has been used in the literature to provide information on the crystalline structure of polymers. This technique is based on Bragg's law of diffraction. When X-rays are scattered by atoms located in periodically ordered planes with a repeat distance, d, as schematized in Figure 49. The incident beam is reflected according to Bragg's law:

where n is an integer, in the wavelength of the incident X-ray beam, d is the distance between planes in the polymer structure and is the angle of the incident and reflected X-ray beam with respect to the plane.

The experimental setup of a X-ray scattering technique, represented in Figure 50-a, contains: an X-ray source that sends a X-ray beam to the sample at a given wavelength, , the beam enters the sample and is scattered. A detector placed behind the sample detects the scattered intensity as a function of the scattering angle 2 .

According to equation 2.7, d is inversely proportional to , meaning that the larger the angle, the smaller the distance d between repeating units. Wide-angle X-ray scattering is the most valuable techniques for the study of ordered systems at the atomic scale. Bragg diffraction peaks are obtained as a function of the scattering angle or equivalently as a function of the scattering vector, q, given by equation 2.7. Combining equations 2.7 and 2.8, the distance d is expressed as a function of the scattering vector, as in equation 2.9.

After integration of the 2D pattern (schematized in Figure 50-b) over all azimuth angles (for an isotropic sample), the variation of intensity as a function of the scattering angle is obtained, as presented in Figure 50-c, for a PA 6,6 sample.

In the case of semi-crystalline polyamides, the diffractogram consists on the superposition of the amorphous halo with the crystalline peaks, as represented in Figure 50-c, whose positions are associated to different crystallographic planes (001, 100, 010 and 110).

In this work, WAXS measurements were performed to verify the amorphous character of polyamides, already verified during DSC measurements by the absence of melting and crystallization peaks. These measurements were performed on an Oxford Xcalibur Mova diffractometer in transmission mode at the Henri Longchambon Diffraction Center of Université de Lyon (R. Vera, E. Jeanneau). Polyamide samples were compression molded 20°C above Tg with a thickness around 1mm. Measurements were performed at room temperature, with the angle varying between 4 to 50°, the detector was at 55.5 cm of the sample and a copper anode was used to generate X-ray of wavelength 1.54 Å. Measured values were adjusted taking into account the absorption coefficient and thickness of the polymer sample. A narrow beam of X-rays with an incident intensity I0, penetrating a film of polymer with mass thickness x (equal to the product of the density and thickness of the polymer material) and density , is transmitted with intensity I given by the exponential attenuation law [START_REF] Hubbell | NIST: X-Ray Mass Attenuation Coefficients[END_REF]:

Values of the mass attenuation coefficient, μ/ρ, were calculated by simple additivity of the mass attenuation coefficients of each atomic element in the molecular structure of polyamides taking into account its weight fraction. For example, PA 6I presents a mass attenuation factor of 5.52 cm 2 •g -1 while copolymers PA 6I/6NaSIPA present higher values due to the presence of the metal cations, as observed in Table 4. The amorphous character of studied polyamides was verified by both DSC and WAXS experiments. As it can be observed in Figure 51-a, the amorphous halo of PA 6I is independent of molecular weight, as expected, since the local structure at the monomer scale is probed. Diffractograms contain two main peaks around which corresponds to the distance between monomers along one chain, while that close to 20° corresponds to the average distance between locally parallel chains. Comparing with PA 6,6, it presents an crystalline lattice characterized by main two distances, the distance between amide groups along the chain (c/2) and the distance between chains forming H-bonds (a), as shown in Figure 52. Similar distances are observed for PA 6I, but the order is not sufficient to create a crystalline phase, due to the stereochemical configuration of the carbonyl groups next to the benzene ring which introduce an inflection point into the polymer chain (see Figure 30 in chapter 1).

Introducing lithium or sodium sulfonate groups in polyamides increases the distance between monomers, while the distance between chains remains constant (Figure 51-b andc). This increase in the distance between monomers (or hydrogen bonds) may be caused by the formation of the complex (Figure 40) that locally increases the quantity of carbonyl groups (thus amide groups), leading to larger average distances between other amide groups. The distances between monomers increase for the PA 10I/10NaSIPA polymers, as the aliphatic chain length increases. When compared to PA 6I, PA 6HIA presents slightly different distance between monomers, as it can be seen in Figure 51-d.The peak corresponding to the distance between monomers is shifted to lower values of , meaning that the distance between monomers increases. In conclusion, these diffraction experiments allowed confirming the amorphous character of polyamides, which was necessary to perform rheological measurements.

Small Angle X-ray Scattering

Small angle X-ray scattering (SAXS) relies on the same technique used for WAXS experiment with a different range of scattering angles. SAXS measurements were performed to investigate the existence of nanophase separation of ion-rich aggregates in polyamides, typically with a characteristic dimension of 2-4 nm [START_REF] Kim | Introduction to Ionomers[END_REF]. In this case, the distance between the sample and the detector is larger to achieve smaller scattering angles, thus to measure higher characteristic distances.

Measurements were carried out in the ID02 beamline at the European Synchrotron Radiation Facility (M. Sztucki, ESRF, Grenoble, France) at a wavelength =0.1 nm and at the CEA Saclay (O. Taché). The accessible q range goes from 0.1 to 8 nm -1 . Samples were dried and prepared by compression molding at 20°C above Tg. Sample thickness was kept around 1 mm. SAXS scattering patterns will be presented as a function of the scattering vector, q.

The empty beam signal was subtracted from scattering data of the samples and data were normalized by the sample thickness, h, as in equation 2.11. To emphasize specifically the scattering from ionic groups, the signal of the PA 6I reference polymer was subtracted to the scattering data of ionic copolymers and the resulting intensity normalized by the volume fraction of ionic groups, , according to equation 2.12.

Rheological measurements

Small strain dynamic oscillatory shear measurements were performed using an ARES (Advanced Rheometric Expansion System) strain-controlled rheometer from TA Instruments, equipped with a Force Rebalance Transducer 2kFRT-N1. Samples were dried down to less than 500 ppm of residual water, as measured using a Karl-Fisher set-up, and compression molded 30°C above Tg under vacuum before being placed between two 4 or 8 mm diameter Invar (alloy with low thermal expansion coefficient) parallel plates. The temperature control was done using a forced convection oven and a liquid nitrogen dewar permitting a control better than ±0.2°C. To insure the complete filling of the geometries, samples were introduced 40°C above Tg and were allowed to equilibrate for 5 minutes, as before each measurement at different temperatures. Temperature steps of 10 K were used at high temperatures, while close to Tg each frequency sweep was done every 5 K. The gap used was kept between 1.2 and 2.0 mm. As the temperature was changed, the plate-plate gap was adjusted to account for thermal expansion of the fixture, maintain a constant sample thickness and keep the normal force as close to zero as possible.

Firstly, strain sweeps were performed at each temperature to determine the linear viscoelastic region for the frequency sweeps (Figure 1), followed by a time sweep to guarantee the linearity of the chosen strain, as well as to verify the stability of the polyamide sample. Thus, frequency sweeps were performed at frequencies from 100 down to 0.1 rad•s -1 , applying the strain selected during the strain sweep and temperatures from Tg + 5°C up to Tg + 100°C. The stability of each sample was guaranteed performing a specific temperature sequence, as schematized in Figure 53: the sample was introduced in the rheometer at Tg + 40°C, temperature was decreased using 10°C or 5°C step, when close to Tg, temperature was increased to Tg + 40°C, and then increased using 10°C step and in between each increase in Tg a measurement at Tg + 40°C was done. This allows verifying that the polyamide sample was stable up to each temperature.

To obtain the complete picture of polymer relaxation, time-temperature superposition was applied to build a master curve, as explained in section 1.1.2.5 and presented in Figure 15. The reproducibility of each polymer was checked at least twice, by performing two independent tests and comparing individual frequency sweeps and then the resulting master curves and shift factors.

Time-temperature superposition was applied by shifting vertically and horizontally frequency sweeps at all temperatures to a reference temperature, Tref. The density variation in the vertical shift factor was taken into account [START_REF] Walsh | Standard Pressure Volume Temperature Data for Polymers[END_REF]. The horizontal shift factors were fitted using a WLF equation (equation 1.27 [START_REF] Williams | The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-forming Liquids[END_REF]) and the parameters C1 and C2 were used to calculate the value of the horizontal shift factor at Tg. Then, master curves were shifted from Tref to Tg by rescaling them using both horizontal and vertical shift factors at Tg. Master curves were shifted to Tg so that all curves overlap in the glassy region, guaranteeing isofrictional conditions, as mentioned in section 1.1.2.5, when comparing the master curves of all samples.

Dielectric spectroscopy measurements

In dielectric spectroscopy, a sample is placed in between two parallel gold-plated electrodes and submitted to a sinusoidal electric field, leading to an alignment of molecular dipoles in the direction of the field depending on polymer mobility, as represented in Figure 54. By varying the frequency of the electric field and/or temperature, the relaxation processes can be identified, by measuring the complex permittivity defined by equation 2.13, where and correspond to the real and imaginary parts of the permittivity. The main result in dielectric spectroscopy is a 3-dimensional map of the permittivity (or modulus) as a function of the frequency and temperature, as presented in Figure 55, for a PA 6,6 sample. As mentioned before, three main relaxation mechanisms have been identified in polyamide: -relaxation, generally related to the motions of the aliphatic parts of the chains, -relaxation, related to the rotation of the amide groups and -relaxation related to segmental motions of the chain, identified as the glass transition [START_REF] Kohan | Nylon Plastics Handbook[END_REF], [START_REF] Beaume | Secondary transitions of aryl-aliphatic polyamides. I. Broadband dielectric investigation[END_REF], [START_REF] Laurati | Dynamics of Water Absorbed in Polyamides[END_REF]. At temperatures above Tg, polymer chains are mobile and the materials start conducting the electric current, especially at low frequencies, as dipoles have more time to align with the electric field, resulting in an apparent increase of the loss permittivity by several orders of magnitude, as observed in Figure 55-a. Results can be expressed in terms of the dielectric modulus, which is the inverse of permittivity (equation 2.14). Considering the loss modulus, the signature of the conductivity is less visible (Figure 55-b) since it corresponds to the inverse of the permittivity (equation 2.14). This representation then allows better observing of the -relaxation, which is often hidden by the conductivity, as represented in Figure 56. It is also possible to observe that the temperature values of the maxima of the E'' and M'' peaks are fairly close to each other.

For the data analysis, isochronal (constant frequency, varying temperature) or isothermal (constant temperature, varying frequency) representation can be used. Two different methods were considered for the analysis of the obtained results. The first method consists in fitting the isothermal loss permittivity using mathematical models, so as to obtain the distribution of relaxation times as a function of temperature. Three main models were considered: Debye, Cole-Cole and Havriliak-Negami [START_REF] Kremer | Broadband Dielectric Spectroscopy[END_REF], [147].

The simplest model is the Debye model, in which a relaxation process has a single relaxation time. The complex permittivity of the polymer when submitted to a time-dependent electric field is given be equation 2.14, where is the frequency, is the relaxation time. This model describes a symmetrical peak having slopes of +1 and -1, as shown in Figure 57-a. Only in rare cases a Debye-like relaxation is observed, since usually the measured dielectric function is much broader than predicted by the Debye function.

Other approaches have been developed, as extensions of the Debye model, in order to account for the broadening of the peaks. The Cole-Cole model is the first empirical expression for the complex dielectric function, proposed by K.S. Cole and R.H. Cole in 1941. This model is presented in equation 2. [START_REF] Fetters | Packing Length Influence in Linear Polymer Melts on the Entanglement, Critical, and Reptation Molecular Weights[END_REF]. The exponent 1 is a parameter that quantifies the symmetrical broadening of the relaxation function, as presented in Figure 57-b. The Debye model is recovered when =1.

The Havriliak-Negami (HN) equation [START_REF] Havriliak | On the equivalence of dielectric and mechanical dispersions in some polymers; e.g. poly(n-octyl methacrylate)[END_REF], presented in Figure 57-c, is an empiric development of the Cole-Cole equation, in which an asymmetry is introduced, by adding a parameter, as demonstrated in equation 2.17, knowing that 1. The Cole-Cole equation in recovered when =1. In this work, a Novocontrol Alpha Analyzer and a Quatro Temperature Control System were used to perform dielectric experiments under a voltage of 2V in the temperature range from -130°C up to 220°C, with 5°C steps. For each temperature, the frequency range was 10 -2 to 10 6 Hz. Polyamide samples were dried down to less than 500 ppm of residual water, as measured using a Karl-Fisher set-up and compression molded in between Teflon sheets with a diameter of 20 mm and the thickness was controlled introducing a Teflon spacer of 500 μm. Polyamide films were introduced in the dielectric apparatus, temperature was increased above Tg, in order to insure a good contact between the sample and the electrodes.

In order to characterize the molecular relaxations of polyamides, the WinFit software was used. It allows fitting the loss permittivity data of a polymer sample following a Havriliak-Negami model. and parameters can be fixed at 1, to reduce to Cole-Cole and Debye functions, respectively. In order to obtain a correct fit, the fit starts from the lowest to the highest temperatures and two relaxations are fitted at the same time. Both and secondary relaxation peaks are generally symmetric, so the parameter in Havriliak-Negami model was set equal to one.

The relaxation peak of the -relaxation was often difficult to fit since it is accompanied by strong conductivity. For non-modified PA samples, it was possible to fit the asymmetrical -relaxation using Havriliak-Negami equation. However, with increasing ionic content, the conductivity was too high to be fit the segmental relaxation. In these cases, the relaxation time of the -relaxation was determined by using the maximum of the loss modulus in isochronal curves (Figure 56). When both isochronal and isothermal representations could be used, it was verified that data coincided

Dynamic mechanical analysis

Dynamic Mechanical Analysis (DMA) is a technique in which mechanical properties of a polymer in the solid state can be measured. A specimen is submitted to a small sinusoidal oscillating deformation at various temperatures as a function of time and temperature. As in rheology, the ratio between stress and strain is given by the dynamic complex modulus that can be decomposed in two parts, the storage modulus (E') and loss modulus (E''). The denomination for the modulus is different from that used in rheology, because in this case, a different mechanical loading is used.

Using this technique, it is possible to determine the polymer relaxation times at different temperatures, as in dielectric spectroscopy experiments. However, in DMA a restricted range of frequencies can be used: 0.1, 1, 10 and 50 Hz. When a polymer relaxation temperature is reached, at a given temperature, there is a small drop in E' for secondary relaxations, while for the main -relaxation a large drop of about 1 or 2 decades in E' is observed. Concomitantly, a peak is observed in E'' and . Figure 58 shows a typical curve obtained by DMA where the three relaxations observed in polyamides are highlighted. Though the range of frequencies accessible using DMA is quite smaller than in dielectric spectroscopy, the results obtained by both techniques are comparable.

In this study, polyamide samples were injected using a Micro-compounder HAAKE™ MiniLab 3. Specimens of 1x10x60 mm 3 were placed in a RSA II or a RSA G2 equipment using a three-point bending geometry, in which a sample is placed between two supports and the mechanical deformation is applied by a cantilever situated at the middle of the sample. Measurements were performed at fixed frequencies (from 0.1 to 50 Hz) as temperature was varied from -150 to 130°C, at a rate of 2°C•min -1 .

Chapter 3 -Dynamics of non-modified polyamides in the molten state

In this chapter, the complete viscoelastic behavior over the complete range of polyamide relaxation of non-modified amorphous PA 6I polyamides will be presented. As explained previously, viscoelastic behavior over the complete range of polyamide relaxation is a topic rarely mentioned in the literature due to the presence of a crystalline phase in classical polyamides and to the lack of chemical stability since post-condensation or hydrolysis can easily take place if water content is different from that at equilibrium. Therefore it is potentially important to find the proper experimental conditions to study the effect of the high density of hydrogen bonds in in polyamides (2 H-bonds per monomer).

Several polyamides of varying molecular weight were synthesized for this study. In a first part, the molecular structure of non-modified polyamides will be presented, followed by the study of their rheological behavior and evolution with molecular weight. In the last section, the fit using well-known models for both unentangled and entangled polymers will be presented.

Molecular architecture

The understanding of the dynamical data obtained by both molten state and solid state measurements cannot be fully achieved without full knowledge of the molecular structure. As evidenced in the first chapter, molecular weight, molecular weight distribution and branching content have a strong influence on the viscoelasticity of polymer melts.

The number-average molecular weight was firstly determined by end group titration, according to equation 2.1, from the measured quantities of amine and acid chain ends (-NH2 and -COOH), and the quantity of blockers (mono functional acid) introduced in the reactor and assuming complete reaction with amine ends. The molecular weight and molecular weight distribution (MWD) were measured by GPC. This measurement allows accessing to both number and weight-average molecular weights, Mn and Mw, respectively, used to calculated the polydispersity index as Ip=Mw/Mn. The complete set of values of Mn by GPC and EGT, Mw and Ip by GPC is presented in Table 5. In this table and during this work, each polyamide will be named after its chemical structure (PA 6I) and the approximate weight-average molecular weight.

The first observation is that the number-average molecular weight obtained by EGT is not always in full quantitative agreement with the value obtained by GPC, especially for low molecular weights, due to the truncation of oligomers during GPC analysis, explained in section 2.2.1.1.

Secondary reactions may take place during polyamide synthesis, possibly leading to a very small fraction of branched chains, explaining small variations in the polydispersity index. In PA 6I synthesis, a single branching reaction was considered, which was the reaction between two diamines and formation of bis(hexmethylene)triamine (BHT), as shown in Figure 28. The content of BHT of each polyamide is reported in Table 5 as measured after hydrolysis of the polyamide.

As observed in Table 5, the values of polydispersity are often larger than 2, the value expected by Flory's theory for linear polycondensation. To better understand the values of polydispersity determined by GPC, the experimental data were fitted using a Flory distribution, as in equation 1.36. The GPC data along with the fit done using one single adjusting parameter, p, are presented in Figure 59 for some polyamides. Generally speaking, Flory's equation for linear polycondensation does not describe well the experimental GPC data, indicating that the synthesis did not result in a statistical assembly of linear chains, which can be explained by the branching reaction previously mentioned. Lastly, glass transition values obtained by modulated DSC are presented, as calculated as the maximum of the derivative of the heat capacity. The glass transition temperatures of PA 6I measured by mDSC are shown in Figure 60 as a function of the number-average molecular weight. Tg increases with increasing molecular weight up to a plateau value for the highest molecular weights, as described by the Fox-Flory equation [START_REF] Flory | Statistical mechanics of chain molecules[END_REF], [START_REF] Fox | Influence of molecular weight and degree of crosslinking on the specific volume and glass temperature of polymers[END_REF], presented in equation 3.1:

The Fox-Flory equation uses K and Tg as parameters. K is a constant that sets a characteristic molar mass and Tg corresponds to the glass transition temperature at infinite molecular weight. For PA 6I, Tg is equal to 129 ± 1°C and K equal to mol•g -1 which is in the same range of values as found by Fox et al. for other polymers, such as polystyrene, polymethylmethacrylate and polyisobutylene, that present a K constant around mol•g -1 [START_REF] Fox | Influence of molecular weight and degree of crosslinking on the specific volume and glass temperature of polymers[END_REF]. One may consider that the samples with the highest molecular weights, namely PA 6I 34 and 41k, have reached the glass transition plateau (within 3 K). M n (g . mol -1 )

T g

Dynamics in the molten state

As explained in section 1.1.2.1, the molecular weight is one of the main structural parameters determining the polymer flow behavior. The master curves of the viscoelastic behavior of unentangled and entangled polyamides were built using time-temperature superposition. Unentangled PA 6I was subsequently fitted using the Rouse model [START_REF] Rouse | A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers[END_REF], while the behavior of entangled polyamides was analyzed using the Reptation model [START_REF] Doi | The Theory of Polymer Dynamics[END_REF].

Time-temperature superposition and master curves

The experimental procedure for rheological measurements was presented in section 2.2.3 and the time-temperature superposition principle was detailed in section 1.1.2.5. Since polyamides may not be stable, besides blocking amine chain ends, a rigorous protocol was used during rheological measurements, as described in section 2.2.3. Using this protocol, it was possible to measure the complete range of frequencies (or temperatures) of polyamide relaxation from nearly the glass transition to the terminal relaxation. The master curves were first obtained at a reference temperature equal to one measured temperature. Then, for comparison at iso-frictional conditions, all master curves were shifted to Tg. To do so, the horizontal shift factor at Tg was determined by extrapolation of the WLF equation to this temperature, using the WLF constants previously determined.

Time-temperature superposition did not work perfectly for the complete temperature range used to build master curves, mostly for low molecular weight polyamides. As explained in section 1.1.2.6, close to the glass transition two phenomena with different temperature dependences take place, resulting in the failure of tTS. This can be easily observed in the tan curves of each polymer, as curves obtained at different temperatures do not overlap. This discrepancy tends to disappear with increasing molecular weight. Thermorheological complexity was previously detected for other polymers, such as polystyrene or polyvinyl acetate [START_REF] Ngai | Identification of different modes of molecular motion in polymers that cause thermorheological complexity[END_REF], thus it is not specific of polyamides or hydrogen bonding polymers.

The master curves of PA 6I polymers with varying molecular weights obtained by applying timetemperature superposition are presented in Figure 61. Classical behavior with increasing molecular weight is observed, with the appearance of an elastic plateau, due to the presence of entanglements, and terminal relaxation times being shifted to lower frequencies. As a consequence of taking Tg (from mDSC measurements) as reference temperature, the G'/G'' crossover points in the glassy region for all curves are located at 10 -1 rad•s -1 , which corresponds to a period of 60 seconds and is perfectly coherent with the time period used for Tg measured by DSC. Near Tg, master curves overlap perfectly, as the mechanisms for relaxation are local and do not depend on the molecular weight. Both PA 6I 34k and PA 6I 41k have molecular weights larger than the molecular weight between entanglements, Me, as a clear elastic plateau of several decades is observed. Since the quantity of branching is slightly higher for PA 6I 41k (Table 5), this can result in a less marked rubbery plateau, since when approaching lower frequencies, smaller chains have already relaxed [START_REF] Trinkle | Van Gurp-Palmen-plot: a way to characterize polydispersity of linear polymers[END_REF], [START_REF] Wasserman | Prediction of linear viscoelastic response for entangled polyolefin melts from molecular weight distribution[END_REF], [START_REF] Cross | Polymer rheology: Influence of molecular weight and polydispersity[END_REF]. Thus, the determination of the molecular weight between entanglements and critical molecular weight was achieved using the experimental data of PA 6I 34k. Since polyamides were not stable at high temperatures, the slopes of 2 and 1 for G' and G'' in the terminal regime, respectively, could not be completely attained especially for higher molecular weights. Me may be determined from the storage modulus G' on the elastic plateau, using equation 1.13 ( ). This parameter may be best determined by using the so-called Van Gurp-Palmen or VGP plot of the phase angle as a function of the complex modulus as shown in Figure 62 [START_REF] Trinkle | Van Gurp-Palmen-plot: a way to characterize polydispersity of linear polymers[END_REF]. The complex modulus at the minimum of the phase angle is around 2.6 MPa, which gives Me equal to 1450 g•mol -1 , for T = Tg and density 1.15 g•cm -3 [START_REF] Cousin | Molecular modelling of polyphthalamides thermal properties: Comparison between modelling and experimental results[END_REF]. Using the equations presented in section 1.1.2.4, the empirical relationship between the molecular weight between entanglements and the critical molecular weight is described. Using equation 1.25, proposed by Fetters et al. [START_REF] Fetters | Packing Length Influence in Linear Polymer Melts on the Entanglement, Critical, and Reptation Molecular Weights[END_REF]- [START_REF] Fetters | Chain dimensions and entanglement spacings[END_REF] the value of the critical molecular weight can be calculated, knowing the packing length, p. The packing length of PA 6I was calculated by molecular dynamics simulation (explained in details in Appendix I). PA 6I has a packing length of 2.3 Å. When one compares the packing length of PA 6I to that of PA 6, equal to 1.976 Å [START_REF] Fetters | Chain dimensions and entanglement spacings[END_REF] there is an increase in the packing length value due to the more rigid and bulky backbone of PA 6I when compared to PA 6. Making the hypothesis that PA 6I behaves in the same way as the polymers reported by Fetters et al. [START_REF] Fetters | Packing Length Influence in Linear Polymer Melts on the Entanglement, Critical, and Reptation Molecular Weights[END_REF]- [START_REF] Fetters | Chain dimensions and entanglement spacings[END_REF], the critical molecular weight Mc for PA 6I was calculated to be equal to 2960 g•mol -1 , using equation 1.25 ( with p in Å).

PA 6I 5k does not show an elastic plateau, though its molecular weight is slightly above the critical molecular weight (Mc). By contrast, PA 6I 34k and PA 6I 41k, which have a molecular weight Mw higher than Mc, shows a clear rubbery plateau. PA 6I 19k is slightly entangled, showing a small incipient rubbery plateau. Thus, PA 6I shows classical behavior as the molecular weight increases, with the onset of a rubbery plateau and terminal relaxation times being shifted to lower frequencies.

In the terminal region, polyamides with a polydispersity close to 2 (and low branching content) show a slope of G' and G'' of 2 and 1, respectively, as expected by the Rouse model. With increasing polydispersity, the slopes of the dynamic moduli are smaller, as the terminal regime is delayed to lower frequencies, due to the existence of a broad distribution of terminal relaxation times. The horizontal shift factors referred to Tg ( ) used to shift each frequency sweep are presented in Figure 63-a together with the fits using WLF equation (equation 1.27). Firstly, a difference in Tg of the PA 6I samples with increasing molecular weight is observed, in accordance to the variation shown in Figure 60. Secondly, a significant difference is observed among the polyamide samples with higher molecular weights (from 32 to 41k). For these four samples, a clear difference in the temperature dependence close to Tg is observed. In fact, the shift factors are steeper, i.e. PA 6I 32 and 41k, which are also the samples with higher content of BHT, have a larger dynamic fragility value. This effect was already observed for other polymers, such as polyglycerol or poly (ether ketone) [START_REF] Schubert | Can Hyperbranched Polymers Entangle? Effect of Hydrogen Bonding on Entanglement Transition and Thermorheological Properties of Hyperbranched Polyglycerol Melts[END_REF], [START_REF] Ahn | Details of Dynamic Mechanical Properties of Dendritic Poly(ether ketone)s in Conjunction with their Highly Branched Structure and Degree of Branching[END_REF], for which it was quantified that branched samples were more fragile than the linear ones, meaning that the fragility increased with increasing degree of branching. This indicates that the branching structure induces a more fragile behavior, as observed as well in the case of PA 6I samples.

For PA 6I samples with approximatively equal BHT contents (PA 6I 5, 19, 33 and 34k), the dynamic fragility index (calculated using Tg from mDSC) increases from 125 to 136, as Mn increases from 2000 to 12000 g•mol -1 , as observed in Table 6 and Figure 63-b. The same trend was observed by Sokolov et al. [START_REF] Kunal | Role of Chemical Structure in Fragility of Polymers: A Qualitative Picture[END_REF] for most non-polar polymers with rigid backbones, such as polystyrene or poly(cyclohexylethylene). Nevertheless, for the same increase in molecular weight, they observed a stronger variation in fragility, of about 40. Dudowicz et al. [START_REF] Dudowicz | Fragility of Glass-Forming Polymer Liquids[END_REF], [START_REF] Dudowicz | The Glass Transition Temperature of Polymer Melts[END_REF] proposed that fragility is related to the packing ability of each polymer, rigid chains and/or bulky side groups inducing difficulties in packing and leaving excess free volume, resulting in higher fragility. One way to understand this effect is by comparing the packing lengths of PA 6I and PS, which are 2.3 Å (result from molecular dynamics simulation) and 3.99 Å [START_REF] Fetters | Chain dimensions and entanglement spacings[END_REF], respectively. Since the packing length increases significantly for PS, it could explain the strong increase in fragility, not observed for PA 6I.

Several studies focused on accounting on Tg variation with increasing molecular weight have been done in the last years [START_REF] Auhl | Linear and nonlinear shear flow behavior of monodisperse polyisoprene melts with a large range of molecular weights[END_REF]. However, besides the variation of Tg with molecular weight, fitted using the Fox-Flory equation, polydispersity and branching also change. It is thus difficult to account for the differences obtained in the WLF variation for different polyamide samples, within a single picture. Based on the 0,0021 0,0022 0,0023 0,0024 0,0025 0,0026 - value of C1 g , the fractional free volume, fg, at Tg can be calculated based on equation 1.43 [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF]. The values of free volume calculated for PA 6I samples are considerably higher than the universal value, generally fg = 0.025, reported in the literature [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF]. Anyway, it is thought that polymers containing double bonds or aromatic rings have a higher free volume at Tg, around 0.030-0.035 [START_REF] Bartoš | Free volume microstructure of amorphous polymers at glass transition temperatures from positron annihilation spectroscopy data[END_REF]. Hence, this difference may possibly be attributed to the aromatic cycle in the PA 6I backbone.

From Rouse model to Reptation model

A polymer with a molecular weight below the critical molecular weight is unentangled and its dynamics in the melt state are described by the Rouse model, for short ideal chains [START_REF] Rouse | A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers[END_REF]. On the other hand, the dynamics of entangled polymers is given by "tube" or reptation models [START_REF] De Gennes | Entangled polymers[END_REF], [START_REF] Doi | The Theory of Polymer Dynamics[END_REF].

The dynamic moduli of unentangled polymers can be described using a polydispersed Rouse model [START_REF] Gray | Dynamic Viscoelastic Behaviour of Low-Molecular-Mass Polystyrene Melts[END_REF], in which a polymer chain is represented as N beads connected by N-1 springs of root mean-square size b.

There is no interaction between the beads, besides that through the connecting springs. To understand more precisely whether hydrogen bonding in polyamide has an effect on the viscoelastic response of polyamides, the obtained master curve of PA 6I 5k was fitted using a regular polydispersed Rouse model (equations 1.31 and 1.32). M0 and l0 were taken as the values of the Kuhn segments which were calculated by molecular dynamics modelling (following the procedure detailed in Appendix I) and are equal to 231 g•mol -1 and 1.2 nm, respectively. The fit with the Rouse model can be done directly with the molecular weight distribution measured by GPC or the Flory distribution (Figure 59) adjusted to best fit the measured distribution. It was verified that using either the experimental or corresponding Flory distribution did not affect the modeled master curves significantly. Fits were performed using a Matlab program, using a least square minimization algorithm and using as a single fitting parameter the reduced friction f = ζlk 2 /MK 2 . By using one single fitting parameter, it means that the absolute value of G' and G'' in the terminal regime are obtained from the Rouse model. Only the data in the terminal region were used, since the Rouse model does not describe the glass transition region. The data were cut off at different frequencies, which provided small variations in the fitting parameter but not in the quality of the obtained fits. This procedure allowed as well defining a standard error. As shown in Figure 64, the Rouse model provides an excellent description of the unentangled PA 6I polymer (PA 6I 5k), which suggests that hydrogen bonds do not have an effect on the viscoelasticity of these polymers in the terminal regime. Finally, it also indicates that the small amount of branching does not influence significantly the rheological behavior of unentangled polyamides. The result of the fitting parameters and the obtained friction coefficients will be presented in Table 9. A more thorough discussion on this subject will be presented along with the Rouse model fits of unentangled ionic copolyamides in section 4.2.2.

For entangled polyamides, the length of the rubbery plateau can be expressed by the ratio between and . The length of the plateau is defined by the molecular weight of the polymer and by its molecular weight between entanglements, as given by equation 1.15, obtained using the Doi and Edwards model (equation 1.15:

) [START_REF] Doi | The Theory of Polymer Dynamics[END_REF], [START_REF] Larson | Definitions of entanglement spacing and time constants in the tube model[END_REF]. It is possible to verify whether PA 6I follows the tube model, by calculating the exponent of the model. In the case of PA 6I 41k the ratio is equal to

. The parameter was taken as the weight-average molecular weight obtained by GPC and the molecular weight between entanglements, Me, as the value calculated by the value of the complex modulus at the minimum of the phase angle. The exponent determined is equal to 3.1, which is in excellent agreement to expectation.

Effect of branching

Throughout the synthesis of polyamides 6I, secondary reactions may occur, leading to the formation of bis(hexamethylene) triamine (BHT), a monomer having three amine sites where the acid monomer can react creating a branched polymer. The activation energy for BHT synthesis, about 32.4 kcal•mol -1 is similar to the activation energy of the condensation reaction, 30.4 kcal•mol -1 [START_REF] Gaillard | Amidification directe[END_REF], which can easily explain the formation of the branched monomer. The triamine monomer is formed between two amide functions, this means that, at high water content in the reactional medium, the condensation reaction is slow and since the reactor is highly charged in NH2 functions (a small excess of diamine is introduced in the reactor to account for the diamine that will leave the reactor along with the distillated water), the BHT formation is favored. One solution to decrease the quantity of branching by BHT formation is by adding a catalyst during polyamide synthesis, which will favor the condensation reaction and thus decrease the quantity of branched functions.

BHT has a strong effect in the obtained GPC spectra, being one of the main sources of the appearance of shoulders in the chromatograms. For example, when comparing two polyamides with similar molecular weights (32 and 34 kg•mol -1 ) but with different BHT contents, the shapes of the two chromatograms are fairly different, as observed in Figure 65. One can easily observe that the chromatogram of the less branched sample (PA 6I 34k) is quite narrower than that of the highly branched sample (PA 6I 32k), which also presents shoulders. The dynamic moduli of two polymers with similar molecular weights and different BHT levels were studied and are presented in Figure 66. In the glassy and transition region, G' and G'' are not significantly affected by the branching content. In these two regions, relaxation is mainly due to local movements, consequently the large scale molecular architecture of polymers does not affect the dynamic modulus, clarifying why branching content does not influence the glassy and transition regions. The rubbery plateau is not affected by the branching and polydispersity (associated to branching). Similarly, the main relaxation time is not significantly affected. One could imagine that the more branched polyamide would have a broadening in the spectrum of relaxation times, due to the broadening of the molecular weight distribution. In any case this was not observed, at least up to 6000 ppm of BHT. It is important to highlight that, after a controlled synthesis, polyamides usually do not contain more than about 4000 ppm of BHT. Thus, we can safely conclude that the BHT level shall not affect the rheological measurements.

Discussion

The viscoelastic behavior over the complete range of polyamide relaxation is a topic rarely mentioned in the literature due to the presence of a crystalline phase in classical polyamides and to the lack of chemical stability during rheological tests [START_REF] Pezzin | Capillary Viscometry of Molten Polycaprolactam[END_REF], [START_REF] Van Ruymbeke | Structure and rheology of branched polyamide 6 polymers from their reaction recipe[END_REF], [START_REF] Acierno | Rheological behavior of polyamide 11 with varying initial moisture content[END_REF], [START_REF] Khanna | New developments in the melt rheology of nylons. I: Effect of moisture and molecular weight[END_REF]. These two problems were prevented by using amorphous and end-blocked polyamides. We were able to develop new amorphous polyamides PA 6I, obtained by the polycondensation of a diamine with an isophthalic acid. To prevent any evolution of the viscosity during rheological measurements, namely by post-condensation, chain ends were blocked during the synthesis step, with a mono functional benzoic acid.

The viscoelastic response of PA 6I shows a typical behavior with increasing molecular weight: a clear rubbery plateau appears and the longest relaxation time is shifted to lower frequencies, as expected by the Rouse and reptation models. The molecular weight between entanglements is estimated at 1340 g•mol -1 as calculated by the value of the plateau modulus. Using empirical equations [START_REF] Fetters | Chain dimensions and entanglement spacings[END_REF], the critical molecular weight was calculated. The determination of the critical molecular weight would not be possible without the realization of molecular dynamics simulation which allowed calculating the value of the packing length. It would be interesting to perform some high temperature viscosity measurements as a function of the polyamide molecular weight to determine experimentally the critical molecular weight of PA 6I and thus confirm whether PA 6I follows the empirical trend developed by Fetters et al [START_REF] Fetters | Packing Length Influence in Linear Polymer Melts on the Entanglement, Critical, and Reptation Molecular Weights[END_REF]- [START_REF] Fetters | Chain dimensions and entanglement spacings[END_REF].

Dynamic moduli of unentangled polyamides were fitted with a polydispersed Rouse model, showing no effect of hydrogen bonds on the shape of the rheological master curves. Entangled PA 6I follows the reptation model with an exponent close to 3.1, which is in excellent agreement to the expectation [START_REF] Van Ruymbeke | Quantitative tube model predictions for the linear viscoelasticity of linear polymers[END_REF].

The effect of branching was also accessed by synthesizing polyamides with different branching contents.

We can conclude that up to 6000 ppm of bis(hexamethylene) triamine, there is no effect on the rheological master curves obtained. Nevertheless, during a controlled synthesis the final BHT content is always smaller than 4000 ppm.

Chapter 4 -Effect of the introduction of ionic interactions on the viscoelasticity of polyamides

In this chapter, the rheology of modified amorphous polyamides with different densities of ionic interactions will be presented. Additionally to interchain hydrogen bonding, ionic interacting groups are added to the polymer backbone. Intermolecular interactions have a strong impact on the rheology of nonpolar or relatively low Tg polymers, as presented in Chapter 1. This was demonstrated in model cases in which strongly interacting groups are diluted and provide a time scale well separated from main chain relaxation time. The case of polyamides is more complex. In systems with a high density of hydrogen bonds, cooperativity of bonds and overlap of time scales raise fundamental issues which have not been described yet. Several polyamides of varying density and strength of interactions were synthesized for this study. In a first part, the influence of the number of ionic groups per chain in unentangled polyamides will be presented, followed by the study of ionic interactions in entangled polyamides and the influence of the counter-ion. The influence of the polymer glass transition temperature will also be discussed. Smallangle X-ray scattering performed in order to investigate the existence of nanophase separation in ionic copolyamides will be presented. Lastly, the viscoelastic behavior of polyamides will be compared to established models for both unentangled and entangled polymers.

Molecular architecture

The molecular weights, as determined by EGT and GPC, and molecular weight distributions by GPC are summarized in Table 7, along with the branching content and glass transition temperatures. For low molecular weights, the number-average molecular weight obtained by EGT is not always in full quantitative agreement with the value obtained by GPC, especially for low molecular weights, due to the truncation of oligomers during GPC analysis.

Ionic copolyamides can be divided into four categories: PA 6I/6LiSIPA samples with nearly equal molecular weight, PA 6I/6LiSIPA with higher molecular weights, PA 6I/6NaSIPA and PA 10I/10NaSIPA. In the first series, it will be possible to observe the influence of increasing ionic content, while keeping the molecular weight constant. The ionic content could not be increased above 20 mol% since the presence of LiSIPA in the reaction media decreases the rate of polycondensation and decreases the molar masses of copolyamides PA 6I/6LiSIPA [START_REF] Cammage | Synthesis of copolyamides based on PA 66 bearing lithium sulfonate groups and having unique thermal properties[END_REF]. In fact, it is assumed that the presence of the sulfonate groups SO3Li or SO3Na slows down polycondensation possibly due to the existence of ionic interactions between -SO3 -and the ammonium moiety -NH3 + (from the polyamide salt), leading to partial deactivation of amine ends [START_REF] Vouyiouka | Solid state postpolymerization of PA 6,6: The effect of sodium 5-sulfoisophthalic acid[END_REF]. It is likely that ionic groups are in fact located at the chain ends as it is difficult to further increase ionic content and/or molecular weight. However, it was not possible to completely corroborate this idea. To achieve a given number of ionic groups per chain, since the LiSIPA content could not be further increased, the molecular weight of ionic copolyamides was increased achieving a minimum of 2 ionic groups per chain in average. Then, the influence of the counter-ion was accessed by synthesizing a copolyamide using a sodium salt of 5-sulfoisophthalic acid. The influence of the polymer glass transition was studied, by synthesizing a polyamide with a longer aliphatic segment (C10), which reduces the glass transition temperature by increasing the overall flexibility of the polymer chain.

The first effect of the introduction of interacting groups was observed on the glass transition temperature.

The Tg of the non-substituted PA 6I varies from 106 to 126°C with increasing molecular weight, while it increases approximately by 10 to 40°C with increasing ionic content, as it can be observed in Figure 67. This result is coherent with results reported in the literature [START_REF] Chen | Ionomer dynamics and the sticky Rouse model[END_REF], [START_REF] Kim | Introduction to Ionomers[END_REF], [START_REF] Weiss | Viscoelastic Behavior of Very Lightly Sulfonated Polystyrene Ionomers[END_REF], [START_REF] Cammage | Synthesis of copolyamides based on PA 66 bearing lithium sulfonate groups and having unique thermal properties[END_REF]. In Figure 67, it is possible to observe that the introduction of ionic groups on the PA 6I backbone induces a nearly linear increase of Tg, at least up to 20 mol%. For the 20 mol% copolyamide, the molecular weight is slightly lower than the other copolyamides in the same series. This could in fact explain why the glass transition does not follow the linear trend of the less ionically modified polyamides, as in this range of molecular weight, a small difference in the molecular weight can induce a relatively large difference in the glass transition temperature, according to the Fox-Flory plot. Furthermore, the increase in Tg seems to be independent of the nature of the counter-ion, as the slopes of the three lines seem to be similar.

As expected, compared to other copolyamides, PA 10I/10NaSIPA presents a lower glass transition temperature. When compared to PA 10I samples, which were not used in this work due to the presence of a crystalline phase, the increase in Tg is proportional to that in PA 6I samples with 10 mol% of LiSIPA monomers, as the slopes of the four lines seem to be similar, as shown in Figure 67.

As it can be observed in Table 7, the values of polydispersity are often larger than 2, particularly for copolyamides with higher molecular weights. As in the previous chapter, the experimental data were fitted using a Flory distribution, as given in equation 1.36. The GPC data along with the fit for linear polycondensation are presented in Figure 68 for some polyamides. Generally speaking, the Flory distribution for linear polycondensation does not well describe the experimental GPC data, indicating that the synthesis did not result in a statistical assembly of linear chains, which can be explained by the branching reactions previously mentioned. Ionic content (%)

Influence of the number of ionic groups per chain in unentangled polyamides

The objective was to introduce interactions which are stronger than the standard amide-amide H-bonds, present in polyamides, by creating a complex structure between amide groups and metal cations. In this section, the viscoelastic behavior of PA 6I/6LiSIPA with increasing ionic content and comparable molecular weight will be presented. The aim is to understand the influence of the increasing number of ionic groups per chain on the viscoelastic behavior in the molten state of unentangled polyamide melts. The understanding of the rheological response of these polymers will be further investigated using the Rouse model, as well as through small-angle X-ray scattering.

Time-temperature superposition and master curves

The copolymers of PA 6I/6LiSIPA with 5, 10, 15 and 20 mol% of LiSIPA with a Mn around 5000 g•mol -1 correspond respectively to 0.3, 0.6, 1 and 1.25 ionic groups per chain on average. The content of ionic co-monomers could not be increased beyond 20 mol%, as explained previously. As stated by Chen et al. [START_REF] Chen | Ionomer dynamics and the sticky Rouse model[END_REF], [START_REF] Huang | Reversible Gelation Model Predictions of the Linear Viscoelasticity of Oligomeric Sulfonated Polystyrene Ionomer Blends[END_REF], [START_REF] Chen | Viscoelasticity of Reversible Gelation for Ionomers[END_REF], the gel point corresponds to one ionic group per chain on average. At this point, changes in the viscoelastic behavior of polyamides should be observed.

The master curves of PA 6I without ionic groups and with similar molecular weight were used as a reference to compare the influence of ionic interactions in PA 6I/6LiSIPA. The master curves are presented in Figure 69. Apart from small differences in the terminal regime coming from minor variations in molecular weight distributions between PA 6I and ionic copolymers, the shapes of the curves were not affected by the presence of the ionic groups, with both sets of G' and G'' curves overlapping in the complete range of frequencies of polyamide relaxation, meaning that the relaxation time distributions (referred to Tref Tg) are essentially identical. Thus, apart from the Tg shift, there is no observable effect on the rheological behavior of these polymers.

However, though glass transition as measured by mDSC was used as reference temperature, the G'/G'' crossover point did not occur exactly at the same frequency. To that end, the reference temperature was took as Tg+3, Tg +5, Tg +8 and Tg +3°C for copolymers with 5, 10, 15 and 20 mol% of ionic groups, respectively. This apparently small variation in the reference temperature is in fact quite important, as it corresponds to a shift of almost one decade in frequency, thus in relaxation time, as schematized in Figure 70. This difference in Tref seems to be correlated to the width of the glass transition, as shown in Figure 72. The width of the glass transition increases from 7 to 13°C with increasing ratio of ionic groups, which reflects a slightly increasing degree of dynamical heterogeneities at the glass transition. Note that for the PA 6I series, the Tg width does not change significantly as the molecular weight increases. For the copolyamide with 20 mol% of ionic groups, it becomes asymmetric and steeper on the high temperature side. For the copolymer with 20 mol% of ionic co-monomers, the difference between Tg and Tref was about 3°C, which seems to deviate from the trend of other results for PA 6I/6LiSIPA up to 15 mol%. One might expect that the variation of Tref -Tg with the LiSIPA content would be linear. However, the molecular weight distribution is quite narrower for the copolymer with 20 mol% of LiSIPA, and that could explain why neither Tg (Figure 67) and its width (Figure 72) follow the linear trend of the less ionically modified polyamides, and might explain as well the deviation of Tref -Tg from the trend of other ionic polymers.

G' (Pa)

As the reference temperature, to compare master curves of different copolyamides, seem to depend on the width of the glass transition, using raw data may be an easier and less misleading way to understand the dynamics of unentangled ionic copolyamides. For instance, using the Cole-Cole plot of the data (G'' as a function of G'), as shown in Figure 71, demonstrates that the relaxation spectrum is not affected by the introduction of ionic groups in the dynamics of unentangled polyamides. This kind of analysis allows comparing rheological data without raising questions about the reference temperature used. It allows as well concluding with more certainty on the lack of effect of the ionic groups on the viscoelastic behavior of polyamides in the molten state.

The polyamide density as a function of temperature used to calculate the vertical shift factors was considered to be independent of the ionic fraction and equal to that of PA 6I. The horizontal shift factors referred to Tg ( ), used to shift each frequency sweep, were fitted using the WLF equation (1.27) and are presented in Figure 73-a. The first remark is the difference in Tg of the polyamide samples with increasing ionic content, as verified by mDSC. The WLF constants are reported for each polyamide sample in Table 8, along with the values of the fractional free volume at Tg. As for PA 6I, the values are slightly superior to the universal value of 0.025. Nevertheless, this difference may possibly be attributed to the aromatic cycle in the polyamide backbone [START_REF] Bartoš | Free volume microstructure of amorphous polymers at glass transition temperatures from positron annihilation spectroscopy data[END_REF] and seems to be independent of the ionic content.

The same data can be plotted as a function of Tg/T as in Figure 73-b, resulting in the so-called Angell plot.

In this plot, the variation of the fragility index m with the fraction of lithium sulfonated groups is observed, as well as in Table 8. For the same molecular weight, the fragility was strongly affected by the introduction of lithium sulfonated groups in the PA 6I backbone. Qualitatively, it is possible to observe in the Angell plot, given in Figure 73-b, that the slope near Tg decreases with increasing ionic content.

Our results may be compared to results reported in the literature for other polymers. Fragility indices may be calculated from reported WLF constants. For example, Eisenberg et al. [START_REF] Kim | Introduction to Ionomers[END_REF], [START_REF] Shohamy | The effect of ion aggregation on some melt properties of styrene ionomers[END_REF] show a similar decrease of the fragility with increasing ionic content in substituted polystyrene ionomers containing up to 7.7 mol% of ionic groups. Similarly, in the case of poly(ethylene oxide)-based ionomers, the authors found that, above 10% ionic groups, the strength parameter, D, increased with increasing ion content. The strength parameter corresponds to the inverse of the fragility index, meaning that higher D corresponds to less fragile, or more Arrhenius-like behavior [START_REF] Fragiadakis | Molecular mobility and Li+ conduction in polyester copolymer ionomers based on poly(ethylene oxide)[END_REF], corresponding to a decrease of dynamic fragility, as observed in ionic modified polyamides.

The origin of this decrease in dynamic fragility by introduction of ionic groups is still not clear. It would be possible to explain this behavior by assuming that ions increase the glass transition temperature by acting as intra-chain, rather than inter-chain, interactions, which would increase the chain rigidity [START_REF] Cruickshank | The role of ionic salts in determining Tg and ionic conductivity in concentrated PEG electrolyte solutions[END_REF]. However, it is not evident why such an increase of chain rigidity would lead to less fragile behavior. As described in Chapter 1, fragility characterizes the sensitivity of molecular dynamics to temperature. A material is categorized as "strong" if its relaxation dynamics change relatively slowly. Conversely, the relaxation dynamics of fragile materials vary rapidly upon cooling through Tg. Sokolov et al. [START_REF] Kunal | Role of Chemical Structure in Fragility of Polymers: A Qualitative Picture[END_REF] attributed changes in fragility to chain rigidity, influenced by side groups rigidity and intermolecular interactions. We may speculate that the formation of a lithium-carbonyl complex would increase packing efficiency, leading to a better cooperativity between chains and thus to slower dynamics close to Tg.

This conclusion is also supported by the fact that in the Angell plot, shown in Figure 18, non-polar glass formers show different behavior when compared to glycerol. For example, o-terphenyl shows higher fragility than glycerol, which may be due to the interactions present in the case of glycerol. Thus, the same effects could then be present in polyamide materials.

The above tentative explanations are quite speculative. The Arrhenius plot (log τ vs 1/T) shows more explicitly the differences in rheological behavior in the various temperature domains (WLF domains close to Tg, Arrhenius at high temperature) than the Angell plot (log τ vs Tg/T). Thus it may show more explicitly whether different mechanisms/interactions are contributing in the various temperature domains. Figure 73-a shows the variation of the horizontal shift factors as a function of 1/T, in which WLF curves were extrapolated to higher temperatures. In the vicinity of Tg the curves are shifted according to the respective values of Tg of the copolymers, with nearly parallel slopes. As temperature increases, curves tend to become closer to each other, which means that the dynamics (in terms of dominant time scales) of the various polymers, which is considerably different in the Tg vicinity (as Tg's are quite different), becomes more similar on increasing temperature.

It appears that all fitted curves tend to merge at high temperature. Our hypothesis is that, at high temperatures, all curves should merge as substituted and non-substituted polymers should present the same viscosity at the same temperature, as interactions should not be effective any longer. Unfortunately, so far it was not possible to perform viscosity measurements at higher temperatures to validate this hypothesis, due to the lack of stability of the materials at higher temperatures.

Fit with Rouse model

As for PA 6I 5k, the dynamic moduli of unentangled PA 6I/6LiSIPA were fitted using a polydispersed Rouse model. For ionic copolymers, it was assumed that neither or change significantly, since the PA 6I backbone was kept constant and only a limited fraction of ionic groups were introduced. As shown in Figure 74, the Rouse model provides an excellent description of the viscoelastic behavior of unentangled polyamides used in this work.

As explained in the first chapter, the Rouse model [START_REF] Rouse | A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers[END_REF] was conceived for ideal chains in which there is no interaction between the monomers, beside that through the connecting springs. Despite the introduction of ionic groups, the Rouse model can still be used to describe the dynamic moduli of PA 6I/6LiSIPA polymers, which suggests that ionic interactions do not have an effect on the viscoelasticity of these polymers. As for PA 6I 5k, fits were performed using a Matlab program, using as a single fitting parameter the reduced friction, f = ζlk 2 /MK 2 , and both GPC and Flory's distribution. It was verified that using either the experimental or corresponding Flory distribution did not affect the modeled master curves significantly.

Using the fitting parameter, it is possible to calculate the friction coefficient, Although diverse interpretations on the differences of the friction coefficient at Tg are still speculative, the estimated monomeric friction coefficient are in the 10 -2 -10 7 range at Tg for various polymers [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF]. In Table 9, it is possible to observe the values of fitting parameter and the corresponding monomeric friction coefficient for the analyzed polymers, using both Flory and GPC molecular weight distribution. At first sight, it is observed that the friction coefficient decreases with the introduction of ionic groups. Taking for example a methacrylate polymer with increasing side group length (methyl, ethyl, … , n-docosyl), as the length of the side group increases, the friction coefficient decreases. This is explained as a function of the efficiency of the local packing, which is less efficient for longer side groups, related to an increase of the specific volume, normally attributed to an increase in the free volume.

The friction coefficients calculated using both GPC and Flory's distribution are quite similar, up to 15 mol% of LiSIPA, while they are quite different for the copolyamide bearing 20 mol% of ionic monomers. One explanation for this result is that, out of the 5 polymers, PA 6I/6LiSIPA with 20 mol% of ionic groups is the one for which Flory's distribution fits the worst the experimental data, even if both Rouse fits (using GPC and Flory's distribution) describe properly the dynamic moduli of this polymers. In this case, the most reliable friction coefficient is the one corresponding to the GPC data.

In Figure 74 (and as it was presented for unentangled polyamides in Figure 64), the curves obtained from the Rouse model overestimate the moduli when reaching about 10 -4 rad/s. This may be due to the fact that the Rouse sub-chain was identified as the Kuhn segment, which is about one monomer. Thus, all Rouse modes down to the monomer scale, contribute to the computed modulus, and this can certainly lead to some overestimation when coming towards high frequencies.

Note that we have used the Rouse model to illustrate the fact that the dynamics of polyamides was not affected by the presence of ionic groups (apart from the Tg shift). We did not attempt to provide a quantitative fit of the whole master curves, which would involve an additional glassy contribution in the high frequency regime, as proposed by Majesté et al.and Gray et al. [START_REF] Majesté | Viscoelasticity of low molecular weight polymers and the transition to the entangled regime[END_REF], [START_REF] Gray | Dynamic Viscoelastic Behaviour of Low-Molecular-Mass Polystyrene Melts[END_REF].

Small Angle X-ray scattering

Small-angle X-ray scattering technique permits to investigate the existence of nanophase separation of ion-rich aggregates in polyamides, typically with a characteristic dimension of 2-4 nm [START_REF] Kim | Introduction to Ionomers[END_REF], [START_REF] Weiss | Viscoelastic Behavior of Very Lightly Sulfonated Polystyrene Ionomers[END_REF]. The results of SAXS data for copolymers of PA 6I/6LiSIPA of 5000 g•mol -1 , with 0, 5, 10, 15 and 20 mol% of ionic groups are shown in Figure 75.

The empty beam signal was subtracted from scattering data of the samples and data were normalized by the sample thickness, as shown in Figure 75-a. The signal of the PA 6I reference polymer was then subtracted and the resulting intensity normalized by the volume fraction of ionic groups (equation 2.11 and 2.12), as described in section 2.2.2.4, obtaining the curves presented in Figure 75-b. Scattering data of low molecular weight copolymers (around 5000 g•mol -1 ), do not show any distinct scattering peak which would be characteristic of ionic group aggregation and phase separation. The increase in the intensity at low q values is attributed to the inhomogeneous distribution of isolated ion pairs at large scales as pointed out in the literature [START_REF] Weiss | Rheological behavior of oligomeric ionomers[END_REF], [START_REF] Yarusso | Microstructure of ionomers: interpretation of small-angle x-ray scattering data[END_REF].

In Figure 75-a, the peak observed at 6 nm -1 corresponds to the structure factor of the amorphous phase, also observed in WAXS experiments, and it is present for all polyamides, modified and non-modified, meaning that it is not due to ionic clustering. It corresponds to the peak observed in WAXS diffractograms at 2θ = 10° (see Figure 51) as both ranges of scattering angles overlap. Also, the intensity of ionic copolyamides is always larger than that of PA 6I, as PA 6I/6LiSIPA samples have higher atomic electron density contrast when compared to PA 6I, which explains the systematic increase in the measured intensity with increasing ionic content. ) q (nm -1 )

Influence of the number of ionic groups per chain in entangled polyamides

Due to the low reactivity of sulfoisophthalic acid lithium salt, it was not possible to further increase the ionic fraction of the unentangled copolyamides presented in the previous section. In order to achieve a higher number of ionic groups per chain, copolyamides with a higher molecular weight were synthesized, while maintaining an ionic fraction of 10 mol %.

The introduction of ionic groups in entangled polymers might be responsible for a behavior like in the Sticky-Reptation model, first described by Stadler et al. [START_REF] Müller | Influence of hydrogen bonding on the viscoelastic properties of thermoreversible networks: analysis of the local complex dynamics[END_REF]. In this case, an additional relaxation time corresponding to the relaxation time of the thermoreversible network of stickers (ionic groups) may be observed.

Time-temperature superposition and master curves

The master curves of a PA 6I with comparable molecular weight distributions were used as references to compare the influence of ionic interactions in the PA 6I/6LiSIPA copolymer with increased molecular weight. The master curve of this polymer is shown in Figure 76. Due to the higher polydispersity of the copolyamide of high molecular weight (Ip = 3.7), it was chosen to compare it with two PA 6I samples of similar Mn and Mw, PA 6I 19k and PA 6I 34k respectively. Apart from small differences in the terminal regime and in the plateau modulus, master curves overlap in the complete range of frequencies, showing no increase in the length of the rubbery plateau due to different relaxation time of the ionic interactions. While the differences in the terminal regime are attributed to variations in the molecular weight distribution, the differences in the plateau modulus are more complex. As mentioned before for unentangled polyamides, both decrease in dynamic fragility and decrease in the monomeric friction coefficient can be related to an increase in the rigidity of the polymer chain. This can as well result in variations in the packing length of the polymer, and thus in the molecular weight between entanglements. In fact, this can be observed in the decrease of the plateau modulus, taking into account that due to high polydispersity the value of the effective rubbery plateau can be affected.

Taking the minimum of the phase angle in the van-Gurp Palmen plot of PA 6I/6LiSIPA 29k, it is possible to determine the value of the plateau modulus (around 2.3 MPa) as shown in Figure 77, which is smaller than that of PA 6I. The value of the molecular weight between entanglements is equal to 1780 g•mol -1 , which compared to the molecular weight between entanglements of PA 6I (determined in section 3.2.1) states an increase of about 30%.

The increase in the molecular weight between entanglements is responsible for the rheological behavior of PA 6I/6LiSIPA 90/10 29k being similar to that of PA 6I 19k. For the same chain length, a polymer with higher molecular weight between entanglements will be less entangled, hence the behavior observed for PA 6I/6LiSIPA 90/10 29k. Since Me increases, also the packing length should slightly increase, as Me p 3 . Nevertheless, since molecular dynamics simulations are quite complex to implement in ionic systems, the value of the packing length was not calculated. Therefore, it was not possible to calculate the critical molecular weight for the 10 mol% PA 6I/6LiSIPA samples. As for PA 6I 19k (presented in the previous chapter), the molecular weight of PA 6I/6LiSIPA 90/10 29k is not sufficiently high, meaning that this copolymer is poorly entangled showing a small incipient rubbery plateau, which cannot be described using the Doi and Edwards model [START_REF] Doi | The Theory of Polymer Dynamics[END_REF], as it is in the transition between the two models for entangled and unentangled polymers.

The values of the WLF constants, obtained by fitting the horizontal shift factors, as demonstrated in Figure 78, referred to Tg, are presented in Table 10, along with the values of free volume at Tg and dynamic fragility. The fragility, as well as the average molecular weight of the ionic copolyamide are in between those of the two PA 6I reference samples. Thus, it is not possible to conclude about a systematic decrease of dynamic fragility with the introduction of ionic groups. The decrease in fragility can be easily identified in the slope of the Angell plot given in Figure 78-b.The free volume at Tg seems to be slightly higher for ionic samples. This can be explained by the increase of packing length linked to the increase in molecular weight between entanglements. As the packing length is proportional to the inverse of density (equation 1.23), a higher packing length implies a decrease in the density, and thus an increase in the free volume at Tg.

Small Angle X-ray Scattering

The analysis of this polymer by small-angle X-ray scattering was performed and is shown in Figure 79, which shows the normalized scattering data from ionic groups, as described in section 2.2.2.4. As for unentangled ionic copolyamides, the intensity of PA 6I/6LiSIPA 90/10 29k is always higher than that of PA 6I, as a result of the higher electron density contrast in ionic copolyamides. The scattering data of PA 6I/6LiSIPA 90/10 29k presents a clear peak approximately located at q 0.8 nm -1 . This corresponds to a distance between ion rich domains estimated as . We may speculate that the appearance of a scattering peak for this polymer may be due to a decreased miscibility as the chain length increases. The strong increase in the scattering data as q 0 is characteristic of ionomers. This may be attributed to a heterogeneous distribution of isolated ion pairs at large scales [START_REF] Weiss | Rheological behavior of oligomeric ionomers[END_REF], [START_REF] Yarusso | Microstructure of ionomers: interpretation of small-angle x-ray scattering data[END_REF].

Yarusso and Cooper [START_REF] Yarusso | Microstructure of ionomers: interpretation of small-angle x-ray scattering data[END_REF] have proposed a model to fit the scattering data obtained for ionomers. This model consists in ion rich spherical domains in which the closest approach radius, RCA, is larger than the particle radius, R. They justify this hypothesis by assuming that each ionic aggregate must certainly be surrounded by a layer of polymer depleted in ions.

Scattering data were analyzed quantitatively to retrieve information about the structure of the ionic copolymer. The upturn at low q values, corresponding to large scale heterogeneities, was ignored in the modelling procedure. The proposed scattering function contains two different contributions, as shown in equation 4.1. The first factor, FB(qR) is the form factor, related to the size and shape of scattering domains. The second factor is a structure factor, related to the spatial distributions and exclusion volume of domains [START_REF] Yarusso | Microstructure of ionomers: interpretation of small-angle x-ray scattering data[END_REF]. The form factor was modeled by a Beaucage function [START_REF] Beaucage | Small-Angle Scattering from Polymeric Mass Fractals of Arbitrary Mass-Fractal Dimension[END_REF] as shown in equation 4.2. This function contains a term which describes the so-called Guinier regime at qR < 1 and a term which describes the Porod regime, at qR > 1. where corresponds to the Thomson factor (electron scattering cross-section), is the squared electron density contrast between ionic clusters and polyamide matrix, corresponds to the volume fraction of ionic clusters, corresponds to the radius of the ionic clusters, erf corresponds to the error function, which is equal to zero for qR < 1 and to one for qR > 1 and BG corresponds to the background value. A schematic representation of these ion domains is shown in Figure 80, where all considered parameters are represented.

Since the fraction of ionic groups inside the ion rich domains is unknown, also the average electron density of the clusters is unknown. It was thus modeled as a function of the fraction of ionic groups in each ion domain, , as given in equation 4.3. The volume fraction of ionic groups in each ion domain, , is related to the volume fraction of ion domains in the polymer, , and to the stoichiometric volume fraction of ionic groups introduced during copolyamide synthesis, , as given in equation 4.4.

Using these two equations, it is possible to combine all unknown parameters into three fitting parameters. Thus, the fit was done by adjusting: the radius R of the particles, the fraction of ion domains in the polymer matrix and which corresponds to the ratio between RCA and R. Isolated ion pairs in the matrix were neglected. Note that this procedure gives a unique set of parameters. ) q (nm -1 ) PA 6I/6LiSIPA 90/10 29k Fit using Yarusso and Cooper model Combining equation 4.1 to 4.4 allowed fitting the scattering data as demonstrated by the black fit curve, as presented in Figure 79, which corresponds to the presence of ionic domains with 4.0 nm of radius with a volume fraction = 0.25. As the volume stoichiometry of SO3Li groups introduced in the reactor is known, it is possible to determine the volume fraction of ionic groups in each ionic domain, equal to 0.07. This value is quite comparable to the reaction volume stoichiometry, equal to 0.0174, which means that phase separation is weak. In reality, ionic aggregation is less visible than in sulfonated polystyrene [START_REF] Weiss | Rheological behavior of oligomeric ionomers[END_REF], [START_REF] Kim | Introduction to Ionomers[END_REF] since polyamide is a highly polar polymer, meaning that the tendency of ionic groups to aggregate will be smaller. It is also possible to determine the number of ionic groups per ionic domain, which is around 400 SO3Li group. These results may be rather interpreted as heterogeneities in the ionic group dispersion, leading to the presence of ion-rich regions without true ionic clustering. Note that a more general model with non-spherical domains could be used as well. It would give semi-quantitatively similar results as regards the characteristic size d and contrasts.

Influence of the counter-ion

The influence of the nature of the counter-ion was also studied by synthesizing a copolyamide using a sodium 5-sulfoisophthalic salt. The choice of the counter-ion was done according to results reported in the literature. Several studies were done in order to study the miscibility of polyamide and non-polar polymers. In references [START_REF] Molnar | Miscibility of polyamide-6 with lithium or sodium sulfonated polystyrene ionomers[END_REF], [START_REF] Rajagopalan | Molecular interpretation of miscibility in polyamide-6 blends with alkali ionomers of sulfonated polystyrene[END_REF], [START_REF] Yoshikawa | Rheological properties of blends of lithiumor sodium-sulfonated polystyrene ionomers with polyamide 6[END_REF]- [START_REF] Feng | Compatibilization of polymer blends by complexation. 1. spectroscopic characterization of ion-amide interactions in ionomer/polyamide blends[END_REF], the authors studied blends of PA 6 with polystyrene, which are non-miscible due to very low entropy of mixing. Miscibility can be enhanced when one or both of the polymers are functionalized with specific interacting groups, such as ionic groups. They studied the effect of adding lithium or sodium sulfonated polystyrene (LiSPS or NaSPS) to the blend with PA 6. In DMA as well as near infrared experiments results reported in the literature [START_REF] Rajagopalan | Molecular interpretation of miscibility in polyamide-6 blends with alkali ionomers of sulfonated polystyrene[END_REF], [START_REF] Yoshikawa | Rheological properties of blends of lithiumor sodium-sulfonated polystyrene ionomers with polyamide 6[END_REF], it is observed that PA 6 is not miscible with non-modified PS or NaSPS, since two loss tangent peaks are observed, while for the blend of PA 6 with LiSPS, one single transition is observed, indicating that the blend becomes miscible. This is attributed to interactions of lithium sulfonate groups with amide groups, which results in a miscible blend.

In conclusion, polyamide and sodium sulfonated polystyrene blend were concluded to be immiscible, due to the lack of interactions between sodium cations and amide functions from the polyamide. The reason stated is that the interaction between Na + and carbonyl group is much weaker than that of Li + , and it is not capable of interacting strongly with the amide groups to leave its sulfonate environment [START_REF] Kim | Introduction to Ionomers[END_REF], [START_REF] Rajagopalan | Molecular interpretation of miscibility in polyamide-6 blends with alkali ionomers of sulfonated polystyrene[END_REF]. In reality, several studies of mixtures of alkali metals with polyamides have presented that the lithium salts give rise to significant interactions, whereas sodium salts do not. Keeping this in mind, it was decided to synthesize polyamides bearing sodium sulfonate functions, as it could, contrarily to lithium groups, form ionic clusters with a well-separated relaxation time and different from that of the chain.

Time-temperature superposition and master curves

Two copolyamides bearing sodium sulfonate groups with different molecular weights were synthesized. The master curves of these two polymers built using time-temperature superposition are presented in Figure 81 and Figure 82, for the copolymers with lower and higher molecular weights, respectively. Master curves of PA 6I samples with a similar molecular weight were used as a reference. For the low molecular weight sample, despite the fact that master curves do not overlap in the entire range of frequencies, due to different molecular weight distributions, there are no significant differences in the viscoelastic response of PA 6I/6NaSIPA samples compared to PA 6I. A second polymer with a high number of ionic groups per chain was synthesized. PA 6I/6NaSIPA 90/10 48k has at least 5 ionic groups per chain on average, being then well above the threshold for presenting a sticky-Rouse behavior. In Figure 82, it is observed that the dynamic moduli of this sample has the same shape as its PA 6I reference with similar molecular weight, without presenting an ion domain relaxation as it would be expected by the reversible gelation or sticky-reptation models [START_REF] Leibler | Dynamics of reversible networks[END_REF], [START_REF] Huang | Reversible Gelation Model Predictions of the Linear Viscoelasticity of Oligomeric Sulfonated Polystyrene Ionomer Blends[END_REF], [START_REF] Chen | Viscoelasticity of Reversible Gelation for Ionomers[END_REF]. In Figure 82, it is also observed that, even though it has a higher molecular weight than the PA 6I reference, PA 6I/6NaSIPA 90/10 48k shows a less entangled behavior, as observed in the values of longest relaxation times of these two samples, referenced to Tg. In fact, as it is the case for lithium sulfonated polyamide samples, the molecular weight between entanglements is slightly different. The dynamic moduli of entangled PA 6I/6NaSIPA 90/10 48k is shown in Figure 82-a, from where it is possible to determine the value of the molecular weight between entanglements for the sodium sulfonate copolyamides, equal to 1820 g•mol-1. Compared to the molecular weight between entanglements of PA 6I (1450 g•mol -1 as determined in section 3.2.1), this is an increase of about 20%, slightly lower than the one observed for PA 6I/6LiSIPA.

The reference temperatures used to shift master curves were close but not identical to Tg of each polymer. Contrarily to lithium sulfonated copolyamides, the reference temperature used to shift PA 6I/6NaSIPA copolymers was slightly below the glass transition temperature measured by DSC. However it is still not clear why this difference was observed.

The values of the WLF constants, obtained by fitting the horizontal shift factors, referred to Tg, are presented in Table 11, along with the values of free volume at Tg and dynamic fragility. The free volume at Tg seems to increase with the introduction of ionic groups. This increase may be explained by the increase of the packing length related to the increase in molecular weight between entanglements and thus a decrease in the polymer density and an increase in the free volume at Tg. The dynamic fragility decreases with the introduction of NaSIPA monomers in the PA 6I backbone, as it can be observed in the Angell plot, given in Figure 83-b. As for lithium sulfonated copolyamides, the explanation for this decrease is still not clear.

Small Angle X-ray Scattering

The analysis of sodium sulfonated polyamides by small-angle X-ray scattering was performed and results are shown in Figure 84, in which the signal of the PA 6I reference polymer was subtracted and the resulting intensity normalized by the volume fraction of ionic groups, as described in section 2.2.2.4.

The intensity was always higher in sodium sulfonate polyamides than in PA 6I, as a result of the higher electron density contrast. Scattering data of PA 6I/6NaSIPA samples present a clear peak, at approximately 1 nm -1 , which corresponds to a distance between ion rich phases estimated calculated as . As for PA 6I/6LiSIPA, the scattering data were fitted using the Yarusso-Cooper model [START_REF] Yarusso | Microstructure of ionomers: interpretation of small-angle x-ray scattering data[END_REF], ignoring the increase in the intensity at low q values, generally associated to heterogeneities in the ionic group dispersion.

Combining equation 4.1 to 4.4, used to prepare the fit presented Figure 84, it was possible to determine the presence of ionic domains between 4.7 to 6.4 nm of radius with a volume fraction in the sample volume from 0.12 to 0.31. Knowing the volume stoichiometry of SO3Na groups introduced in the reactor, it is possible to determine the volume fraction of ionic groups in each ionic domain, around 0.16 and 0.07. There is an increase in ionic aggregation when compared to lithium substituted copolyamides, which may be possibly related to the different interactions of alkali metal cations with carbonyl groups. If sodium cations tend to less interact with polyamide backbone, as expected by the miscibility studies presented above [START_REF] Molnar | Miscibility of polyamide-6 with lithium or sodium sulfonated polystyrene ionomers[END_REF], [START_REF] Rajagopalan | Molecular interpretation of miscibility in polyamide-6 blends with alkali ionomers of sulfonated polystyrene[END_REF], [START_REF] Yoshikawa | Rheological properties of blends of lithiumor sodium-sulfonated polystyrene ionomers with polyamide 6[END_REF]- [START_REF] Feng | Compatibilization of polymer blends by complexation. 1. spectroscopic characterization of ion-amide interactions in ionomer/polyamide blends[END_REF], thus it can explain the increasing aggregation tendency. Nevertheless, this value is comparable to the reaction volume stoichiometry, meaning that phase separation remains weak, due to the polarity of the polyamide backbone. It is also possible to determine the number of ionic groups per ionic domain, around 600 SO3Na groups. I(q) (mm - 1) q (nm -1 ) PA 6I/6NaSIPA 90/10 16k PA 6I/6NaSIPA 90/10 48k Fit using Yarusso and Cooper model

Influence of the polymer glass transition temperature

In the literature, the effect of ionic interactions was so far mostly observed in low Tg polymers such as phosphonium siloxanes or poly(ethylene oxide) or polystyrene samples, which presented Tg values lower than 100°C. The polyamides studied in this work have high glass transition temperatures, above 120°C in the case of unentangled polyamides and above 150°C in the case of entangled polyamides. It is possible that, at high temperatures T > Tg interactions are no longer active, which could explain the results obtained so far. In fact, the lack of an ion domain relaxation in the dynamic moduli of polyamides can be explained by the short lifetime of ionic association at temperatures above the glass transition of ionic copolyamides.

Time-temperature superposition and master curves

To test this hypothesis, polyamides with a lower glass transition temperature were synthesized, while maintaining the amorphous character of polyamides. To do so, the diamine chain segment was increased, by replacing the hexamethylene diamine by decamethylene diamine (C10N2H24). The increase in the aliphatic length provided a decrease in Tg of about 30°C, as verified by DSC measurements. PA 10I samples were synthesized in order to compare the effect of interactions in polymers with the same backbone, i.e. to prevent changes in the rheological behavior related to the polymer backbone. It was observed that PA 10I samples were able to crystallize during the rheological measurements, making it not possible to obtain master curves in the complete range of polyamide relaxation. The crystallization behavior was studied by differential scanning calorimetry experiments. Above the glass transition, PA 10I samples crystallized, crystallization being more pronounced for small cooling rates and smaller molecular weights. Taking into account the average value of 200 J•g -1 as the enthalpy of melting [START_REF] Guérin | Polyamides PA[END_REF], [START_REF] Preda | Dynamics of polyamide in the solid state in presence of solvents and in the molten state[END_REF] of a polyamide sample, PA 10I exhibited a crystalline degree up to 10%. To evaluate the effect of ionic groups on the rheological behavior of PA 10I/10NaSIPA, PA 6I master curves with similar molecular weight were thus used as a reference. The master curves of PA 10I/10NaSIPA copolymer were built using timetemperature superposition principle, using as reference temperature the glass transition of each polymer, and the resulting dynamic moduli are presented in Figure 85.

The understanding of polymer relaxation is thus a bit complex, since in this case, the polymer backbone could not be kept constant. One could suppose that master curves present the same shape, apart from differences in the glassy and rubbery plateau values and in the slopes of G' and G'' in the terminal regime. While the differences in the terminal regime are attributed to variations in the molecular weight distributions, the differences in the rubbery and glassy modulus are more complex. As mentioned before a variation on the rigidity of the polymer chain can affect the packing length of the polymer and thus the molecular weight between entanglements. The increase in the aliphatic chain length may lead to a decrease in both packing length and molecular weight between entanglements. This hypothesis was confirmed by molecular dynamics simulation. Nevertheless, a decrease in the plateau modulus is observed in Figure 85, as a result of an increase in the molecular weight between entanglements. This increase seems to be purely related to the introduction of the ionic groups in the PA 10I backbone. In fact, it is possible to determine the value of the molecular weight between entanglements for the PA 10I/10NaSIPA 90/10 which is equal to 1660 g•mol -1 and thus smaller than that of PA 6I/6LiSIPA with the same ionic content. This result is in agreement with a decrease in the molecular weight between entanglements of C10 based polyamides when compared to C6 based polyamides. The values of the WLF constants referred to Tg, obtained by fitting the horizontal shift factors, as shown in Figure 86-a, were used to calculate the values of the fractional free volume at Tg and dynamic fragility index. A small decrease is observed in these two parameters, which are equal to 0.037 and 116, for the fractional free volume at Tg and the dynamic fragility respectively. This decrease is related to the rigidity of the polymer chain, which is also decreased to the longer aliphatic segment, which enhances chain flexibility. The decrease in the glassy modulus is explained as well by the decrease in chain rigidity. The decrease in fragility can be easily identified in the slope of the Angell plot given in Figure 86-b. Concerning the viscoelastic behavior, there is no obvious ion domain relaxation present, as timetemperature superposition worked perfectly for PA 10I/10NaSIPA 90/10 sample, as well as the Doi and Edwards model, which correctly describes the ratio of the two relaxation times ( rep and e), with an exponent close to 3.

Small Angle X-ray Scattering

The normalized small-angle X-ray scattering PA 10I based sodium sulfonated polyamides is presented in Figure 87, in which the signal of the PA 6I reference was subtracted and the resulting intensity normalized by the volume fraction of ionic groups. Scattering data of PA 10I/10NaSIPA sample present a clear peak, at approximatively 0.7 nm -1 , which corresponds to a distance between ion rich phases estimated as . As for other ionic copolyamides, the scattering data was fitted using the Yarusso-Cooper model, disregarding the increase in intensity at low q values, generally associated to heterogeneities in the ionic group dispersion.

Combining equations 4.1 to 4.4, it was possible to determine the presence of ionic domains with 5.8 nm of radius with a volume fraction in the sample volume of 0.14. As the volume stoichiometry of SO3Na groups introduced in the reactor is known, it is possible to determine the volume fraction of ionic groups in each ionic domain, equal to 0.12. It is possible to observe than there is an increase in ionic aggregation when compared to other samples, which may be possibly related to the lower polarity of the backbone and also to the different interactions of sodium cations with carbonyl groups. It is also possible to determine the number of ionic groups per ionic domain, around 800 SO3Na groups. I(q) (mm - 1) q (nm -1 ) PA 10I/10NaSIPA 90/10 44k Fit using Yarusso and Cooper model

Discussion on the effect of ionic interactions on the viscoelasticity of polyamides

The effect of ionic interactions on the viscoelasticity of polyamide melts was studied using molten state rheology. The first observed feature was the increase in the glass transition temperature with the introduction of alkali metal sulfonated groups in a PA 6I backbone. The increase in Tg is linear with increasing ionic content and seems to be independent of the counter-ion introduced.

One of the aims of this work was to understand the effect of ionic interactions on the viscoelastic behavior of unentangled polyamides. As LiSIPA monomer decreases the rate of polycondensation due to the existence of ionic interactions between -SO3 -and the ammonium -NH3 + from the polyamide salt, it was not possible to further increase the content of ionic groups while maintaining the same molecular weight. Thus, it was not possible to be in the exact conditions to observe a sticky Rouse-like behavior.

For unentangled copolyamides, not only the glass transition temperature was shifted to higher values, but also the transition was broader, for higher ionic contents. Apart from this shift in Tg, the shape of the rheological master curves was not affected. Master curves of the dynamic moduli overlapped in the complete range of frequencies, apart some variances in the molecular weight distribution and probably in the molecular weight between entanglements (later determined for entangled PA 6I/6LiSIPA 90/10 29k). Tg was chosen to be the reference temperature for all master curves, which permits to compare measurements guaranteeing iso-frictional conditions. Nonetheless, for copolymers PA 6I/6LiSIPA, a small deviation of the rheological reference temperature Tref from their own Tg was observed. In order to guarantee the superposition of the G'/G'' crossover point at low temperatures, Tref deviated from Tg in a systematic way, correlated to the broadening of the glass transition. For samples up to 15 mol% of ionic groups, the difference between Tref -Tg varied linearly with the width of the glass transition. Our hypothesis to describe this behavior relates to the experimental temperature sequence used in rheology. As the glass transition is approached from the high temperature side (up to Tg + 5°C), as the glass transition becomes broader for each polymer, as we approach Tg the dynamics are slower. Thus the glass transition measured by rheology is increasing with increasing glass transition width.

The variation deviated from its linearity for the PA 6I/6LiSIPA 80/20. We may speculate that this is due to the molecular weight distribution which is quite lower for the copolymer with 20 mol% of LiSIPA, and that could explain why neither Tg (Figure 63) and its width (Figure 68) follow the linear trend of the less ionically modified polyamides. Nevertheless, to better understand this difference between Tref -Tg additional experiments should be performed in other polymers with increasing glass transition width. This would not only help understand the deviation in the reference temperature, but also, why the PA 6I/6LiSIPA 80/20 does not follow the same trend.

In order to remove any influence from the reference temperature, we chose to represent all master curves of the dynamic moduli of unentangled copolyamides with 0, 5, 10, 15 and 20 mol% of ionic groups using a Cole-Cole representation. This allowed analyzing data independently of the reference temperature and it allowed concluding that the relaxation spectrum of all five samples is not modified by the introduction of an increasing amount of ionic groups.

The Rouse model was used to fit the dynamic moduli of substituted or not-substituted unentangled polyamides, which provided a good description of the dynamic moduli of copolymers of both PA 6I and 110 PA 6I/6LiSIPA, confirming that the hydrogen bonds between polyamide chains and ionic interactions introduced in the PA 6I backbone do not play a role in the dynamic moduli of polyamides. The calculated friction coefficients seem to slightly decrease with the introduction of ionic groups. We may speculate that this decrease is due to the introduction of side groups which induces a decrease in packing efficiency and thus a decrease in the friction coefficient. However, in order to possibly better understand this effect, the Kuhn parameters of copolyamides with increasing ionic content should be determined.

As described in Chapter 1, the dynamic fragility is a parameter that characterizes the sensitivity of molecular dynamics to temperature, close to the glass transition temperature. A material is considered "strong" if its relaxation dynamics change relatively slowly, while, "fragile" materials' relaxation dynamics vary rapidly upon cooling through Tg. Sokolov et al. [START_REF] Kunal | Role of Chemical Structure in Fragility of Polymers: A Qualitative Picture[END_REF] and Dudowicz et al. [START_REF] Dudowicz | Fragility of Glass-Forming Polymer Liquids[END_REF] attributed changes in the dynamic fragility to chain rigidity, influenced by side groups rigidity and intermolecular interactions. A strong effect of the introduction of ionic groups was observed on this parameter. Nevertheless, this effect was already observed in other studies. Eisenberg et al. [START_REF] Kim | Introduction to Ionomers[END_REF], [START_REF] Shohamy | The effect of ion aggregation on some melt properties of styrene ionomers[END_REF] found the same fragility diminution behavior with increasing ionomer content in styrene ionomers containing up to 7.7 mol % ionomer. Also, in the case of poly(ethylene oxide)-based ionomers with increasing lithium sulfonate fraction, studied by Fragiadakis et al. [START_REF] Fragiadakis | Molecular mobility and Li+ conduction in polyester copolymer ionomers based on poly(ethylene oxide)[END_REF], it was found that, above 10% ionic groups, the strength parameter 'D', which is the inverse of the fragility parameter, increases with increasing ion content (higher 'D' corresponds to less fragile, or more Arrhenius-like behavior). In other words, they observed a decrease in the dynamic fragility parameter, as observed in the case of the modified polyamides in this work. However, this decrease in dynamic fragility by introduction of ionic groups is still not clear. It may be possible that the formation of a lithium-carbonyl complex would increase packing efficiency, leading to a better cooperativity between chains and thus to slower dynamics close to Tg.

For unentangled polyamides, it seems that ionic groups only have an effect on the glass transition of the polymer as observed in Arrhenius plot (log τ vs 1/T). The use of the Arrhenius plot instead of Angell's plot (log τ vs Tg/T) is preferred, as it shows more clearly the differences in rheological behavior in the different temperature domains (WLF domains close to Tg and Arrhenius-like at high temperature). Therefore it may show more explicitly whether different mechanisms/interactions are contributing in the various temperature domains. As observed in Figure 73-a, in the vicinity of Tg the curves are shifted according to the respective values of Tg of the copolymers, with nearly parallel slopes. The fitted horizontal shift factors (WLF curves) were extrapolated to higher temperatures. It appears that all fitted curves tend to join at high temperature. Our hypothesis is that, at high temperatures, all curves should merge as substituted and non-substituted polymers should present the same viscosity at the same temperature, as interactions should not be effective any longer. Unfortunately, so far we were not able to perform viscosity measurements at higher temperatures to validate this hypothesis, due to the lack of polyamide stability at higher temperatures.

In SAXS analysis, low molecular weight copolymers (around 5000 g•mol -1 ), do not show any scattering peak which would be characteristic of ionic group aggregation and phase separation. Concerning the viscoelastic behavior, there is no obvious ion domain relaxation present and time-temperature superposition principle can be effectively applied to the dynamic moduli for all unentangled ionomers. The glass transition temperature strongly depends on ion content, at least up to 20 mol% of LiSIPA monomers, regardless of the fact that ionic clusters were not observed in SAXS results and that no effect on the rheological master curves was observed. This suggests that the carbonyl groups prevent nanophase separation of the ionic groups, promoting complexation of lithium cations.

Entangled PA 6I/6LiSIPA copolymers present a lower plateau modulus, which raised some questions about the effect of ionic groups on the molecular weight between entanglements. In effect, the molecular weight between entanglements increases with the introduction of 10 mol% of LiSIPA monomers in the PA 6I backbone, which is related to an increase in the rigidity of the polymer chain that affects the packing length of the polymer and thus the molecular weight between entanglements. This increase in packing ability can thus explain the decrease in the fragility by the improvement of the cooperativity between chains and thus to slower dynamics close to Tg. To verify this packing length increase molecular dynamics of PA 6I/6LiSIPA samples should be performed. However, molecular dynamics modeling is very complex in these systems and has not been addressed during this study. It would also be difficult to verify the increase of the packing length in another way. This would require, for example, being able to measure the size of the chains of different molecular weights in a theta solvent. The same conclusions are applied to PA 6I/6NaSIPA polymers as well as PA 10I/10NaSIPA polymers.

Concerning the viscoelastic behavior of different entangled copolyamides with different counter-ions, time-temperature superposition can be successfully applied to the dynamic moduli for all ionomers. One single glass transition temperature is observed in the mDSC analysis, regardless of the observation of a broad scattering peak in SAXS results.

In SAXS measurements, both PA 6I/6LiSIPA 90/10 29k, PA 6I/6NaSIPA and PA 10I/10NaSIPA copolymers presented a scattering peak around 0.7 to 1 nm -1 , corresponding to a distance between objects of about 6 to 9 nm. We may speculate that the appearance of a scattering peak for this polymer may be due to a decreased miscibility as the chain length increases. Scattering data were fitted with a weak segregation model with spherical ion-rich domains [START_REF] Yarusso | Microstructure of ionomers: interpretation of small-angle x-ray scattering data[END_REF]. Using the values obtained for the quantitative fit of the scattering data, it was possible to observe that the fraction of ionic groups inside the ionic domains only slightly higher than the volume fraction of ionic groups introduced in the reactor. This means that ionic groups are inly weakly aggregated probably due to the polarity of polyamides, contrarily to polystyrene samples where ionic aggregation was observed. Also, it is possible to observe some differences between lithium and sodium sulfonated polyamides. The polymers bearing NaSIPA groups showed a slightly higher ionic aggregation, as sodium groups tend to less interact with the polyamide backbone. Also, as the backbone polarity is decreased in the case of PA 10I/10NaSIPA, ionic groups have a stronger tendency to aggregate. Therefore, the increase in Tg suggests that ionic interactions exist and have an effect on the dynamics in the Tg region while at high temperatures the viscoelastic behavior shows no influence of the relaxation of these ionic interactions, which leads to the idea that the ionic interactions dissociate close to the glass transition [START_REF] Kim | Introduction to Ionomers[END_REF].

Contrarily to unentangled copolyamides, extrapolation of the WLF fits to higher temperatures did not merge, as it can be observed in Figure 78, Figure 83 and Figure 86. Our suggestion is that contrarily to unentangled polyamides, where differences in the molecular weight do not have a big repercussion in the viscosity of the resulting polymers (for M < Mc η~M), in high molecular weight copolyamides, since molecular weight distributions are more dispersed, resulting in bigger differences in viscosity, as for M > Mc η~M 3.4 . So far, alkali metal sulfonated groups were mostly studied in non-polar or relatively low Tg polymers, up to 100°C. The absence of effect on the dynamic moduli of polyamides may be due to the overly short lifetime of ionic interactions when compared to the elevated glass transition temperature of the studied copolyamides. To study this hypothesis, copolyamides with lower glass transition were synthesized, without success on observing an ion domain relaxation. Some other amorphous copolyamides with lower glass transition (Tg ~ 80°C) were synthesized but due to the low reactivity of the co-monomers, molecular weights could not be increased to achieve one or two ionic groups per polymer chain and no interesting effect was observed for copolyamide with less than one ionic group. Consequently it is not possible to efficiently conclude on the viscoelastic behavior of copolyamides and how it is different from other polymers studied in the literature. Some questions remain to be answered such as: Is it the glass transition and the short lifetime of ionic interactions that preclude an ion domain relaxation namely two distinct time scales that cannot be observed? or Are both used alkali metals in interaction with the amide groups from the polyamide backbone? To have a better vision on this, it would be necessary to synthesize polyamides with low glass transition while keeping a sufficient number of sodium ionic groups per chain. It would also be interesting to analyze ionic copolyamides by far infrared spectroscopy and observe how the metal cation-carbonyl complexation varies when the alkali metal is changed. It would also be interesting to synthesize methylated polyamides in order to preclude the formation of lithium-carbonyl complex, reducing the polarity of the polymer backbone.

Chapter 5 -Effect of the introduction of stronger hydrogen bonding groups on the viscoelasticity of unentangled polyamides

The effect of hydrogen bonding groups in polyamides was studied by adding a hydroxyl group capable of forming additional stronger hydrogen bonds in the PA 6I backbone. In fact, the interaction of an amide group with the -OH from the phenol group is estimated at 40 kJ•mol -1 (energy values estimated by simulation in vacuum) [START_REF] Hartikainen | Structure and Morphology of Polyamide 66 and Oligomeric Phenolic Resin Blends: Molecular Modeling and Experimental Investigations[END_REF], while an amide-amide standard H-bond (as in Figure 29) has an energy of interaction around 25 kJ•mol -1 [START_REF] Jorgensen | Optimized intermolecular potential functions for amides and peptides. Structure and properties of liquid amides[END_REF], [START_REF] Klotz | Hydrogen Bonds between Model Peptide Groups in Solution[END_REF].

Molecular dynamics simulation allowed better understanding on the conformation of PA 6HIA when compared to non-modified polyamide PA6I monomer conformations around the phenyl ring, as observed in Figure 93. Molecular dynamics simulation allowed obtaining the values of the end-to-end distance from which the values of the Kuhn segment and thus the packing length can be calculated.

Molecular architecture

The molecular weights, as determined by EGT and GPC, and molecular weight distributions by GPC are summarized in Table 12, along with the branching content and glass transition temperatures. As it can be observed in Table 12, the values of polydispersity are often larger than 2. As in the previous chapters, the experimental data were fitted using a Flory distribution, as given in equation 1.36. The GPC data along with the fit for linear polycondensation are presented in Figure 88. As it can be observed, the Flory distribution for linear polycondensation does not well describe the experimental GPC data, indicating that the synthesis did not result in a statistical assembly of linear chains, which can be explained by the branching reactions previously mentioned.

The chromatogram of PA 6HIA and PA 6I 5k, which have a similar molecular weight Mn and similar BHT contents, around 4000 ppm are shown in Figure 89. When considering the chromatogram of PA 6HIA obtained by GPC, it is clear that another type of branching is necessary to understand the molecular structure of this polyamide.

Contrary to other polyamides in this work, PA 6HIA presents two possible types of branching. The first one by formation of BHT has been discussed previously. The second branching reaction, through the formation of phenolic ester, is specific to PA 6HIA. In fact, two acid monomers can react by an esterification reaction (-OH function from phenol + -COOH function from isophthalic acid), leading to an ester, in this case a phenolic ester (Figure 43-a). Branching by phenolic ester can be measured by NMR measurements, as described in Chapter 2.

Time-temperature superposition and master curves

As for other polymers, the viscoelastic behavior of PA 6HIA samples presented in Figure 90 was obtained by time-temperature superposition. The master curves of a PA 6I with comparable molecular weight were used as a reference to understand the influence of stronger hydrogen bonding in the viscoelastic behavior of polyamides. The shape of the master curves was not affected by the presence of the stronger hydrogen bonding groups, with both sets of G' and G'' curves overlapping in the entire range of frequency of polyamide relaxation, meaning that the relaxation time distributions (referred to Tref = Tg) are essentially identical. Thus, apart from the Tg shift, there was no observable effect on the rheological behavior. Also, it is possible to conclude that branching by phenolic ester has no influence on the shape of master curves, at least up to 2.6 mol% (see Table 7). -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 a T (rad/s)

PA 6I 5k PA 6HIA T ref = T g G', G'' (Pa) (a)
-7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0

10 -1 PA 6I 5k PA 6HIA T ref = T g a T (rad/s) tan (b) 116
Regarding the molecular weight distributions of these two polymers, it is observed that PA 6HIA has a higher weight-average molecular weight. However, PA 6HIA presents the same rheological behavior as PA 6I. This may be due to an increase in packing length and thus molecular weight between entanglements, induced by the stronger hydrogen bonds in PA 6HIA. While the packing length of PA 6I is equal to 2.3 Å, this value increases up to 2.9 Å in PA 6HIA. This difference is in fact associated to the molecular weight between entanglements, as . The ratio of the molecular weights between entanglements can be determined as proportional to the ratio of the packing lengths to the cube. In fact, the ratio of is equal to 0.5 which is approximately the ratio of molecular weights between the two polymers presenting the same rheological behavior. This is consistent with the fact that for the same chain length, a polymer with higher molecular weight between entanglements will present a less entangled behavior, which is effectively observed for PA 6HIA.

The polyamide density as a function of temperature used to calculate the vertical shift factors was considered equal to that of PA 6I. The horizontal shift factors referred to Tg used to shift each frequency sweep were fitted using a WLF equation (equation 1.27) and are presented in Figure 91-a. The first observation is the difference in Tg of the polyamide samples with the introduction of hydroxyl group, as verified by DSC. The WLF constants are detailed for each polyamide sample in Table 13, along with the values of the dynamic fragility index and the fractional free volume at Tg. As in the case of PA 6I, also PA6HIA presents a fractional free volume which is slightly superior to the universal value of 0.025 [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF], as a result of the presence of the aromatic cycle in the polymer backbone.

T 1/T (K -1 ) PA 6I 5k PA 6HIA 0, 0,0026 -10 -8 -6 -4 -2 0 (a) log a
T T g /T PA 6I 5k PA 6HIA (b) 75 0,80 0,85 0,90 0,95 1,00 -10 -8 -6 -4 -2 0 log a
The variation of the dynamic fragility index between PA 6I 5k and PA 6HIA can be observed in the Angell plot presented in Figure 91-b. Sokolov et al. [START_REF] Agapov | Effect of Polar Interactions on Polymer Dynamics[END_REF] found that the influence of polar groups on the fragility index of polymers depends strongly on the position of the polar substituent. In the case of propylene or poly(isoprene), as the polar groups are directly attached to the polymer backbone, fragility increased upon the introduction of polar groups in the non-polar polymer backbone. On the other hand, when the polar group is separated from the backbone, for example by an aromatic cycle, this behavior is not trivial. Furthermore, Osterwinter et al. [START_REF] Osterwinter | Rheological Consequences of Hydrogen Bonding: Linear Viscoelastic Response of Linear Polyglycerol and Its Permethylated Analogues as a General Model for Hydroxyl-Functional Polymers[END_REF] concluded that the fragility of polyglycerol samples was smaller than that of their methylated analogue. The same behavior was observed in hydroxyl-terminated polydimethylsiloxane samples [START_REF] Xing | Impact of Hydrogen Bonding on Dynamics of Hydroxyl-Terminated Polydimethylsiloxane[END_REF]. In other words, the introduction of hydrogen bonding groups induced a decrease of the dynamic fragility index.

In PA 6HIA samples, the dynamic fragility seems to be larger than that of the non-modified PA 6I. However, it is not completely clear whether the groups can be considered directly attached to the polymer backbone or not.

However, we may speculate that, with increasing temperature, the hydrogen bond strength, as well as the number and the lifetime of intermolecular associations decrease. The intermolecular associations are weaker at the glass transition temperature, which in the case of PA 6HIA is quite high (around 145°C).

For instance, polyglycerol samples present a glass transition between -30 and -10°C. Above the glass transition temperature of polyglycerol, hydrogen bonds are still active and can increase cooperativity between chains and thus the polymer sample present a more Arrhenius like behavior, with a dynamic fragility value close to 50. In the case of polyamides, with a glass transition close to 150°C, hydrogen bonds are weakened and thus above this temperature do not contribute to the cooperativity between chains, which results in a higher fragility value.

In Figure 91-a, it can be observed that at high temperature, both WLF extrapolations seem to merge, independently of the presence of hydrogen bonding groups. Our hypothesis is that, at high temperatures, both polymers should present the same viscosity at the same temperature, since interactions should not be effective any longer.

Fit with the Rouse model

As for unentangled PA 6I and ionic copolymers, the dynamic moduli of PA 6HIA was fitted using a polydispersed Rouse model. Both Kuhn segment parameters, Mk and lk,, change significantly, due to the significantly different hydrogen bonding network and chain conformation in PA 6HIA. Thus these values were calculated by molecular dynamics simulation, as described in Appendix I. As shown in Figure 92, the Rouse model provides a good description of the viscoelastic behavior of unentangled PA 6HIA, independently of hydrogen bonding. As explained before, the Rouse model [START_REF] De Gennes | Entangled polymers[END_REF] was created for ideal chains in which there is no interaction between the monomers. Despite the introduction of stronger hydrogen bonding groups, this model can still be used to describe the dynamic moduli of polyamides, which suggests that hydrogen bonds do not have an effect on the viscoelasticity of these polymers in the terminal region. As for PA 6I 5k, fits were performed using a Matlab program, using as a single fitting parameter the reduced friction and both GPC and Flory's distribution. As Flory's distribution does not correctly describe the molecular weight distribution obtained by GPC, the fit was improved when using 118 the experimental distribution. Both fitting parameter and friction coefficient obtained using both experimental and Flory distributions are presented in Table 14.

Firstly, it is possible to observe that independently of the quality of the fit using the Flory distribution for the molecular weight, both distributions result in similar fitting parameters, thus in similar friction coefficients. Furthermore, when comparing the friction coefficients obtained for unentangled PA 6I and PA 6HIA, it is observed that both polymers present similar friction coefficients, meaning that hydrogen bonding groups introduced in PA 6HIA do not play a role in the viscoelastic behavior of polyamides in the terminal relaxation regime. G', G'' (Pa) a T (rad/s)

Discussion on the effect of stronger hydrogen bonding groups

Stronger hydrogen bonds were introduced in the PA 6I backbone, by adding a hydroxyl group in the aromatic cycle. In fact, the hydrogen bond strength increases from 25 kJ•mol -1 [START_REF] Jorgensen | Optimized intermolecular potential functions for amides and peptides. Structure and properties of liquid amides[END_REF], [START_REF] Klotz | Hydrogen Bonds between Model Peptide Groups in Solution[END_REF] to 40 kJ•mol -1 [START_REF] Hartikainen | Structure and Morphology of Polyamide 66 and Oligomeric Phenolic Resin Blends: Molecular Modeling and Experimental Investigations[END_REF] with the interaction evolving from standard amide-amide to a phenol-amide interaction. Molecular dynamics simulations have shown that the introduction of a stronger H-bond may possibly allow a different conformation around the phenyl ring, which does not happen in PA 6I samples.

The molecular weight distribution of PA 6I was in fact almost twice broader than that of PA 6I, as determined by GPC measurements. However, the master curves of both polymers overlap in the complete range of frequencies when Tg is taken as the reference temperature. For the same chain length, a polymer with higher molecular weight between entanglements will be less entangled, as it is observed in the case of PA 6HIA. This is a result of the increase in the packing length, induced by a different hydrogen bond network in PA 6HIA, which allowed for different chain conformation. The packing length of PA 6I was equal to 2.3 Å, while it increased up to 2.9 Å in PA 6HIA. Since , the ratio of molecular weight between entanglements can be estimated as proportional to the ratio of the packing lengths to the cube. In effect, the ratio of is around 0.5, which is approximately the ratio of molecular weights between the two polymers presenting the same rheological behavior.

The Rouse model was used to fit the dynamic moduli in the terminal regime of both polymers, PA 6I and PA 6HIA. Despite the difference in the Kuhn segment (length and molecular weight), both polymers present the same monomeric friction coefficient. We may thus speculate that hydrogen bonds do not play a role in the viscoelastic behavior of polyamides in the terminal regime.

The dynamic fragility seems to increase with the introduction of hydroxyl groups in the polyamide backbone. Osterwinter et al. [START_REF] Osterwinter | Rheological Consequences of Hydrogen Bonding: Linear Viscoelastic Response of Linear Polyglycerol and Its Permethylated Analogues as a General Model for Hydroxyl-Functional Polymers[END_REF] and Sokolov et al. [START_REF] Xing | Impact of Hydrogen Bonding on Dynamics of Hydroxyl-Terminated Polydimethylsiloxane[END_REF] observed that the introduction of hydrogen bonding groups induced a decrease on the dynamic fragility index in polyglycerol or PDMS samples. However, the opposite is observed in PA 6HIA samples, in which the dynamic fragility seems higher than that of PA 6I. This behavior can also be compared to the case of molecular glasses in which it is observed that hydrogen bonded liquids such as glycerol are stronger than non-polar liquids, such as o-terphenyl [START_REF] Schoff | Rheological measurements[END_REF], [START_REF] Angell | Relaxation in liquids, polymers and plastic crystals-strong/fragile patterns and problems[END_REF], [START_REF] Ito | Thermodynamic determination of fragility in liquids and a fragile-to-strong liquid transition in water[END_REF] (see Figure 18). Our hypothesis is that, due to the higher glass transition temperature of PA 6HIA samples, the intermolecular associations are weaker at the glass transition temperature. In polyglycerol, above the glass transition temperature (around -30 and -10°C), hydrogen bonds are active and increase the cooperativity between chains, which results in a more Arrhenius like behavior (dynamic fragility close to 50). In the case of polyamides, the glass transition is close to 150°C. At these high temperatures, hydrogen bonds are weakened and thus are no long active resulting in a higher dynamic fragility value.

Molecular dynamics simulation allowed better understanding on the conformation of PA 6HIA when compared to non-modified polyamide. PA6I monomer conformations around the phenyl ring (in terms of dihedral angles as shown in Figure 93-a) show a preference for the conformation where amide groups are in the same plane as phenyl rings and with carbonyl groups in different directions. In this case it is said that the preferred conformation is that where the dihedral angles are (0°, 180°) or (180°, 0°). PA 6HIA shows different conformation behavior when compared to PA 6I, as shown in Figure 93-b. In fact, the possibility of forming a pair of hydrogen bonds in between two phenol and two amide groups promotes a 120 modification in the probability density of the dihedral angles, with a small preference for the angle 0°, as it allows the formation of a pair of hydrogen bonds.

Chapter 6 -Dynamics of amorphous polyamides in the solid state

So far, it was found that the intermolecular interactions introduced in the PA semi-aromatic backbone did not have any pronounced effect on the viscoelastic properties of polyamides, apart having an influence on the glass transition temperature, dynamic fragility index and in the geometrical parameters, such as the packing length, the Kuhn parameters or the molecular weight between entanglements. To extend the study of the influence of intermolecular interactions in polyamide from the molten state dynamics to those in the solid state, dielectric spectroscopy measurements were performed on the semi-aromatic polyamides. Furthermore, physical aging experiments were performed in order to understand the influence of the interacting groups on the dynamics close to and below the glass transition temperature. This technique was considered so as to understand if interactions were responsible for the creation of zones where the dynamics were different from those of the polyamide matrix. This chapter is divided in two main sections. In the first the results obtained in the physical aging experiments on polyamide samples will be presented followed by the results obtained by dielectric spectroscopy measurements, for which a comparison with the results obtained by dynamic mechanical analysis (DMA) was performed.

Physical aging as a measure of the dynamical heterogeneities

Physical aging refers to the time-dependent changes in the thermodynamics properties of polymers in the glassy state, which are not in equilibrium. In the literature, several ionomer samples presented nanophase separation, hence the non-polar polymer phase and the ionic clusters may exhibit two distinct glass transitions or a gradient of mobility. Aging experiments may provide a way to probe the in between the two glass transitions mobility [START_REF] Chun | The development of the ionic microphase in sulfonated poly(ethylene-co-propylene-co-ethylidene norbornene) ionomers during physical aging above Tg[END_REF], [START_REF] Maurer | Influence of polymer composition on the thermal characteristics of sulfonated EPDM ionomers[END_REF]- [START_REF] Tsagaropoulos | Dynamic Mechanical Study of the Factors Affecting the Two Glass Transition Behavior of Filled Polymers. Similarities and Differences with Random Ionomers[END_REF].

In the case of ionic copolyamides, one single glass transition was observed in DSC, although entangled copolyamides exhibited an ion rich phase as detected by SAXS experiments that shall not be mistaken with the nanophase separation observed in other ionomers. To better understand the influence of the introduction of interacting groups in the vicinity of glass transition and on its width, a new protocol in Differential Scanning Calorimetry (DSC) was developed. The objective was to understand if interacting groups introduced in the polymer backbone played a role of dynamical heterogeneities, i.e. zones where the local dynamics are heterogeneous due to different local compositions of the interacting groups in the modified polyamides, as schematized in Figure 36.

Aging experiments

The protocol used for these experiments was detailed in section 2.2.2.2. After complete erase of sample thermal history by performing a first heating cycle, temperature was decreased to an aging temperature, comprised between Tg -5 and Tg -40°C, and an isothermal step was performed at this temperature for 5h, during which the polymer sample was allowed to age and to relax towards equilibrium. This relaxation process can be named enthalpy relaxation, physical aging or structural recovery. The enthalpy relaxation can be determined by comparing two heating cycles of an aged and an unaged sample. To do so, after the isothermal aging step, a first heating cycle was performed, in which the glass transition of the aged sample was measured. The heating step was performed up to a temperature well above Tg in order to effectively erase aging history. A second cooling step was thus performed followed by a heating step in which the enthalpy of the unaged sample was measured. In Figure 94, it is possible to observe the differences between the two heat flow curves of the aged and unaged sample for different aging temperatures. The enthalpy gain achieved during aging is calculated as the integral of the difference between this two curves, as given in equations 2.3 to 2.5. Below the glass transition temperature, polymer dynamics are slow, as molecular motions are frozen. In Figure 95, three aging temperatures were considered: one close to Tg (e. g. Tg -5°C), one far from Tg (e. g. Tg -40°C) and one in between (e. g. Tg-15°C). Considering the highest aging temperature (Tg -5°C), since temperature is close to the glass transition temperature, chains still present some degrees of freedom and are able to rapidly relax, achieving equilibrium. Nevertheless, due to the proximity with the glass transition temperature, the equilibrium line is close to the non-equilibrium glassy state line, meaning that the enthalpy gain will be small. With decreasing aging temperature (e. g. Tg-15°C), the equilibrium line is farther from the glassy state line. Though molecular motions are slower at lower aging temperatures, during 5h, the sample may still be able to achieve equilibrium. In the last example, for the lowest aging temperature (e. g. Tg -40°C), molecular motions are entirely frozen and the equilibrium line is far from the glassy state line, meaning that a long time, longer than 5h, is needed for sample to achieve equilibrium. This tendency can be observed in the experimental data presented in Figure 94, where a clear increase in the enthalpy gain during aging (given by the integral of the difference of the two curves) of samples is observed down to 20°C below Tg, while it decreases for lower aging temperatures.

As mentioned previously, the protocol presented in Figure 48 was performed for different aging temperatures, from Tg-5 to Tg-40°C. Higher Tg polyamides present a glass transition temperature up to 40°C above that of non-modified PA 6I. In this case, if there are some zones with a low content in interacting groups, the dynamics may be similar to those of PA 6I, so a different behavior should be observed for aging temperatures close to the glass transition temperature of non-modified PA 6I. In other words, near the glass transition of PA 6I, the dynamics of these zones rich in PA 6I, should be faster than the dynamics of the zones with higher content of interacting groups, which present a high Tg.

Physical aging of ionic copolyamides

The gain in enthalpy at different temperatures for each polymer will be considered, as the integral of the difference of the two heating curves, the one corresponding to the aged sample and the following concerning the unaged sample (Figure 94). The differences, the so-called overshoots, are presented in Figure 96, for PA 6I 5k. By calculating the integral of each curve, the enthalpy gain during isothermal aging can be calculated and plotted as a function of the aging temperature or the difference between aging temperature and glass transition temperature, Tg -Ta, as presented in Figure 97.

In Figure 97, no significant difference in the shape of the curves with increasing ionic content is observed.

If ion containing copolyamides presented zones rich in non-substituted polyamide, one could expect that close to the glass transition of PA 6I (around 106°C), the dynamics would be faster, as the shape of the curves of ion containing samples should present a certain asymmetry on the low temperature side, which is not observed in Figure 97-a. This is thus confirmed by plotting the same curves as a function of the difference between the aging temperature and glass transition temperature for each polymer, as in Figure 97-b, where it is possible to observe that all curves present the same shape with close amplitude and width, within experimental uncertainties.

124 As demonstrated in the previous chapter, unentangled copolyamides present no scattering peak, generally associated to ionic clustering, or moreover, ion rich phases, which is in agreement with the results obtained during physical aging experiments. In fact, dynamical heterogeneities, corresponding to the presence of an ion rich phase and a polymer rich phase, are not observed in SAXS data or physical aging results.

Physical aging experiments were also performed on entangled lithium-containing copolyamides and presented in Figure 98. It is possible to observe in Figure 98-b that the enthalpy gain for PA 6I/6LiSIPA is inferior to that of PA 6I. Our hypothesis is that since this polymer presents an ion rich phase (as observed in SAXS experiments), the fraction of polyamide present in this ion rich phase may possibly be constrained and not able to relax during this time, which results in a decrease of the enthalpy gain. In other words, polyamide chains inside ion rich phase and even those close to these regions may have their mobility limited due to the presence of the constraint imposed by ionic groups. This results in a smaller fraction of chains that can relax hence the decrease in the enthalpy gain observed for PA 6I/6LiSIPA 90/10 29k, when compared to PA 6I 5k. However, it was not possible to further confirm this hypothesis. Note that aging experiments were performed down to an aging temperature that corresponds to the glass transition of PA 6I (Tg = 126°C).

For PA 6I/6NaSIPA samples, the same experiments were performed at temperatures between Tg -5 and Tg -40°C. The results obtained after integration of the overshoot peaks at the different aging temperatures are presented in Figure 99. PA 6I/6NaSIPA samples seem to have a higher structural recovery during aging. However this effect is still not well understood to date and should be studied in details.

Physical aging of polyamides with stronger hydrogen bonds

The same aging experiments were performed for PA 6HIA samples at temperature between Tg -5 and Tg -40°C. The overshoot peaks of PA 6HIA, as shown in Figure 100, seem to narrower than those of PA 6I, which could mean that the dynamics in the vicinity of the glass transition temperature are more homogeneous in PA 6HIA.The results obtained after integration of the overshoot peaks at the different aging temperatures are presented in Figure 101.

The enthalpy relaxation due to physical aging of PA 6HIA seems to be smaller than that of PA 6I. Jong et al. studied the physical aging of poly(ether sulfone) samples with increasing molecular weight [START_REF] Jong | Physical aging of poly(ether sulfone) from enthalpy relaxation measurements[END_REF].

Contrary to what has been observed for polystyrene samples, poly(ether sulfone) shows a decreasing enthalpy relaxation with increasing molecular weight at comparable Tg-Ta intervals, as it is affected by the hydrogen bonding through the terminal -OH. We may speculate that the same effect is observed in the case of PA 6HIA. As stronger hydrogen bonding groups are introduced, it is possible that the enthalpy relaxation dynamics are affected by this slower dynamics in the vicinity of the hydrogen bonds. (a)

Dielectric spectroscopy

Dielectric spectroscopy measurements allowed a better understanding of the dynamics in the solid state of modified and non-modified polyamides and the effect of the introduction of interacting groups on the secondary relaxations of polyamides. So far, three relaxations have been identified in polyamides, as described in the first chapter: the -relaxation, generally attributed to motion of the aliphatic -CH2sequences of polyamide; β-relaxation, mostly attributed to the rotation of the amide groups -NHCO-and -relaxation, a segmental relaxation associated to the glass transition.

Influence of the introduction of ionic groups in polyamide relaxations

Unentangled polyamides with increasing ionic content were analyzed by dielectric spectroscopy. The secondary relaxations were fitted using the Cole-Cole equation. However, the characteristic times as a function of temperature for the -relaxation were difficult to obtain, especially with increasing ionic content. Since this relaxation is shifted to higher temperatures with increasing ionic content, it could not be clearly separated from sample conductivity.

The results obtained during a dielectric spectroscopy test for the PA 6I 5k sample are presented in Figure 102. The main relaxation process was fitted using an asymmetric Havriliak-Negami equation, though it may be difficult to correctly identify the asymmetry of the peak, as the -relaxation is accompanied by polymer conductivity. When sample conductivity increases strongly, mostly in ionic copolyamides, the characteristic times of the -relaxation were determined using the loss dielectric modulus, M''. In this case, isochronal tests were analyzed and the maximum of the -relaxation peak was taken as the temperature corresponding to a certain relaxation frequency.

The temperature dependences of the characteristic times of all relaxations in unentangled polyamides with increasing ionic content are shown in the relaxation map presented in Figure 103. Two β-relaxations were observed in the relaxation spectra for all polymers, with and without the introduction of ionic groups. The interpretation of the appearance of these two relaxations will be discussed later.

The -relaxation is observed in the typical temperature range from -130 to -50°C (Figure 102-a). A first β-relaxation (β1) was observed in the temperature range from -70 to 0°C (Figure 102-b), followed by a second low amplitude β-relaxation (β2) which was observed in the temperature range from 10 to 60°C (Figure 102-c). Lastly, the -relaxation was observed at high temperature, close to the glass transition temperature measured by mDSC for each polyamide. Note that the increase in the loss permittivity around , independently of temperature, observed in Figure 102-a corresponds to a measurement artefact due to a non-zero contact resistance and cannot be attributed to a secondary relaxation. Firstly it is observed that none of the secondary relaxations are affected by the introduction of ionic groups, as experimental points for all polymers overlap in the complete temperature range. The characteristic times of the β2-relaxation as a function of temperature are quite dispersed in copolyamides. Polymer conductivity is enhanced by the introduction of ionic groups, starting at higher frequencies (or equivalently, at lower temperatures). The appearance of the conductivity at higher frequencies influences the quality of the fit of the β2-relaxation.

-relaxation

Secondary relaxations were fitted using an Arrhenius law (equation 1.38). The resulting activation energy for the -relaxation, EA, and characteristic relaxation times at infinite temperature, , are reported in Table 15. It is observed in Table 15 that the activation energy associated with the -relaxation is not significantly affected by the introduction of ionic groups. The activation energy is in the usual range of 40 kJ•mol -1 as reported by Laurati et al. for PA 6,6/6I copolymers and Laredo et al. for PA 6 polymers [START_REF] Laredo | Water Absorption Effect on the Dynamic Properties of Nylon-6 by Dielectric Spectroscopy[END_REF], [START_REF] Laurati | Dynamics of Water Absorbed in Polyamides[END_REF], [START_REF] Starkweather | The effect of water on the secondary dielectric relaxations in nylon 66[END_REF].

However, the prefactor is much smaller than the value expected for this relaxation time, which should be close to the Debye time ( ). This may be explained by the presence of confined water in polyamides. Indeed, Laurati et al. discovered a new relaxation process in the same temperature range as the -relaxation, which they associated to water molecules motion [START_REF] Laurati | Dynamics of Water Absorbed in Polyamides[END_REF], as it can be observed in Figure 34. This relaxation process presents an activation energy and a characteristic relaxation time at infinite temperature, (

) different from those of the -relaxation. We may speculate that the presence of water molecules influences the characteristic relaxation time at infinite temperature. In fact, the value determined for PA 6I samples or copolyamides is similar to that found by Laurati et al.. T g (DSC) 132 However the activation energy is not strongly affected. As the water concentration is small, it may be possible that the overall measured activation energy is not significantly affected. Note that it is difficult to further decrease the water content in polyamides below 500 ppm of water, obtained after several days in vacuum at 100°C. This hypothesis is further supported by the decreasing width of the -relaxation with increasing temperature, as schematized in Figure 104, since the same behavior was observed by Laurati et al. [START_REF] Laurati | Dynamics of Water Absorbed in Polyamides[END_REF]. Note that the same behavior was observed by Laredo et al. [START_REF] Laredo | Water Absorption Effect on the Dynamic Properties of Nylon-6 by Dielectric Spectroscopy[END_REF]. However they do not give an explanation of the phenomenon.

Using the values obtained in the Arrhenius fit, the entropy related to each relaxation can be calculated, according to equation 5.1. From this value, and as the activation energy is merely related to the enthalpy changes, the free energy, ΔG can be calculated for each relaxation, at 300 K, from equation 5.2.

The values of ΔG (about 18 kJ•mol -1 ) are similar to the rotational barrier for the C-C rotation [START_REF] Smith | Quantum Chemistry Study of Conformational Energies and Rotational Energy Barriers in n -Alkanes[END_REF], [START_REF] Wiberg | Rotational barriers. 2. Energies of alkane rotamers. An examination of gauche interactions[END_REF]. As the value is less significant than the free energy barrier, ΔG, the differences observed in the characteristic relaxation times at infinite temperatures do not strongly influence the final ΔG value, which may explain why the determined value is in agreement with the literature.

The free energy, ΔG, is independent of the LiSIPA content, as observed in Table 15. Since therelaxation is attributed to the rotation of the aliphatic chain, which should not be affected by ionic groups introduction. Thus, it is clear why this relaxation is not affected by LiSIPA introduction.

β-relaxations

Two β-relaxations are observed in Figure 103, though none of these relaxations is affected by the introduction of ionic groups, as experimental points for all polymers overlap in the complete range of temperatures. However, it is still not clear why two β-relaxations are observed. It may be speculated that β-relaxation is strongly affected by the presence of water, as it is related to the rotation of amide groups that can interact with water molecules. Thus the presence of confined water in polyamide samples may explain the observation of two β relaxations [START_REF] Laurati | Dynamics of Water Absorbed in Polyamides[END_REF].

For the β2-relaxation, as the points are rather dispersed, it is not possible to properly fit the experimental data. Nevertheless, it can be concluded that this relaxation is not associated with the introduction of ionic groups, as it is as well observed in PA 6I and PA 6HIA samples (described later).

The characteristic relaxation times of the β1-relaxation as a function of temperature were fitted using the Arrhenius equation. The Arrhenius fitting results along with the entropy and free energy at 300 K for this relaxation are presented in Table 16. It is possible to observe that the activation energy of β1 relaxation (around 70 kJ•mol -1 ) is not affected by the introduction of lithium sulfonate groups. For PA 6I, the free energy associated to this relaxation, ΔG, is 45.4 kJ•mol -1 which is in agreement to the value determined by Bocahut et al. for the highest free energy barrier associated to amide rotation (ΔG=40 ± 5 kJ•mol -1 ) [START_REF] Bocahut | Modeling Molecular Relaxation Mechanisms in Amorphous Polymers: Application to Polyamides[END_REF]. Nevertheless, the free energy barrier seems to slightly increase with the introduction of ionic groups, though this increase may not be significant. However, it is possible that due to the Li-(C=O)4 complex formation, the energy barrier associated with the amide group rotation increases. The characteristic relaxation times at infinite temperature, , are similar to the values found in the literature for the β-relaxation. The -relaxation, as observed in Figure 103, follows the trend of the increasing glass transition temperature, as previously verified by mDSC. Since sample conductivity increases strongly with increasing ionic content, the characteristic times of the -relaxation were determined using the loss dielectric modulus, M'', by taking the maximum temperature of the -relaxation peak in the isochronal measurements. However, when this relaxation was distinguishable from the increase in conductivity on the isothermal loss permittivity data, characteristic times of this relaxation were determined using both techniques (M'' and '').

The characteristic relaxation times of the -relaxation can be fitted using the WLF equation, as mentioned previously in section 2.2.4. The WLF constants referred to Tg obtained in the fit are presented in Table 17, along with the dynamic fragility values calculated following equation 1.30. It is possible to observe that the dynamic fragility index, m, presents the same trend to decrease with increasing ionic content, though the values are not identical to those determined during the rheological measurements. The difference between fragility values and WLF coefficients obtained by rheology and dielectric spectroscopy measurement is not yet understood. However, the overall trend of decreasing dynamic fragility index with increasing ionic content is the same.

Influence of the nature of counter-ion

The effect of the introduction of sodium sulfonate groups in the polyamide matrix was also considered in terms of solid state dynamics. In this case, polymer conductivity started at even higher frequencies, thus it was not possible to determine the characteristic relaxation times of both and β2 relaxations.

Conductivity in NaSIPA substituted copolymers takes place at a lower temperature when compared to LiSIPA substituted samples. For instance, the conductivity of PA 6I/6LiSIPA 90/10 5k (Tg = 128°C) appeared around 400 K, as the glass transition was approached, while it appeared around 373 K for PA 6I/6NaSIPA samples. As lithium cations interact with carbonyl groups, as shown in Figure 40, which may not take place for sodium cations (as presented in several works in the literature [START_REF] Molnar | Miscibility of polyamide-6 with lithium or sodium sulfonated polystyrene ionomers[END_REF], [START_REF] Rajagopalan | Molecular interpretation of miscibility in polyamide-6 blends with alkali ionomers of sulfonated polystyrene[END_REF], [START_REF] Yoshikawa | Rheological properties of blends of lithiumor sodium-sulfonated polystyrene ionomers with polyamide 6[END_REF]), we may speculate that sodium cations are "free" in the polyamide matrix, increasing its conductivity even at lower temperatures (or higher frequencies). Lithium cations are complexed with carbonyl groups, thus their movement is restricted and a certain energy barrier needs to be crossed in order to conduct. In the case of PA 6I/6NaSIPA, cations are randomly dispersed in the polymer, which helps to enhance polymer conductivity. Nevertheless, conductivity tests should be performed in order to corroborate this hypothesis.

Furthermore, both and β1 relaxations are not affected by the introduction of sodium sulfonate groups, as the characteristic relaxation times of both relaxations overlap in the complete range of temperatures, as it can be observed in Figure 105. Both relaxations were fitted with an Arrhenius equation, which allowed obtaining the activation energy and characteristic relaxation time at infinite temperature. For therelaxation no significant difference, due to the introduction of NaSIPA groups, is observed in the activation energy and characteristic relaxations times, hence no difference as well in the free energy barrier for this relaxation. For the β1 relaxation, both activation energies are similar, independently of the introduction of sodium ionic groups, as in the case of PA 6I/6LiSIPA copolyamides. Nevertheless, the small differences observed in the free energy barrier for lithium sulfonated copolyamides are not observed in this sample. Thus, we may suppose that the complex formation does not take place. This is in agreement with the fact that sodium sulfonate groups do not enhance the miscibility of polystyrene and PA 6 blends, as reported in the literature [START_REF] Molnar | Miscibility of polyamide-6 with lithium or sodium sulfonated polystyrene ionomers[END_REF], [START_REF] Rajagopalan | Molecular interpretation of miscibility in polyamide-6 blends with alkali ionomers of sulfonated polystyrene[END_REF].

Influence of molecular weight

Non-modified polyamides were synthesized with several molecular weight distributions, from unentangled to entangled behavior, as presented in section 3.1. One single polyamide PA 6I was analyzed by dielectric spectroscopy with the aim of obtaining the relaxation map of PA 6I 5k. The results obtained by dielectric spectroscopy can then be compared with the results of higher molecular weight polyamides analyzed by DMA. It was observed in Chapter 3, that the increase in molecular weight in polyamides incites an increase in Tg until reaching a plateau for high molecular weights, as expected by Fox-Flory. This increase in Tg will be reflected in the -relaxation as it is related to the glass transition of the polymer. Secondary relaxations should not be affected by the increase in molecular weight since they are related to local motions along the polymer backbone.

As previously mentioned, secondary relaxations were fitted using the symmetric Cole-Cole equation (equation 2.16), while the main relaxation ( -relaxation) was fitted using the asymmetric Havriliak-Negami, though it may be difficult to correctly identify this asymmetry as the -relaxation is accompanied by polymer conductivity.

Comparison with DMA results

Dynamic mechanical analysis allows obtaining these same relaxations maps at a smaller frequency range, by determining the characteristic relaxation temperatures as a function of a constant frequency. In this case, isochronal measurements were performed at frequencies of 0.1, 1, 10 and 50 Hz. The temperature corresponding to each relaxation was taken as the maximum of the relaxation peak observed in E'', as described in the standard ASTM D5279 (see Figure 58).

To correctly determine the characteristic temperature associated to each relaxation, a Gaussian function was used to fit each relaxation peak and to determine the peak maximum. As the polyamides studied in this work are amorphous, the peak of the -relaxation is not well defined, as the polymer starts to flow near Tg. Thus, it was not possible to determine the characteristic temperatures of the -relaxation by DMA experiments. Two tests were performed for each polymer at each frequency, as a way to calculate the error bar associated to each characteristic relaxation temperature. The relaxation map obtained for three PA 6I samples with increasing molecular weight is shown in Figure 106.

Both and β relaxations are in the same temperature range as determined by dielectric spectroscopy results. These relaxations were fitted with the Arrhenius law (equation 1.38), which allowed determining the activation energy of each relaxation, EA, and the characteristic times at infinite temperature, , which are resumed in Table 20. It is possible to observe that the activation energies calculated for PA 6I with different molecular weights are similar, as expected since both and β are related to local motions, which do not depend on the size of the polymer chain. Note that the temperature range of the -relaxation are quite disperse due to the proximity of these temperatures with the lowest experimental temperature (around -150°C). In conclusion, both DMA and dielectric spectroscopy techniques are similar in terms of the results obtained. Nevertheless, the dielectric spectroscopy allows obtaining points over a wider range of frequencies, which improves the quality of the fits. Also, dielectric spectroscopy is more sensitive, as it allowed observing a second β relaxation, which was not observed in DMA results. As the polyamides used are amorphous, it was not possible to observe this relaxation in DMA, which is another advantage of the dielectric spectroscopy technique.

Influence of the introduction of hydrogen bonding groups in polyamide relaxations

The influence of the introduction of hydrogen bonding groups in polyamides was also analyzed in dielectric spectroscopy measurements. As for other polyamides, secondary relaxations were fitted using the Cole-Cole equation, while the Havriliak-Negami equation was used to determine the characteristic times of the -relaxation. The characteristic times of the -relaxation were as well determined using the loss dielectric modulus, M'', by analyzing the isochronal tests and taking the maximum of the -relaxation peak as the temperature relaxation corresponding to a certain relaxation frequency.

The relaxation map of PA 6HIA samples is shown in Figure 107, in which PA 6I with comparable molecular weight was used as reference. It is observed that all relaxations are shifted to higher temperatures. We may speculate that, due to the network of stronger H-bonds, between phenol and amide groups, there is a better cooperativity between polymer chains, which leads to a shift to higher temperatures of all relaxations. This will be looked into in terms of activation energy of each relaxation and free energy barrier for each relaxation.

Secondary relaxations were fitted with an Arrhenius equation (equation 1.38). As mentioned before, and β relaxations are shifted to higher temperatures, but their activation energies are not significantly changed, as it can be observed in Table 21 and Table 22. The free energy barrier of -relaxation is not affected by the introduction of hydroxyl groups in polyamide. However, the free energy barrier of β1-relaxation is slightly higher than that of PA 6I. We may speculate that due to the stronger hydrogen bonds between the hydroxyl function and amide groups (40 kJ•mol -1 , compared to 25 kJ•mol -1 in standard amide-amide H-bonds), the rotation of the amide groups is somehow more difficult, meaning that the energy needed to the amide group to rotate is higher.

The -relaxation is shifted accordingly to the glass transition temperature measured by mDSC. The characteristic times of this relaxation as a function of the temperature can be fitted using the WLF equation 1.27. The WLF constants referred to Tg as well as the calculated dynamic fragility parameter are presented in Table 23. A significant difference is observed in the WLF constants of these two polymers, though no significant difference was observed in the WLF constants obtained in the rheological measurements. However, it is still not clear why this difference is observed in the case of PA 6HIA.

Discussion on the dielectric spectroscopy results

Dielectric spectroscopy results allowed better understanding of polyamide dynamics in the solid state and the effect of the introduction of ionic groups or hydrogen bonding groups.

In PA 6I samples, as expected no difference was observed in the results by both DMA and dielectric spectroscopy. As secondary relaxations are related to local motions of the polymer backbone and thus the size of the polymer chains has no influence on these relaxations. In the dielectric spectroscopy results, a second β-relaxation was observed. However, the appearance of this relaxation is still not clear, but may be possibly related to the presence of water in the polyamide samples. To better understand this relaxation, dielectric spectroscopy measurements should be performed in samples with different water contents, as to understand the link between the water presence and the intensity and temperature dependence of the βrelaxations. Nevertheless, in the context of this work, we may conclude that this second β-relaxation is not associated to the presence of ionic groups or stronger hydrogen groups, as it is observed for all polymers, including PA 6I.

Furthermore, no significant difference was observed in the relaxation maps of ionic copolyamides when compared to PA 6I, apart from the shift in the -relaxation, as expected by the results observed in mDSC.

The conductivity of the polymer samples also increased with increasing ionic content. However, conductivity measurements should be carried out in these samples in order to quantify the conductivity increase in PA 6I/6LiSIPA samples and to compare it to PA 6I/6NaSIPA samples. This could be helpful to better understand the interactions between lithium or sodium ions with the polyamide backbone, as in a first glance, the conductivity of sodium substituted samples is higher than that of lithium substituted polyamides, since the sodium ion is not in interaction with the amide groups and mobile within the polymer and thus present a better conductivity. It would be helpful as well to perform far infrared spectroscopy, as way to understand this kind of complexation and the differences using different alkali metals, in our case, lithium and sodium.

Small differences in the free energy barrier were observed in the case of ionic copolyamides or PA 6HIA. However, these differences were quite small, which makes it difficult to know whether they are significant. Nevertheless, it could be possible that the lithium complexation or stronger H-bonds add a certain energy barrier to that of the amide rotation in non-modified PA 6I. In order to better understand this, molecular dynamics simulation could be carried out as well as more detailed dielectric spectroscopy measurements, after complete knowledge on the influence of water in secondary relaxations.

Although it is not possible to comment more on the observed relaxations, the most important conclusion is that no additional relaxation was observed above the relaxation, as it was the case in other systems in the literature [START_REF] Chen | Ionomer dynamics and the sticky Rouse model[END_REF], [START_REF] Chen | Linear Viscoelastic and Dielectric Properties of Phosphonium Siloxane Ionomers[END_REF], [START_REF] Müller | The molecular dynamics of thermoreversible networks as studied by broadband dielectric spectroscopy[END_REF]. The interchain interactions may then break during the or β relaxations, which makes it impossible to observe their effect above the glass transition of the polyamides.

Conclusions

The main objective of this PhD thesis was to characterize and understand the effects in both molten and solid states of adding strong interactions in polymers, within the framework of a Solvay project which deals with supramolecular materials. In this work, the fundamental understanding of the dynamics of differently modified polyamides in both solid and molten states was performed.

The viscoelastic behavior over the complete range of polyamide relaxation is a topic not often mentioned in the literature due to the presence of a crystalline phase in classical polyamides and to the lack of chemical stability during rheological tests [START_REF] Pezzin | Capillary Viscometry of Molten Polycaprolactam[END_REF], [START_REF] Van Ruymbeke | Structure and rheology of branched polyamide 6 polymers from their reaction recipe[END_REF], [START_REF] Acierno | Rheological behavior of polyamide 11 with varying initial moisture content[END_REF], [START_REF] Khanna | New developments in the melt rheology of nylons. I: Effect of moisture and molecular weight[END_REF]. In our case, these two problems were prevented by using amorphous and end-blocked polyamides. Since most polyamides are semi-crystalline, this study begins with the synthesis of new amorphous polyamides PA 6I, obtained by the polycondensation of a diamine with an isophthalic acid. The choice of the isophthalic acid insures not only the amorphous character of the studied samples, but also permits to easily introduce stronger interactions, while keeping the polymer backbone intact. To prevent any evolution of the viscosity during rheological measurements, namely by post-condensation, chain ends were blocked during the synthesis step, with a mono functional benzoic acid.

The viscoelastic response of PA 6I shows a typical behavior with increasing molecular weight. As the molecular weight increases, a clear rubbery plateau appears and the longest relaxation time is shifted to lower frequencies, as expected by the Rouse and reptation models. The molecular weight between entanglements is estimated at 1340 g•mol -1 as calculated by the value of the plateau modulus. Using empirical equations [START_REF] Fetters | Packing Length Influence in Linear Polymer Melts on the Entanglement, Critical, and Reptation Molecular Weights[END_REF]- [START_REF] Fetters | Chain dimensions and entanglement spacings[END_REF], the critical molecular weight was calculated to be 2960 g•mol -1 . The determination of the critical molecular weight would not be possible without the realization of molecular dynamics simulations which allowed obtaining the value of the packing length. The packing length is used to denote the number of individual chains present in a given volume of the melt. It would be interesting to perform some high temperature viscosity measurements as a function of the polyamide molecular weight to determine experimentally the critical molecular weight of PA 6I and thus confirm that polyamides PA 6I follow the empirical trend developed by Fetters et al [START_REF] Fetters | Packing Length Influence in Linear Polymer Melts on the Entanglement, Critical, and Reptation Molecular Weights[END_REF]- [START_REF] Fetters | Chain dimensions and entanglement spacings[END_REF].

Dynamic moduli of unentangled polyamides were fitted with the Rouse model, showing no effect of hydrogen bonds between amide groups (2 per monomer) on the shape of the rheological master curves. Concerning the entangled PA 6I, the exponent on the reptation model was determined as equal to 3.1, which is in excellent agreement to expectation.

The effect of branching was also accessed by synthesizing polyamides with different branching contents. We can conclude that up to 6000 ppm of BHT bis(hexamethylene) triamine, there is no effect on the rheological master curves obtained.

Ionic interactions were added to the PA matrix by co-polymerizing PA 6I's monomers with a salt of lithium or sodium sulfonated isophthalic acid (LiSIPA or NaSIPA), in molar fractions from 5 to 20%. The glass transition temperature of the copolymers increased about 10 to 40°C with increasing LiSIPA or NaSIPA content. No pronounced difference was observed in the increase of the Tg associated with the different alkali metals used.

For the same Mw, master curves of PA 6I and substituted polyamides overlap in the complete range of frequencies when using an appropriate reference temperature, which was close to, but not identical to Tg. The rheological analysis of entangled copolyamides allowed a better understanding on the effect of these ionic groups on the molecular weight between entanglements and thus on the packing length of these polymers. The modeling of the ionic copolymers has not been carried out since molecular dynamics simulations are quite complex to implement in ionic systems. However, it would be interesting to compare the experimental results with the values obtained by molecular dynamics simulation.

Furthermore, we show that adding ionic groups, besides shifting the Tg, indeed has an effect on the Angel's dynamical fragility, i.e. on the temperature variation of the rheological response. The Arrhenius (log τ vs 1/T) plot shows more explicitly the differences in rheological behavior in the various temperature domains than the Angell plot (log τ vs Tg/T). In the Arrhenius plot, in the vicinity of Tg, the curves are shifted according to the respective values of Tg of the copolymers, with nearly parallel slopes. As temperature increases, curves tend to become closer to each other, which means that the dynamics (in terms of dominant time scales) of the various polymers becomes more similar on increasing temperature. Thus, ionic interactions only have an effect near Tg.

In other studied ionomer reported in the literature, the relaxation of the ion clusters by the so-called ion hopping process corresponds to a distinct time scale. In our systems, the ionic groups may have the effect of stickers, through their complexation with the amide groups (see Figure 40). However, even though the polyamide backbone is polar, it would be possible to still have a phase separations. We did not observe a distinct ion domain relaxation. Altogether, ion dissociation does not provide a time scale distinct from the time scale of the main polymer relaxation. Three hypotheses were proposed to explain the lack of an effect of the introduction of ionic groups:1. The number of interactions was not sufficiently high in unentangled polyamides? 2. The interactions are not strong enough and are diluted in the polyamide matrix? 3. The lifetime of ionic associations is too small to influence the dynamics of polyamides above Tg? The synthesis of PA 6I/6NaSIPA and PA 10I/10NaSIPA copolyamides was an important step in the comprehension of these questions. However, it would be necessary to synthesize polymers with lower glass transition temperature in order to see an effect. A way to do this is by increasing the length of the aliphatic parts of the chain, without losing the amorphous character, or by methylating the amide groups, which should also help decreasing the polarity of the backbone and thus maybe enhance phase separation. Also, far infrared spectroscopy could be used to better understand the interactions between alkali metal cations and the polyamide backbone and possibly study its temperature dependence as a way to target a glass transition temperature that would enable this supramolecular behavior, as expected by the Sticky-Rouse or Sticky-Reptation models.

Polyamides with additional stronger hydrogen bonds, PA 6HIA, exhibited a higher molecular weight between entanglements. For the same chain length, a polymer with higher molecular weight between entanglements will present a less entangled behavior, as it is observed in the case of PA 6HIA. This is a result of the increase in the packing length, incited by a different hydrogen bond network in PA 6HIA, which allowed for different chain conformation. The increase in packing length was confirmed by molecular dynamics simulation.

Contrary to polyglycerol samples, that exhibit a less fragile behavior due to hydrogen bonding, PA 6HIA samples show the inverse behavior. Our hypothesis is that the hydrogen bonds are weaker at the glass transition temperature of PA 6HIA (Tg = 145°C). In polyglycerol samples (-30 < Tg <-10°C), hydrogen bonds are active above Tg and have an effect on the cooperativity between chains, which results in a more Arrhenius like behavior (dynamic fragility close to 50).

Another peculiarity of this polymer is that, additionally to branching by BHT formation, two diacid monomers can react, forming a phenolic ester. The branching content in PA 6HIA is thus of 4000 ppm (~0.3 %) of BHT and around 3% of phenolic ester. However, no clear difference was observed in the rheological master curves of this polymer.

To better understand the effect of the introduction of interacting groups in the polyamide backbone, dielectric spectroscopy and physical aging experiments were performed. The dynamics in the solid state seems to be independent of the introduction of ionic groups, as ionic groups have no influence in the secondary relaxations of polyamide. Conversely, in the presence of stronger hydrogen bonding groups, due to additional rotational restrictions, all secondary relaxations seem to be shifted to higher temperatures.

To conclude, some new developments were made on the viscoelasticity of polymer with various interactions, such as hydrogen bonds and ionic groups. To obtain stronger modification of the dynamic behavior of polyamides, new systems with stronger interactions, active up to higher temperatures (at least above Tg) should be investigated.

The interactions between atoms are modeled by the sum of: (1) covalent bond potentials; (2) valence angle potentials used to describe the angular deformation energy; (3) Van der Waals potentials; (4) electrostatic potentials and (5) torsion potentials.

These potentials can be separated in two categories: those that only take place between atoms connected by at least one covalent bond (Ebond, Eangle and Etorsion) and those that exist both within the molecule and with neighbor molecules (EvdW and Eelec).

All parameters in molecular dynamics simulation rely on this function, which is determined in the simulations thank to the force field, described in equation I. [START_REF] Chen | Ionomer dynamics and the sticky Rouse model[END_REF]. The values of the different parameters are obtained from an experimental data base or from the results from quantum mechanics. The systems studied are modeled as a "All atom" system as each atom is considered as a single particle. In this case, the force field includes a series of parameters that depend on the bonds in which each atom is present. The systems need to be built as a way to have a statistical group of samples, with a size large enough to be representative of the studied phenomenon.

I.1 Adaptation of the force field OPLS

A large number of force fields more or less specialized in one or more fields of application are proposed. With regard to the modeling of the condensed matter, the most used are AMBER and OPLS-AA.

-AMBER (Assisted Model Building with Energy Refinement) in which atomic charges have been optimized to reproduce the electrostatic potential, having an energy term for explicitly treating the hydrogen bonds.

-OPLS-AA (Optimized Potentials for Liquid Simulations -All Atoms), using some of AMBER's formalism for bound terms, originally developed to replicate liquid densities of organic molecules.

In this work, the force field used is OPLS-AA. To describe the electrostatic interactions, the partial charges of each atom of the polymers must be calculated. For that, a trimer is built by condensation of a monomer (Figure I.2), created in Materials Studio, a software distributed by BIOVIA, which allows the creation of the monomers, their condensation, and the optimization of the geometry of the chains. The trimer then undergoes geometric optimization thanks to the Forcite tool present in Materials Studio.

The Hartree Fock method [11], which is a method of approximate resolution of the Schrödinger equation for the wave function, makes it possible to calculate the Mulliken charges for each atom of this trimer. Two bases are used for the calculation of the charges, SV (P) for the four atoms of the amide group (Figure I.2), and SVP for the rest of the atoms (based on previous internal work of T. Cosnier and A. Bocahut). The advantage of using a trimer is to be able to differentiate terminal (Begin and End) monomers and monomers in the middle of the chain. Subsequently, for a higher degree of polymerization DP, the polymer charges will be the combination of the charges of these three monomers: the begin of the chain, B, N-2 middle monomers M, and a chain end E. The monomer M charges are symmetrized along the aliphatic chain, and around the phenolic ring, including between amide groups. A unique name is assigned to each atom of the three monomers considered as well as an OPLS number needed to calculate the different interaction potentials and a partial charge. The atoms are separated by monomer (B, M and E), and ordered by atom group number. The intra-and inter-monomer bonds are also detailed for each atom, without forgetting to connect the last atom of monomer Begin with the first atom of the monomer Middle and the last atom of the monomer Middle and the first atom of monomer End. Also in Materials Studio, it is possible to create a linear polymer chain with a polymerization degree of 40. The amorphous simulation box is then created using the Amorphous Cell Calculation tool, containing 16 chains of DP = 40 at a temperature of 800 K. At this point, the force fields calculated previously are implemented for each monomer of each chain.

I.2 Implementation of the molecular dynamics simulation

It is necessary to generate a box of simulation containing a system representative of the experimental conditions. The chosen procedure allows generating the system and reaching a density close to its experimental density by means of dynamics in NPT set (the number of molecules, the pressure and the temperature are constant, the volume of the cell can change), and NVT (number of molecules, volume and temperature are fixed, pressure adjusts). Since the simulated system does not undergo thermal degradation, the high temperature used, 800K, brings a larger molecular mobility so as to reduce the simulation time. A simulated annealing coupled with a Monte Carlo algorithm makes it possible to quickly explore the minima of the hypersurface of the potential energy , to find the global minimum of this function, and thus produce a simulation box with an energetically stable configuration, at a temperature of 300K. The data from steps 11 is then processed to calculate the desired average statistical or geometrical quantities.

Step 11 allows obtaining a sufficient statistical population of simulation box configurations, to average the results on a time scale of 10 ns. Firstly, the compactness of the chains is evaluated thanks to the end-to-end distance R0, calculated as the distance between the first terminal atom present on the monomer B and the last terminal atom present on the monomer E and the radius of gyration Rg, which is a way of estimating the radius of a sphere that contains the polymer chain. Finally, the Kuhn parameters and the packing length of each polymer are determined using the equations presented in section 1.1.2.4.

Step 12 may be used to estimate the glass transition temperature of the simulated system. However, the obtained Tg's are measured over a short time scale (a few ns) and do not correspond thus directly to experimental Tg values.

I.3 Results

Molecular dynamics allowed simulating two different polymers: PA6I and PA6HIA. The modeling of the ionic copolymers has not been carried out since molecular dynamics simulations are quite complex to implement in ionic systems.

The data treatment script used uses as input the trajectory of the simulation box during step 11, the degree of polymerization, the number of chains, the number of covalent bonds on the main chain per monomer, the molar mass of the monomer (without the terminal hydrogens), and the length of the monomer. These parameters are entered by the user and the length and weight of the monomer are calculated in the BIOVIA software. Table I.1 shows the geometrical parameters calculated for each polymer.

The Kuhn length (lk) decreases and the packing length (p) increases for PA6HIA when compared to PA6I. This means that PA 6HIA chains are more compact than PA 6I chains. To complete the measurements of the geometrical parameters in step 11, it is possible to measure the distributions and average values of certain characteristic angles, such as the torsion angles ψ1 and ψ2 (Figure I.4) between the aromatic cycle and the amide functions, and to trace their distribution.

The distributions of angles ψ1 and ψ2 are similar. Two preferential conformations are observed around 0 ° and 180 °, which means that the amide groups are in the same plane as the aromatic ring. By integrating the probability distribution between -90 and + 90 °, we obtain the proportion of the monomers in the cis conformation (ψ1 around 0°), and between + 90 ° and + 270 ° for the trans conformation (ψ1 around 180°). Due to the symmetry of the molecule, distributions of ψ1 and ψ2 are equivalent. For PA6I, the cis/trans proportions are very similar, being 47/53. For PA6HIA, a small increase in the proportion of cis conformations is noted for both angles ψ1 and ψ2, rising to about 53/47%. This change lay be due to the 2 H-bonds that can be formed in the case of PA 6HIA as shown in Figure I.7.

To study the number of hydrogen bonds formed at each instant of the simulation, it is considered that a hydrogen bond is formed when the distance between polar groups is less than a certain characteristic distance. For each polymer, the average number of hydrogen bonds formed is calculated and then reported to the density of the polymer (Table I.2). The number of donor and acceptor atoms increases for PA 6HIA, due to the addition of a -OH group which leads to an increase in the number of possible sites for hydrogen bond formation.

Résumé étendu en français Introduction

Cette thèse de doctorat a été réalisée au sein du Laboratoire Polymères et Matériaux Avancés UMR5268, unité mixte de recherche entre le CNRS et Solvay. Ce travail s'inscrit dans le cadre d'un projet Solvay qui s'intéresse aux effets de l'ajout d'interactions fortes dans les polymères, dans le but de développer de nouveaux matériaux qui auraient des propriétés supramoléculaires. Dans ce travail, la compréhension fondamentale de la dynamique des polyamides modifiés a été étudiée, à la fois dans les états solide et fondu.

Les polyamides appartiennent à la famille des polymères thermoplastiques largement utilisés dans les industries du textile, de l'automobile et de l'emballage alimentaire, en raison de leurs excellentes propriétés mécaniques et barrières [START_REF] Kohan | Nylon Plastics Handbook[END_REF]. Les polyamides les plus utilisés sont aliphatiques et semicristallins tels que le PA 6, le PA 66 et le PA 6,10. Cependant, d'autres structures ont été développées ces dernières années pour des applications spécifiques, telles que les polyamides semi-aromatiques ou aromatiques, ainsi que les polyamides amorphes [START_REF] Kohan | Nylon Plastics Handbook[END_REF].

Le but de cette étude est de comprendre le rôle de la force et de la densité des interactions intermoléculaires dans les polyamides. Pour cela, un ensemble de polyamides a été conçu et synthétisé pour cette étude afin d'éviter les limitations inhérentes à l'étude rhéologique des polyamides semicristallins. Différents groupes en interaction ont été utilisés, avec différentes forces d'interaction, tout en maintenant constant le squelette du polymère. L'effet de ces interactions sera examiné par mesures dynamiques à la fois à l'état fondu et à l'état solide, en utilisant des mesures de rhéologie et de spectroscopie diélectrique.

Chapitre 1 -Etude Bibliographique

Le premier chapitre de ce manuscrit présente une étude bibliographique qui traite, d'abord de la rhéologie et de l'ensemble des paramètres qui peuvent influencer la réponse dynamique d'un matériau, suivi du cas spécifique de l'effet des interactions intermoléculaires dans le comportement rhéologique des polymères à l'état fondu. La packing length est un paramètre lié à l'architecture locale du polymère et s'avère être un paramètre caractéristique contrôlant un nombre important de propriétés physiques fondamentales du polymère, telles que la masse entre enchevêtrements, le module au plateau, etc… [START_REF] Fetters | Packing Length Influence in Linear Polymer Melts on the Entanglement, Critical, and Reptation Molecular Weights[END_REF]- [START_REF] Fetters | Chain dimensions and entanglement spacings[END_REF].

synthétisés. L'influence de la température de transition vitreuse du polymère a également été étudiée, en synthétisant des copolyamides avec une chaîne aliphatique plus longue et dont la Tg est plus basse. La DSC a permis d'obtenir les températures de transition vitreuse mais aussi de vérifier si l'introduction de différentes interactions influençait la dynamique proche mais au-dessous de Tg par des mesures de vieillissement physique. La diffraction de rayons-X aux grands angles (WAXS) a confirmé le caractère amorphe de tous les polyamides utilisés au long de cette étude, tandis que la diffraction des rayons-X aux petits angles (SAXS) a permis d'observer si les groupements ioniques étaient ségrégés dans la matrice polyamide.

Les mesures de rhéologie ont permis d'étudier la dynamique au-dessus de la transition vitreuse et l'effet de l'ajout des différents groupements latéraux. Les courbes maîtresses du module dynamique des polymères sont obtenues par le principe de superposition temps-température, en utilisant l'équation de Williams-Landel-Ferry (WLF) (équation 1.27). La température de référence est choisie parmi l'une des températures mesurées pendant l'essai de rhéologie. Cependant il est possible de reporter les courbes maîtresses et les constantes de WLF à la température de transition vitreuse, Tg, ce qui permet de comparer des polymères avec des dynamiques différentes dans des conditions iso-frictionnelles (ce qui permet que les courbes maîtresses se superposent au voisinage de Tg).

La Spectroscopie Diélectrique (BDS) et l'Analyse Dynamique Mécanique (DMA) nous ont permis de caractériser la dynamique des polyamides et l'influence de l'ajout des différents groupements latéraux audessous de la transition vitreuse. Tous les échantillons étudiés sont présentés dans le Tableau 3, p. 50 dans le manuscrit.

Chapitre 3 -Dynamique à l'état fondu de polyamides non-modifiés

Dans ce chapitre, la réponse viscoélastique sur toute la gamme de fréquences de relaxation des polyamides PA 6I amorphes est présentée. Différents polyamides avec des masses molaires (Mw) comprises entre environ 5000 et 40000 g•mol -1 ont été synthétisés. Les indices de polymolécularité sont un peu supérieurs à la valeur attendue par la théorie de la polycondensation linéaire de Flory. Cela est sans doute dû à la formation de ramifications lors de la synthèse des polyamides.

Les courbes maîtresses des polyamides avec différentes masses molaires sont présentées dans la Figure II. L'augmentation de la masse molaire des PA 6I se traduit par une évolution du comportement rhéologique : apparition d'un plateau caoutchoutique et décalage du temps de relaxation terminal vers les plus basses fréquences. À partir des courbes maîtresses des polymères enchevêtrés, la masse molaire entre enchevêtrements a été déterminée à partir de la valeur du plateau caoutchoutique ( ) et est égale à 1340 g•mol -1 .La masse critique a également été calculée (en utilisant le modèle de Fetters et al. avec p en Å [START_REF] Fetters | Packing Length Influence in Linear Polymer Melts on the Entanglement, Critical, and Reptation Molecular Weights[END_REF]- [START_REF] Fetters | Chain dimensions and entanglement spacings[END_REF]), la valeur de la packing length p étant préalablement déterminée par modélisation de la dynamique moléculaire. La masse critique est égale à 2960 g•mol -1 .

Il a également été montré que les polyamides non-enchevêtrés suivent le modèle de Rouse indépendamment des liaisons hydrogène entre chaînes de polyamide. Les polyamides enchevêtrés, quant à eux, suivent le modèle de la reptation.

Chapitre 4 -Effet des interactions ioniques sur la dynamique à l'état fondu de polyamides

La synthèse des ionomères a été effectuée en utilisant différents taux de sel d'acide 5-sulfoisophtalique de lithium ou sodium (LiSIPA ou NaSIPA). Ces copolymères (PA 6I/6LiSIPA ou PA 6I/6NaSIPA) ont été synthétisés avec un taux croissant de monomères ioniques allant jusqu'à 20%, la faible réactivité des monomères ioniques ne permettant pas d'obtenir des taux supérieurs lors de la synthèse des copolymères. Il a été constaté que la température de transition vitreuse augmente linéairement et significativement avec l'introduction de groupements ioniques (jusqu'à 20 mol%) et indépendamment du contre-ion.

Les groupements sulfonate de lithium peuvent interagir avec les groupements amides en formant un complexe entre un cation Li + et quatre groupements amide. Cependant les groupements sulfonate de sodium n'interagissent pas avec les groupements amide [START_REF] Molnar | Miscibility of polyamide-6 with lithium or sodium sulfonated polystyrene ionomers[END_REF], [START_REF] Rajagopalan | Molecular interpretation of miscibility in polyamide-6 blends with alkali ionomers of sulfonated polystyrene[END_REF], [START_REF] Yoshikawa | Rheological properties of blends of lithiumor sodium-sulfonated polystyrene ionomers with polyamide 6[END_REF]- [START_REF] Feng | Compatibilization of polymer blends by complexation. 1. spectroscopic characterization of ion-amide interactions in ionomer/polyamide blends[END_REF].

L'influence du nombre de groupements ioniques par chaîne a été étudiée dans des polyamides nonenchevêtrés et dans des polyamides enchevêtrés. Dans le cas des copolyamides ioniques non-enchevêtrés, 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 La température de référence choisie n'est pas identique à Tg. En effet, la température de référence s'écarte systématiquement de la Tg du polymère avec l'introduction de groupements ioniques, jusqu'à 15 mol% de LiSIPA. Cela semble être lié à la largeur de la Tg qui augmente avec l'introduction de groupements ioniques, qui pourraient en effet jouer le rôle d'hétérogénéités dynamiques et donc d'élargir le spectre des temps de relaxation du polyamide. Ce phénomène est étudié en termes de vieillissement physique dans le chapitre 6. La fragilité dynamique (pente de la courbe de log aT en fonction de Tg/T à Tg) diminue avec l'augmentation du taux de groupements ioniques, comme observé sur la figure IV, ce qui pourrait être dû à une augmentation de la rigidité de la chaîne. Le même comportement est observé dans le diagramme d'Angell où le glycérol, par exemple, a une fragilité inférieure à celle de l'o-terphenyl, à cause de la présence d'interactions hydrogènes entre les molécules. Cependant, à haute température toutes les courbes semblent se superposer, indiquant que les groupements ioniques n'ont aucune influence à haute température. L'analyse par diffusion des rayons-X aux petits angles a montré qu'il n'y a aucune ségrégation de phase dans ces systèmes. En effet, on n'observe aucun pic de diffraction pouvant correspondre à une taille caractéristique de nano-domaines. L'allure des courbes maîtresses des copolyamides ioniques enchevêtrés ne se distingue pas de celles des polyamides non-modifiés, même si la masse entre enchevêtrements est affectée d'environ 30% par l'introduction de 10 mol% de groupements ioniques. Cela est directement lié à l'augmentation de la packing length (Me p 3 ), qui est également liée à l'augmentation de la rigidité de la chaîne. Cependant, la modélisation de la dynamique moléculaire est très complexe dans ces systèmes et n'a pas été abordée. Il également difficile de vérifier l'augmentation de la packing length d'une autre manière. Il faudrait pour cela, par exemple, pouvoir mesurer la taille des chaînes de différentes masses en solvant theta.

L'influence du contre-ion utilisé a également été étudiée. Des groupements sulfonate de sodium ont été ajoutés dans des polyamides enchevêtrés, les groupements sodium interagissant moins avec les groupements amides, et peuvent donc s'agréger et former des clusters ioniques responsables par l'apparition d'une relaxation avec une dynamique différente de celle de la chaîne.

Les résultats obtenus montrent que l'allure des courbes maîtresses n'est pas significativement affectée par l'ajout de groupements sulfonate de sodium dans les polyamides enchevêtrés. L'effet de la diminution de la température de transition vitreuse a aussi été étudié, en synthétisant des polymères avec une chaîne aliphatique plus longue, sans effet dans l'allure des courbes maîtresses.

Tous les copolyamides enchevêtrés ont également été analysés en SAXS. L'analyse quantitative de la diffusion montre qu'une faible ségrégation ionique est observée avec des domaines faiblement enrichis en groupements ioniques (Figure V). Cette faible tendance à la ségrégation peut être expliquée par la polarité de la chaîne de polyamide qui, contrairement à des polymères non-polaires, permet la dispersion des charges ioniques dans la matrice polymère. D'après la littérature, les groupes sulfonate de métaux alcalins ont été principalement étudiés dans des polymères non-polaires ou à Tg relativement faibles (jusqu'à 100°C). L'absence d'effet sur les modules dynamiques des polyamides peut être due à une durée de vie des interactions ioniques trop courte vis-àvis de la température de transition vitreuse élevée des copolyamides étudiés. Pour étudier cette hypothèse, un copolyamide avec une transition vitreuse plus basse a été synthétisé, sans succès non plus dans l'observation de relaxations distinctes liées à la présence de domaines riches en ions. D'autres copolyamides amorphes avec une transition vitreuse encore inférieure (Tg ~ 80°C) ont été synthétisés. En raison de la faible réactivité des comonomères, les masses molaires n'étaient pas suffisamment élevées pour avoir un ou deux groupements ioniques par chaîne de polymère. Aucun effet particulier n'a donc été observé pour le copolyamide de basse Tg avec moins d'un groupement ionique. Par conséquent, il est possible de conclure que le comportement viscoélastique des copolyamides se distingue des autres polymères non-polaires étudiés dans la littérature. Cependant, certaines questions restent sans réponse, telles que : la transition vitreuse et la courte durée de vie des interactions ioniques empêchent-elles l'observation d'une relaxation des domaines riches en ions (deux échelles de temps distinctes qui ne peuvent pas être observées) ? Les deux métaux alcalins utilisés sont-ils en interaction avec les groupements amides ? Pour avoir une meilleure vision à ce sujet, il serait nécessaire de synthétiser des copolyamides à transition vitreuse plus basse tout en conservant un nombre suffisant de groupements ioniques par chaîne. Il serait également intéressant d'analyser les copolyamides ioniques par spectroscopie infrarouge lointain et d'observer la variation de la complexation cation-carbonyle selon le métal alcalin utilisé. Il serait également intéressant de synthétiser des polyamides méthylés afin d'éviter la formation de complexe lithium-carbonyle, réduisant ainsi la polarité du squelette du polymère (ce qui aurait aussi pour effet de diminuer la Tg). Les courbes maîtresses des PA 6I et PA 6HIA se superposent dans toute la gamme de fréquences. Cependant la masse molaire du PA 6HIA est environ le double de celle du PA 6I. La modélisation de la dynamique moléculaire a permis de déterminer la packing length du PA 6HIA et en effet, celle-ci est supérieure à celle du PA 6I, ce qui est probablement la conséquence des liaisons hydrogène plus fortes dans le PA 6HIA. Alors que la packing length du PA 6I est égale à 2.3 Å, elle augmente à 2.9 Å dans le PA 6HIA. Etant donné que Me p 3 , le rapport des masses molaires entre enchevêtrements Me (PA 6I) / Me (PA 6HIA) est proportionnel au rapport des packing length au cube. Le rapport des Me calculé de cette façon est égal à 0.5, ce qui correspond approximativement au rapport des masses molaires entre les deux polymères présentant le même comportement rhéologique. Cependant, les courbes maîtresses du PA 6HIA restent bien décrites par le modèle de Rouse, ce qui montre qu'il n'y a pas d'effet de l'ajout de liaisons hydrogène plus fortes. La fragilité dynamique du PA 6HIA est supérieure à celle du PA 6I, contrairement à ce qui est observé dans la littérature, par exemple, pour le polyglycérol et le polyglycérol méthylé (Osterwinter et al. [45]). Il est fort probable que, lorsque la température augmente, le nombre et la durée de vie des associations intermoléculaires par liaison hydrogène diminuent. Les associations intermoléculaires sont plus faibles à température de transition vitreuse élevée comme dans le cas du PA 6HIA (environ 145°C). Par exemple, les échantillons de polyglycérol présentent une transition vitreuse comprise entre -30 et -10°C. Au-dessus de la température de transition vitreuse du polyglycérol, les liaisons hydrogène sont toujours actives et Des expériences de vieillissement physique (ou relaxation enthalpique) ont été réalisées afin de comprendre l'influence des interactions ajoutées sur la dynamique au voisinage et au-dessous de la température de transition vitreuse. Cette technique a été considérée afin de comprendre si les interactions étaient responsables de la création de zones où la dynamique était différente de celle de la matrice polyamide.

À masses molaires équivalentes, l'augmentation du taux de groupements ioniques de 0 à 20 mol% entraîne une augmentation de Tg de plus de 40°C. De plus, les mesures de Tg en DSC modulée révèlent effectivement une augmentation de la largeur de la transition vitreuse suite à l'ajout de groupements ioniques (voir Figure 72, p.89 dans le manuscrit).

Les gains d'enthalpie pendant l'étape de vieillissement sont déterminés et représentés en fonction de la température de vieillissement pour chaque polymère, comme il est montré sur la Figure VIII, pour les polymères non-enchevêtrés. Aucun élargissement des courbes de relaxation enthalpique en fonction de la température de vieillissement n'est observé (notamment dans la région de la température de transition vitreuse du PA 6I), malgré l'ajout de monomères ioniques. La présence de régions pauvres en groupements ioniques (dus à la ségrégation de ceux-ci) devrait se traduire par la présence de vieillissement au voisinage de la Tg du PA 6I. Nous estimons donc que ces expériences indiquent que la dynamique reste essentiellement homogène dans les systèmes ioniques, en tous cas aussi homogène que dans le PA 6I. Afin d'élargir l'étude de l'influence des interactions intermoléculaires dans les polyamides sur la dynamique à l'état solide, des mesures de spectroscopie diélectrique ont été effectuées. Trois processus de relaxation ont été identifiés dans le polyamide : la relaxation γ est en général attribuée à la rotation de la chaîne aliphatique, la relaxation β à la rotation du groupement amide ; la relaxation α est causée par les mouvements coopératifs de segments entiers des chaînes polymères et équivalente à la Tg.

L'introduction de groupements ioniques n'a pas d'effet sur les relaxations secondaires du polyamide, comme l'indique la carte des relaxations présentée sur la Figure IX. Pour déterminer les temps de relaxation de la relaxation nous avons étudié l'évolution du module de perte diélectrique, M'', en fonction de la température à fréquence constante (courbes isochrones), afin de réduire la contribution de la conductivité. La relaxation peut être décrite par l'équation de WLF indépendamment du taux de groupements ioniques. Deux relaxations β ont été observées. Cependant, l'apparition de la deuxième relaxation 2, qui pourrait être liée à la présence d'eau dans les échantillons de polyamide, reste à étudier plus en détails. Pour mieux comprendre cette relaxation, les mesures de spectroscopie diélectrique devraient être effectuées sur des échantillons avec différentes teneurs en eau. Néanmoins, dans le cadre de cette étude, nous pouvons conclure que cette seconde relaxation β2 n'est pas associée à la présence de groupements ioniques, dans la mesure où elle est observée pour tous les polymères, y compris le PA 6I.

En outre, aucune différence significative n'a été observée dans les cartes de relaxation des copolyamides ioniques par rapport au PA 6I, à l'exception du décalage de la relaxation vers les plus hautes températures, déjà vérifié par DSC et rhéologie. La conductivité des copolyamides augmente selon la teneur en ions. Cependant, des mesures de conductivité devraient être effectuées sur ces échantillons afin de quantifier l'augmentation de conductivité dans les échantillons de PA 6I/6LiSIPA et de comparer à l'augmentation sur les échantillons de PA 6I/6NaSIPA. Cela permettrait d'approfondir la compréhension des interactions entre les ions lithium ou sodium et le polyamide. En effet, la conductivité des échantillons substitués au sodium semble supérieure à celle des polyamides substitués au lithium, ce qui confirmerait que l'ion sodium n'interagit pas avec les groupements amides et est mobile au sein du polymère, présentant donc une meilleure conductivité.

Bien qu'il ne soit pas possible de commenter davantage sur les relaxations observées, la conclusion la plus importante est qu'aucune relaxation supplémentaire n'a été observée au-dessus de la relaxation , comme c'était le cas dans certains systèmes connus de la littérature [START_REF] Chen | Ionomer dynamics and the sticky Rouse model[END_REF], [START_REF] Chen | Linear Viscoelastic and Dielectric Properties of Phosphonium Siloxane Ionomers[END_REF], [START_REF] Müller | The molecular dynamics of thermoreversible networks as studied by broadband dielectric spectroscopy[END_REF]. Les interactions entre chaînes peuvent alors se rompre lors des relaxations ou , empêchant l'observation de leur effet au-dessus de la transition vitreuse des polyamides. T g (DSC)

f

  AbbreviationsBDSBroadband dielectric spectroscopy

Figure 1 -

 1 Figure 1 -Schematic illustration of the strain sweep test at a fixed frequency. Two regions are visible: linear for SAOS (small amplitude oscillatory shear) and non-linear for LAOS (large amplitude oscillatory shear). From [10].

Figure 2 -

 2 Figure 2 -(a) Common representation of Hooke model using a spring; (b) Common representation of Newton model using a dashpot (c) Common representation of Maxwell model.

Figure 3 -

 3 Figure 3 -Dynamic moduli as expected by the Maxwell moduli (with G0 = 1) for a system with a relaxation time, . From [12].

Figure 4 -

 4 Figure 4 -Behavior of an elastic and viscoelastic material when submitted to a stress relaxation experiment. Behavior of an entangled polymer melt with the appearance of a rubbery plateau and terminal relaxation region. Adapted from [13].

Figure 5 -

 5 Figure 5 -(a) Experimental modulus data on a polydimethylsiloxane sample to a five-constant generalized Maxwell model from [12] (b) Generalized Maxwell model.

Figure 6 -

 6 Figure 6 -Elastic modulus obtained by time-temperature superposition of linear monodisperse poly(cyclohexylethylene) samples with increasing molecular weight. From [19].

Figure 7 -

 7 Figure 7 -Variation of zero shear viscosity of a polymer melt as a function of molecular weight. Adapted from [12].

Figure 8 -

 8 Figure 8 -Rouse chain: monomers are represented as beads connected by springs.

Figure 9 -

 9 Figure 9 -Viscoelastic response of an unentangled poly(cyclohexylethylene) melt with a Mw=5000 g•mol -1 and fit by the Rouse model. Reference temperature was taken as 187°C. From [12], [20].

Figure 10 -

 10 Figure 10 -Experimental data and theoretical curves of the dynamics moduli of polystyrene (Mw=13000 g•mol -1 ). The solid line corresponds to the fit using the modified Rouse model while the dashed line corresponds to the simple Rouse model. From [23].

Figure 11 -

 11 Figure 11 -Relaxation moduli for an entangled monodisperse 1,4-polybutadiene melt with a Mw=130000 g•mol -1 . Reference temperature was taken as 25°C. Adapted from [27].

Figure 12 -

 12 Figure 12 -(a) Schematic representation of a chain in the tube in a polymer melt. (b) Contour length fluctuations (CLF) process from [22]. (c) Constraint release (CR) process from [28].

Figure 13 -

 13 Figure 13 -Freely jointed chain model.

Figure 14 -

 14 Figure 14 -(a) Determination of plateau modulus as the minimum of G''. From [44]. (b) Van Gurp-Palmen plot used to determine the plateau modulus. From [34].

Figure 15 -

 15 Figure 15 -Illustration of the time-temperature superposition principle, using oscillatory shear data of a polyisobutylene sample. The plot on the left shows the data acquired at 11 different temperatures from 192K to 298K. Reference temperature was taken as 298K. From [44].

Figure 16 -

 16 Figure 16 -Tan curves of a monodisperse atactic polystyrene (Mn= 98000 g•mol -1 ), where thermorheological complexity effect is present. From [40] 1.1.2.7 Dynamic Fragility

Figure 17 -

 17 Figure 17 -Structures studied by Sokolov et al. [62] to understand the influence of polar interactions on the fragility index (a) para (b) orto (c) meta polystyrene.

Figure 18 -

 18 Figure 18 -Strong and fragile behavior in glass-forming liquids. Inset shows the corresponding heat capacity jump at Tg (Cp, liq/Cp,glass) at each temperature. From [50].

Figure 19 -

 19 Figure 19 -Computed differences observed in dynamic moduli in the regime of terminal relaxation (using equation 1.31 and 1.32) when polydispersity increases. Inset shows the differences in the molecular weight distribution. High frequency modulus was taken to be 10 6 Pa. Full curves G'; dashed curves G''.

Figure 20 -

 20 Figure 20 -(a) Schematic representation of the introduction of urazole groups on a linear polymer chain. (b) and (c) Master curves of the storage and loss moduli at Tref=273 K for 0, 1 and 2% of interacting groups. Solid lines indicate behavior in the terminal zone. Adapted from [2].

Figure 21 -

 21 Figure 21 -Relaxation map of polybutadiene/stickers systems evidencing a relaxation at higher temperature than the glass transition. From [83].

Figure 22 -

 22 Figure 22 -(a) and (b) Master curves of the storage and loss moduli at Tref=233K for polybutadiene with 0, 2, 3 and 4 mol% of urazole groups. The master curves are shifted consecutively half a decade for clarity. (c) Comparison of the relaxation moduli, G(t) for an entangled linear polymer with (solid line) and without stickers (dashed line). From [84].

Figure 23 -

 23 Figure 23 -(a) SAXS curves of HSPS and PS-SO3X, where X=H or metal (Li, Na, K, Rb, Cs) with a degree of substitution (DS) of 4.8 mol%. Note that peak amplitude does not strictly reflect the degree of segregation, since different metal cations were used and as intensities obtained were not normalized by the electron density contrast. (b) and (c) G' and G'' moduli of PS and PS-SO3X with DS=4.8 mol%. Reference temperatures are Tg=45°C. From [88].

Figure 24 -

 24 Figure 24 -(a) Chemical structure of polysiloxane-based phosphonium ionomers. (b) Master curves of dynamic modulus G(t) for phosphonium ionomers with different ionic fractions (0, 5, 11, 22 and 26 mol%). Lines correspond to the fit using Sticky-Rouse or KWW equation. From [47].

Figure 25 -

 25 Figure 25 -(a) Chemical structure of PEO600-Na copolymer ionomers with ionic fraction varied from 0 to 1. (b) Chemical structure of PTMO650-fNa ionomer with ionic fraction f= 0 or 1. (c) Master curves of dynamic moduli for PEO600-fNa samples having different ionic fractions (f= 0, 49 and 100 mol%). Reference temperature is 20°C. (d) Master curves of dynamic moduli for PTMO650-f Na (f= 0 or 100 mol%). Reference temperature is 20°C. From [3].

Figure 26 -

 26 Figure 26 -Master curves of dynamic moduli (a) G' and (b) G'' for NaSPS ionomers with different ionic contents. Solid lines correspond to the model prediction. From [92].

Figure 27 -

 27 Figure 27 -Polycondensation reaction for polyamide synthesis.

Figure 28 -

 28 Figure 28 -Secondary reaction during polyamide synthesis leading to branching.

Figure 29 -

 29 Figure 29 -Hydrogen bond network in polyamides in PA 66.

Figure 30 -

 30 Figure 30 -PA 6I structure showing point of inflection of carbonyl groups adjacent to benzene ring [107].

Figure 31 -

 31 Figure 31 -Evolution of the complex viscosity at 230°C of polyamide 11 samples with different water concentrations. From [104].

Figure 32 -

 32 Figure 32 -Schematic representation of PA6,6 molecular relaxations.

Figure 33 -

 33 Figure 33 -Relaxation map of a PA 6 sample showing both secondary relaxations, and , fitted using Arrhenius equation, and -relaxation fitted using VFT equation for (a) wet sample (b) dry sample. Adapted from [122].

Figure 34 -

 34 Figure 34 -Relaxation map of both dry (full symbols) and wet (open symbols) PA 66/6I copolymer. Letters in the plot distinguish the different processes. Adapted from [124].

Figure 35 -

 35 Figure 35 -Enthalpy (or specific volume) as a function of temperature in the glass transition region. Ta corresponds to the annealing (or aging) temperature. Adapted from [125].

Figure 36 -

 36 Figure 36 -Schematic illustration of a miscible polymer blend composed of polymer A (red) and B (blue). From [131].

Figure 37 -

 37 Figure 37 -SAXS data for zinc sulfonated poly(ethylene-co-propylene-co-ethylidene norbene samples as a function of the aging time. From [133].

Figure 38 -

 38 Figure 38 -Schematic variation of the free volume with temperature, for an amorphous material.. Adapted from [15].

Figure 39 -

 39 Figure 39 -Chemical structure of amorphous polyamides used in this work.

Figure 40 -

 40 Figure 40 -Schematic representation of the different stages of interaction between lithium sulfonate and carbonyl groups of the polyamide backbone. From [139].

Figure 41 -

 41 Figure 41 -Chemical structure of the C10 based polyamide.

Figure 42 -

 42 Figure 42 -(a) Chromatogram of PA 6I 5k sample, where the peaks of the oligomers are visible. (b) Chromatogram of PA 6I 5k sample in the Flory distribution units.

Figure 43 -

 43 Figure 43 -(a) Chemical structure of the phenolic ester molecule formed by esterification reaction between two di-acid monomers (b) Chemical structure of the ester molecule formed by esterification reaction between one acid monomer and one blocker molecule (mono-functional benzoic acid).

Figure 44 -

 44 Figure 44 -Chemical shifts observed in 1 H NMR spectra, values in brackets correspond to the shifts in 13 C spectra.

Figure 45 -

 45 Figure 45 -(a) Illustration of temperature modulation in mDSC; (b) comparison of reversible and non-reversible heat flows. The curves were offset for clarity.

Figure 46 -

 46 Figure 46 -Thermogram obtained by mDSC with special focus on the heat capacity jump during glass transition.

Figure 47 -

 47 Figure 47 -Schematic representation of (a) enthalpy or volume as a function of temperature and (b) heat capacity as a function of the temperature, obtained as the derivative of the enthalpy. The point A is obtained by simple cooling at 10°C min -1 while the point B is obtained after an isothermal step at the aging temperature (Ta).

Figure 48 -

 48 Figure 48 -DSC sequence used in the physical aging experiments. All heating and cooling rates were equal to 10 °C min -1 , aging time was kept constant and equal to 5h at different temperatures Ta.

Figure 49 -

 49 Figure 49 -Representation of the planes of the ordered layers in a polymer and representation of both incident and reflected beam.

Figure 50 -

 50 Figure 50 -(a) Schematic representation of X-ray set up in transmission mode (b) an example of the resulting 2D pattern for PA 6,6. WAXS diffractogram of a PA 6,6 sample. From [143].

Figure 51 -

 51 Figure 51 -WAXS diffractogram of (a) PA 6I samples of different MW (b) copolymers PA 6I/6LiSIPA (5000 g•mol -1 ) (c) copolymers PA 6I/6LiSIPA, PA 6I/6NaSIPA and PA 10I/10NaSIPA with high MW (d) PA 6HIA.

Figure 52 -

 52 Figure 52 -Schematic representation of an crystalline lattice in PA 6,6 (a) triclinic unit cell (chain axis is along c; (b) chain arrangement in the a-c plane where H-bonds are formed; (c) chain arrangement in plane perpendicular to chain axis (view from the top). Adapted from [1].

Figure 53 -

 53 Figure 53 -Temperature sequence used in the rheological measurements.

Figure 54 -

 54 Figure 54 -Schematic representation of dipole orientation (polarization) when an electric field is applied.

Figure 55 -

 55 Figure 55 -PA 6,6 molecular relaxations as observed in (a) loss permittivity and (b) loss modulus M'' 3D maps. From [146].

Figure 56 -Figure 57 -

 5657 Figure 56 -Comparison between loss permittivity and modulus spectra at about 6 Hz for a dry PA 6I/6LiSIPA copolymer.

Figure 58 -

 58 Figure 58 -PA 6I DMA curve showing E', E'' and tan δ and the different polyamide relaxations.

Figure 59 -

 59 Figure 59 -Experimental molecular weight distributions obtained by GPC (points) and corresponding Flory's fit (solid curves) (a) PA 6I 5k (b) PA 6I 34k.

Figure 60 -

 60 Figure 60 -Plot of Tg as a function of PA 6I molecular weight. Fit corresponds to the Fox-Flory equation.

Figure 61 -Figure 62 -

 6162 Figure 61 -(a) Dynamic moduli (filled symbols, G' and Empty symbols, G'') and (b) corresponding loss trangent of PA 6I with varying molecular weights. The reference temperature corresponds to Tg for all polyamides. Solid lines in (a) correspond to the slope of 2 and 1 for G' and G'', respectively.

Figure 63 -

 63 Figure 63 -Horizontal shift factors with reference temperature equal to Tg for all PA 6I polymers as a function: (a) 1/T and (b) Tg/T (Angell plot). Symbols are experimental data, curves correspond to WLF fits.

Figure 64 -

 64 Figure 64 -Experimental data (black points, full symbols; G', empty symbols: G'') obtained by time-temperature superposition and the corresponding fit with the Rouse model.

Figure 65 -

 65 Figure 65 -GPC chromatograms of two different polyamides (PA 6I 32k and 34k).

Figure 66 -

 66 Figure 66 -Dynamic moduli of polymers containing different BHT levels. Inset shows a zoom on the dynamic moduli in the chain terminal relaxation region.

Figure 67 -

 67 Figure 67 -Tg as measured by mDSC as a function of the ionic fraction of PA 6I/6LiSIPA, PA 6I/6NaSIPA and PA 10I/10NaSIPA with different molecular weights (0% corresponds to PA 6I 5K, PA 6I 19k, PA 6I 41k and PA 10I samples).

Figure 68 -

 68 Figure 68 -Molecular weight distributions obtained by GPC (points) and corresponding Flory's fit (solid curves) (a) PA 6I/6LiSIPA 85/15 5k (b) PA 6I/6LiSIPA 80/20 5k (c) PA6I/6LiSIPA 90/10 29k (d) PA6I/6NaSIPA 90/10 16k.

Figure 69 -

 69 Figure 69 -(a) Dynamic moduli (filled symbols: G'; empty symbols: G'') and (b) tan obtained by time-temperature superposition of PA 6I and PA 6I/6LiSIPA copolymers with ~5000 g•mol -1 . The reference temperatures were considered close to, but not identical to Tg. Solid lines in (a) correspond to the slope of 2 and 1 for G' and G'', respectively.

Figure 70 -

 70 Figure 70 -Master curves of PA 6I 5k and unentangled polyamides using as reference temperature the glass transition as determined from DSC.

Figure 71 -

 71 Figure 71 -Cole-Cole plot (G'' as a function of G') in double log-log scale for unentangled copolyamides from 0 to 20 mol% of LiSIPA. Gray line represents the G'=G'' line.

Figure 72 -

 72 Figure 72 -Plot of the derivative of the reversible heat capacity (Cp) during glass transition obtained by mDSC for PA 6I and PA 6I/6LiSIPA copolymers with ~ 5000 g•mol -1 . Curves represent Gaussian fits used to calculate the widths of the transitions. Note that the peak around 150°C cannot be related to a second glass transition temperature associated to ionic domains as it is presented in the PA 6I curve. Both shape, amplitude and temperature of this peak are compatible with vaporization of the catalyst used during synthesis.

Figure 73 -

 73 Figure 73 -Horizontal shift factors for PA 6I 5k and all copolymers PA 6I/6LiSIPA with 5000 g•mol -1 as a function: (a) 1/T and (b) Tg/T (Angell plot). Symbols are experimental data, curves correspond to WLF fits.

Figure 74 -

 74 Figure 74 -Dynamic moduli (points) and fit with Rouse model (black curve) of: (a) PA 6I/6LiSIPA 95/5 5k; (b) PA 6I/6LiSIPA 90/10 5k; (c) PA 6I/6LiSIPA 85/15 5k; (d) PA 6I/6LiSIPA 80/20 5k.

Figure 75 -

 75 Figure 75 -(a) Raw scattering data normalized by sample thickness and empty beam signal and (b) normalized scattering data (using equations 2.11 and 2.12) for copolymers of PA 6I/6LiSIPA, with 0, 5, 10, 15 and 20 mol% of ionic groups.

Figure 76 -Figure 77 -

 7677 Figure 76 -(a) Dynamic moduli (filled symbols: G'; empty symbols: G'') and (b) tan obtained by time-temperature superposition of PA 6I 19k and 34k and PA 6I/6LiSIPA 90/10 29k. The reference temperature was taken as Tg.

Figure 78 -

 78 Figure 78 -Horizontal shift factors for PA 6I 34k and PA 6I/6LiSIPA 90/10 29k as a function: (a) 1/T and (b) Tg/T. Symbols are experimental data, curves correspond to WLF fits.

Figure 79 -

 79 Figure 79 -Normalized scattering data for copolymers of PA 6I/6LiSIPA 29k with 10 mol% of ionic groups.

Figure 80 -

 80 Figure 80 -Schematics of the ion rich domains in copolyamides.

Figure 81 -Figure 82 -

 8182 Figure 81 -(a) Dynamic moduli (filled symbols: G'; empty symbols: G'') and (b) tan obtained by time-temperature superposition of PA 6I 19k and PA 6I/6NaSIPA 90/10 16k. The reference temperature was taken close to Tg.

Figure 83 -

 83 Figure 83 -Horizontal shift factors for PA 6I 34k and PA 6I/6NaSIPA copolymers as a function: (a) 1/T and (b) Tg/T. Symbols are experimental data, curves correspond to WLF fits.

104 Figure 84 -

 10484 Figure 84 -Normalized scattering data for copolymers of PA 6I/6NaSIPA with 10 mol% of ionic groups.

Figure 85 -Figure 86 -

 8586 Figure 85 -(a) Dynamic moduli (filled symbols: G'; empty symbols: G'') and (b) tan obtained by time-temperature superposition of PA 6I 34k and PA 10I/10NaSIPA 90/10 44k. The reference temperature was taken as Tg.

Figure 87 -

 87 Figure 87 -Normalized scattering data for copolymer of PA 10I/10NaSIPA with 10 mol% of ionic groups.

Table 12 - 114 Figure 88 -

 1211488 Figure 88 -Molecular weight distributions obtained by GPC (points) and corresponding Flory's fit (solid curves) for the PA 6HIA sample.

Figure 89 -

 89 Figure 89 -GPC chromatogram revealing differences in PA 6HIA due to branching by phenolic ester formation.

Figure 90 -

 90 Figure 90 -(a) Dynamic moduli (filled symbols: G'; empty symbols: G'') and (b) tan obtained by time-temperature superposition of PA 6I 5k and PA 6HIA. The reference temperature was taken as Tg.

Figure 91 -

 91 Figure 91 -Horizontal shift factors for PA 6I 5k and PA 6HIA as a function: (a) 1/T and (b) Tg/T. Symbols are experimental data, curves correspond to WLF fits.

Figure 92 -

 92 Figure 92 -Dynamic moduli (points) and fit with Rouse model (black curve) of PA 6HIA.

Figure 93 -

 93 Figure 93 -(a) Schematic representation of the different dihedral angles in semi-aromatic polyamides. From [109]. (b) Most probable conformation in the case of PA 6HIA, obtained by molecular dynamics simulation.

Figure 94 -

 94 Figure 94 -Comparison of the heat flow curves for the aged and unaged samples of PA 6I 5k at different aging temperatures. Black curves represent the unaged heat flow curves, corresponding to each aged sample. Each curve was shifted for clarity.

Figure 95 -

 95 Figure 95 -Comparison of different aging temperatures in terms of the gain of enthalpy or volume.

Figure 96 -

 96 Figure 96 -Curves of calorimetric overshoots at various aging temperatures of PA 6I 5k (isothermal aging of 5h).

Figure 97 -Figure 98 -

 9798 Figure 97 -Enthalpy gain during 5h isothermal aging step as a function of (a) the aging temperature or (b) Tg -Ta for different unentangled polyamides with increasing content of ionic groups.

Figure 99 -

 99 Figure 99 -Enthalpy gain during 5h isothermal aging step as a function of (a) the aging temperature or (b) Tg -Ta for PA 6I/6NaSIPA 90/10 copolyamides. PA 6I 5k was used as reference.

Figure 100 -

 100 Figure 100 -Comparison of the overshoot peaks at various aging temperatures of PA 6I 5k and PA 6HIA (isothermal aging during 5h).

Figure 101 -

 101 Figure 101 -Enthalpy gain during 5h isothermal aging step as a function of (a) the aging temperature or (b) Tg -Ta for PA 6HIA. PA 6I 5k was used as reference.

Figure 102 -

 102 Figure 102 -Isotherms of the loss permittivity, '', in the region of the (a) (b) β1 (c) β2 and (d) -relaxations in PA 6I 5k (continues in the next page).

Figure 102 -

 102 Figure 102 -Isotherms of the loss permittivity, '', in the region of the (a) (b) β1 (c) β2 and (d) -relaxations in PA 6I 5k (continuation from previous page).

Figure 103 -

 103 Figure 103 -Relaxation map evidencing the characteristic , β and relaxation times as a function of temperature for PA 6I 5k and PA 6I/6LiSIPA 5k with increasing ionic content. -relaxation was fitted using the WLF equation.

Figure 104 -

 104 Figure 104 -Cole-Cole exponent (representing the broadness of the relaxation) as a function of the temperature for therelaxation.

Figure 105 -

 105 Figure 105 -Relaxation map evidencing the characteristic , β and relaxation times as a function of temperature for PA 6I 5k and PA 6I/6LiSIPA 5k with increasing ionic content. -relaxation was fitted using the WLF equation.

Figure 106 -

 106 Figure 106 -Relaxation map obtained by DMA experiments for PA 6I samples with increasing molecular weight. Curves correspond to Arrhenius fits. PA 6I 5k results were obtained by BDS.

Figure 107 -

 107 Figure 107 -Relaxation map evidencing the characteristic , β and relaxation times as a function of temperature for PA 6I 5k and PA 6HIA with similar molecular weight. Experimental points of the -relaxation were fitted using the WLF equation.

Figure I. 2 -

 2 Figure I.2 -Trimer used to model PA 6I, with representation of monomer Begin, Middle and End.

Figure I. 3 -

 3 Figure I.3 -Different stages of molecular dynamics simulation.

Figure I. 5 -

 5 Figure I.5 -Schematics of the two angles in polyamides.

Figure I. 6 -

 6 Figure I.6 -Density of probability of the angle ψ1 for PA 6I and PA 6HIA.

Figure I -

 I Figure I -Structure des polyamides amorphes utilisés dans cette étude. L'architecture moléculaire des polyamides a été déterminée par titration des groupements terminaux (EGT) et par chromatographie d'exclusion stérique (GPC). La caractérisation dynamique des polyamides a été réalisée par calorimétrie différentielle à balayage (DSC) et la caractérisation structurale par diffraction des rayons-X.

Figure II -

 II Figure II -(a) Module dynamique (symboles pleins, G' et symboles vides, G'') et (b) angle de perte des PA 6I avec différentes masses molaires. La température de référence correspond à la Tg de chaque polymère. Les lignes en (a) représentent les pentes de 2 et 1 attendues dans le régime terminal pour G' et G''.

  rad/s) présenté sur la Figure III, il n'y a pas d'effet sur l'allure des courbes maîtresses avec l'introduction de groupements ioniques. Les courbes maîtresses étant décrites par le modèle de Rouse, l'ajout de groupements ioniques n'a aucun effet sur la dynamique des polyamides à l'état fondu, à part le décalage de la Tg.

Figure III -

 III Figure III -(a) Module dynamique (symboles pleins, G' et symboles vides, G'') et (b) angle de perte des PA 6I/6LiSIPA avec différents taux de groupements ioniques. La température de référence est proche de la Tg de chaque polymère. Les lignes en (a) représentent les pentes de 2 et 1 pour G' et G''.

Figure IV -

 IV Figure IV -Facteurs de shift horizontaux pour les copolyamides non-enchevêtrés avec 5000 g•mol -1 en fonction de : (a) 1/T and (b) Tg/T (Diagramme d'Angell). Les points représentent les données expérimentales et les lignes l'équation de WLF.

Figure V -

 V Figure V -Donnés de diffraction de rayons-X aux petits angles des copolyamides ioniques enchevêtrés. Le pic de diffraction est décrit avec un modèle pour des domaines ioniques sphériques [158].

Chapitre 5 -

 5 Effet de liaisons hydrogène plus fortes sur la dynamique à l'état fondu de polyamides L'ajout des groupements phénols se traduit tout d'abord par l'augmentation de la Tg du polymère, de 106 à 145°C. Contrairement aux autres polymères étudiés, le PA 6HIA présente deux types de ramifications possibles (bis-hexaméthylène diamine et ester phénolique), mais aucune influence sur l'allure des courbes maîtresses, présentées sur la Figure VI, n'est observée. Il est vraisemblable que le taux de ramification reste faible.

Figure VI -

 VI Figure VI -(a) Module dynamique (symboles pleins, G' et symboles vides, G'') et (b) angle de perte de PA 6HIA. La température de référence est proche de la Tg de chaque polymère. Les lignes en (a) représentent les pentes de 2 et 1 attendues dans le régime terminal pour G' et G''.

  coopérativité entre les chaînes. Le polymère présente donc un comportement Arrhénien, avec une fragilité dynamique proche de 50. Dans le cas des polyamides, avec une transition vitreuse proche de 150°C, les liaisons hydrogène sont affaiblies ou ont une durée de vie extrêmement courtes aux échelles de temps d'observation, ce qui induit la non-coopérativité entre chaînes au-dessus de cette température et se traduit par une augmentation de la fragilité dynamique.Par modélisation de la dynamique moléculaire, nous avons cherché à comprendre si le PA6HIA présente des conformations préférentielles, éventuellement différentes de celle du PA6I. L'idée initiale est qu'une configuration 'cis' alternée des angles dièdres ψ1 et ψ2 pour le PA6HIA pourrait permettre de former une paire de liaisons hydrogène couplées (FigureVII). La dynamique moléculaire semble indiquer la présence de telles paires de liaison couplées. Ce résultat reste cependant à confirmer. De plus, les liaisons hydrogène phénol-amide formées ainsi ont une énergie de liaison plus élevée que les liaisons amideamide présentes dans les PA6I.

Figure VII -Chapitre 6 -

 VII6 Figure VII -Paire de liaisons hydrogènes couplées présentes dans le PA 6HIA par rotation d'un des angles dièdres.

Figure VIII -

 VIII Figure VIII -Gains d'enthalpie pendant l'étape de vieillissement en fonction de la température de vieillissement (Tg-5°C à Tg-40°C) pour tous les polymères non-enchevêtrés.

Figure IX -

 IX Figure IX -Carte de relaxations des PA 6I 5k et PA 6I/6LiSIPA 5k avec un taux croissant de groupements ioniques. Les courbes représentent l'équation de WLF ajusté sur les points expérimentaux de la relaxation .

Table 1 -Literature overview of chemistry and main rheological observation in hydrogen bonding polymers.

 1

	Author	Polymer + interacting groups	Tg shift (°C)	Entangled	H-Bond density	Main results
	Stadler et al. [2], [73]-[75]	Polybutadiene + 0-2% urazole groups	-110 to -100	Yes	2	Two relaxations observed in G''
						Presence	of	ordered
		Polybutadiene or				clusters	with	melting
	Dardin et	polyisobutylene +				temperature above 100°C;
	al. [85],	0-2% 4-(3,5-dioxo-	----	No	2	relaxation	of		U4A
	[86]	1,2,4-triazolidin-4-yl)				multiplets ( *-relaxation)
		benzoic acid				detected with dielectric
						spectroscopy.
	Osterwinter et al. [45]	Polyglycerol and its methylated analogue (PG-OH and PG-OMe)	-75 to -25	From unentangled to entangled	Each monomer	Longer relaxation time for PG-OH
	Lewis et al. [77]	Poly(n-butylacrylate) + 0-23 mol % interacting groups (AP, AA, CEA, UPy)	For AP: -50 to -12°C; For others : -50 to -40°C	No	1 to 4, from 25 to 70 kJ•mol -1	UPy-substituted polymer present a plateau and gel-like behavior in terminal region
						Gel-like	behavior	in
	Feldman et al. [78]	Poly(n-butylacrylate) +0-13 mol %UPy	-50 to -10°C	No	4	terminal relaxation time above region, with
						experimental times
	Shabbir et al. [79]	Poly(n-butylacrylate) + 0-38 mol % of hydrolyzed PnBA	-----	Yes	1	Gel-like terminal region for 38% behavior in hydrolyzed sample
		Poly(n-butylacrylate-				UPy-substituted polymer
	Jangizehi et al. [80]	ran-hydroxyethyl acrylate) copolymers	No	Yes	4	present a plateau and gel-like behavior in terminal
		+ 1-4 mol % UPy				region
	Véchambre et al. [81], Callies et al. [82]	Poly(n-butylacrylate) +0-6 mol %bis-urea or tri-urea	No	From unentangled to entangled	4 to 6	tTS works except for entangled polymers at high temperature, gel-like behavior in terminal region

Table 2 -Literature overview of chemistry and main rheological observation in ionomers. Author (Ref.) Polymer + interacting groups Tg shift (°C) Entangled Cluster Number of ionic groups per chain Main results

 2

							New	low-frequency
							relaxation not present in the
	Weiss et al. [88]	PS+SO3Na (0-1.71 mol%)	No	Yes	No	4	unmodified effect of ionic groups can be polystyrene, varied by weakening the
							ion-dipole strength using
							NH4 + instead of Na +
	Weiss et al. [90]	PS+SO3X (X= H, Li, Na, K, Rb, Cs) (0, 2.5 and 4.8 mol%)	80 to 90	No	Yes, 4 to 6 nm clusters	1 or 2	tTS fails, comparison with HSPS shows no influence of H-bonds
		Polysiloxane with					No plateau due to weak
	Chen et al. [47]	phosphonium fraction (0-26	-84 to -70	No	No	---	ionic association, master curves fitted with Sticky-
		mol%)					Rouse and KWW
	Chen et al. [3]	Poly(ethylene oxide) + SO3Na (0-100 mol%)	-45 to 0	No	----	Up to 1 each monomer	No plateau due solvation of ionic groups, master curves fitted with Sticky-Rouse and KWW
	Chen et al. [3]	Poly (tetramethylene oxide) + SO3Na (0-100 mol%)	No	No	---	Up to 1 each monomer	tTS fails on plateau, master curves fitted with Sticky-Rouse
							Different behaviors obtained
	Huang et al. [92]	PS+SO3X (X=Na, K, Rb, Cs)(0.066 to 2.7 mol%)	Less than 10°C	No	Yes	Up to 4	depending on the number of ionic groups per chain, development of the
							reversible gelation model
	Kim et al. [96]	Poly(ethyl acrylate-co-Na acrylate) (up to 13 mol%)	Linear increase with content ionic	----	Yes	----	Ionomers show a cluster loss tangent peak and ionic modulus

Table 3 -List of studied semi-aromatic amorphous polyamides and corresponding weight-average molecular weights.

 3

	Polymer	Sample characteristics
	PA 6I	Increasing MW: Mw = 4700, 19300, 31600, 33400, 34500, 41600 g•mol -1
	PA 6HIA	Mw = 9000 g•mol -1

PA 6I/6LiSIPA

Mw = 5000 g•mol -1 and increasing ionic content: 95/5, 90/10, 85/15 and 80/20 PA 6I/6LiSIPA 90/10 Mw = 29000 g•mol -1 PA 6I/6NaSIPA 90/10 Two different molecular weights: 16000 and 48000 g•mol -1 PA 10I/10NaSIPA 90/10 Mw = 44000 g•mol -1

Table 4 -Mass attenuation coefficients calculated for each polymer sample. Polymer Mass attenuation coefficient (cm 2 •g -1 )

 4

	PA 6I

Table 5 -Molecular weight and molecular weight distribution as obtained by EGT and GPC analysis, as well as quantity of branching formed during reaction by formation of BHT and glass transition temperature, as obtained by mDSC.

 5

	Polymer	Mn (g•mol -1 ) EGT (±1%)	Mn (g•mol -1 ) GPC (±10%)	Mw (g•mol -1 ) GPC (±5%)	Ip= Mw/Mn (±10%)	ppm	BHT mol% BHT/ amide group	Tg ± 1 (°C)
	PA 6I 5k	1950	2650	4760	1.8	4400	0.25	106
	PA 6I 19k	6100	7500	19300	2.6	4150	0.23	121
	PA 6I 32k	9530	11700	31600	2.7	6400	0.36	126
	PA 6I 33k	10000	10300	33450	3.3	1970	0.11	126
	PA 6I 34k	12220	12120	34520	2.8	2000	0.11	126
	PA 6I 41k	10190	14300	41600	2.9	7060	0.40	126

Table 6 -WLF parameters at a reference temperature equal to Tg, fractional free volume at Tg and dynamic fragility index values for PA 6I samples with varying molecular weight.

 6

	Polymer	C1 g (±5%)	C2 g (K) (±5%)	fg	m
	PA 6I 5k	12.4	37.5	0.035	125
	PA 6I 19k	13.5	39.8	0.032	128
	PA 6I 32k	13.1	34.3	0.033	152
	PA 6I 33k	12.9	38.0	0.034	122
	PA 6I 34k	12.4	36.3	0.035	135
	PA 6I 41k	13.4	35.5	0.032	151

Table 7 -Molecular weight and molecular weight distribution as obtained by EGT and GPC analysis, as well as quantity of branching formed during reaction by formation of BHT and glass transition temperature, as obtained by mDSC.

 7

	Polymer	Mn (g mol -1 ) EGT (±1%)	Mn (g mol -1 ) GPC (±10%)	Mw (g mol -1 ) GPC (±5%)	Ip= Mw/Mn (±10%)	BHT (ppm)	Tg ± 1 (°C)
	PA 6I/6LiSIPA 95/5 5k	1740	2620	5260	2.0	2330	117
	PA 6I/6LiSIPA 90/10 5k	1750	2590	5250	2.0	2810	128
	PA6I/6LiSIPA 85/15 5k	---	2720	5760	2.1	3040	142
	PA6I/6LiSIPA 80/20 5k	1520	1520	4590	2.5	2560	148
	PA 6I/6LiSIPA 90/10 29k	5700	7770	28960	3.7	---	155
	PA 6I/6NaSIPA 90/10 16k	3780	5850	16110	2.8	---	147
	PA 6I/6NaSIPA 90/10 48k	10530	13340	48250	3.6	3360	156
	PA 10I/10NaSIPA 90/10 44k	8470	9170	44640	4.9	n.a.	123

Table 8 -WLF fit parameters at reference temperature equal to Tg, fractional free volume at Tg and dynamic fragility index values for PA 6I/6LiSIPA samples with varying ionic content (PA 6I 5k was introduced as the reference).

 8

	Polymer	C1 g (±5%)	C2 g (K) (±5%)	fg	m
	PA 6I 5k	12.4	37.5	0.035	125
	PA 6I/6LiSIPA 95/5 5k	12.9	43.7	0.034	115
	PA 6I/6LiSIPA 90/10 5k	13.0	47.6	0.034	109
	PA 6I/6LiSIPA 85/15 5k	14.0	58.3	0.031	100
	PA 6I/6LiSIPA 80/20 5k	13.3	56.2	0.039

Table - Fitting parameter obtained by fitting the dynamic moduli of unentangled polyamides using both Flory and GPC distribution and the calculated friction coefficient.

 -

	Polymer	Fitting parameter, f, Flory	Fitting parameter, f, GPC	(kg•s -1 ) Flory	(kg•s -1 ) GPC
	PA 6I 5k	(8.88 ± 0.19) 10 -16	(8.40 ± 0.17) 10 -16	32.9 ± 0.7	31.1 ± 0.6
	PA 6I/6LiSIPA 95/5 5k	(3.59 ± 0.09) 10 -16	(3.37 ± 0.08) 10 -16	13.3 ± 0.4	12.5 ± 0.3
	PA 6I/6LiSIPA 90/10 5k	(3.82 ± 0.15) 10 -16	(3.46 ± 0.12) 10 -16	11.4 ± 0.6	12.8 ± 0.5
	PA 6I/6LiSIPA 85/15 5k	(3.20 ± 0.02) 10 -16	(3.36 ± 0.02) 10 -16	11.9 ± 0.1	12.5 ± 0.1
	PA 6I/6LiSIPA 80/20 5k	(7.21 ± 0.28) 10 -16	(1.73 ± 0.02) 10 -16	17.4 ± 0.3	6.41 ± 0.04

Table 10 -WLF fit parameters at reference temperature equal to Tg, fractional free volume at Tg and dynamic fragility index values for PA 6I/6LiSIPA 90/10 29k (PA 6I 19k was introduced as the reference).

 10

	Polymer	C1 g (±5%)	C2 g (K) (±5%)	fg	m
	PA 6I 19k	12.4	37.5	0.035	125
	PA 6I 34k	12.4	36.3	0.035	135
	PA 6I/6LiSIPA 90/10 29k	10.4	35.1	0.041	126

Table 11 -WLF fit parameters at reference temperature equal to Tg, fractional free volume at Tg and dynamic fragility index values for PA 6I/6NaSIPA 90/10 samples (PA 6I 19 and 34k were used as reference).

 11

	Polymer	C1 g (±5%)	C2 g (K) (±5%)	fg	m
	PA 6I 19k	12.4	37.5	0.035	125
	PA 6I 34k	12.4	36.3	0.035	135
	PA 6I/6NaSIPA 90/10 16k	10.7	49.9	0.041	90
	PA 6I/6NaSIPA 90/10 48k	10.1	42.1	0.043	103

Table 13 -WLF fit parameters at reference temperature equal to Tg, fractional free volume at Tg and dynamic fragility index values for PA 6I/6LiSIPA 90/10 29k (PA 6I 19k was introduced as the reference).

 13

	0,0020	0,0022	0,0024
	Polymer		C1 g (±5%)	C2 g (K) (±5%)	fg	m
	PA 6I 5k		12.4	37.5	0.035	125
	PA 6HIA		12.9	38.2	0.034	142

Table 14 -Fitting parameter obtained by fitting the dynamic moduli of unentangled polyamides using both Flory and GPC distribution and the calculated friction coefficient.

 14

	Polymer	Fitting parameter, f, (Flory)	Fitting parameter, f, (GPC)	(kg•s -1 ) (Flory)	(kg•s -1 ) (GPC)
	PA 6I 5k	(8.88 ± 0.19) 10 -16	(8.40 ± 0.17) 10 -16	32.9 ± 0.7	31.1 ± 0.6
	PA 6HIA	(7.21 ± 0.28) 10 -16	(6.84 ± 0.24) 10 -16	29.9 ± 1.2	28.4 ± 1.0

Table 15 -Arrhenius fitting parameters, entropy and free energy (at 300K) of relaxation for relaxation in PA 6I 5k and PA 6I/6LiSIPA 5k with increasing ionic content. Polymer (all 5k) -relaxation

 15

		(kJ•mol -1 )	(	ΔS (J•K -1 mol -1 )	ΔG (kJ•mol -1 )
	PA 6I	44.4	2.9	86.8	18.3
	PA 6I/6LiSIPA 95/5	44.1	3.0	86.5	18.2
	PA 6I/6LiSIPA 90/10	44.8	1.9	90.3	17.7
	PA 6I/6LiSIPA 85/15	44.8	2.0	89.9	17.8
	PA 6I/6LiSIPA 80/20	43.2	3.5	85.3	17.6

Table 16 -Arrhenius fitting parameters, entropy and free energy (at 300K) of β1 relaxation in PA 6I 5k and PA 6I/6LiSIPA 5k with increasing ionic content.

 16

					β1-relaxation
		Polymer
		(all 5k)	(kJ•mol -1 )	(	ΔS (J•K -1 mol -1 )	ΔG (kJ•mol -1 )
		PA 6I	72.5	0.19	90.4	45.4
	PA 6I/6LiSIPA 95/5	70.5	0.9	77.5	47.3
	PA 6I/6LiSIPA 90/10	68.6	2.2	70.0	47.6
	PA 6I/6LiSIPA 85/15	73.8	0.6	80.8	49.5
	PA 6I/6LiSIPA 80/20	66.2	37	46.5	52.2
	6.2.1.3	-relaxation

Table 17 -WLF fitting parameters for the -relaxation and calculated dynamic fragility for PA 6I and PA 6I/6LiSIPA copolymers with a molecular weight around 5000 g•mol -1 .

 17

	Polymer (all 5k)	C1 g	C2 g (K)	m
	PA 6I	17.3	73.3	89
	PA 6I/6LiSIPA 95/5	24.4	142.9	67
	PA 6I/6LiSIPA 90/10	19.1	133.6	57
	PA 6I/6LiSIPA 85/15	27.1	229.9	49
	PA 6I/6LiSIPA 80/20	19.1	164.4	49

Table 18 -Arrhenius fitting parameters, entropy and free energy (at 300K) of relaxation in PA 6I 5k and PA 6I/6NaSIPA 90/10 16k.

 18

				-relaxation
	Polymer
		(kJ•mol -1 )	(	ΔS (J•K -1 mol -1 )	ΔG (kJ•mol -1 )
	PA 6I 5k	44.4	2.9	86.8	18.3
	PA 6I/6NaSIPA 90/10 16k	46.3	1.1	95.0	17.8

Table 19 -Arrhenius fitting parameters, entropy and free energy (at 300K) of β1relaxation in PA 6I 5k and PA 6I/6NaSIPA 90/10 16k.

 19

				β1-relaxation
	Polymer
		(kJ•mol -1 )	(	ΔS (J•K -1 mol -1 )	ΔG (kJ•mol -1 )
	PA 6I 5k	72.5	0.19	90.4	45.4
	PA 6I/6NaSIPA 90/10 16k	68.6	1.1	75.8	45.8

Table 20 -Arrhenius fitting parameters for and β relaxations in PA 6I increasing molecular weight, as obtained by DMA measurements.

 20

		γ-relaxation	β-relaxation
	Polymer
		(kJ•mol -1 )		(kJ•mol -1 )
	PA 6I 5k (BDS)	44.4	2.9 10 -17	72.5	1.9 10 -17
	PA6I 19k (DMA)	40.6	2.3 10 -16	62.5	2.0 10 -15
	PA6I 32k (DMA)	42.1	2.8 10 -16	60.0	1.1 10 -14

Table 21 -Arrhenius fitting parameters, entropy and free energy (at 300K) of relaxation for -relaxation in PA 6I 5k and PA 6HIA.

 21

				-relaxation
	Polymer
		(kJ•mol -1 )	(	ΔS (J•K -1 mol -1 )	ΔG (kJ•mol -1 )
	PA 6I 5k	44.4	2.9	86.8	18.3
	PA 6HIA	47.3	1.0	95.5	18.6

Table 22 -Arrhenius fitting parameters, entropy and free energy (at 300K) of relaxation for β-relaxations in PA 6I 5k and PA 6HIA.

 22

				β1-relaxation
	Polymer
		(kJ•mol -1 )	(	ΔS (J•K -1 mol -1 )	ΔG (kJ•mol -1 )
	PA 6I 5k	72.5	1.9	90.4	45.4
	PA 6HIA	83.1	0.31	105.5	51.4

Table 23 -WLF fitting parameters for the -relaxation and calculated dynamic fragility for PA 6I and PA 6HIA samples.

 23

	Polymer	C1 g	C2 g (K)	m
	PA 6I 5k	17.3	73.3	89
	PA 6HIA	27.9	151.1

Table I .1 -Results obtained by molecular dynamics simulation for PA 6I and PA 6HIA.

 I

	Parameter	PA6I	PA6HIA
	Density (g•cm -3 )	1.14	1.18
	lk (nm)	1.2	1.04
	Mk (g•mol -1 )	231	212
	Packing length, p (	2.3	2.9
	I.4 Hydrogen bonding in PA 6I and PA 6HIA

  La rhéologie d'un polymère dépend tout d'abord de la masse molaire. Quand la masse du polymère est inférieure à la masse molaire dite critique, Mc, celui-ci est non-enchevêtré et son comportement est décrit par le modèle de Rouse. À l'inverse, quand la masse molaire est supérieure à Mc, le polymère est enchevêtré, c'est-à-dire que des contraintes physiques existent et retardent la relaxation du polymère. Cela se traduit par l'apparition d'un plateau caoutchoutique et un décalage des temps de relaxation vers les plus basses fréquences. Dans ce cas, la viscoélasticité du polymère est décrite par le modèle de la reptation (aussi appelé modèle du tube ou modèle de Doi et Edwards).

Remerciements

Appendix

I.

Molecular dynamics simulations [START_REF] Bocahut | Modeling Molecular Relaxation Mechanisms in Amorphous Polymers: Application to Polyamides[END_REF], [START_REF] González | Force fields and molecular dynamics simulations[END_REF] The molecular dynamics simulations allows modelling an amorphous box of simulation containing as many polymer chains as wanted in a thermodynamic state which is close to the experimental reality. Then, by following the evolution of this box at several temperatures, it is possible to access average values of geometrical parameters, such as the end-to-end distance, Kuhn segment length and molecular weight, packing length, etc. The box is built using periodic boundary conditions, as a way to model an infinite system using a finite simulation box.

Simulations consist in modelling the evolution of a system of particles through time and recording the position and speed of the particles at each time step. The molecular mechanics uses simple physical models to calculate the potential energy as the sum of all terms that describe the considered interaction between atoms. The movement of each particle (atom) is calculated numerically by integrating classical equation of dynamics in translation.

The total force, , that is applied to a certain particle, i, is defined as:

The potential interactions is a function of the three-dimensional coordinates of each particle, i.e. 3N variables for a system with N particles and it associates to each particle of position a potential energy value defined by the modeled interactions. In Figure I.1, the particle in A is placed in a local (metastable) minimum and has enough thermal energy (kBT) to overcross the energy barrier in B and thus reach the global minimum C. The molecular dynamics will then allow the particle to reach the energy minimum. The number of average hydrogen bonds is slightly higher for PA6HIA. However, relative to the number of donor atoms total, this ratio is constant for the 2 polymers. By relating the number of hydrogen bonds to the volume of the simulation box, this gives a density of bonds which increases significantly for PA6HIA compared to PA6I.